Formulas of Acoustics

Formulas of Acoustics F. P. Mechel (Ed.) Formulas of Acoustics Second Edition With contributions by: M. L. Munjal, ...

Author: F.P. Mechel | M. L. Munjal | Michael Vorlander | Peter Koltzsch | Martin Ochmann | A. Cummings | W. Maysenholder | Walter K. Arnold

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Formulas of Acoustics

F. P. Mechel (Ed.)

Formulas of Acoustics Second Edition

With contributions by: M. L. Munjal, M. Vorl ander, ¨ P. Koltzsch, ¨ M. Ochmann, A. Cummings, W. Maysenholder, ¨ W. Arnold

123

Prof. Dr. Fridolin P. Mechel Landhausstraße 12 71120 Grafenau Germany

Library of Congress Control Number: 2008922894

ISBN: 978-3-540-76832-6 This publication is available also as: Electronic publication under ISBN 978-3-540-76833-3 and Print and electronic bundle under ISBN 978-3-540-76834-0 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. c Springer-Verlag Berlin Heidelberg New York 2008 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Springer is part of Springer Science+Business Media springer.com Editor: Dr. Christoph Baumann, Kerstin Kindler, Heidelberg, Germany Development Editor: Lydia Mueller, Heidelberg, Germany Typesetting and Production: le-tex publishing services oHG, Leipzig, Germany Cover Design: Frido Steinen-Broo, Girona, Spain Printed on acid-free paper

SPIN: 12190720 2109 — 5 4 3 2 1 0

Preface to the first edition, abbreviated Modern acoustics is more and more based on computations, and computations are based on formulas. Such work needs previous and contemporary results. It consumes much time and effort to search needed formulas during the actual work. Therefore, fundamentals and results of acoustics that can be expressed as formulas will be collected in this book. The formula collection is subdivided into fields of acoustics (Chapters). For some fields, in which this author is not expert enough, he invited co-authors to contribute. Most colleagues contacted for possible contributions were convinced of the project and agreed spontaneously. The material within a field of acoustics is subdivided in Sections which deal with a defined task. Some overlap of Sections should be tolerated; but the subdivision into well-defined Sections will be helpful to the reader to find a particular topic of interest. The present formula collection should not be considered a textbook in a condensed form. Derivations of a presented result will be described only as far as they are helpful in understanding the problem; the more interested reader is referred to the “source” of the result. Useful principles and computational procedures will also be included, even if they need more describing text. Symbols and quantities will be defined in the Section, and wherever useful a sketch will help to explain the object and the task. One of the advantages of a formula collection is seen in uniform definitions, notations and symbols for quantities. A strict uniformity in the form of a central list of symbols used never works, according to this author’s observation. Therefore, only commonly used symbols (such as medium density, speed of sound, circular frequency, etc.) are collected in a central list of symbols (see Conventions); other symbols are defined in the relevant chapter. Most Sections contain, below their title or in the text, a reference to the literature. It cannot be the task and intention of this book either to indicate time priorities of publications concerning a topic or to give a survey of the existing literature.The reference quoted is the source of more information, which the author has used. Higher transcendental functions used in the formulas will be explained by reference to mathematical literature, if necessary. If functions are used with different definitions in the literature, the definition applied here will be presented. The authors think that the book in its present form contains most of traditional and modern results of both fundamental and special character so that the book can be

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helpful to researchers and engineers in the fields of physical acoustics, noise control, and room acoustics. The manuscript was written in a camera ready form (in order to avoid proof reading). So printing errors are the responsibility of the editing author. He would be grateful for indications of such errors. The author gratefully acknowledges the support given to the project by the co-authors and by the publisher. Grafenau, October 2001

Preface to the second edition The book was out of print in 2004. The need of reprint gave a first opportunity to apply some corrections to (rather harmless) misprints and to a few more serious formula errors (the positions of the errors are marked by a footnote ∗) ). Some of the shown diagrams were generated by the computing program Mathematica ; this program unfortunately has lost its ability to write axes and plot labels so that they can be understood by receiving text programs. Therefore transscriptions to plot labels are enumerated near the diagrams, where necessary. This second edition is moderately enlarged by some additional topics in new Sections. Grafenau, May 2008

Contents Preface to the first edition, abbreviated . . . . . . . . . . . . . . . . . . . . . . . . .

V

Preface to the second edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VI

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VII

Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIX A Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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B General Linear Fluid Acoustics . . . . . . . . . . . . . . . . . . . . . . F.P. Mechel B.1 Fundamental Differential Equations . . . . . . . . . . . . . . . . B.2 Material Constants of Air . . . . . . . . . . . . . . . . . . . . . . . B.3 General Relation for Field Admittance and Intensity . . . . . . . B.4 Integral Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . B.5 Green’s Functions and Formalism . . . . . . . . . . . . . . . . . B.6 Orthogonality of Modes in a Duct with Locally Reacting Walls B.7 Orthogonality of Modes in a Duct with Bulk Reacting Walls . . B.8 Source Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . B.9 Sommerfeld’s Condition . . . . . . . . . . . . . . . . . . . . . . . B.10 Principles of Superposition . . . . . . . . . . . . . . . . . . . . . B.11 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . B.12 Adjoint Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . B.13 Vector and Tensor Formulation of Fundamentals . . . . . . . . B.14 Boundary Condition at a Moving Boundary . . . . . . . . . . . B.15 Boundary Conditions in Liquids and Solids . . . . . . . . . . . . B.16 Corner Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . B.17 Surface Wave at Locally Reacting Plane . . . . . . . . . . . . . . B.18 Surface Wave Along a Locally Reacting Cylinder . . . . . . . . . B.19 Periodic Structures, Admittance Grid . . . . . . . . . . . . . . . B.20 Plane Wall with Wide Grooves . . . . . . . . . . . . . . . . . . . . B.21 Thin Grid on Half-Infinite Porous Layer . . . . . . . . . . . . . . B.22 Grid of Finite Thickness with Narrow Slits on Half-Infinite Porous Layer . . . . . . . . . . . . . . . . . . . . B.23 Grid of Finite Thickness with Wide Slits on Half-Infinite Porous Layer . . . . . . . . . . . . . . . . . . . .

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5 8 11 12 13 19 20 21 22 22 25 26 26 38 39 40 41 42 44 48 50

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C Equivalent Networks . . . . . . . . . . . . . . . . . . . . F.P. Mechel C.1 Fundamentals of Equivalent Networks . . . . . . C.2 Distributed Network Elements . . . . . . . . . . C.3 Elements with Constrictions . . . . . . . . . . . . C.4 Superposition of Multiple Sources in a Network C.5 Chain Circuit . . . . . . . . . . . . . . . . . . . . . C.6 Partition Impedance of Orifices . . . . . . . . . .

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D Reflection of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.P. Mechel D.1 Plane Wave Reflection at a Locally Reacting Plane . . . . . . . . . . D.2 Plane Wave Reflection at an Infinitely Thick Porous Layer . . . . . D.3 Plane Wave Reflection at a Porous Layer of Finite Thickness . . . . D.4 Plane Wave Reflection at a Multilayer Absorber . . . . . . . . . . . D.5 Diffuse Sound Reflection at a Locally Reacting Plane . . . . . . . . D.6 Diffuse Sound Reflection at a Bulk Reacting Porous Layer . . . . . D.7 Sound Reflection and Scattering at Finite-Size Local Absorbers . . D.8 Uneven, Local Absorber Surface . . . . . . . . . . . . . . . . . . . . . D.9 Scattering at the Border of an Absorbent Half-Plane . . . . . . . . . D.10 Absorbent Strip in a Hard Baffle Wall, with Far Field Distribution . D.11 Absorbent Strip in a Hard Baffle Wall, as a Variational Problem . . D.12 Absorbent Strip in a Hard Baffle Wall, with Mathieu Functions . . D.13 Absorption of Finite-Size Absorbers, as a Problem of Radiation . . D.14 A Monopole Line Source Above an Infinite, Plane Absorber; Integration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.15 A Monopole Line Source Above an Infinite, Plane Absorber; with Principle of Superposition . . . . . . . . . . . . . . . . . . . . . D.16 A Monopole Point Source Above a Bulk Reacting Plane, Exact Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.17 A Monopole Point Source Above a Locally Reacting Plane, Exact Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.18 A Monopole Point Source Above a Locally Reacting Plane, Exact Saddle Point Integration . . . . . . . . . . . . . . . . . . . . . D.19 A Monopole Point Source Above a Locally Reacting Plane, Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.20 A Monopole Point Source Above a Bulk Reacting Plane, Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E Scattering of Sound . . . . . . . . . . . . . . . . . . . . F.P. Mechel E.1 Plane Wave Scattering at Cylinders . . . . . . . . E.2 Plane Wave Scattering at Cylinders and Spheres E.3 Multiple Scattering at Cylinders and Spheres . . E.4 Cylindrical Wave Scattering at Cylinders . . . . .

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E.5 E.6 E.7 E.8 E.9 E.10 E.11 E.12 E.13 E.14 E.15 E.16 E.17 E.18 E.19 E.20 E.21 E.22 E.23 E.24 F

Cylindrical or Plane Wave Scattering at a Corner Surrounded by a Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plane Wave Scattering at a Hard Screen . . . . . . . . . . . . . Cylindrical or Plane Wave Scattering at a Screen with an Elliptical Cylinder Atop . . . . . . . . . . . . . . . . . . . . . . . Uniform Scattering at Screens and Dams . . . . . . . . . . . . Scattering at a Flat Dam . . . . . . . . . . . . . . . . . . . . . . Scattering at a Semicircular Absorbing Dam on Absorbing Ground . . . . . . . . . . . . . . . . . . . . . . . . Scattering in Random Media, General . . . . . . . . . . . . . . Function Tables for Monotype Scattering . . . . . . . . . . . . Sound Attenuation in a Forest . . . . . . . . . . . . . . . . . . . Mixed Monotype Scattering in Random Media . . . . . . . . Multiple Triple-Type Scattering in Random Media . . . . . . . Plane Wave Scattering at Elastic Cylindrical Shell . . . . . . . Plane Wave Backscattering by a Liquid Sphere . . . . . . . . . Spherical Wave Scattering at a Perfectly Absorbing Wedge . . Impulsive Spherical Wave Scattering at a Hard Wedge . . . . . Spherical Wave Scattering at a Hard Screen . . . . . . . . . . . Spherical Wave Scattering at a Cone . . . . . . . . . . . . . . . Polar Mode Numbers at a Soft Cone . . . . . . . . . . . . . . . Polar Mode Numbers at a Hard Cone . . . . . . . . . . . . . . Scattering at a Cone with Axial Sound Incidence . . . . . . . .

Radiation of Sound . . . . . . . . . . . . . . . . . . . . . . . . . F.P. Mechel F.1 Definition of Radiation Impedance and End Corrections F.2 Some Methods to Evaluate the Radiation Impedance . . F.3 Spherical Radiators . . . . . . . . . . . . . . . . . . . . . . F.4 Cylindrical Radiators . . . . . . . . . . . . . . . . . . . . . F.5 Piston Radiator on a Sphere . . . . . . . . . . . . . . . . . F.6 Strip-Shaped Radiator on Cylinder . . . . . . . . . . . . . F.7 Plane Piston Radiators . . . . . . . . . . . . . . . . . . . . F.8 Uniform End Correction of Plane Piston Radiators . . . F.9 Narrow Strip-Shaped, Field-Excited Radiator . . . . . . . F.10 Wide Strip-Shaped, Field-Excited Radiator . . . . . . . . F.11 Wide Rectangular, Field-Excited Radiator . . . . . . . . . F.12 End Corrections . . . . . . . . . . . . . . . . . . . . . . . . F.13 Piston Radiating Into a Hard Tube . . . . . . . . . . . . . F.14 Oscillating Mass of a Fence in a Hard Tube . . . . . . . . F.15 A Ring-Shaped Piston in a Baffle Wall . . . . . . . . . . . F.16 Measures of Radiation Directivity . . . . . . . . . . . . . F.17 Directivity of Radiator Arrays . . . . . . . . . . . . . . . . F.18 Radiation of Finite Length Cylinder . . . . . . . . . . . . F.19 Monopole and Multipole Radiators . . . . . . . . . . . . . F.20 Plane Radiator in a Baffle Wall . . . . . . . . . . . . . . .

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287 289 291 295 297 299 301 309 309 311 313 316 327 328 329 330 330 335 337 339

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F.21 Ratio of Radiation and Excitation Efficiencies of Plates . . . . . . . . . . F.22 Radiation of Plates with Special Excitations . . . . . . . . . . . . . . . . . G Porous Absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.P. Mechel G.1 Structure Parameters of Porous Materials . . . . . . . . . . . . . G.2 Theory of the Quasi-homogeneous Material . . . . . . . . . . . G.3 Rayleigh Model with Round Capillaries . . . . . . . . . . . . . . G.4 Model with Flat Capillaries . . . . . . . . . . . . . . . . . . . . . G.5 Longitudinal Flow Resistivity in Parallel Fibres . . . . . . . . . . G.6 Longitudinal Sound in Parallel Fibres . . . . . . . . . . . . . . . G.7 Transversal Flow Resistivity in Parallel Fibres . . . . . . . . . . . G.8 Transversal Sound in Parallel Fibres . . . . . . . . . . . . . . . . G.9 Effective Wave Multiple Scattering in Transversal Fibre Bundle G.10 Biot’s Theory of Porous Absorbers . . . . . . . . . . . . . . . . . G.11 Empirical Relations for Characteristic Values of Fibre Absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . G.12 Characteristic Values from Theoretical Models Fitted to Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . H Compound Absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . F.P. Mechel H.1 Absorber of Flat Capillaries . . . . . . . . . . . . . . . . . . . . H.2 Plate with Narrow Slits . . . . . . . . . . . . . . . . . . . . . . . H.3 Plate with Wide Slits . . . . . . . . . . . . . . . . . . . . . . . . H.4 Dissipationless Slit Resonator . . . . . . . . . . . . . . . . . . . H.5 Resonance Frequencies and Radiation Loss of Slit Resonators H.6 Slit Array with Viscous and Thermal Losses . . . . . . . . . . . H.7 Slit Resonator with Viscous and Thermal Losses . . . . . . . . H.8 Free Plate with an Array of Circular Holes, with Losses . . . . H.9 Array of Helmholtz Resonators with Circular Necks . . . . . . H.10 Slit Resonator Array with Porous Layer in the Volume, Fields H.11 Slit Resonator Array with Porous Layer in the Volume, Impedances . . . . . . . . . . . . . . . . . . . . . H.12 Slit Resonator Array with Porous Layer on Back Orifice . . . . H.13 Slit Resonator Array with Porous Layer on Front Orifice . . . H.14 Array of Slit Resonators with Subdivided Neck Plate . . . . . H.15 Array of Slit Resonators with Subdivided Neck Plate and Floating Foil in the Gap . . . . . . . . . . . . . . . . . . . . H.16 Array of Slit Resonators Covered with a Foil . . . . . . . . . . H.17 Poro-elastic Foils . . . . . . . . . . . . . . . . . . . . . . . . . . H.18 Foil Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.19 Ring Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . H.20 Wide-Angle Absorber, Scattered Far Field . . . . . . . . . . . . H.21 Wide-Angle Absorber, Near Field and Absorption . . . . . . . H.22 Tight Panel Absorber, Rigorous Solution . . . . . . . . . . . .

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H.23 Tight Panel Absorber, Approximations . . . . . . . . . . . . . . . . . . . . H.24 Porous Panel Absorber, Rigorous Solution . . . . . . . . . . . . . . . . . .

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Sound Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.P. Mechel I.1 “Noise Barriers” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.2 Sound Transmission through a Slit in a Wall . . . . . . . . . . . . . . . . I.3 Sound Transmission through a Hole in a Wall . . . . . . . . . . . . . . . I.4 Hole Transmission with Equivalent Network . . . . . . . . . . . . . . . . I.5 Sound Transmission through Lined Slits in a Wall . . . . . . . . . . . . . I.6 Chambered Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.7 “Noise Sluice” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.8 Sound Transmissionßindexsound transmission through plates through Plates, Some Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . I.9 Sound Transmission through a Simple Plate . . . . . . . . . . . . . . . . I.10 Infinite Double-Shell Wall with Absorber Fill . . . . . . . . . . . . . . . . I.11 Double-Shell Wall with Thin Air Gap . . . . . . . . . . . . . . . . . . . . . I.12 Plate with Absorber Layer Behind . . . . . . . . . . . . . . . . . . . . . . I.13 Sandwich Panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.14 Finite-Size Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.15 Single Plate across a Flat Duct . . . . . . . . . . . . . . . . . . . . . . . . . I.16 Single Plate in a Wall Niche . . . . . . . . . . . . . . . . . . . . . . . . . . I.17 Strip-Shaped Wall in Infinite Baffle Wall . . . . . . . . . . . . . . . . . . . I.18 Finite-Size Plate with a Front Side Absorber Layer . . . . . . . . . . . . . I.19 Finite-Size Plate with a Back Side Absorber Layer . . . . . . . . . . . . . I.20 Finite-Size Double Wall with an Absorber Core . . . . . . . . . . . . . . . I.21 Plenum Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.22 Sound Transmission through Suspended Ceilings . . . . . . . . . . . . . I.23 Office Fences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.24 Office Fences, with Second Principle of Superposition . . . . . . . . . . I.25 Infinite Plate Between Two Different Fluids . . . . . . . . . . . . . . . . . I.26 Sandwich Plate with an Elastic Core . . . . . . . . . . . . . . . . . . . . . I.27 Wall of Multiple Sheets with Air Interspaces . . . . . . . . . . . . . . . .

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Duct Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.P. Mechel J.1 Flat Capillary with Isothermal Boundaries . . . . . . . . . . J.2 Flat Capillary with Adiabatic Boundaries . . . . . . . . . . . J.3 Circular Capillary with Isothermal Boundary . . . . . . . . . J.4 Lined Ducts, General . . . . . . . . . . . . . . . . . . . . . . . J.5 Modes in Rectangular Ducts with Locally Reacting Lining . J.6 Least Attenuated Mode in Rectangular, Locally Lined Ducts J.7 Sets of Mode Solutions in Rectangular, Locally Lined Ducts J.8 Flat Duct with a Bulk Reacting Lining . . . . . . . . . . . . . J.9 Flat Duct with an Anisotropic, Bulk Reacting Lining . . . . . J.10 Mode Solutions in a Flat Duct with Bulk Reacting Lining . .

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J.11 J.12 J.13 J.14 J.15 J.16 J.17 J.18 J.19 J.20 J.21 J.22 J.23 J.24 J.25 J.26 J.27 J.28 J.29 J.30 J.31 J.32 J.33 J.34 J.35 J.36 J.37 J.38 J.39 J.40 J.41 J.42 J.43 J.44 J.45 J.46

Flat Duct with Unsymmetrical, Locally Reacting Lining . . . . Flat Duct with an Unsymmetrical, Bulk Reacting Lining . . . . Round Duct with a Locally Reacting Lining . . . . . . . . . . . . Admittance of Annular Absorbers Approximated with Flat Absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . Round Duct with a Bulk Reacting Lining . . . . . . . . . . . . . Annular Ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Duct with a Cross-Layered Lining . . . . . . . . . . . . . . . . . Single Step of Duct Height and/or Duct Lining . . . . . . . . . . Sections and Cascades of Silencers, no Feedback . . . . . . . . . A Section with Feedback Between Sections Without Feedback . Concentrated Absorber in an Otherwise Homogeneous Lining Wide Splitter-Type Silencer with Locally Reacting Splitters . . . Splitter-Type Silencer with Locally Reacting Splitters in a Hard Duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Splitter Type Silencer with Simple Porous Layers as Bulk Reacting Splitters . . . . . . . . . . . . . . . . . . . . . . Splitter-Type Silencer with Splitters of Porous Layers Covered with a Foil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lined Duct Corners and Junctions . . . . . . . . . . . . . . . . . Sound Radiation from a Lined Duct Orifice . . . . . . . . . . . . Conical Duct Transitions; Special Case: Hard Walls . . . . . . . Lined Conical Duct Transition, Evaluated with Stepping Duct Sections . . . . . . . . . . . . . . . . . . . . . Lined Conical Duct Transition, Evaluated with Stepping Admittance Sections . . . . . . . . . . . . . . . . . Mode Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mode Excitation Coefficients . . . . . . . . . . . . . . . . . . . . Cremer’s Admittance . . . . . . . . . . . . . . . . . . . . . . . . . Cremer’s Admittance with Parallel Resonators . . . . . . . . . . Influence of Flow on Attenuation . . . . . . . . . . . . . . . . . . Influence of Temperature on Attenuation . . . . . . . . . . . . . Stationary Flow Resistance of Splitter Silencers . . . . . . . . . . Non-linearities by Amplitude and/or Flow . . . . . . . . . . . . Flow-Induced Non-linearity of Perforated Sheets . . . . . . . . Reciprocity at Duct Joints . . . . . . . . . . . . . . . . . . . . . . Mode Sets in Flat Ducts with Unsymmetrical, Locally Reacting Lining . . . . . . . . . . . . . . . . . . . . . . . . Mode Sets in Annular Ducts with Unsymmetrical, Locally Reacting Lining . . . . . . . . . . . . . . . . . . . . . . . . Mode Sets in Annular Ducts via Mode Sets in Flat Ducts with Unsymmetrical Lining . . . . . . . . . . . . . Bent, Flat Ducts with Locally Reacting Lining . . . . . . . . . . . Lined Bow Duct Between Lined Straight Ducts . . . . . . . . . . Zero-Order and First-Order Transmission Loss of Turning-Vane Splitter Silencers . . . . . . . . . . . . . . . . .

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625 628 629

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712 715 718 720 725 731 740 741 742 748 750

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754

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J.47 Bent and Straight Ducts with Unsymmetrical Linings . . . . . . . . . . . J.48 Silencer with Rectangular Turning-Vane Splitters . . . . . . . . . . . . . K Muffler Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.L. Munjal, F.P. Mechel K.1 Acoustic Power in a Flow Duct . . . . . . . . . . . . . . . . . . K.2 Radiation from the Open End of a Flow Duct . . . . . . . . . . K.3 Transfer Matrix Representation . . . . . . . . . . . . . . . . . . K.4 Muffler Performance Parameters . . . . . . . . . . . . . . . . . K.5 Uniform Tube with Flow and Viscous Losses . . . . . . . . . . K.6 Sudden Area Changes . . . . . . . . . . . . . . . . . . . . . . . . K.7 Extended Inlet/Outlet . . . . . . . . . . . . . . . . . . . . . . . . K.8 Conical Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.9 Exponential Horn . . . . . . . . . . . . . . . . . . . . . . . . . . K.10 Hose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.11 Two-Duct Perforated Elements . . . . . . . . . . . . . . . . . . K.12 Three-Duct Perforated Elements . . . . . . . . . . . . . . . . . K.13 Three-Duct Perforated Elements with Extended Perforations K.14 Three-Pass (or Four-Duct) Perforated Elements . . . . . . . . K.15 Catalytic Converter Elements . . . . . . . . . . . . . . . . . . . K.16 Helmholtz Resonator . . . . . . . . . . . . . . . . . . . . . . . . K.17 In-Line Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.18 Bellows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.19 Pod Silencer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.20 Quincke Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.21 Annular Airgap Lined Duct . . . . . . . . . . . . . . . . . . . . K.22 Micro-Perforated Helmholtz Panel Parallel Baffle Muffler . . K.23 Acoustically Lined Circular Duct . . . . . . . . . . . . . . . . . K.24 Parallel Baffle Muffler (Multipass Lined Duct) . . . . . . . . . L

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793

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793 795 796 796 798 799 801 803 804 804 806 814 820 825 828 830 831 831 832 833 834 836 837 839

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843

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843 847 853 857 861 865 869

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873

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873 876 877 879 882

Capsules and Cabins . . . . . . . . . . . . . . . . . . . . . . . . . . . F.P. Mechel L.1 The Energetic Approximation for the Efficiency of Capsules L.2 Absorbent Sound Source in a Capsule . . . . . . . . . . . . . L.3 Semicylindrical Source and Capsule . . . . . . . . . . . . . . L.4 Hemispherical Source and Capsule . . . . . . . . . . . . . . . L.5 Cabins, Semicylindrical Model . . . . . . . . . . . . . . . . . L.6 Cabin with Plane Walls . . . . . . . . . . . . . . . . . . . . . . L.7 Cabin with Rectangular Cross Section . . . . . . . . . . . . .

M Room Acoustics . . . . . . . . . . . . . . . . . . . . . . M. Vorländer, F.P. Mechel M.1 Eigenfunctions in Parallelepipeds . . . . . . . . M.2 Density of Eigenfrequencies in Rooms . . . . . . M.3 Geometrical Room Acoustics in Parallelepipeds M.4 Statistical Room Acoustics . . . . . . . . . . . . . M.5 The Mirror Source Model . . . . . . . . . . . . .

785 787

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M.5.1 Foundation of Mirror Source Approximation . . . . . . . . M.5.2 General Criteria for Mirror Sources . . . . . . . . . . . . . . M.5.3 Field Angle of a Mirror Source . . . . . . . . . . . . . . . . . M.5.4 Multiple Covering of MS Positions . . . . . . . . . . . . . . M.5.5 Convex Corners . . . . . . . . . . . . . . . . . . . . . . . . . M.5.6 Interrupt Criteria in the MS Method . . . . . . . . . . . . . M.5.7 Computational Parts of the MS Method . . . . . . . . . . . M.5.8 Inside Checks . . . . . . . . . . . . . . . . . . . . . . . . . . M.5.9 What Is Needed in the Traditional MS Method? . . . . . . . M.5.10 The Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.5.11 A Concave Model Room, as an Example . . . . . . . . . . . M.5.12 The MS Method in Rooms with Convex Corners . . . . . . M.5.13 A Model Room with Convex Corners . . . . . . . . . . . . . M.5.14 Other Grouping of Mirror Sources . . . . . . . . . . . . . . M.5.15 Combination of Corner Fields to Obtain the Room Field . M.5.16 Collection of the MSs of a Wall Couple in a Corner Source M.5.17 A Kind of Reciprocity in the MS Method . . . . . . . . . . . M.5.18 Limit Case of Parallel Walls . . . . . . . . . . . . . . . . . . M.5.19 The Second Principle of Superposition (PSP) . . . . . . . . M.5.20 The PSP for Unsymmetrical Absorption . . . . . . . . . . . M.5.21 A Global Application of the PSP . . . . . . . . . . . . . . . . M.5.22 Reverberation Time with Results of the MS Method . . . . M.5.23 A Room with Concave Edges as an Example . . . . . . . . . M.6 Ray-Tracing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.7 Room Impulse Responses, Decay Curves and Reverberation Times . . . . . . . . . . . . . . . . . . . . . . . . . M.8 Other Room Acoustical Parameters . . . . . . . . . . . . . . . . . . . N Flow Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Költzsch N.1 Concepts and Notations in Fluid Mechanics, in Connection with the Field of Aeroacoustics . . . . . . . . . . . . . . . . . N.1.1 Types of Fluids . . . . . . . . . . . . . . . . . . . . . . N.1.2 Properties of Fluids . . . . . . . . . . . . . . . . . . . N.1.3 Models of Fluid Flows . . . . . . . . . . . . . . . . . . N.2 Some Tools in Fluid Mechanics and Aeroacoustics . . . . . . N.2.1 Averaging . . . . . . . . . . . . . . . . . . . . . . . . . N.2.2 Decomposition (in General) . . . . . . . . . . . . . . N.2.3 Decomposition of the Physical Quantities in the Basic Equations . . . . . . . . . . . . . . . . . . N.2.4 Correlations . . . . . . . . . . . . . . . . . . . . . . . N.2.5 Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . N.3 The Basic Equations of Fluid Motion . . . . . . . . . . . . . . N.3.1 Continuity Equation, Momentum Equation, Energy Equation . . . . . . . . . . . . . . . . . . . . . N.3.2 Thermodynamic Relationships . . . . . . . . . . . .

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882 883 884 885 886 887 888 888 889 890 891 896 899 903 906 907 910 910 913 920 921 922 924 935

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N.4 N.5 N.6

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N.9

N.10

Non-linear Perturbation Equations, non-linear Euler Equations . . . . . . . . . . . . . . . . . . . . . . N.3.4 Formulation of Euler Equations to Use in Computational Aeroacoustics (CAA) . . . . . . . . . . . . . . The Equations of Linear Acoustics . . . . . . . . . . . . . . . . . . . . . . Inhomogeneous Wave Equation, Lighthill’s Acoustic Analogy . . . . . . N.5.1 Lighthill’s Inhomogeneous Wave Equation . . . . . . . . . . . . N.5.2 Solutions of Inhomogeneous Wave Equation . . . . . . . . . . . Acoustic Analogy with Source Terms Using Pressure . . . . . . . . . . . N.6.1 Lighthill’s Representation of the Source Term with Use of Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . N.6.2 Pressure-Source theory (Ribner) . . . . . . . . . . . . . . . . . . N.6.3 Pressure-Source Theory (Meecham) . . . . . . . . . . . . . . . . Acoustic Analogy with Mean Flow Effects, in the Form of Convective Inhomogeneous Wave Equation . . . . . . . . . . . . . . . N.7.1 Phillips’s Convective Inhomogeneous Wave Equation . . . . . . N.7.2 Lilley’s Convective Inhomogeneous Wave Equation . . . . . . . N.7.3 Lilley’s Wave Equation with a New Lighthill Stress Tensor . . . N.7.4 Convected Wave Equation for the Dilatation (Legendre) . . . . N.7.5 Goldstein’s Third-Order Inhomogeneous Wave Equation . . . . N.7.6 Goldstein-Howes Inhomogeneous Wave Equation . . . . . . . . N.7.7 Ribner’s Recent Reformulation of Lighthill’s Source Term . . . N.7.8 Inhomogeneous Wave Equation Including Stream Function (Albring/Detsch) . . . . . . . . . . . . . . . . . . . . . . . . . . . Acoustic Analogy in Terms of Vorticity, Wave Operators for Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N.8.1 Powell’s Theory of Vortex Sound . . . . . . . . . . . . . . . . . . N.8.2 Howe’s Formulation of Acoustic Analogy Equation for Total Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . N.8.3 Möhring’s Equation with Source Term Linearly Dependent on Vorticity Field . . . . . . . . . . . . . . . . . . . . . . . . . . . N.8.4 Convected Wave Operators for Total Enthalpy in Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N.8.5 Doak’s Theory of Aerodynamic Sound Including the Fluctuating Total Enthalpy as a Basic Generalised Acoustic Field for a Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acoustic Analogy with Effects of Solid Boundaries . . . . . . . . . . . . . N.9.1 Ffowcs Williams–Hawkings (FW-H) Inhomogeneous Wave Equation, FW-H Equation in Differential and Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . . N.9.2 Curle’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acoustic Analogy in Terms of Entropy, Heat Sources as Sound Sources, Sound Generation by Turbulent Two-Phase Flow . . . . . . . . . . . . . . N.10.1 Acoustic Analogy in Terms of Entropy, Sound Generation by Fluctuating Heat Sources (Dowling, Howe) . . . . . . . . . .

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956 958 960 963 963 965 967 967 968 969 970 970 971 972 972 973 973 974 975 976 976 977 980 980 981 984 984 988 988 988

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N.10.2 Acoustic Analogy in Terms of Heat Release, Turbulent Density Fluctuations and Turbulent Velocity Fluctuations on Outer Flame Surface (Strahle) . . . . . . . . . . . . . . . . N.10.3 Sound Power Radiated by a Turbulent Flame . . . . . . . . . N.10.4 Sound Generation by Turbulent Two-Phase Flow . . . . . . . N.11 Acoustics of Moving Sources . . . . . . . . . . . . . . . . . . . . . . . N.11.1 Sound Field of Moving Point Sources . . . . . . . . . . . . . . N.11.2 Formulation of Equation of Sound Sources in Motion Based on Ffowcs Williams–Hawkings Equation . . . . . . . . . . . . N.11.3 Moving Kirchhoff Surfaces . . . . . . . . . . . . . . . . . . . . N.12 Aerodynamic Sound Sources in Practice . . . . . . . . . . . . . . . . N.12.1 Jet Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N.12.2 Rotor Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N.13 Power Law of the Aerodynamic Sound Sources . . . . . . . . . . . . . O Analytical and Numerical Methods in Acoustics . . . . . . . . . . . . . . M. Ochmann, F.P. Mechel O.1 Computational Optimisation of Sound Absorbers . . . . . . . . . . O.2 Computing with Mixed Numeric-Symbolic Expressions, Illustrated with Silencer Cascades . . . . . . . . . . . . . . . . . . . O.3 Five Standard Problems of Numerical Acoustics . . . . . . . . . . . O.3.1 The Radiation Problem . . . . . . . . . . . . . . . . . . . . . O.3.2 The Scattering Problem . . . . . . . . . . . . . . . . . . . . . O.3.3 The Sound Field in Interior Spaces . . . . . . . . . . . . . . O.3.4 The Coupled Fluid–Elastic Structure Interaction Problem O.3.5 The Transmission Problem . . . . . . . . . . . . . . . . . . . O.4 The Source Simulation Technique (SST) . . . . . . . . . . . . . . . . O.4.1 General Description of the Source Simulation Technique . O.4.2 Spherical Wave Functions and Symmetry Relations . . . . O.4.3 Variants of the SST with Spherical Wave Functions . . . . . O.4.4 Position of Sources and Their Optimal Choice . . . . . . . O.4.5 Numerical Aspects . . . . . . . . . . . . . . . . . . . . . . . . O.4.6 A Numerical Example: Sound Scattering from a Non-Convex Cat’s-Eye Structure . . . . . . . . . . . O.4.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . O.5 The Boundary Element Method (BEM) . . . . . . . . . . . . . . . . O.5.1 Boundary Integral Equations . . . . . . . . . . . . . . . . . O.5.2 Discretization of the Boundary Integral Equation . . . . . O.5.3 Solution of the Linear System of Equations . . . . . . . . . O.5.4 Critical Frequencies and Other Singularities . . . . . . . . O.5.5 The Interior Problem: Sound Fields in Rooms and Half-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . O.5.6 The Scattering and the Transmission Problem . . . . . . . O.6 The Finite Element Method (FEM) . . . . . . . . . . . . . . . . . . . O.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The Sound Field in Irregular Shaped Cavities with Rigid Walls . . . . . . . . . . . . . . . . . . . . . . . O.6.3 Supplementary Aspects and Fluid–Structure Coupling O.7 The Cat’s Eye Model . . . . . . . . . . . . . . . . . . . . . . . . . . O.7.1 Cat’s Eye Model and General Fundamental Solutions∗) O.7.2 Mode Orthogonality . . . . . . . . . . . . . . . . . . . . . O.7.3 Remaining Boundary Conditions . . . . . . . . . . . . . O.7.4 Mode Coupling Integrals . . . . . . . . . . . . . . . . . . O.7.5 Reduction of the System of Equations . . . . . . . . . . O.8 The Orange Model . . . . . . . . . . . . . . . . . . . . . . . . . . O.8.1 Elementary Solutions and Field Formulations . . . . . O.8.2 Orthogonality of Modes . . . . . . . . . . . . . . . . . . O.8.3 Field Matching . . . . . . . . . . . . . . . . . . . . . . . . O.8.4 Mode Coupling Integrals and Mode Norms . . . . . . . O.8.5 Reduction of the Systems of Equations . . . . . . . . . . O.8.6 Numerical Examples . . . . . . . . . . . . . . . . . . . .

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P Variational Principles in Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . A. Cummings P.1 Eigenfrequencies of a Rigid-Walled Cavity and Modal Cut-on Frequencies of a Uniform Flat-Oval Duct with Zero Mean Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . P.2 Sound Propagation in a Uniform Narrow Tube of Arbitrary Cross-Section with Zero Mean Fluid Flow . . . . . . . . . . P.3 Sound Propagation in a Uniform, Rigid-Walled, Duct of Arbitrary Cross-Section with a Bulk-Reacting Lining and no Mean Fluid Flow: Low Frequency Approximation . . . . . . . . . P.4 Sound Propagation in a Uniform, Rigid-Walled, Rectangular Flow Duct Containing an Anisotropic Bulk-Reacting Wall Lining or Baffles . . . . . P.5 Sound Propagation in a Uniform, Rigid-Walled, Flow Duct of Arbitrary Cross-Section, with an Inhomogeneous, Anisotropic Bulk Lining . . . . P.6 Sound Propagation in a Uniform Duct of Arbitrary Cross-Section with one or more Plane Flexible Walls, an Isotropic Bulk Lining and a Uniform Mean Gas Flow . . . . . . . . . P.7 Sound Propagation in a Rectangular Section Duct with four Flexible Walls, an Anisotropic Bulk Lining and no Mean Gas Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Q Elasto-Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Maysenhölder, F.P. Mechel Q.1 Fundamental Equations of Motion . . . . . . . . . . . . . . . . . Q.2 Anisotropy and Isotropy . . . . . . . . . . . . . . . . . . . . . . . Q.3 Interface Conditions, Reflection and Refraction of Plane Waves Q.4 Material Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . Q.5 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q.5.1 General Relations . . . . . . . . . . . . . . . . . . . . . .

1133 1135 1139 1140 1143 1143

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XVIII

Table of Contents

Q.6 Q.7 Q.8 Q.9 Q.10

Q.11 Q.12 Q.13 Q.14 Q.15 Q.16 Q.17 Q.18 Q.19 Q.20

Q.5.2 Surface Intensity . . . . . . . . . . . . . . . . . Q.5.3 Time-Harmonic Wavefields . . . . . . . . . . Q.5.4 Rayleigh’s Principle . . . . . . . . . . . . . . . Q.5.5 Energy Velocity and Group Velocity . . . . . Random Media . . . . . . . . . . . . . . . . . . . . . . Periodic Media . . . . . . . . . . . . . . . . . . . . . . . Homogenisation . . . . . . . . . . . . . . . . . . . . . Q.8.1 Bounds on Effective Moduli . . . . . . . . . . Q.8.2 Effective Moduli for Particular Structures . . Plane Waves in Unbounded Homogeneous Media . . Q.9.1 Anisotropic Media . . . . . . . . . . . . . . . . Q.9.2 Isotropic Media . . . . . . . . . . . . . . . . . Waves in Bounded Media . . . . . . . . . . . . . . . . . Q.10.1 Plate Waves . . . . . . . . . . . . . . . . . . . . Q.10.2 Rayleigh Waves . . . . . . . . . . . . . . . . . . Q.10.3 Waves in Thin Plates . . . . . . . . . . . . . . . Q.10.4 Waves in Thin Beams . . . . . . . . . . . . . . Moduli of Isotropic Materials and Related Quantities Modes of Rectangular Plates . . . . . . . . . . . . . . . Partition Impedance of Plates . . . . . . . . . . . . . . Partition Impedance of Shells . . . . . . . . . . . . . . Density of Eigenfrequencies in Plates, Bars, Strings, Membranes . . . . . . . . . . . . . . . . . . . . Foot Point Impedances of Forces . . . . . . . . . . . . Transmission Loss at Steps, Joints, Corners . . . . . . Cylindrical Shell . . . . . . . . . . . . . . . . . . . . . . Similarity Relations for Spherical Shells . . . . . . . . Sound Radiation From Plates . . . . . . . . . . . . . .

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1143 1144 1144 1145 1145 1146 1148 1148 1149 1151 1151 1153 1154 1154 1159 1160 1163 1165 1170 1174 1176

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1178 1179 1184 1186 1190 1191

R Ultrasound Absorption in Solids . . . . . . . . . . . . . . . . . . W. Arnold R.1 Generation of Ultrasound . . . . . . . . . . . . . . . . . . . R.2 Ultrasonic attenuation . . . . . . . . . . . . . . . . . . . . . R.3 Absorption and Dispersion in Solids Due to Dislocations . R.4 Absorption Due to the Thermoelastic Effects, Phonon Scattering and Related Effects . . . . . . . . . . . . R.5 Interaction of Ultrasound with Electrons in Metals . . . . R.6 Wave Propagation in Piezoelectric Semiconducting Solids R.7 Absorption in Amorphous Solids and Glasses . . . . . . . R.8 Relation of Ultrasonic Absorption to Internal Friction . . R.9 Gases and Liquids . . . . . . . . . . . . . . . . . . . . . . . R.10 Kramers-Kroning Relation . . . . . . . . . . . . . . . . . . .

. . . . . . . . 1197 . . . . . . . . 1198 . . . . . . . . 1199 . . . . . . . . 1204 . . . . . . .

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1206 1208 1210 1210 1211 1211 1211

Chapter Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215 General Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1251

Contributors Prof. Dr. M. L. Munjal Dept. Mechanical Engineering Indian Institute of Science Bangalore 560 012 India Prof. Dr. M. Vorländer Institut f u¨ r Technische Akustik RWTH Aachen Templergraben 55 52056 Aachen Germany Prof. Dr. Peter Költzsch Jägerstraße 17 01099 Dresden Germany Prof. Dr. M. Ochmann Technische Fachhochschule Berlin Luxemburger Straße 10 13353 Berlin Germany Prof. Dr. A. Cummings Trenwith Ludlow Road, Little Stretton, Church Stretton Salop SY6 6RB UK Prof. Dr. W. Maysenhölder Altenbergstraße 33 70180 Stuttgart Germany Prof. Dr. W. Arnold Frauenhofer Institut f u¨ r Prüfverfahren Universität Saarbrücken, Gebäude 37 66123 Saarbrücken Germany

A Conventions The following conventions will be used in the book. Exceptions will be clearly noted in the respective Sections.

Time Factor

√ • The time factor for harmonic oscillations and waves is ej – t ; j = −1. This choice implies that the imaginary part of impedances with mass reaction are positive, with spring reaction negative, and the imaginary part of admittances with mass reaction are negative, with spring reaction positive. • If not stated differently, the time function is assumed to be ej – t ; the time factor then is dropped, mostly.

Impedance and Admittance • The term impedance is used for the ratio of sound pressure p to the vector component v of particle velocity in some specified direction, Z = p/v. • Mechanical impedance is used for the ratio of a vector component of force F to particle velocity v in that direction, Zm = F/v. • Flow impedance is used for the ratio of sound pressure p to volume flow q = S · v through a surface S, with v the velocity component normal to S. • Admittance is the ratio of the vector v of particle velocity to sound pressure p. The admittance is is a true vector (in contrast to the reciprocal of an impedance). • Mechanical admittance is the reciprocal of mechanical impedance. • Flow admittance is the ratio of the flow vector q to sound pressure; it is a true vector.

Sound Intensity and Power Sound intensity is the vector I = p · v ∗ (where the asterisk indicates the complex conjugate); I stands for the oscillating sound power in the direction of v through a unit surface. The effective (or active) intensity is the real part thereof in the time average; the reactive intensity is the imaginary part of the time average. (The formally possible definition I∗ = p∗ · v would produce conflicts at sound sources, and should be avoided, therefore.) Sound power is the integral of the scalar product of sound intensity with the surface element vector ds over a surface S: ¢ = I · ds. S

Dimensions Mostly in mks units.Where necessary, the dimension of a quantity is indicated in brackets [. . . ].

A

2

Conventions

Complex Quantities Field quantities, such as sound pressure p, particle velocity v, oscillating parts of density , and temperature T, etc. are mostly complex. If one records such a quantity in an oscillogram, one may take either the real or the imaginary part of a complex expression, after multiplication by the dropped time factor. If one records the amplitude, this corresponds to taking the (absolute) magnitude of the complex quantity.

Symbol “Decorations” Unnecessary symbol decorations, such as hats for amplitudes, underbars for complex quantities, etc. are avoided. If necessary in the local context, an arrow indicates a vector v ; a star is used for the complex conjugate p∗ ; primes are used either for the derivative of functions, f (x) , f (x), or (where no ambiguity is possible) for the real and imaginary parts of complex quantities, p = p + j · p .

Commonly Used Symbols The following symbols are commonly used in most sections of the book. If a section uses the same symbol with a different definition, it will be noted. c0 f j k0 p q t v Z0 ‰ Š Š0 0 – ˜ œ ¢ ¡

∗)

adiabatic sound speed in the medium (e.g. in air) [m/s]; frequency [Hz]; √ = (−1) imaginary unit; = –/c0 free field wave number of a plane wave [1/m]; sound pressure [Pa] = [N/m2 ]; ∗) volume flow [m3 /s]; ∗) time [s]= [sec]; velocity [m/s]; = 0 c0 wave impedance of free plane wave [Pa·s/m]; sound absorption coefficient; adiabatic exponent of the medium; wavelength [m]; wavelength of free plane wave; mass density [kg/m3 ]; mass density of the medium; = 2 · f angular (or circular) frequency [1/s]; ∗) polar angle of cylindrical or spherical co-ordinates; azimuthal angle of spherical co-ordinates; power; p/(d · v) flow resistivity of porous material [Pa·s/m2 ] (flow resistance per unit thickness d)

See Preface to the 2nd edition

Conventions

A

3

Numbering of Equations Equations are numbered, beginning with number (1) in each Section. Reference to an equation is made as, e.g.,“Eq. (x)” to an equation with number (x) in the same Section, or as, e.g.,“Eq. (K.y.x)” to the equation with number (x) in the Section with number y of the Chapter K.

Conversions in Plot Labels Some diagrams were generated by the computing program Mathematica . This program has lost its ability to write plot labels in a suitable form for exportation to text or graphic programs. Equivalences between general notations and plot label forms will be marked near the plots, if necessary.

B General Linear Fluid Acoustics The medium does not support shear stresses, except viscous shear. The medium parameters are constant in time; stationary flow does not exist, or its velocity is low enough, to be neglected in its influence on the sound field; see > Ch. N, “Flow Acoustics”, for sound fields in flows.

B.1

Fundamental Differential Equations

No viscous and/or caloric losses: Conservation of mass: Conservation of impulse:

∂ + 0 div v = 0 q · ƒ(r − rq ). ∂t ∂v 0 = −grad p. ∂t

Equation of state:

p = c20 · .

Relation between pressure and particle velocity:

v =

Homogeneous wave equation for a harmonic wave:

p + k02 · p = 0.

j grad p. k0Z0

Helmholtz’s wave equation for harmonic wave with monopole source at rq : + k02 p = −j k0 Z0 · q · ƒ(r − rq ). Adiabatic sound velocity:

c20 =

‰ P0 . 0

p = sound pressure; v = particle velocity; = density; = Laplace operator; P0 = atmospheric pressure; q = volume flow density of monopole source; r = space co-ordinate; rq = source position; ƒ = Dirac delta function; for other symbols, see “Conventions” Boundary conditions: (on both sides of boundary)

• Matching of sound pressures, • Matching of normal particle velocities;

(1) (2) (3) (4) (5) (6) (7)

B

6

General Linear Fluid Acoustics

or: (for waves on both sides)

• Matching of phase velocities parallel to boundary and • Matching of normal field admittances on both sides of boundary.

Medium with viscous and caloric losses: See also: Mechel (1995) Field quantities, pressure p, density and (absolute) temperature T, are composed of stationary parts (with subscript 0 ) and oscillating parts (with subscript 1 ). Velocities v are oscillating particle velocities. The sound field is composed of three coupled waves: the density wave (index ), the viscous shear wave (index Œ) and the heat wave (index ). 1 1 ∂v grad p1 − Œ v − Œ grad div v = 0. (8) + Impulse equation: ∂t 0 3 ∂T1 + (‰ − 1) T0 div v − T1 = 0. Heat balance: (9) ∂t ∂1 Conservation of mass: + 0 div v = 0. ∂t p1 1 T1 Equation of state: − − = 0. (10) p0 0 T0 Heat conduction inside a ∂Ti1 − i Ti1 = 0. (11) bordering medium (index i): ∂t p0 0 T0 c0 ‰ Œ

= = = = = = = = = cp =

atmospheric pressure; stationary density; absolute temperature; adiabatic speed of sound; adiabatic exponent; kinematic viscosity; temperature conductivity /(0 cp ); heat conductivity; specific heat at constant pressure

Field composition with potentials (according to Rayleigh) ¥ is a scalar potential; ¦ is a vector potential with

rot grad ¥ ≡ 0

With vector identity

= grad div − rot rot

one gets:

−grad[j– ¥ −

Both terms vanish individually (Rayleigh’s postulate): Equivalent to two wave equations:

v = −grad ¥ + rot ¦ ;

(12) div ¦ ≡ 0.

(13) (14)

p1 4 ≡ 0. (15) − Œ ¥]+rot[j– ¦ −Œ ¦] 0 3

p1 4 − Œ ¥ = 0 ; j– ¦ − Œ ¦ = 0. (16) 0 3 ( + kŒ2 ) ¦ = 0 ; ( + k2 ) ( + k2 ) ¥ = 0. (17)

j– ¥ −


B

Characteristic (plane) wave numbers: – • for viscous wave: kŒ2 = −j ; Œ • for density wave k and thermal wave k : 2 2 2 2 4 4 4 c0 c0 c0 +j + Œ ± +j + Œ + jŒ − 4j − – 3 – 3 ‰– 3 k2 = j– . 2 k2 4 c0 + jŒ 2 ‰– 3

k2 – ‰– −1 ± 1 − 4j 2 , ≈j Approximations to wave numbers: 2 ‰c0 k2 k2 ≈ (–/c0 )2

or with lower degree of precision:

;

7

(18)

(19)

(20)

k2 ≈ −j‰–/ = ‰Œ/ · kŒ2.

Decomposition of scalar potential for density wave ¥ and thermal wave ¥ : ¥ = ¥ + ¥ ( + k2 ) ¥ = 0 ; ( + k2 ) ¥ = 0. (21)

1 j 2 k ¥ + k2 ¥ . (22) = 0 – p1 = ¢ ¥ + ¢ ¥. p0 2 2 j k, ‰– − jk, ‰ 4 2 . ¢, = 2 j– + Œ k, = 2 3 – – − jk, c0

with wave equations: Relative variation of density: Relative variation of pressure: with sound pressure coefficients:

(23) T1 = Ÿ ¥ + Ÿ ¥ T0 2 2 j(‰ − 1)k, 4 ‰Œ 2 ‰ k, = Ÿ, = k +j – 2 − . 2 , 2 3 c0 – – − jk, c0

Relative variation of temperature: with temperature coefficients:

(24) Approximations to wave numbers and coefficients: 2 – ‰– – 2 ; (25) ; kŒ2 = −j ; k0 = −j with wave number definitions: k02 = c0 Œ 1 1 1 4 ‰ 4 ‰ 2 4‰ 4

+ + 2 + + + 2 − 2 + k2 k02 3kŒ2 k0 k02 3kŒ2 k0 k0 ‰k02 3kŒ2 . = 1 4 2‰ k2 + 2 (26) k0 ‰k02 3kŒ2

≈

1 2 k 2 0

k2 · 1 + 1 − 4 20 k0

≈

k02 2 2 k0 · (1 − k02/k0 )

≈

k02 2 k0

B

8


Approximations: 2 k0 4 Pr 1 − 1+‰ 1+ 1 + 1, 2165 2 3 ‰ k 0 2 2 2 k ≈ k0 = k0 4‰2 Pr k02 1+ 1 + 1, 8259 2 3 k0

k02 2 k0 , k02 2 k0

(27)

Ÿ k2 ≈ −(‰ − 1) 20 , Ÿ k0 ¢ ≈ ‰ j

k2

≈‰j

–

jk 2 ¢ ≈ 0 –

k02 , –

2 k0 4‰ Pr 1− 2 3 4‰ Pr jk02 k0 ‰ 1 − , ≈ – 3 k02 1+‰ 2 k0

‰

(28)

(29)

4‰ Pr ¢ ≈1− = −0.3033. ¢ 3

(30)

Pr = Œ/ Prandtl number Boundary conditions with vt = tangential velocity, vn= normal velocity, Ti= temperature behind the boundary; = heat conductivity of the medium with the sound wave, i = heat conductivity of the medium behind the boundary: vt = 0

;

T1 = T1,i

;

vn = vn,i , ∂ ∂ T1 = i T1,i . ∂n ∂n

(31)

Isothermal boundary condition:

T1 = 0.

(32)

Adiabatic boundary condition:

∂ T1 = 0. ∂n

(33)

B.2

Material Constants of Air

See also: Mechel (1995); VDI-Wärmeatlas (1984)

For definitions of symbols see

> Sect. B.1 of this chapter and Table 1.Regressions (range

see Fig. 1) using measured data are given in the form f (T) =

4

i=−4

ai · Ti/2 for the material

constants of dry air as functions of (absolute) temperature T (in Kelvin degrees K). The atmospheric pressure is assumed to be P0 =1 [bar]=105 [Pa].The range of application of the regressions is 100 K ≤T≤ 1500 K.


B

9

Table 1 Material constants of air at standard conditions (20ı C; 1 bar) Quantity

Symbol

Value

Dimension

Remark

Molekular weight

M

28.96

kg/kmol

Dry air

Gas constant

R

287.10

J/kgK

Ideal gas

Density

0

1.1886

kg/m3

Sound velocity

c0

343.30

m/s

Dynamical viscosity

†

17.9910−6

Ns/m2

Kinematic viscosity

Œ

15.1310−6

m2 /s

Œ = †=0

Adiabatic exponent

‰

1.401

–

‰ =cp /cv

J/kgK

P const.

3

c20 =‰P0 /0

Specific heat

cp

1.00710

Temperature expansion

3.42110−3

1/K

Heat cconductivity

0.02603

W/mK

Temperature conductivity

/

21.7410−6

m2 /s

/= =(0 cp)

Prandtl number

Pr

0.6977

–

Pr=Œ//

Interrelations are: Prandtl number: Specific heat at constant volume: Isothermal compressibility:

Pr = Œ/. ß2 T . K0 1 ∂ 1 ∂V = . K=− V ∂P T ∂P T cv = cp −

(1) (2) (3)

V = volume; P = static pressure; = coefficient of thermal volume expansion Temperature dependence of Prandtl number: Pr = 0.66000 + 6.5853 · 10−6 · (T − 700) + 3.97457 · 10−7 · (T − 700)2 −1.43416 · 10−12 · (T − 700)4 + 3.05114 · 10−18 · (T − 700)6. Sound velocity: 1 ‰ P0 p˜ v 2 ‰R T = = = ≈ c0 = ˜ 0 3 M K 0 0 T ≈ 333m/s + 0.6 · Ÿ˚C ≈ 20, 05 T˚K . = 108.28 M

(4)

(5)

10

B


Sound velocity of a mixture of two gas components (x is the concentration of the component with primes): c2x =

x · cp + (1 − x) · cp RT . x · M + (1 − x) · M x · cp /‰ + (1 − x) · cp /‰

‰ = adiabatic exponent; P0 = atmospheric pressure; 0 = atmospheric density; p˜ = sound pressure; ˜ = oscillating density; = compressibility; K0 < v 2 > = average square of molecular velocities; R = universal gas constant; M = molecular weight, Ÿ = temperature in Celsius; cp = specific heat at constant pressure Example for measured data (points) and regression (curve):

Figure 1 Adiabatic exponent ‰ as function of absolute temperature T. Points: measured; curve: regression

(6)


B

11

Table 2 Regression coefficients for material data as functions of (absolute) Temperature T Quantity

a0

a˙1

a˙2

a˙3

a˙4

0 kg=m3

−29:2987

† Ns=m2

−3:30199 10−4 1:39487 10−5 4:35462 10−3

−2:29854 10−7 −0:0294172

1:43167 10−9 0:0740619

4:55963 10−12 0:03768996

Œ m2 /s

1:04734 10−4

−1:00547 10−5 −2:16340 10−4

4:03090 10−7 −3:69703 10−3

−3:87707 10−9 0:0183863

6:20832 10−11 9:00314 10−3

‰ –

25:9651

−1:08207 ) −313:593

0:0273543 2:04477 103

−3:79526 10−4 −4:89956 103

2:26518 10−6 −2:50299 103

cp J/kgK

1:66918 104 )

−983:174 −1:29648 105

32:7843 4:55412 105

−0:540032 −1:55411 105

3:48332 10−3 −1:17469 105

W/mK

7:13849

−0:400186 −72:5736

0:0129070 386:078

−2:17156 10−4 −778:310

1:4793566 10−6 −403:616

/ m2 /s

0:0128841

−7:09636 10−4 −0:133306

2:21478 10−5 0:723019

−3:54709 10−7 −1:49085

2:33895 10−9 −0:771509

1/K

0:0762123

4:36358 10−3 −0:695016

1:37872 10−4 3:55119

−2:28121 10−6 0:516673

1:54530 10−8 −0:098786

B.3

1:38519 363:205

−0:0384181 −2:08219 103

10−4

5:78952 6:48716 103

−3:65858 10−6 3:25451 103

General Relation for Field Admittance and Intensity

See also: Mechel, Vol. I, Ch. 3 (1989)

The vector component Gn in a direction n of the field admittance G j ∂p/∂n vn is defined by Gn = = . p k0Z0 p

(1)

If the sound pressure is described by magnitude and phase p(r) = |p(r)| · ej œ(r) , (2) ∂ ∂ 1 ln |p(r)| . (3) the field admittance is given by Gn (r) = − œ(r) + j · k0Z0 ∂n ∂n Near an absorbing wall the reactance of the wall admittance determines the slope of sound pressure level by the term ln(|p(r)|) (“admittance rule”). The time-averaged intensity of a harmonic wave is With the admittance relation follows:

1 1 p · vn∗ = G∗n · |p|2 . 2 2 ∂ |p(r)|2 ∂œ(r) +j· ln |p(r)| ; · In = − 2 k0Z0 ∂n ∂n

In =

(4) (5)

the real part of In is the effective intensity, and the imaginary part the reactive intensity. ∗)

See Preface to the 2nd edition.

B

12


G(r) =

In vector notation:

1 −grad œ(r) + j · grad ln |p(r)| , k0Z0

|p(r)|2 I = − · grad œ(r) + j · grad ln |p(r)| . 2 k0 Z0

System of two coupled differential equations for magnitude and phase of sound pressure (with the relations from above):

(6)∗)

|p(r)| + k02 1 − Z20 · Re{G(r)} · |p(r)| = 0, œ(r) − 2 k02Z20 · Re{G(r)} · Im{G(r)} = 0.

If a sound field has no sources or sinks, then div Re{I} = 0. The effective intensity Ieff = Re{I} has the rotation

rot Ieff

(7) −1 = |p(r)| · grad œ(r) × grad |p(r)| k0 Z0

(with × for the cross product of vectors). It follows that rot Ieff =0 if phase œ(r) and magnitude |p(r)| have parallel gradients (as in a plane wave).

B.4

Integral Relations

See also: Pierce (1981) and others

Consider two different sound fields p1 , p2 in a volume V with a bounding surface S (with outwards directed surface element ds ). Green’s integral is then 2 − p2 · ∇p 1 · ds. (1) p1 · p2 − p2 · p1 dr = p1 · ∇p V

S

The fields may differ either by different source strengths and/or locations, and/or by different boundary conditions on S, and/or are different forms (modes) for the same sources and boundaries. The surface S is either soft (p(S)=0) or hard (∂p/∂n=0) on parts S0 or locally reacting on parts Sa with surface admittance G, or parts S∞ are at infinity, where the fields obey Sommerfeld’s condition. With the fundamental relations of > Sect. B.1 it follows that p1 · q2 · ƒ(r − r2) dr− p2 · q1 · ƒ(r−r1 ) dr (2)

p1 · v2 · ds − p2 · v1 · ds = S

S

V

V

if field p1 has a source with volume flow q1 at r1 and field p2 has a source with volume flow q2 at r2 . Integration over the Dirac delta functions gives

p1 · v2 · ds − p2 · v1 · ds = p1 (r2 ) · q2 − p2 (r1 ) · q1 . (3) S

S

The reciprocity principle follows if both p1 and p2 everywhere satisfy the same boundary conditions: p1 (r2) · q2 = p2 (r1 ) · q1 . ∗)

See Preface to the

2nd

edition.

(4)


◦

If both fields are source free, then If, additionally, they satisfy the same boundary condition on a part, e.g. Sa , of the surface S, then

S

p1 · v2 · ds −

Sa +S∞

13

(5)

p1 · v2 · ds −

If Sa is hard for p1 and/or soft for p2 , then and if they obey the same far field conditions, then

◦ p2 · v1 · ds = 0. S

B

p2 · v1 · ds = 0. (6)

Sa +S∞

p1 · v2 · ds − Sa +S∞

p2 · v1 · ds = 0, (7) S∞

p1 · v2 · ds = 0.

(8)

Sa

If one or both fields have sources, the relevant source terms appear on the right-hand sides.

B.5

Green’s Functions and Formalism

See also: Skudrzyk (1971)

In a loss-free medium it is convenient to formulate the wave equation for the sound j grad p. (1) pressure field p. The particle velocity is then v = k0 Z0 Let r be a general co-ordinate. (2) The homogeneous wave equation is p(r) + k02 · p(r) = 0. The inhomogeneous wave equation with a source of volume flow q(r) is

p(r) + k02 · p(r) = −j k0 Z0 q(r).

(3)

Here q(r) is the rate of volume generation per unit volume and unit time. Green’s formalism uses a potential function g for the field (instead of the sound pressure function) ∂g i.e. v = −grad g ; p = 0 . (4) ∂t The Green’s function g(r|rq , –) is the solution of the inhomogeneous wave equation for a time harmonic excitation by a point source in rq of unit strength, which satisfies specified boundary conditions, with the Dirac delta function: Any solution h(r) of the homogeneous wave equation, satisfying the boundary conditions, can be added to give a solution G(r|rq ):

g(r|rq , –) + k02 g(r|rq , –) = − ƒ(r − rq )

(5)

ƒ(r − rq ) = ƒ(x − xq ) · ƒ(y − yq ) · ƒ(z − zq ). (6) G(r|rq ) = g(r|rq ) + h(r).

The Green’s function of a point source in free space is e−j k0 R g(r|rq , –) = ; R = (x − xq )2 + (y − yq )2 + (z − zq )2 . 4R

(7)

14

B


The volume flow of the source is given by From this it follows in three dimensions:

∂g . lim −4R R→0 ∂R lim g(r|rq , –) = 2

r→rq

(8) 1 ; 4R

1 lim g(r|rq , –) = ln |r − rq |; 2 ∂g ∂g − = −1, ∂x xq +— ∂x xq −—

in two dimensions:

r→rq

in one dimension:

(9a) (9b) (9c)

i.e., the one-dimensional Green’s function has a discontinuity in slope at x=xq . (10) Green’s functions are reciprocal: g(r|rq , –) = g(rq |r, –). The sound pressure field p(r,–) in a finite space with given boundary conditions and a volume source distribution f(r,–) has to be a solution of the wave equation p(r, –) + k02 p(r, –) = −f (r, –).

(11)

The solution can be expressed with Green’s functions of the infinite space as p(r, –) = f (rq , –) · g(r|rq , –) dVq ∂ ∂ + g(r|rq , –) p(rq , –) − p(rq , –) g(r|rq , –) dSq , ∂nq ∂nq

(12)

where the subscript q indicates the variable for differentiation and integration. This is the Helmholtz–Huygens equation. The surface integral simplifies if either g(r|rq , –) or its normal derivative vanishes. Green’s functions may be defined also for non-harmonic sources but with time functions with unit spectral density, i.e. for time the function ƒ(t–t0 ): ∞ g(r|rq , t − t0 ): = g(r, t|rq , t0 ): =

g(r|rq , –) · ej –(t−t0 )

0

d– . 2

(13)

t Then:

p(r, t) =

f (rq , t0) · g(r|rq , t − t0 ) dt0 ,

(14)

−∞

where f (rq , t0) is the time function of the source having the spectrum f (r, –). e−j k0 r+j –t (15a) 4R 1 ƒ t − r c0 . belongs to the time function (15b) 4R Green’s functions in closed spaces can be expanded in modes ¦ n ; these are solutions of the homogeneous wave equation

The Green’s function

¦n + kn2 ¦n = 0

g(r|rq , –) =

(16)


B

15

satisfying the boundary conditions (and Sommerfeld’s far field condition if the space is infinite in one dimension). The wave number kn (instead of k0 ) recalls that in a finite size space harmonic solutions exist only if the frequency is a resonant frequency. The modes are orthogonal, and may be made orthonormal, i.e. 0 if n = m, (17) ¦n · ¦m dV = ƒnm = 1 if n = m. assuming the boundary conditions have one of the following forms: ¦n = 0 or ∂¦n ∂n = 0 or ¦n = − · ∂¦n ∂n.

(18)

Then Green’s function can be expanded: g(r|rq , –) =

A n ¦n =

n

¦n (r) · ¦n (rq ) n

kn2

− k02

;

An =

¦n (rq ) . kn2 − k02

(19)

1 ¦n (r) · ¦n (rq ). (20) 2 kn If the space is infinite, complex modes are convenient. The orthogonality integral then should be [instead of (17)]: 0 if n = m, (21) ¦n · ¦m∗ dV = ƒnm = 1 if n = m, The residues at the poles

k0 = ±kn

are

±

(the asterisk indicates the complex conjugate). The Green’s function then is ¦n (r) · ¦ ∗ (rq ) n g(r|rq , –) = . 2 − k2 k n 0 n

(22)

If one sets the condition that the physical solution should be real, the relations follow: ∗ (r), ¦n (r) = ¦−n

¦n (r) + ¦−n (r) = ¦n (r) + ¦n∗ (r) = 2 Re{¦n (r)}.

(23)

The eigenvalues kn need not be a discrete set of values, but may be continuous. Then the Green’s function is 1 g(r|rq , –) = 2

+∞ −∞

¦ (r) · ¦ ∗ (rq ) kn2 − k02

∂kn ∂n

−1 dkn .

(24)

The form of Green’s functions for continuous eigenvalues is similar to integral transforms: SF (x) = F(z) · ¦ ∗ (x, z) · w(z) dz, (25) F(z) = SF (x) · ¦ ∗ (x, z) · w(x) dx.

16

B


The weight function w(z) is often introduced by the co-ordinate system; generally it represents the density of eigenvalues in z space. The following orthogonality and normalizing relations are used: w(k) ¦ (k, z) · ¦ ∗ (x, z) · w(z) dz = ƒ(k − x), (26) w(k) ¦ (k, z) · ¦ ∗ (x, …) · w(k) dk = ƒ(z − …). In particular, the Dirac delta function is represented by +∞ ¦ (k, r) · ¦ ∗ (x, rq ) · w(k) · w(rq ) dk; ƒ(r − rq ) =

(27)

−∞

SF (x) =

thus

−w(rq ) · ¦ ∗ (x, rq ) , k 2 − x2

(28)

and the Green’s function becomes +∞ w(rq ) · g(r, rq |–) = F(r) = −∞

¦ (k, r) · ¦ ∗ (x, rq ) · w(k) · w(rq ) dx. k 2 − x2

(29)

Some examples of Green’s functions are given below. A set of plane waves: Substitute above x → ‰ indicating a wave number vector; denote with r the co-ordinate vector of a point. A set of plane waves is represented by ¦ (‰, r) = A(‰) · e−j ‰·r

(30)

(with the scalar product ‰ · r in the exponent). The density w(‰) is unity. The amplitudes are A(‰) = 1 (2) (for normalisation). In a two-dimensional space (x,y) with ‰ · r = ‰x x + ‰y y the Green’s function becomes +∞ +∞ g(r, rq |–) = −∞ −∞

e−j ‰·(r−rq ) d‰x d‰y 42 ‰2 − k02

;

‰2 = ‰x2 + ‰y2 .

(31)

It can be shown that the integral goes over to the Hankel function of the second kind: g(r, rq |–) =

−j (2) H (k0 R) 4 0

;

R = |r − rq | .

(32)

Cylindrical waves: wave A set of eigenfunctions of the Bessel differential equation is •n (r) = Jm (n r/a) kn = (n/a)

;

;

Jm (n) = 0,

n = 0, 1, 2, . . . ,

(33)


B

17

where n are zeros of the Bessel function Jm (z) of order m. The orthogonality relation is

a Jm (n r/a) · Jm ( r/a) r dr = 0

0 ; = n, 2 − a2 Jm+1 (n ) · Jm−1 (n )

;

= n.

(34)

If a →∞ the eigenvalues become continuous. One sets ¦ (k, z) = A · Jm (kz)

;

A = 1,

(35)

where A=1 follows from the normalisation. Two-dimensional infinite space in polar co-ordinates: Two-dimensional eigenfunctions of the wave equation satisfying the normalisation conditions (26) in polar co-ordinates (r,œ) are ‰r Jm (‰r) · e−j mœ . w(‰) w(r)¦ (‰, z) = (36) 2 The Green’s function becomes ∞ Jm (‰r) · Jm (‰rq ) 1 ƒm cos (m(œ − œq )) ‰ d‰. g(r, rq |–) = 82 m=0 ‰2 − k02 +∞

(37)

−∞

One gets after evaluation of the integral

⎧ ∞ ⎨Jm (k0 r) · H(2) m (k0 rq ) −j g(r, rq |–) = ƒm cos (m(œ − œq )) ⎩ 4 m=0 Jm (k0 rq ) · H(2) m (k0 r)

;

r ≤ rq ,

;

r ≥ rq .

(38)

Three-dimensional infinite space: The Green’s function is g(r, rq |–) =

e−j k0 R 4R

;

R = |r − rq |.

(39)

Green’s function in spherical harmonics: In the spherical co-odinates r, œ, ˜ the Green’s function is g(r, rq |–) =

j k0 (2) e−j k0 R =− h (k0 R) 4R 4 0 =−

∞ n (n − m) ! j k0 cos (m(œ − œq )) (2n + 1)· ƒm 4 n=0 (n + m) ! m=0

m ·Pm n (cos ˜q ) · Pn (cos ˜ )

·

⎧ ⎨jn (k0r) · h(2) n (k0 rq )

;

r < rq ,

⎩

;

r > rq ,

jn (k0rq ) · h(2) n (k0 r)

(40)

B

18


m where jm (z) , h(2) m (z) are spherical Bessel and Hankel functions and Pn (x) are associated Legendre functions.

If the source distance rq goes to infinity, one gets for the plane wave incident from the spherical directions œq , ˜ q (in the spherical angles œ, ˜ )

e−j k0 ·r =

∞ n=0

(−j)n (2n + 1)·jn (k0 r)

n

(n − m) ! m × ƒm cos (m(œ − œq )) · Pm n (cos ˜q ) · Pn (cos ˜ ). (n + m) ! m=0

(41)

Green’s function in cylindrical co-ordinates: In the cylindrical co-odinates r, ˜ , z the Green’s function is −1 g(r, rq |–) = 83

2

+∞ +∞ d

0

−∞ −∞

e+j ‰r r cos (−˜ )−j ‰r rq cos (−˜q )−j ‰z (z−zq ) ‰r d‰r d‰z . k02 − ‰r2 − ‰z2

(42a)

Performing the integration over ‰z with ‰z = ± k02 − ‰r2 = ± j g(r, rq |–) = 82

2

∞ d

0

0

‰r −j ‰r r cos (−˜ )+j ‰r rq e

cos (−˜q )−j ‰z (z−z0 )

· e−j |z−zq | d‰r .

(42b)

With the exponentials expressed by Bessel functions, one gets −j ‰r Jm (‰r r) · Jm (‰r rq ) · e−j |z−zq | d‰r ƒm cos (m(˜ − ˜q )) g(r, rq |–) = 4 m≥0 ∞

(43)

0

with ⎧ 2 ⎨ k0 − ‰r2 if 0 < ‰r < k0 , = ⎩ 2 −j ‰r − k02 if 0 < k0 < ‰r .

(44)

For rq =0 (43) reduces to the term with m=0. Point source above hard or soft plane: The Green’s function for a hard plane is

g(r, rq |–) =

e−j k0 r e−j k0 r , + 4r 4r

(45)

where r is the distance from the source to the field point and r is the distance from the image source (in a mirror-reflected position relative to the plane) to the field point. If the plane is soft, then

g(r, rq |–) =

e−j k0 r e−j k0 r − . 4r 4r

(46)


B

19

Point source above a locally reacting plane: The plane is at x=0; the source at xq , yq , zq ; the image source at –xq , yq , zq . Let n · r = nx x + ny y + nz z be the scalar product of the wave direction vector n with the co-ordinate vector r. A plane wave, reflected at the plane, can be represented as pr = R · e−j k0 (nx x+ny y+nz z)

;

R=

… nx − 1 , … nx + 1

(47)

where … is the normalised surface impedance of the plane. The Green’s function in Cartesian co-ordinates is 1 g(r, rq |–) = 83 ‰ 2 = ‰x2 + ‰y2 +

+∞

−∞ ‰z2 ;

ej [‰x x+‰y (y−yq )+‰z (z−zq )] −j ‰x xq + R e+j ‰x xq d3 ‰ e ‰2 − k02

(48)

d3 ‰ = d‰x · d‰y · d‰z

and in cylindrical co-ordinates: g(r, rq |–) =

−j ƒm cos (m(˜ − ˜q )) 4 m≥0 ∞ ‰ × Jm (‰ r) · Jm (‰ rq ) e−j ‰x |x−xq | + R e−j ‰x |x+xq | d‰ ‰x

(49)

0

where ‰ 2 = k02 − ‰x2 . An approximate expression (if field point and/or source are distant to the plane) is 1 g(r, rq |–) = 4

! e+j k0 |r−rq | e−j k0 |r−rq | +R , |r − rq | |r − rq |

(50)

where r is the vector of the field point, rq the vector to the original source, and rq the vector to the mirror source.

B.6

Orthogonality of Modes in a Duct with Locally Reacting Walls

See also: Mechel, Vol. III, Ch. 26 (1998)

Consider a duct whose interior contour follows a co-ordinate surface of a separable system of co-ordinates and whose contour surface is either totally or in parts locally reacting with an admittance G (the other parts are either hard or soft). Let the crosssection normal to the axial co-ordinate x be A, and let r be the one- or two-dimensional co-ordinate normal to x.

B

20


Let pm (x, r) = Tm (r) · Rm (x) be a mode in the duct, i.e. a field which satisfies the homogeneous wave equation and the boundary conditions,with the transversal function Tm (r) and the axial function Rm (x). Such modes are orthogonal over the cross section A, i.e.

Tm (r) · Tn (r) · g(r) dr = ƒm,n · Nm ,

(1)

A

where g(r) is the weight function induced by some co-ordinate systems; it is independent of the mode order m; ƒm,n is the Kronecker symbol, and Nm is the norm of the mode. The orthogonality of modes, under the conditions mentioned, holds whatever the value of G is, and also if the medium in the duct has losses (i.e. k0, Z0 complex). They form a complete set of solutions (see Morse/Feshbach 1953, part I, Sect. 6.3, pp. 738 et sqq.) if the defining boundaries normal to r are either hard or soft or locally reacting, and if in this case the derivative ∂p/∂r does not appear in the separated wave equation of the co-ordinate r. Modes may be one-, two-, or three-dimensional according to the number of pairs of walls that define the boundary conditions.

B.7

Orthogonality of Modes in a Duct with Bulk Reacting Walls

See also: Mechel, Vol. III, Ch. 27 (1998); Cummings (1989)

Assume a duct like that in > Sect. B.6, but whose duct lining is laterally (bulk) reacting, and whose outer wall (behind the lining) is hard. The field in the interior volume of the duct, with cross section A1, is marked with an index i=(1), the field in the lining with an index i=(2), and its cross section is A2 . Let the characteristic wave number and wave impedance in A1 be, respectively, k0 and Z0 , and let the characteristic propagation constant and wave impedance of the lining material in A2 be, respectively, a , Za . The transversal functions of a mode T(i) m (r) are different in the two areas; its axial function Rm (x) is the same. The modes are orthogonal over the cross section A1+A2 with the mode norm in the case of a single homogeneous layer of the lining in a cylindrical duct: (1) 2 (2) 2 1 1 Tm (r) dr + Tm (r) dr. (1) Nm = j k0 Z0 a Za A1

A2

In the case of multiple layers, an integral must be added for each layer.


B.8 See

B

21

Source Conditions > Sects. B.1, B.4, B.5.

A special form of the boundary conditions, the source condition, must be satisfied if the sound field p(r) is excited by a sound source. Commonly used are volume flow sources q(rq ) either as a point source in three-dimensional fields, or as a line source in twodimensional fields, or, more generally, as a source distribution on a surface Sq , or as a source distribution in a volume Vq . In the case of distributed sources, q(rq ) is the spatial density of emanating volume flow. The source condition requires that the integral of the outward normal velocity over a small spherical surface around a point source, or over a narrow cylindrical surface around a line source,or on Sq around distributed sources,equal the given source strength q. This form of the source condition is sometimes difficult to evaluate. A form more suitable to evaluation shall be given.

First, consider a point source or a line source. This case is illustrated with a line source (for simplicity); a point source is treated similarly. Let the source be located at (rq , ˜q ) in a cylindrical co-ordinate system (r, ˜ ). In general ˜ stands for a co-ordinate over which orthogonal modes exist (i.e. the modes satisfy the homogeneous wave equation, Sommerfeld’s condition, and the boundary conditions at the surfaces normal to ˜ ). The line source is located at Q. This defines two zones: zone (a) with 0≤r≤rq and zone (b) with rq ≤ r < ∞. The modes have the form

pm (r, ˜ ) = T(†m˜ ) · Rm (r).

(1)

They are orthogonal over (0,˜ 0 ) with norms Nm . The radial functions Rm (r) are formulated so that they are continuous at r=rq , but discontinuous in their radial derivatives. The source condition can be written in the form

k0 Z0 q Z0 vr (rq +0)−vr (rq − 0) =! · ƒ(˜ −˜q ). (2) k 0 rq

The Dirac delta function may be expanded in modes:

ƒ(˜ − ˜q ) =

m≥0

bm · cos (†m ˜ ).

(3)

˜0 . . . · cos (†m ˜ ) d˜ ,

By application on both sides of 0

(4)

B

22


follow the bm from

ƒ(˜ − ˜q ) =

1 cos (†m˜q ) · cos (†m ˜ ). (5) (2) ˜0 m≥0 Nm

Factor (2) is applied if the source is on a boundary, else (2) → 1. If the sources q(˜ ) are distributed over the surface at r = rq , this distribution is synthesised with the modes having norms as above.

B.9

Sommerfeld’s Condition

See also: Skudrzyk (1971)

If the field extends to infinity, it must approach zero there, unless it is a plane wave in a loss-free medium. A sufficient condition is a medium with losses. ∂p + j k0 p = 0. (1) Otherwise: lim r r→∞ ∂r A weaker but simpler condition is, with A an arbitrary constant,

lim |r p| < A.

r→∞

B.10 Principles of Superposition

See also: Ochmann/Donner (1994); Mechel (2000)

Some principles of superposition may help to reduce more general problems to a repetition of simpler standard tasks. First principle of superposition (by Mechel): (unsymmetry superposition)

Two opposite walls, normal to the same co-ordinate, locally react with different admittances G1 , G2. The sound fields at the walls have the corresponding indices 1,2.


The boundary conditions at these surfaces are (with normal particle velocity components vi ):

v1

=

G1 · p1 ;

v2

=

G2 · p2 .

B

23

Set (with G1, G2 selected so that Re{Ga }≥0): Gs is the symmetrical and Ga the antisymmetrical part of the boundary conditions.

1 (G1 + G2 ) , 2 1 Ga = (G1 − G2 ) . 2

(1)

Suppose the sound fields ps , pa are known for the two symmetrical linings Gs , Ga , respectively, on each side, i.e. with the boundary conditions at both flanks: ps is the symmetrical solution belonging to Gs , pa the antisymmetrical solution belonging to Ga .

1 (G1 + G2 ) · ps , 2 1 va = Ga · pa = (G1 − G2 ) · pa . 2

(2)

vs1,2 + va1,2 = G1 · ps1,2 + pa1,2 , vs1,2 − va1,2 = G2 · ps1,2 − pa1,2 .

(3)

It follows immediately that Comparing this with the boundary conditions of the original task, one sees the correspondence:

Gs =

vs = Gs · ps =

p1 = ps1,2 + pa1,2 , p2 = ps1,2 − pa1,2 .

The desired solution is evidently p=ps +pa , because both lines formally merge at the walls. Second principle of superposition (by Ochmann): (symmetry superposition) Suppose the object has a plane of symmetry. The medium is steady across the plane of symmetry, and no sound transmissive foil or sheet is in that plane. Let a co-ordinate z be normal to the plane of symmetry, directed from the side of incidence to the side of transmission, with z=0 in the plane of symmetry. Co-ordinate transversals to z are represented by x.

An index 1 marks the half-space with the incident wave pe and a reflected and/or backscattered wave prs ; an index 2 marks the half-space with the transmitted wave pt . The fields in the two half-spaces are p1 (x, z) = pe (x, z) + prs (x, z), p2 (x, z) = pt (x, z). Replace the original task by two subtasks; in the first one the sound transmissive parts of the plane of symmetry are assumed to be hard, in the second one they are assumed

24

B


to be soft. Both conditions are marked by upper indices (h), (s), respectively. Solve the problems of reflection and/or backscattering for the two subtasks. The sound field components of the original task then are 1 (h) prs (x, z) + p(s) z ≤ 0, rs (x, z) ; 2 1 (x, −z) − p(s) z ≥ 0. pt (x, z) = p(h) rs (x, −z) ; 2 rs prs (x, z) =

(4)

Third principle of superposition (by Mechel): (hard-soft superposition) The task: Find the sound field pa with (part of) the boundaries absorbent with local reaction, described by a wall admittance G. Suppose the solutions are known for the same source and geometry, but all walls are ideally reflecting, i.e. either hard or soft or mixed with both types. The third principle of superposition composes pa for the absorbent boundary with such solutions.

The example assumes a line source at Q in a wedge-shaped space with one hard flank at ˜ = 0, and one locally absorbing flank at ˜ = ˜0 . The standard situation with a soft flank at ˜ = 0 is treated similarly; other situations are treated after application of the first and second principles of superposition. It is assumed that the field ph is known, for which the flank at ˜ = ˜0 is hard, and that ps is known, with a soft flank at ˜ = ˜0 . Both fields satisfy the source condition at Q individually (see > Sect. B.8). 1 ph (r, ˜ ) + G · X(r) · ps (r, ˜ ) The desired field pa then is pa (r, ˜ ) = 1 + G · X(r) with the “cross impedance”

X(r) =

ph (r, ˜0 ) ph (r, ˜0 ) = −j k0 Z0 . vsn (r, ˜0) gradn ps (r, ˜0 )

(5)

The index n indicates the vector component normal to the absorbing wall and directed into it. X(r) is an impedance, formed with the sound pressure if the flank at ˜ = ˜0 is hard divided by the normal particle velocity if the flank is soft. The third principle of superposition returns an exact solution if X is constant with respect to the co-ordinate on the absorbing wall (r in the example); otherwise an approximation to pa is obtained.


B

25

B.11 Hamilton’s Principle

See also: Cremer/Heckl (1995); Morse/Feshbach (1953)

Let Ekin be the kinetic (effective) energy of a vibrating system, associated with oscillating masses, and Epot its (effective) potential energy, associated with displacements against stresses; further let W be the (effective) work done by external forces on the system. Lagrange function:

L = Ekin − Epot

Hamilton’s principle: If the system starts to oscillate from reasonable initial conditions, the form of oscillation which it assumes is such that the time average of its Lagrange function is an extreme if the form of the oscillation is varied (ƒ stands for such variations): t2 t2 (1) ƒ L dt + ƒW dt = 0. t1

t1

If the work W of external forces is constant over time intervals, and time average values of L and W are used, then ƒ L + ƒ W = 0. If the system is adiabatic, i.e. W = 0, Hamilton’s principle requires ƒ L = 0. The form of the system’s oscillation is governed by amplitudes either of system elements or of field components, such as modes. The variation is applied to these amplitudes am . On the other hand, many systems have to obey boundary conditions, which are constraints in terms of variational methods. These boundary conditions are formulated as equations gk (am ) = 0, and they are introduced into Hamilton’s principle using the Lagrange multipliers Šk (see Morse/Feshbach, 1953, part I, Sect. 3.1), leading to the form of Hamilton’s principle suited for application to mechanical systems: 1 T

T

Ekin − Epot dt + Šk∗ · gk + Šk · gk∗ = min .

(2)

k

0

The Šk are treated in the application of the principle like the amplitudes am , i.e. they are parts of the variation. This expression is formulated as a function f(am , Šk ). The energies will be sums with products am · an∗ as factors. The minimum is found where the following equations hold: ∂f =0 ∂ an∗

;

∂f = 0. ∂ Šk∗

(3)

This gives a set of linear equations for the am , Šk . In distributed systems and/or wave fields, the integration is not only over time but also over space. If the sound field is described by a velocity potential function •(r), the Lagrange density is L = (r) dr, V

0 1 |grad •|2 − 2 (r) = Ekin (r) − Epot (r) = 2 c0

∂• ∂t

2 !

0 |grad •|2 +k02 •2 . = 2

(4)

26

B


B.12 Adjoint Wave Equation L and here must not be confused with these symbols in The wave equation is a secondorder linear differential equation, with p,q possibly functions of r: The adjoint wave equation is Both satisfy the identity P(g,f) is the bilinear concomitant. If (g)=0 can be solved, then solutions of L(f)=0 are The general solution is In the special case q(r)=dp(r)/dr,

> Sect. B.11.

L (f (r)) = f (r) + p · f (r) + q · f (r) = 0.

(1)

g(r) = g (r) − p · g (r) + (q − p ) · g(r) = 0. (2)

d P(g, f ) . dr f1(r) = g(r) · e− p dr , − p(s) ds e f2(r) = f1 (r) · dr. g2 f (r) = a · f1 · (r) + b · f2 (r). g(r) = e− p(s) ds dr, g · L(f ) − f · (g) =

f1(r) =

g(r) g (r)

;

f2(r) =

1 . g (r)

(3) (4)

(5)

B.13 Vector and Tensor Formulation of Fundamentals Co-ordinate systems: Let (x1 , x2 , x3 ) be a rectilinear, orthogonal coordinate system. The vector components → = OP = [x1 , x2 , x3 ]. of a point P are given by R

Let (u1, u2, u3) be a curvilinear, orthogonal coordinate system. The co-ordinate surfaces are given by u1 (x1 , x2, x3 ) = const, u2 (x1 , x2, x3 ) = const, u3 (x1 , x2, x3 ) = const. The intersection of two co-ordinate surfaces is a co-ordinate line.


B

27

Tangent vectors at co-ordinate lines: ∂R ∂x1 ∂x2 ∂x3 = , , , ∂u1 ∂u1 ∂u1 ∂u1 ∂R ∂x1 ∂x2 ∂x3 2 = R = , , , ∂u2 ∂u2 ∂u2 ∂u2 ∂R ∂x1 ∂x2 ∂x3 3 = . = , , R ∂u3 ∂u3 ∂u3 ∂u3

1 = R

Normal vectors on co-ordinate surfaces: ∂u1 ∂u1 ∂u1 1 , N = grad u1 = , , ∂x1 ∂x2 ∂x3 2 = grad u2 = ∂u2 , ∂u2 , ∂u2 , N ∂x1 ∂x2 ∂x3 ∂u3 ∂u3 ∂u3 3 N = grad u3 = . , , ∂x1 ∂x2 ∂x3

(1)

(2)

i form the basis vectors of a system of co-ordinates, then the N i are the basis of If the R the “reciprocal” system, with 1; i=k k Ri • N = ƒi,k ; ƒi,k = with the “dot product” or “scalar product”. 0 ; i = k Unitary tensors: i • R k = gik gik = R

covariant co-ordinates ,

i • N k gik = R

mixed co-ordinates ,

i • N k = g ki gik = N

contravariant co-ordinates

(3)

with gij • gjk = gi1 · g1k + gi2 · g2k + gi3 · g3k = ƒi,k . The determinant of gik is the square 1R 2R 3. of the scalar triple product g = det(gik ) = R

B

28


Vector components of a vector a: i; ai = a • R i; ai = a • R

covariant components:

(4) 2 3 i i; a = a · R1 + a · R2 + a · R3 = a • R 1 + a2 · R 2 + a3 · R 3 = ai • R i’ a = a1 · R

contravariant components:

1

vector representation in a covariant basis: vector representation in a contravariant basis: It follows that ai = gi1 a1 + gi2 a2 + gi3 a3 = gij aj ,

(5)

ai = gi1 a1 + gi2 a2 + gi3 a3 = gij aj ,

where the last notations use the “summation rule” (summation over multiple indices). i = gij N j j. i = gij R ; R (6) Thus: N Transformation between systems of co-ordinates U(u1, u2, u3 ) → V(v1, v2 , v3): With definitions:

" " " 1 2 3 "" " ∂ v ,v ,v = "" = 1 2 3 ∂ (u , u , u ) " " " " Aik =

and

i = R

k

ai =

∂v i ∂u k

;

j=1

i = Bki R

i • N k Aki = R and:

∂v 1 ∂u 2 ∂v 2 ∂u 2 ∂v 3 ∂u 2

∂v 1 ∂u 3 ∂v 2 ∂u 3 ∂v 3 ∂u 3

Bki =

" " " " " " " = det Ai = 0 k " " " " "

∂u k ∂v i

k

k Aki R

i = N

;

k

k = Aik N

(9)

k

k , Bik N

k • N i Bki = R

;

Aik ak =

j=1

k

Bik ak

k

;

ai =

k

Bki ak =

(7)

(8)

−1 = det Bki ; 3 3 j j Aij · Bk = Bij · Ak = ƒi,k ,

follows that

∂v 1 ∂u 1 ∂v 2 ∂u 1 ∂v 3 ∂u 1

Aki ak .

(10)

(11)

k

Vector algebra: Consider the vectors i = ai · R i; a = ai · R i

b =

i

c =

i

i

i = bi · R i = ci · R

i

i

i; bi · R i. ci · R

(12)


B

Scalar product:

" " " " −1 i i i k ik " a • b= a bi = ai b = gik a b = g ai bk = g "" i i i,k i,k "

g11 g21 g31 a1

g12 g22 g32 a2

g13 g23 g33 a3

b1 b2 b3 0

29

" " " " " . (13) " " "

Length of a vector: a = |a| =

√

a • a =

gik ai ak =

i,k

ai ak =

i,k

gik ai ak .

Cosine of the angle between two vectors: gik ai bk a • b i,k = cos (a, b) . = ab gik ai ak gik bi bk i,k

Vector (cross) product: " " R 1 R 2 " √ 1 a × b = g "" a a2 1 " b b2

(14)

i,k

(15)

i,k

3 R a3 b3

Vector triple product: " 1 " a a2 a3 √ "" 1 abc = g " b b2 b3 " c1 c2 c3

" " " " = √1 " g "

" " " " = √1 " g "

" " R " 1 " a1 " " b1

" " a1 " " b1 " " c1

2 R a2 b2

a2 b2 c2

3 R a3 b3

a3 b3 c3

" " " ". " " " " " ". " "

(16)

(17)

Derivatives of basis vectors: Notation:

= =R R ik

ki

∂ 2R ∂ui ∂uk

;

= R = R

mn

nm

Transformation: ∂Bs n = + R Rs . Bim Bkn R mn ik ∂u m s

∂ 2R . ∂vm ∂vn

(18)

(19)

i,k

Christoffel symbols of second kind: = R ik

j j R ik j

or

j ik

j ik

:

•R j. =R ik

Transformation: s ∂Bn j r s . + Bim Bkn Bjr R = ik mn ∂um i,j,k

s

(20)

(21)

B

30


Christoffel symbols of first kind: {ikj}: # $ # $ = •R k. j or R ikj = R ikj R ik

(22)

ik

j

Transformation: # $ ∂Bsn t Bim Bkn Bjr ikj + gs t B. {mnr} = ∂um r s,t

(23)

i,j,k

Relations with unitary tensors of the co-ordinate systems: j s sj g {iks} ; {ikj} = gsj = , ik ik s

j ik

=

(24)

s

1 sj g 2 s

∂gis ∂gis ∂gik + − . ∂ui ∂uk ∂us

(25)

Derivative of a vector along a curve: Let a curve be defined by the equations u i = u i(‘) ; i = 1, 2, 3 , with the parameter ‘ varying along the curve. i = ai R i Let further be a vector a = ai R i

i

ai = ai (u1(‘), u2(‘), u2(‘))

with functions

;

ai = ai (u1 (‘), u2(‘), u2(‘)).

The complete derivative of the vector components is duk duk D ai ∂ai i = = ∇k ai , + aj jk d‘ ∂u d‘ d‘ k j,k k duk duk D ai ∂ai j − aj = = ∇k ai ik d‘ ∂uk d‘ j,k k d‘ with the notation ∇k ai =

∂ai + ∂uk j

i jk

aj

;

∇k ai =

(26)

∂ai − ∂uk

Derivative of a tensor: ∂aik i k ∇j aik = + ai s , as k + js js ∂uj s k k ∂a s i + ask , ais − ∇j aik = ji js ∂uj s ∂ai k s s ∇j ai k = as k − − ai s . ji jk ∂uj

j

j ik

aj . (27)

(28)

s

It holds that

∇j a i b k = ∇j (ƒik ) =

∇j ai bk + ai ∇j (bk ) , ∇j gik = ∇j gik = ∇j gik = 0 .

(29)


B

31

Orthonormal basis vectors: Orthonormal basis vectors are called

ei

;

i=1,2,3.

i = Hi ei ; R i = hi ei , The basis vector components are R with 1/2 ∂xk 2 ; Hi = |Ri | = ∂ui k 2 1/2 ∂u k i| = hi = |R = 1/Hi ; ei • ej = ƒi,j ; ∂xi k gij

= Hi Hj ƒi,j ⎛

H21

⎜ ⎜ # $ ⎜ 0 gij = ⎜ ⎜ ⎜ ⎝ 0

; 0

gij =

0

Vector components:

1 ƒ ; Hi Hj i,j

⎞

0

H22

⎟ ⎟ 0 ⎟ ⎟ ⎟ ⎟ 2 ⎠ H3

a =

(31) g = det(gij ) = H1 H2H3 .

;

i

ai∗ ei

a • b =

Scalar product:

(30)

;

ai∗ = a • ei =

ai = Hi ai . Hi

ai bi ∗ ∗ = ai bi . 2 i Hi i

(33)

Vector (cross) product: " " H1e1 " 1 " a1 a × b = H1H2 H3 "" b

H2e2 a2 b2

H3 e3 a3 b3

" " " " e1 " " ∗ "=" a " " 1∗ " " b 1

e2 a2∗ b∗2

Vector triple product: " " a1 a2 " 1 " b1 b2 abc = H1 H2H3 "" c c2 1

a3 b3 c3

" " ∗ " " a " " 1∗ "=" b " " ∗1 " " c 1

a2∗ b∗2 c∗2

" " " ". " "

1

(32)

a2∗ b∗3 c∗3

e3 a3∗ b∗3

" " " ". " "

(34)

(35)

Differential operators: The gradient of a scalar function is a vector: = grad œ = ∇œ

1 ∂œ ei = (gradi œ) ei . Hi ∂ui i

Nabla operator (a vector):

(36)

i

= ∇

1 ∂ 1 ∂ 1 ∂ , , H1 ∂u1 H2 ∂u2 H3 ∂u3

.

32

B


The divergence of a vector is a scalar: • a = div a = ∇

∂ ∂ H 1 H2 H3 1 1 i ∗ · ai . (37) H 1 H2 H 3 · a = H1 H2 H3 ∂ui H 1 H2 H 3 ∂ui Hi i

i

The rotation of a vector is a vector: ⎛ H1e1 1 × a = ⎝ ∂/∂u1 rot a = ∇ H1 H 2 H 3 a1 ⎛

H1 e1 1 ⎝ ∂/∂u1 = H 1 H 2 H3 H1 a1∗

H2e2 ∂/∂u2 H2 a2∗

H2e2 ∂/∂u2 a2

⎞ H3 e3 ∂/∂u3 ⎠ a3

⎞ H3e3 ∂/∂u3 ⎠ H3a3∗

(38)

The Laplacian of a scalar function: • ∇)œ = œ = (∇

∂ H1 H2 H3 ∂œ 1 H 1 H2 H3 ∂ui ∂ui H2i

(39)

i

The Laplacian of a vector is a vector:

a = grad (div a) − rot (rot a).

(40)

Identities: grad (U1 U2) = U1 · grad U2 + U2 · grad U1, 1 • grad V 2 = V 2 + V 2 • grad V 1 + V 1 × rot V 2 + V 2 × rot V 1, 1 • V grad V = U · div V +V · grad U, div U · V 2 • rot V 1 × V 2 = V 1 − V 1 • rot V 2, div V = U · rotV + grad U × V, rot U · V 2 • grad V 1 × V 2 = V 1 − V 1 • grad V 2 + V 1 div V 2 − V 2 div V 1, rot V • ∇ ×V = div rot V = 0, ∇ × ∇U ∇ = rot grad U = 0, • ∇U ∇ = div grad U = U. Some co-ordinate systems (see a scalar function: U. A vector: V;

> Sects. B.10, B.13 for more systems):

(41)


B

33

Cartesian co-ordinates: [x,y,z] Line, surface, and volume elements: (ds)2 = dx2 + dy 2 + dz2 , dFx = dy · dz ; dFy = dz · dx ; dFz = dx · dy,

(42)

dV = dx · dy · dz, ai

= ai = ai∗ .

Differential operators: grad U =

∂U ∂U ∂U ex + ey + ez , ∂x ∂y ∂z

∂Vx ∂Vy ∂Vz + + , ∂x ∂y ∂z " " ex ey ez " rot V = "" ∂/∂x ∂/∂y ∂/∂z " Vx Vy Vz

(43)

= div V

2

U = ∂∂xU2 +

∂2U ∂y 2

+

" " " ", " "

(44)

∂2 U . ∂z 2

(45)

Circular cylindrical co-ordinates: [r, œ, z] = [Vr , Vœ , Vz ] ; a scalar function U. A vector: V Line, surface, and volume elements:

Transformation:

x = r · cos œ r = x2 + y 2 y = r · sin œ œ = arctan (y/x) z=z z=z

(ds)2 = dr2 + r2 dœ2 + dz2 dFr = r dœ · dz ; dFœ = dr · dz ; dFz = r dr dœ dV = r dr · dœ · dz Hr

=1

;

Hœ = r

;

(46)

Hz = 1

Differential operators: ∂U 1 ∂U ∂U er + eœ + ez ∂x r ∂œ ∂z ∂Vz 1 ∂ (rVx ) ∂Vœ + + div V = r ∂r ∂œ ∂z 1 ∂(rVœ ) ∂Vr ∂Vr ∂Vz 1 ∂Vz ∂Vœ = − er + − eœ + − ez rot V r ∂œ ∂z ∂z ∂r r ∂r ∂œ ∂U 1 ∂ 2U ∂ 2U 1 ∂ r + 2 U = + 2 r ∂r ∂r r ∂œ2 ∂z

grad U =

(47)

34

B


Spherical co-ordinates: [r, œ, ˜ ] = [Vr , Vœ , V˜ ] ; a scalar function U. A vector: V Transformation:

2 + z2 , r = x2 + y ˜ = arctan ( x2 + y 2 /z) , œ = arctan (y/x) .

x = r · sin ˜ · cos œ ; y = r · sin ˜ · sin œ ; z = r · cos ˜ ;

(48)

Line, surface, and volume elements: (ds)2 = dr2 + r2d˜ 2 + r2 sin2 ˜ dœ2 , dFr = r2 sin ˜ dœ · d˜ ; dFœ = r dr · d˜ ; dF˜ = r dr dœ , dV = r2 sin ˜ · dr · dœ · d˜ , Hr = 1 ; Hœ = r sin ˜ ; H˜ = r .

(49)

Differential operators: 1 1 ∂U ∂U ∂U er + eœ + e˜ , ∂r r sin ˜ ∂œ r ∂˜ ∂Vœ 1 ∂ r 2 Vr 1 ∂ (sin ˜ V˜ ) 1 = div V + + , 2 r ∂r r sin ˜ ∂˜ r sin ˜ ∂œ ∂ sin ˜ Vœ ∂V˜ 1 = − er rot V r sin ˜ ∂˜ ∂œ 1 ∂(rV˜ ) ∂Vr 1 ∂Vr 1 ∂(rVœ ) − e˜ + − eœ , + r sin ˜ ∂œ r ∂r r ∂r ∂˜ ∂2U 1 ∂ 1 ∂U 1 ∂ 2 ∂U . U = 2 r + 2 sin ˜ + 2 2 r ∂r ∂r r sin ˜ ∂˜ ∂˜ r sin ˜ ∂œ2

grad U =

(50)

Differential relations of acoustics: Field variables (overbar: total quantity; index 0: stationary value) density: pressure: temperature: entropy: velocity:

¯ p¯ T¯ S¯ v¯

= = = = =

0 + p0 + p T0 + T S0 + S v0 + v

† = dynamic viscosity; ‹ = volume viscosity; = heat conductivity Total time derivative: Equation of continuity:

D... ∂ ... = + (¯v • grad) . . . . Dt ∂t D ¯ + ¯ · div v¯ = 0. Dt

(51) (52)


linearised and v0 =0: Navier-Stokes equation:

B

∂ + 0 div v = 0. ∂t D v¯ 4 ¯ = −grad (¯p + ¯ ¥ − (‹ + †) div v¯ ) Dt 3 −† rot (rot v¯ ),

35

(53) (54)

with ¥ = potential of an external force per unit mass (e.g. gravity);

Apply and compose: ‰ Cv Cp †

= = = = =

∂v ∂t

4 −grad p + (‹ + †) grad (div v) (55) 3 −† rot (rot v) . ∂ ∂ux ∂uy ∂uz + + = vx + [rot (rot v)]x (56) grad (div v) x = ∂ x ∂x ∂y ∂z rot v = 0 (57) v = v + vt with div vt = 0

linearised and v0 =0; ¥ =0:

0

=

molecular mean free path length; adiabatic exponent; specific heat at constant volume; specific heat at constant pressure; dynamic viscosity

This leads to the following two differential equations: 0

∂ v = −grad p + (‹ + 43 †) · v , ∂t

(58)

∂ vt = −† · rot (rot vt ) . 0 ∂t Energy equations: A) Heat conduction: Jh = − · grad T with = heat conductibility. 0 c0 Cv 5 ≈ †Cv . √ 3 ‰

From molecular gas dynamics:

= 1.6

Energy balance with heat conduction:

∂T = div grad T. ∂t 0 Cp

(59) (60)

B) Viscous energy loss per unit volume:

D Di i

Shear stresses by viscosity Dik

∂ vi Dik . i,k ∂ xk ∂vi 2 = − ‹ − † div v − 2† 3 ∂xi ∂vi ∂vk = −† + ∂xk ∂xi

=

(61)

(62)

36

B


C) Balance of internal energy E per unit mass: dE 1 dU 1 = = div grad T + D − P · div u dt 0 dt 0 with: E U D P

= = = =

(63)

internal energy per unit mass, internal energy per unit volume, viscous energy loss, static pressure

D) Balance of entropy S per unit mass: D dS D 1 = + div grad T = + 2 |grad T|2 + div dt T T T T

grad T . T

(64)

E) Balance of heat Q per unit volume: dQ dS = 0 T = D + div grad T = D + |grad T|2 + T · div dt dt T Equation of state: For an ideal gas: with R0 = gas constant.

¯ p¯ = ¯ · R0 · T,

grad T . T

(65)

(66)

Equation of state for the mass density variation ˜ in a sound wave (sound field quantities with a ∼ , stationary quantities with a − , atmospheric values with 0 ): ∂ ¯ ∂ ¯ ˜ = ‰ 0 KS p˜ − T ˜ −−−−−−→ 0 p˜ − 0 T ˜ ≈ 0 p˜ . ˜ = · p˜ + ·T (67) ideal gas P0 ∂ p¯ T T0 P0 ∂ T¯ p Equation of state for the entropy variation S˜ in a sound wave: ˜ ∂ S¯ ∂ S¯ ˜ − ‰ − 1 p˜ −−−−−−→ Cp T − ‰ − 1 p˜ . (68) ˜ = Cp T S˜ = · p˜ + ·T ideal gas ∂ p¯ T T0 ‰ T0 ‰ P0 ∂ T¯ p Thermodynamic relations: 1 ∂V KT = − isothermal compressibility; V ∂P T 1 ∂V isotrope compressibility; KS = − V ∂p S 1 ∂V ß= thermal expansion coefficient; V ∂T p ß ∂p = = thermal pressure coefficient; ∂T V KT

(69) (70) (71) (72)


Cp Cv

‰=

B

adiabatic exponent;

K T = ‰ Ks ; ‰−1=

T ß2 ; Ks Cp

(75)

1 1 = adiabatic sound velocity c0; 0 Ks ‰0KT 1 1 1 ∂ Ks = = ; −−−−−−→ ‰ 0 ∂P T 0 c20 ideal gas ‰P0

=

1 0

∂p ∂T

(73) (74)

c20 =

ß=−

37

∂ ∂T

= V

P

−−−−−−→ − ideal gas

1 ; T0

(76) (77)

(78)

P0 ß ß 0 c20 = ; −−−−−−→ ideal gas T0 Kt ‰

(79)

≈ 1.6 √ 0 c0 Cp ‰

characteristic mean free molecular path length for heat conduction effects; mean free path

(80)

† ≈ √ 0 c0 ‰ ‹ + 4/3† 4 ‹ = + v v = 0 c0 3 †

characteristic mean free molecular path length for shear viscosity effects;

(81)

characteristic mean free molecular path length for shear and bulk viscosity;

(82)

h = v =

Linearised fundamental equations for a density wave (time factor e+j– t ): 2 – – – 1 + v − j (‰ − 1) h ; k2 ≈ c0 c0 c0 with ∂ p˜ 1 ˜p = −k2 p˜ ; ; = j – · p˜ ; c20 = ∂t 0 Ks ‰−1 – ˜ temperature variation: T= 1 + j h · p˜ ; ‰ c0 – 1 density variation: ˜ = 2 1 − j (‰ − 1) h · p˜ ; c0 c0 entropy variation: longitudinal particle velocity:

Cp ‰ − 1 – h · p˜ ; T0 ‰ c0 v j · grad p˜ . − v = –0 0 c0

S˜ = j

(83)

(84) (85) (86) (87)

B

38


Linearised fundamental equations for a temperature wave: kT2 ≈

with

−j – ; c0 h

pressure variation: density variation: entropy variation:

˜; T˜ = −kT2 T

˜ ∂T ˜ = j – · T; ∂t

j ‰– ˜ v − h · T; c0 j ‰– −‰ ˜ h − v · T; ˜ = 2 1 + c0 c0 Cp – ˜ S˜ = 1 + j (‰ − 1) h − v · T. T0 c0 p˜ =

(88) (89) (90) (91)

B.14 Boundary Condition at a Moving Boundary

See also: Kleinstein/Gunzburger (1976)

A boundary separates two media with density and sound velocity 1, c1 and 2 , c2 , respectively (other quantities are distinguished with the same indices). The boundary moves with a velocity U0 normal to its surface. A wave is incident from the side with 1 , c1 . The co-ordinate normal to the surface is x, directed 1→ 2. One-dimensional wave equation (in fixed co-ordinates) for density i on both sides i=1,2: 2 (1) ∂ ∂t + U0 ∂ ∂x = c2i ∂ 2 ∂x2 with general solutions –1 x – ¯ 1x ¯ 1t + 1 = F1 –1 t − + G1 – c1 (1 + M1 ) c1 (1 − M1) –2 x 1 = F2 –2 t − c2 (1 + M2 ) M1 M2 F1 G1 F2

= = = = =

(2)

U0/c1; U0/c2; incident wave; reflected wave; transmitted wave

Boundary conditions:

ƒv

=

0 → ∂ ∂t + U0 ∂ ∂x ƒ v = 0

ƒp

=

0 → ∂ ∂t + U0 ∂ ∂x ƒ p = 0

(3)

This leads to the Doppler shifted frequencies: – ¯1 –2 –1 = = . 1 + M 1 1 − M 1 1 + M2

(4)


B

39

Wave numbers: k1 = –1 (c1 + U0), k¯ 1 = −– ¯ 1 (c1 − U0 ),

(5)

k2 = –2 (c2 + U0). Rule of conservation of wave numbers: ∂k ∂t + ∂– ∂x = 0

or

− U0 · ƒk + ƒ– = 0

or

U0 =

ƒ– . ƒk

(6)

Applications: A) The boundary is a shock front in the undisturbed medium: i.e. –2 =k2 =0; it follows that U0 = –1 /k1 = phase velocity in the medium i = 1. B) A shock front with a jump in the state of the medium: Shock front equation of gas dynamics:

U20 = ƒ p + v 2 ƒ ,

(7)

in the limit of small amplitudes, i.e.

2 ≈ 1 ; p2 ≈ p1 ; v1 ≈ v2,

(8)

it follows that

U0 =

d– ƒ– → = group velocity. ƒk dk

(9)

C) Stationary shock front, i.e. U0 = 0, follows with –1 = –2 ;

k2 = k 1

c1 1 + M1 . c2 1 + M2

(10)

D) Shock front with velocity U0 and U2 = flow velocity behind shock: –1 = k1c1

;

–2 = (c2 + U2) k2 ,

U 2 1 + M1 –2 = 1+ –1 c2 1 + M2

;

k2 c 1 1 + M 1 = . k1 c 2 1 + M 2

(11)

B.15 Boundary Conditions in Liquids and Solids

See also: Gottlieb (1975)

Let a plane pressure front be parallel to a plane boundary. Let the density and sound velocities in a solid on both sides be i , ci , respectively, and tensions and velocities i , ui ; i = 1,2; i = 1 input side. In a liquid let pi = −i and vi be the sound pressure and particle velocity, respectively.

B

40


A) In a homogeneous solid with 1 , c1 :

1 − 2 = 1 c1 (u2 − u1).

(1)

B) In a homogeneous liquid with 1 , c1 :

p1 − p2 = 1 c1 (v1 − v2).

(2)

C) At a solid-liquid interface v 4 = u3 =

−1 − p2 + 1 c1 · u1 + 2 c2 · v2 , 1 c1 + 2 c2

2 c2 · 1 − 1 c1 · p2 − 1 c1 · 2 c2 · (u1 − v2) . −p4 = 3 = 1 c1 + 2 c2

(3)

B.16 Corner Conditions

See also: Felsen/Marcuvitz (1973)

Two-dimensional corner: Consider a field f (r) · g(œ, z). The condition in the corner at r = 0 is R

|f (r)|2 · r · dr = finite,

(1)

0

from which follows |f (r)|2 ≤

1 , r2(1−)

for

r → 0 , small, positive.


B

41

Three-dimensional corner: Consider a field f(r). The condition in the corner at r = 0 is R

|f (r)|2 · r2 · dr = finite,

(2)

0

from which follows |f (r)|2 ≤

1 , r3(1−2/3)

for r → 0 , small, positive.

B.17 Surface Wave at Locally Reacting Plane


Surface waves are well known in elastic bodies (e.g. as Rayleigh waves). Here surface waves in a fluid are considered, but not those which, as a consequence of a surface wave in an elastic boundary, are produced in the fluid. Synonyms are “guided wave”, because surface waves may follow curved boundaries,“creeping wave” in the scattering at cylinders and spheres as they are slow waves propagation around the scattering objects. Consider a plane boundary in the x, z plane, with air in the half-space y ≥ 0. Let the surface be characterised either by a surface impedance Z or by surface admittance G = 1/Z. A wave of the form

p(x, y) = P0 · e− x x · e− y y · ej –t

(1)

satisfies the wave equation if

x2 + y2 = − k02 ,

(2)

the radiation condition if

x : = Re{ x } ≥ 0 ; y : = Re{ y } ≥ 0

(3)

and the boundary condition if

Z0 G=Z0 Gy : = −

!

" Z0 vy "" j y − y + j y = = . p " y=0 k0 k0

(4)

42

B


If one compares the last relation with the general admittance relation of > Sect. B.3, in " " which the sound field is written as p(x, y) = " p(x, y) " · ej œ(x,y) , ∂ ∂ 1 ln |p(x, y)| , (5) which reads Z0 G = − œ(x, y) + j · k0 ∂n ∂n " " ∂ ∂ one gets y = œ(x, y) ; y = ln " p(x, y) ". (6) ∂y ∂y These are just the definitions of y , y as phase and level measures. Thus a surface wave is a wave type which satisfies the fundamental equations and the boundary condition “by definition”. The graph shows curves Re{ x /k0} = const (nearly horizontal lines) and curves Im{ x /k0} = const (nearly vertical lines) in the complex plane of Z0 G = Z0 G + j · Z0 G. The parameter steps Re{ x /k0}, Im{ x /k0} are 0.2 over the values 0, . . ., 3. The curve Im{ x /k0} = 1 is thick.Values Im{ x /k0} < 1 are on the left of the curve for Im{ x /k0} = 1. The waves there are “fast”; the waves on the right of that curve are “slow”. Because Re{ x /k0} > 0, the waves are attenuated along the surface.

B.18 Surface Wave Along a Locally Reacting Cylinder


The topic here is a surface wave along a cylinder, not around a cylinder. The cylinder has a diameter 2a and is locally reacting at its surface with the normalised radial impedance


B

43

W= Z/Z0 = 1/(Z0 G) (G = admittance). The wave is supposed to have an axial symmetry. It is formulated as p(r, z) = P0 · K0 ( r r) · e− z z Z0 vr (r, z) =

;

r2 + z2 = −k02 ,

j −j r gradr p = P0 · K1( r r) · e− z z k0 k0

(1)

with the modified Bessel function K0(z) of the second kind of zero order. The boundary condition at r = a leads to the characteristic equation for r a: r a ·

K1 ( r a) = −j k0 a · Z0 G = −j · U. K0 ( r a)

(2)

Start values for the numerical solution are r a k0a ≈ −j Z0 G. With its solution the axial propagation costant z is evaluated from z a =j k0 a

2 1 + r a k0a .

(3)

Lines for constant real or imaginary parts of z = r a = z + j · z in the plane of U = k0a · Z0 G; for z , z = 0 to 5; z = 0.2 (The parameter values at the lines are approximately equal to the co-ordinate values of U)

44

B


B.19 Periodic Structures, Admittance Grid


An object with a periodic surface is a special case of an object with an inhomogeneous surface (other inhomogeneous surfaces which are amenable to analysis are those in which either the scale of the inhomogeneities and their distances is small compared to Š0 , then the average admittance is relevant, or the inhomogeneities are at large distances from each other, then scatter matrices can be set up). The method to be applied with periodic structures will be displayed in this and the next sections with some typical examples. In principle, the quantities that describe the periodic surface, such as its surface admittance or the sound field at the surface, are synthesised with a Fourier series. The Fourier terms are waves which have different names in the literature: “spatial harmonics” (used here), “Hartree harmonics” (often used in microwave technology), or “Bloch waves” (used in solid state physics). The most important quality of these waves is their orthogonality over a period, which makes them suited for the synthesis of field quantities. Plane surface with periodic admittance function G(z) and incident plane wave: Consider a plane with a periodic surface admittance G(z) and a plane wave pe incident with a polar angle ˜ (the wave vector in the x,z plane).

The plane wave pe is pe (x, z) = Pe · e−j (kx x+kz z) kx

= k0 cos ˜

;

(1) kz = k0 sin ˜ .

The field in the half-space x ≤ 0 is written p(x,z)= pe (x,z)+ps (x,z) with the scattered wave ps formulated as ps (x, z) =

+∞

An · e‚n x · e−j ßn z

;

‚n2 = ß2n − k02

;

Re{‚n } ≥ 0.

(2)

n=−∞

The relation for ‚n ensures the (term-wise) satisfaction of the wave equation, and the condition for ‚n the satisfaction of Sommerfeld’s far field condition. The scattered field ps can be written as a product ps (x, z) = e−j ß0 z · S(x, z) of a propagation factor e−j ß0 z and a factor S(x,z) which must be periodic in z: S(x, z) = S(x, z + T). This gives for the wave numbers in the z direction ßn = ß0 + n

2 T

;

n = 0, ±1, ±2, . . . .

(3)


B

45

The spatial harmonic with the order n=0 evidently must agree in its z pattern with the trace of the incident wave at the surface: ß0 = kz = k0 sin ˜ .

2 (4) Thus: ßn = k0 sin ˜ + n Š0 T ; ‚n2 = k02 sin ˜ + n Š0 T − 1 , and the sound field in x ≤ 0: p(x, z) = Pe · e−j k0 x·cos ˜ + A0 · e+j k0 x·cos ˜ + (5) √ k0 x (sin ˜ +n Š0 /T)2 −1 −j (2n/T) z ·e−j k0 z sin ˜ . An · e ·e + n=0

The second term in the brackets is a homogeneously reflected plane wave; the terms in the sum are higher scattered waves. The exponent of the exponential factor with x under the sum must be zero or imaginary if the spatial harmonic should extend to infinity, i.e. the harmonic is “radiating”. The condition for radiating harmonics (order ns ) is T T − (1 + sin ˜ ) ≤ ns ≤ (6) (1 − sin ˜ ) . Š0 Š0 At (and near) the lower limit the harmonic propagates in the opposite z direction of the incident wave; at (and near) the upper limit the harmonic propagates in the same z direction as the incident wave (if the limits are reached exactly, the harmonic propagates as a plane wave parallel to the surface). The lower limit is attained (or surpassed) the first time for ns < 0 with 1/2 ≤ T/Š0 ≤ 1; the upper limit for ns > 0 with 1 ≤ T/Š0 < ∞. A radiating harmonic does not exist for T/Š0 < 1/2. The non-radiating harmonics shape the near field at the surface. The amplitudes An are determined from the boundary condition Z0 vx (0, z) =! G(z)·p(0, z) at the surface. One expands: +∞

G(z) =

−j (2n /T) z

gn · e

n=−∞

;

1 gn = T

+T/2

G(z) · e+j (2n/T) z dz,

(7)

−T/2

or, alternatively, ∞ a0 an · cos 2n z T + j · bn · sin 2n z T ; + 2 n=1

G(z) = 2 an = T bn =

+T/2

G(z) · cos 2n z T dz; (8)

−T/2 +T/2

2 T

G(z) · sin 2n z T dz;

−T/2

1 a0 an + j · bn ; n = ±1, ±2, . . . ; g0 = . gn = 2 2 The boundary condition gives for m = 0 A0 g0 + cos ˜ + An · g−n = Pe · cos ˜ − g0 n=0

(9)

46

B


and for m=0 A 0 gm +

2 An · gm−n − j ƒm,n sin ˜ + m Š0 T − 1 = −Pe · gm ,

(10)

n=0

with the Kronecker symbol ƒm,n . This is a linear, inhomogeneous system of equations for the amplitudes An . The special case G(z) = const leads to An=0 = 0 and the known reflection factor A0 cos ˜ − g0 = r0 = . Pe cos ˜ − g0

(11)

The absorbed effective power (on a period length) is ⎡ ⎤ 2 T ⎣ 2 2 2 |Ans | 1 − sin ˜ + ns Š0 T ⎦ . |Pe | − |A0| cos ˜ − ¢ = 2Z0

(12)

ns =0

Referring this to the incident effective power

¢e =

T |Pe |2 · cos ˜ 2Z0

gives the absorption coefficient: " " " "2 " A0 " 2 ¢ 1 "" Ans "" 2 " " (˜ ) = = 1 − " " − 1 − sin ˜ + ns Š0 T . " " ¢e Pe cos ˜ n =0 Pe

(13)

(14)

s

The last term is a correction for the absorption ofa homogeneous surface (represented by the first two terms) due to the structured surface; only radiating spatial harmonics enter into this correction. This is plausible because only the radiating harmonics transport energy into the far field. Grooved wall with narrow, absorber-filled grooves: Consider, as a simple example, a plane wall with rectangular grooves, width a, distance T, and depth t, the grooves being filled with a porous material with characteristic values a , Za . A plane wave pe is incident under a polar angle ˜ (the wave vector in the x,z plane).

The grooves are narrow (a < Š0 /4) so that only a plane wave can be assumed to exist in the grooves. Then the grooves can be characterised by an admittance Gs in the groove


B

47

orifice, and the admittance of the arrangement is G(z) = 0 in front of the ribs between the grooves: 1 Gs = tanh (k0 t · an ) ; an = a /k0 ; Zan = Za /Z0. (15) Zan The Fourier coefficients of G(z) are a a sin (ma/T) g0 = Gs ; gm = g−m = Gs ; m = 1, 2, 3, . . . . (16) T T ma/T The system of equations for the amplitudes An of the spatial harmonics becomes (with Pe = 1) m = 0: T T A0 · s(0) + Zs cos (˜ ) + (An + A−n ) · s(n) = Zs cos (˜ ) − s(0) a a n≥1 m = ±1, ±2, . . . : ⎧ ⎡ 2 ⎪ ⎪ ⎨ 1 − (sin ˜ + ms Š0 /T) ⎢ T An ⎢ A0 · s(m)+ ⎣s(m − n)+ƒm,n a Zs ⎪ 2 n=±1, ±2,... ⎪ ⎩−j (sin ˜ + ms Š0 /T) − 1

⎤

(17)

⎥ ⎥ = −s(m) ⎦

with Zs = 1/Gs and the abbreviations: sin (ma/T) ; s(0) = 1 ; s(−m) = s(m). (18) s(m) = ma/T The upper form after the brace holds if m ≤ ms , with ms the limit of orders of radiating harmonics; otherwise the lower form holds. For a/T → 1 it follows that An=0 = 0 and A0 = (Zs cos ˜ − 1)/(Zs cos ˜ + 1). This is the analytic justification for making a homogeneous (bulk reacting) absorber layer locally reacting by thin partition walls with small distances.

Magnitude of the reflection factor |r| of wall with grooves for normal incidence. Full line: periodic surface; dashed: homogeneous

48

B


Magnitude of the reflection factor |r| of wall with grooves for oblique incidence, as a periodic surface for a list of ˜ values (dashes become shorter for increasing list place). a/T = 0.5; T/t = 1; R = 1 The examples shown below use the parameters F = f · t/c0 = t/Š0 ; R = ¡ · t/Z0 with the flow resistivity ¡ of the porous material (glass fibres) in the grooves; a/T; T/t. The first graph shows the magnitude of the reflection factor |r| for a homogeneous surface (dashed) and a periodic surface (full).

B.20 Plane Wall with Wide Grooves


In contrast to the previous may exist in them.

> Sect. B.19 the grooves are no longer narrow; higher modes

The ground of the grooves is supposed to be terminated with an admittance Gs (e.g. produced by a porous layer there). Where possible, the relations are taken from the previous > Sect. B.19.


B

49

The grooves are numbered Œ= 0,±1,±2,. . . and a co-ordinate zŒ = z–Œ·T is used in the Œth groove with –a/2 ≤zŒ ≤ +a/2. The field in the groove is formulated as zŒ 1 −j ß0 ·ŒT −j ‰m x +j ‰m x pk (x, zŒ ) = e Bm · e · cos m − (1) + Cm · e a 2 m≥0 with ‰m =

⎧ 2 ⎨ k0 − (m/a)2 ≥ 0

;

⎩ −j (m/a)2 − k02 ≥ 0

;

k0 ≥ m/a ,

(2)

k0 < m/a .

The amplitudes Cm of the groove modes reflected at the ground are Cm = −Bm ·

Gs − ‰m /k0 −2j ‰m t ·e : = Bm · Rm . Gs + ‰m /k0

(3)

The Rm are modal reflection factors “measured” in the groove orifice.

Magnitude of the reflection factor |r| of a wall with wide grooves. The grooves are completely filled with glass fibre material; t = groove depth; R = ¡ · t/Z0 . Full: with spatial harmonics; dashed: homogeneous The boundary conditions in the plane x=0 lead to the inhomogeneous linear system of equations (m= 0,±1,±2,. . . ) for the amplitudes An of the spatial harmonics, which exist in the half-space x< 0 (with ƒn.m = Kronecker symbol; ƒ0 =1; ƒm>0 =2 ): ⎡ ⎤ ∞ +∞ ƒ‹ ‰‹ 1 − R‹ a j ‚ m ⎦ An ⎣ (−1)‹ s−‹,n · s‹,m − ƒn,m 2T ‹=0 2 k0 1 + R‹ k0 n=−∞ (4) ⎡ ⎤ ∞ ƒ‹ ‰‹ 1 − R‹ a = Pe · ⎣ƒ0,m · cos ˜ − (−1)‹ s−‹,0 · s‹,m ⎦ 2T ‹=0 2 k0 1 + R‹

B

50


with the abbreviation j m/2 sin (m/2 − ßn a/2) m sin (m/2 + ßn a/2) sm,n = e + (−1) . m/2 − ßn a/2 m/2 + ßn a/2

(5)

The amplitudes Bm follow with the An from

! (−1)m ƒm An · s−m,n . Pe · s−m,0 + Bm = 2 (1 + Rm ) n=0,±1,±2... The reflection factor r of the arrangement follows with the An as in the previous B.19.

(6) > Sect.

20 lg|p| dB, wide-slit comb plate

0 dB 2

-10

1.5 -1

1

-0.8 -0.6 x/T

z/T

0.5

-0.4 -0.2

00

Sound pressure level 20 · lg|p| in front of a wall (at x/T = 0) with wide grooves, completely filled with glass fibre material. F = T/Š0 = 0.75; ˜ = 45◦ ; a/T = 0.5; T/t = 0.25; R = ¡ · t/Z0 = 1. One spatial harmonic is radiating, therefore the periodicity extends to far distances

B.21 Thin Grid on Half-Infinite Porous Layer


A thin grid with slits of width a at mutual distance T covers a half space of porous absorber material with flow resistivity ¡ and characteristic values a , Za (or, in a normalised form, an = a /k0 , Zan = Za /Z0 ). A plane wave pe is incident at a polar angle ˜ (wave vector in the x,z plane).


B

51

Field formulation in the zone I (x ≤ 0): pI (x, z) = pe (x, z) + ps (x, z) = Pe · e−j (kx x+kz z) +

+∞

An · e‚n x · e−j ßn z ;

(1)

n=−∞

; kx2 + kz2 = k02 , ß0 = kz = k0 sin ˜ ; ßn = ß0 + 2n T = k0 (sin ˜ + nŠ0 /T) , ; ‚0 = j k0 cos ˜ ; ‚n = k0 (sin ˜ + nŠ0 /T)2 − 1. ‚n2 = ß2n − k02 kx = k0 cos ˜

; kz = k0 sin ˜

(2)

Radiating spatial harmonics with order ns in the limits: −

T T (1 + sin ˜ ) ≤ ns ≤ (1 − sin ˜ ). Š0 Š0

(3)

Field formulation in the zone III (x≥ 0): pIII (x, z) =

+∞ n=−∞

vIIIx (x, z) =

Dn · e−—n x · e−j ßn z = e−j ß0 z

+∞ n=−∞

Dn · e−—n x · e−j (2n/T)z ,

+∞ k0 −j ß0 z —n e Dn · e−—n x · e−j (2n/T)z , a Za k0 n=−∞

—2n = ß2n + a2

;

—n = k0

2 2. sin ˜ + n Š0 T + an

(4)

(5)

The boundary conditions on the front and back side of the grid, together with the orthogonality of the spatial harmonics, lead to the following linear inhomogeneous system of equations (m = 0, ±1, ±2, . . .): —n —0 —0 T T an Zan cos ˜ + an Zan cos ˜ − m=0: A0 An s(n) = Pe + ; (6) a k0 k0 a k0 n=0 m=0:

A0 ·

—n —0 T —0 ‚m s(m) + An s(m − n) − ƒm,n · an Zan j = −Pe · s(m) ; (7) k0 k a k k 0 0 0 n=0

with the Kronecker symbol ƒm,n and the abbreviation s(n): =

sin (na/T) . na/T

(8)

52

B


The amplitudes Dn follow from D0 = Pe + A0; Dn=0 = An . Special case a/T → 0: (i.e. hard plane) it follows that A0 = Pe ; An=0 = 0; special case a/T → 1: (i.e. open absorber) it follows that An=0 = 0 and ⎛ ⎞2⎛ ⎞ A0 cos ˜ cos ˜ = ⎝ an Zan − 1⎠ ⎝ an Zan + 1⎠, Pe 2 2 sin2 ˜ + an sin2 ˜ + an

(9)

which is just the reflection factor at a semi-infinite absorber layer.

Reflection coefficient |r|2 for a thin grid on an infinite glass fibre layer for different ratios a/T (dashes are shorter with increasing values). ˜ = 0; R = ¡T/Z0 = 1 Special case: Ignore all higher spatial harmonics, i.e. An=0 =0: ⎛ ⎞2⎛ ⎞ T A0 ⎝ T cos ˜ cos ˜ = − 1⎠ ⎝ an Zan + 1⎠. an Zan Pe a a 2 2 2 sin ˜ + an sin2 ˜ + an Special case: The material in zone III is air: i.e. an = j; Zan = 1; —n /k0 → ‚ n /k0 : ‚n T An · j s(m − n) + ƒm,n · k0 a n T = Pe cos ˜ · s(m) + ƒ0,m · ; m = 0, ±1, ±2, . . . . a The reflection coefficient |r|2 is evaluated by " " " "2 " An " 2 " A0 " 1 2 " " " 1 − (sin ˜ + nŠ0 /T)2 . " |r| = " " + Pe cos ˜ n=n =0 " Pe " s

Parameters in the examples shown below are ˜ , a/T, R = ¡ · T/Z0, F = T/Š0 .

(10)

(11)

(12)


B

53

Sound pressure level in front of (x/T < 0) and in (x/T > 0) the absorber, covered with a thin grid. ˜ = 45◦ ; T/Š0 = 0.75; a/T = 0.5; R = ¡T/Z0 = 1

B.22 Grid of Finite Thickness with Narrow Slits on Half-Infinite Porous Layer


In contrast to the object in the previous > Sect. B.21, the grid now has a finite thickness d, and the slits of the grid are assumed to be narrow so that only a plane wave must be assumed in the slits. The slits form the new zone II. The slit and grid period at z = 0 can be taken as the representatives for the other slits. The field formulations in zones I and III remain as in

> Sect. B.21.

54

B


The field in the Œth slit, with z = Œ · T + zŒ ; Œ = 0, ±1, ±2, . . ., is formulated as pII (x, zŒ ) = e−j ß0 zŒ B · e−jk0 x + C · e+jk0 x Z0 vxII (x, zŒ ) = e−j ß0 zŒ B · e−jk0 x − C · e+jk0 x

(1)

The boundary conditions lead to B sin (ß0a/2) (1 + Sa ) e+j k0 d = , Pe ß0a/2 (S + Sa ) cos (k0d) + j (1 + SSa ) sin (k0 d) sin (ß0 a/2) C (1 − Sa ) e−j k0 d =− Pe ß0 a/2 (S + Sa ) cos (k0 d) + j (1 + SSa ) sin (k0d)

(2)

with the abbreviations +∞ a k0 sin (ßn a/2) 2 j S: = T n=−∞ ‚n ßn a/2

+∞ a a Za k0 sin (ßn a/2) 2 Sa : = . T k0 Z0 n=−∞ —n ßn a/2

;

(3)

With B and C it follows that A0 = Pe − (B − C)

a sin (ß0a/2) , T cos ˜ ß0 a/2

a k0 sin (ßm a/2) Am = −j (B − C) T ‚m ßm a/2 Dm =

;

(4) m = ±1, ±2, . . . ,

k0 sin (ßm a/2) a a Za B · e−jk0 d − C · e+jk0 d T k0Z0 —m ßm a/2

;

m = 0, ±1, ±2, . . . .

(5)

The reflection coefficient |r|2 follows from "2 " " " a/T sin (ß0 a/2) (B − C) "" |r|2 = "" 1 − cos ˜ ß0 a/2 2 ! a/2) a/T a/T sin (ß 0 + |B − C|2 Re{S} − . cos ˜ cos ˜ ß0 a/2

(6)

In the special case of air, instead of a porous material, behind the grid sin (ß0 a/2) (1 + S) e+j k0 d B =2 , 2 Pe ß0 a/2 (1 + S) e+j k0 d − (1 − S)2 e−j k0 d C sin (ß0 a/2) (1 − S) e−j k0 d = −2 . Pe ß0 a/2 (1 + S)2 e+j k0 d − (1 − S)2 e−j k0 d

(7)

The parameters in the following examples are ˜ , a/T, d/T, R = ¡ · T/Z0 , F = T/Š0 (equivalences: |r|∧ 2 → |r|2 ; lam0 → Š0 ).


B

55

Reflection coefficient |r|2 of a porous half-space covered with a grid with finite thickness for some ratios a/T (the dash becomes shorter for larger a/T). ˜ = 0; d/T = 0.25; R = ¡T/Z0 = 1

Sound pressure level in front of, in, and behind the grid. ˜ = 45◦ , F = T/Š0 = 0.75, a/T = 0.5, d/T = 0.25, R = 1. The ratio a/T = 0.5 is too large for the assumption of only a plane wave in the slits. This assumption produces just a least square error matching at the orifices

B

56


As above, but for ˜ = 45◦

B.23 Grid of Finite Thickness with Wide Slits on Half-Infinite Porous Layer


The object is the same as in the previous > Sect. B.22, but the slit channels are no longer assumed to be narrow, i.e. higher modes are assumed in the slits. The field formulations remain the same as in

> Sect. B.21.

The field in the Œth slit, Œ = 0, ±1, ±2, . . ., with zŒ = z − Œ · T, is formulated as Bm · e−j ‰m x + Cm · e+j ‰m x · cos m(zŒ a − 1/2) pII (x, zŒ ) = e−j ß0 ŒT m≥0

‰m

⎧ 2 ⎨ k0 − (m/a)2 = ⎩−j (m/a)2 − k02

; ;

m ≤ mg m > mg

(1) ;

mg = Int (k0 a/)

For the auxiliary amplitudes Xm , Ym (m≥0) Xm : = Bm − Cm Bm =

;

Ym : = Bm · e−j ‰m d − Cm · e+j ‰m d

Xm · e+j ‰m d − Ym e+j ‰m d − e−j ‰m d

;

Cm =

Xm · e−j ‰m d − Ym e+j ‰m d − e−j ‰m d

(2)


B

57

a combined system of equations (m = 0, 1, 2. . .) is derived from the boundary conditions: ! −2j ‰m d k0 ‰n m ƒm,n 4T 1 + e Xn j · s−m,Œ · sn,Œ + (−1) k0 Œ=0,±1,... ‚Œ ƒm a 1 − e−2j ‰m d n≥0 4T 2(−1)m e−j ‰m d = Pe · s−m,0 + · Y m , a ƒm 1 − e−2j ‰m d (3) ! −2j ‰m d ƒ 4T Z 1 + e k ‰n k0 m,n 0 0 Yn · s−m,Œ · sn,Œ + (−1)m k0 Œ=0,±1,... —Œ ƒm a a Za 1 − e−2j ‰m d n≥0 =

8T k0 Z0 (−1)m e−j ‰m d · Xm a a Za ƒm 1 − e−2j ‰m d

with ƒm,n the Kronecker symbol; ƒ0 = 1, ƒn>0 = 2; and the abbreviation sin (m/2 − ßn a/2) sin (m/2 + ßn a/2) sm,n : = ej m/2 + (−1)m , m/2 − ßn a/2 m/2 + ßn a/2

(4)

with sm,n = s−m,n and: sm,n = 2

ßn a/2 sin (ßn a/2) · −j cos (ßn a/2) (ßn a/2)2 − (m/2)2

; ;

m = even . m = odd

(5)

Reflection coefficient of a thick layer of glass fibres, covered with a grid with wide slits for some ratios a/T (the dashes are shorter for higher a/T). ˜ = 45◦ ; d/T = 0.25; R = ¡ · T/Z0 = 1 The amplitudes An , Dn (n=0, ±1, ±2,. . . ) follow from

! k0 a ‰m An = j ƒ0,Œ · Pe · cos ˜ − · sm,n , (Bm − Cm ) ‚n 2T m≥0 k0

‰m a a Za k0 +—n d Dn = e · sm,n . Bm e−j ‰m d − Cm e+j ‰m d 2T k0 Z0 —n k0 m≥0

(6)

58

B


The reflection coefficient |r|2 is evaluated as in the previous

> Sect. B.22.

The parameters in the shown examples are ˜ , a/T, d/T, R = ¡ · T/Z0, F = T/Š0 (equivalences: |r|∧ 2 → |r|2; lam → Š0 ).

References Cremer, L., Heckl, M.: Koerperschall, 2nd edn. Springer, Berlin (1995)

Mechel, F.P.: Schallabsorber,Vol. III, Ch. 26: Rectangular duct with local lining.Hirzel,Stuttgart (1998)

Cummings, A.: Sound Generation in a Duct with a Bulk-Reacting Liner. Proc. Inst. Acoust. 11 part 5, 643–650 (1989)

Mechel, F.P.: Schallabsorber, Vol. III, Ch. 27: Rectangular duct with lateral lining Hirzel, Stuttgart (1998)

Felsen, L.B., Marcuvitz, N.: Radiation and Scattering of Waves, p. 89 Prentice Hall, London (1973)

Mechel, F.P.: A Principle of Superposition. Acta Acustica (2000)

Gottlieb: J. Sound Vibr. 40, 521–533 (1975)

Moon, P., Spencer, D.E.: Field Theory Handbook, 2nd edn. Springer, Heidelberg (1971)

Kleinstein, Gunzburger: J. Sound Vibr. 48, 169–178 (1976) Mechel, F.P.: Schallabsorber, Vol. I, Ch. 3: Sound fields: Fundamentals. Hirzel, Stuttgart (1989) Mechel, F.P.: Schallabsorber, Vol. I, Ch. 11: Surface Waves. Hirzel, Stuttgart (1989) Mechel, F.P.: Schallabsorber, Vol. I, Ch. 12: Periodic Structures. Hirzel, Stuttgart (1989) Mechel, F.P.: Schallabsorber, Vol. II, Ch. 3: Field equations for viscous and heat conducting media. Hirzel, Stuttgart (1995) Mechel, F.P.: Schallabsorber, Vol. III, Ch. 18: Multilayer finite walls. Hirzel, Stuttgart (1998)

Morse, P.M., Feshbach, H.: Metheods of Theoretical Physics, part I McGraw-Hill, New York (1953) Ochmann,M.,Donner,U.: Investigation of silencers with asymmetrical lining; I: Theory. Acta Acustica 2, 247–255 (1994) Pierce, A.D.: Acoustics, Ch. 4: McGraw-Hill, New York (1981) Skudrzyk,E.: The Foundations of Acoustics,Springer, New York (1971) VDI-Waermeatlas, 4th edition, VDI, Duesseldorf (1984)

C Equivalent Networks The application of equivalent networks is a useful method for the solution of many tasks in acoustics. The method is applicable if the sound field at any value of the x co-ordinate in the “direction of propagation” has the same lateral distribution. Plane waves are just a special case. The conception of end corrections, or, equivalently, oscillating mass, extends the range of application even to a space with contractions. The method of equivalent networks is based on the analogies (electro-acoustic analogies) with electrical circuits.

C.1 Fundamentals of Equivalent Networks

See also: Mechel, Vol. II, Ch. 2 (1995)

Electromagnetic quantities and their relations (A is a cross-sectional area, or Ampere):

Table 1 Electromagnetic quantities and their relations Quantities electric Quantity

Relation

magnetic Dimension

Quantity

Relation

Dimension

Voltage

U

Volt, V

Current

I

Ampere, A

El. field strength

E

V/m

Magn. field strength

H

A/m

El. induction

D = q=A

A s=m

Magn. induction

B = ¥m =A

V s=m

Charge, flow

q = s I dt = D A = ¥e

As

Flow

Vs

Voltage

U = s E ds

V

Current

¥m = s U dt = BA I = H ds

Current

I = dq=dt

A

Voltage

U = d¥m =dt

V

Capacity

C = q=U = ¥e =U

A s=V

Inductivity

L = U=(dI=dt) = ¥m =(dI=dt)

V s=A

A

60

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Equivalent Networks

Table 2 & 3 Passive electrical and mechanical circuit components Element Resistor

Quantity

Symbol

Letter

Resistance

U I s I dt C= U

R

Capacitor Capacity

C

Coil

L

Inductivity

Complex Impedance

Z

Complex Admittance

G

Definition R=

U dI=dt U Z= I I 1 G= = U Z L=

U = 0; I = 0

Connection I

Transformer

I

1

U

U

1

2

I I1

U

U

u=

2

w2 U2 I1 u= = w1 U1 I2

2

2

1

Element

Quantity

Symbol

Letter

Resistor

Friction

R

Spring

Compliance

C

Mass

Inertance

M

Complex Impedance

Z

Complex Admittance

G

F v vdt C= F F M= dv=dt F Z= v R=

G=

v 1 = F Z

F = 0; v = 0

Rigid Connection Lever

Definition

l2 l1 F1 F1 F2 l2 l1 F2

u=

I2 I1

u=

F2 v1 = F1 v2

Equivalent Networks

Table 4 Defining relations for passive mechanical circuit elements Element

Co-ordinates

Friction

F1

Relations F2

R

F = F1 − F2 v = v1 − v 2

v1 x

v2 x

1

2

C

Spring

F=Rv

1 = (x1 − x10 ) 2 = (x2 − x20 ) = 1 − 2

− F1

F1 x 10

x1

Mass

F = F1 =

1 v j–C

x 2 x 20

F = j–M v

F

~

M v x

Lever

20

00 = x0 − x00

x2 F2

l

2 = 20 − 00

10

F0 = −(F1 + F2 )

F1 l

x

1

1

u= x

= = = = = = = =

20 = x2 − x20 1 = 10 − 00

2

x

F v x R M C l

10 = x1 − x10

F0 00

x

force; velocity; position; deformation; friction factor; mass; compliance; length

0

l

F2 = (l1 =l2 ) F1 2

l1

2 = (l2 =l1 ) 1

C

61

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Equivalent Networks

The velocity of a resistance is the relative velocity of both ends of the resistance. The velocity of a spring is the relative velocity of both ends of the spring. The velocity of a mass is its velocity relative to the point on which the force source is supported. The force acting on a resistance or a spring is the force difference at both ends of the element. Boundary conditions: Node theorem: The sum of all forces acting on an immaterial node point is zero. Mesh theorem: The sum of the velocities in a closed mesh is zero. A spring is supposed to have no mass; a mass is supposed to be incompressible. A hard (or rigid) termination with v = 0 corresponds in electrical circuits • to an open termination in the UK analogy (see below), • to a short-circuited termination in the Uv analogy; a soft (or pressure release) termination with F = 0 (or p = 0) corresponds in electrical circuits • to a short-circuited termination in the UK analogy, • to an open termination in the Uv analogy. Rules: • There is no force difference across a spring. • There is no velocity difference across a mass. • The second pole of a mass is at the point on which the driving force is supported. ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ u 0 v v 1 ⎠ Relations for levers with a leverage u: ⎝ 2 ⎠ = ⎝ . 1 ⎠·⎝ 0 F2 F1 u Sources: Helmholtz theorem:

Uo ~

Zi Uo = Zi·Is

Is

Zi

Zi

is the internal source impedance;

Uo is the open-circuit source voltage; Is

is the short-circuit source current

Equivalent Networks

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63

Reciprocal networks: A reciprocal network is composed of elements Zr which follow from the elements Z of the original network by the rule Z → Zr = r2/Z∗) with the reciprocal invariant r. With suitably normalised impedances, one can take r = 1. Voltage sources change to current sources, and vice versa. In both networks voltage transfer ratios ↔ current transfer ratios correspond to each other and have the same frequency response curves. An advantage of the reciprocal network possibly is its easier conception and realisation. The shape of the reciprocal network changes: a mesh changes to a node; a node changes to a mesh (see below for a more precise rule). Table 5 Reciprocal electrical elements Reciprocal Resistance Inductivity Capacity Impedance Admittance

2

R

r /R

L

L/r

C

r C

Z

r 2/ Z

G

r 2G

2

Capacity 2

Inductivity Admittance Impedance

U/ r

Voltage source

U

Current source

Reciprocal resistance

~

Current source

Zi

r 2/ Z i

I

Voltage source

Z

i

r I

~

r 2/ Z i

Rule for the construction of a reciprocal network of networks, which can be drawn in one plane: • Draw a point into every mesh of the original network and one point outside the network. • Connect all pairs of points with each other by lines which cross circuit elements. • Replace the crossed elements with their reciprocal elements. • If necessary, redraw the reciprocal network in a better form. ∗)


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Equivalent Networks

Below is an example illustrating the application of this rule.

L1

R1 U

L2

~

C

R2

U/ r L 1/ r

r 2/ R 1

2

L /r 2

2

r 2/ R

2

2

r C

Figure 1 Example of reciprocal networks Electro-acoustic UK analogy: Table 6 Corresponding elements in the UK-analogy electrical

mechanical

Voltage

U

Force

F

Current

I

Velocity

v

Resistance

Re

U = Re I

Resistance

Rm

F = Rm v

Coil

L

M

Condensator

Ce

U = j–L I Mass 1 U= I Spring j–Ce

F = j–M v 1 F= j–Cm v

Impedance

Ze

Impedance

Zm

Admittance

Ge

U = Ze I 1 U= I Ge

Admittance

Gm

Voltage source

Current source

Uo

~

Fource source

Zi

Is

Velocity source

Zi Node Mesh

cm

Fo

F = Zm v 1 F= v Gm

~

Zi

vs Zi

I=0

Mesh

U=0

Node

v=0 F=0

Equivalent Networks

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65

Electro-acoustic Uv analogy: Table 7 Corresponding elements in the Uv analogy Electrical

Mechanical

Voltage

U

Velocity

v

Current

I

Force

F

Resistance

Re

U = Re I

Resistance

1=Rm

v=

Coil

L

U = j–L I

Spring

Cm

v = j–Cm F

Condensator

Ce

U=

M

v=

Impedance

Ze

U = Ze I

Admittance

Gm

v = Gm F

Admittance

Ge

U=

Impedance

Zm

v=

1 I Mass j–Ce

1 I Ge

1 F Rm

1 j–M F

1 F Zm

vo

Voltage source

Uo

~

Velocity source

Zi

Zi

Is Current source

Node Mesh

Zi

Force source

I=0

Node

U=0

Mesh

Fo

~

Zi

F=0 v=0

Networks in the UK and Uv analogy, respectively, are reciprocal to each other.

C.2 Distributed Network Elements


One distinguishes between “lumped” elements, as in > Sect. C.1, with no sound propagation within one element, and “distributed” elements with internal sound propagation. Distributed elements are homogeneous, i.e. without change in the cross section and/or material. They are introduced into network analysis as four poles, whereas lumped elements are two poles.

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Equivalent Networks

Four poles themselves can be represented as equivalent networks, either as T networks or as ¢ networks. The four-pole representation is used for duct sections and/or layers with internal axial sound propagation. In the following formulas t is the duct section length or layer thickness t. The axial propagation constant a is either the characteristic propagation constant of the medium in the duct or layer for a plane wave propagating in the axial direction, or the axial component for oblique propagation. Correspondingly, Za is the characteristic impedance of the medium, or the axial component. If the medium in the duct or layer is air, then a = j · k0; Za = Z0 . Four-pole equations: p1 = cosh (a t) · p2 + Za sinh (a t) · v2

(1)

Za · v1 = sinh (a t) · p2 + Za cosh (a t) · v2 v1

v2 Γa , Za

p1

p2

t

Equivalent T-circuit impedances: Z1 = Za · coth (a t) − Z2 = Za Z2 =

(2)

cosh (a t) − 1 sinh (a t)

Za sinh (a t)

(3)

v1

v2 Z1

p1

Z1 Z2

p2

Equivalent ¢-circuit impedances: Z1 = Za · sinh (a t) Za sinh (a t) cosh (a t) − 1 1 1 1 − = Z2 Za tanh (a t) Z1

(4)

Z2 =

(5)

Equivalent Networks

v1

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67

v2 Z1

p1

Z2

p2

Z2

Some simple systems with distributed-network elements are displayed below. Tube section with hard termination: p Z = = −j cot (k0t), Z0 vx Z −−−−→ tŠ0

C=

1 0 c20 1 = , j– t j–C

(6)

t . 0 c20

Z

k0, Z0 p

vx

k0, Z0

t Remarks: • For t Š0 a spring-type reactance; • First resonance at k0t = /2 ; t = Š0 /4; • For /2 < k0 t < a mass-type reactance. Tube section with hard termination, filled with porous material: p Za Z = = coth (a t), Z0 vx Z0 Za 1 0 c20 ≈ . Z −−−−→ tŠa a t j – ‰ t

(7)

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Equivalent Networks

Remarks: • ‰ = adiabatic exponent of air; • = porosity of porous material. Tube section with open termination: Z p = = j tan (k0 t), Z0 vx (8)

Z −−−−→ j –0 t = j – M, tŠ0

M = 0 t. Remarks: • The assumption p = 0 at the orifice is an approximation for narrow tubes; • Without load of radiation impedance !; • t Š0 mass-type reactance; • /2 < k0t < spring-type reactance. Z

k0, Z0 p

vx

k0, Z0

p=0

t

Tube section with open termination, filled with porous material: p Za Z = = tanh (a t) Z0 vx Z0 j Z −−−−→ a Za t ≈ Z0 k0t · (a /k0)2 ; tŠa ‰

¡t −−−→ E1

;

j – 0 t −−−→ E1

(9)

Equivalent Networks

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69

Remarks: • The assumption p = 0 at the orifice is an approximation for narrow tubes; • Without load of radiation impedance !; • E = 0 f /¡ absorber parameter; • ¡ = flow resistivity of porous material; • = porosity of porous material; • Ša = wavelength in absorber material; • t Ša and E 1: Z ≈ resistance, and E > 1 : Z ≈ mass reactance. Tube terminated with Helmholtz resonator with thin resonator plate: Z=

1 0 c20 p Sb , ≈ −j Z0 cot (k0t) −−−−→ tŠ0 j – vx Sb t

1 Sa p Sa 0 c20 . ≈ · Z −−−−→ Zs = tŠ0 j – vx Sa V Z Sa k0, Z0

p

k0, Z0

Zs vx

Sb

t

Remarks: • End corrections of orifices neglected !; • Sa · Sb Š2 0 ; • Z = “homogeneous” impedance; • Zs = interior orifice impedance; • s = Sa /Sb = resonator plate porosity; • V = t · Sb = resonator volume.

(10)

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Equivalent Networks

Perforated plate in a tube: 1 Zi = Zt + Ze = Zs + Ze , 1 Zt = Zs .

∗)

(11)

Remarks: • d Š0 ; • Ze = “homogeneous” load impedance; • Zi = “homogeneous” input impedance; • Zt = “homogeneous” partition impedance of plate; • Zs = partition impedance of perforations. A layer of air (transformation of impedances by a layer): Zi j tan (k0t) + Ze /Z0 , = Z0 1 + j Ze /Z0 · tan (k0t) Zi −−−−→ tŠ0

(12)

j –0 + Ze . t 1 + Ze · j 0 c20

Zi

k0, Z0

pi

k0, Z0

t

∗)

see Preface to the 2nd edition.

Ze

pe

Equivalent Networks

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71

Remarks: • Zi = “homogeneous” input impedance; • Ze = “homogeneous” load impedance. A layer of porous material (transformation of impedances by a layer): Zi tanh (a t) + Ze /Z0 . = Z0 1 + j Ze /Z0 · tanh (a t)

(13)

Remarks: • Zi = “homogeneous” input impedance; • Ze = “homogeneous” load impedance.

C.3 Elements with Constrictions


Equivalent networks can be used for sound in multilayer absorbers or in ducts without constrictions because the lateral sound distribution functions can be divided out. Nevertheless, constrictions can also be represented with equivalent networks if their lateral dimensions and the lateral dimensions in front of and behind the constriction are small compared to the wavelength or, more precisely, if no higher modes can propagate in the wide cross sections. Then the constrictions produce only near fields. These can be represented by equivalent oscillating masses Mi at the orifice on the side of sound incidence and Me at the orifice of sound exit. The equivalent oscillating mass is proportional to the end correction. The following examples show two Helmholtz resonators, one excited by an incident wave pi on the resonator, the other excited by a sound pressure pi at the back side of the resonator volume.

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Equivalent Networks

v Sa pi

Sb

v

d

Rr

Rr

V

Mi + M + M e p

2pi

F

t

represents the radiation resistance of the orifice near the incidence;

Mi is the equivalent oscillating mass on the outer side; mass); Me is the equivalent oscillating mass on the interior side mass)

v Ri

V pi

Sb

v, p

2pi

Mi + M + M e F

Sa t

Ri

d

represents the interior source resistance;

Mi is the equivalent oscillating mass on the outer side; Me is the equivalent oscillating mass on the interior side

Rr

p

Equivalent Networks

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73

C.4 Superposition of Multiple Sources in a Network Helmholtz’s theorem of superposition for multiple sources: If a network is excited by more than one voltage source (current sources) with the same frequency, the state of the network with common excitation is a superposition of states in which only one source is active, and the network terminals at the other sources are short-circuited (open).

C.5 Chain Circuit


A chain circuit is a useful representation of multilayer absorbers (see

> Sect. D.4).

A chain network consists of longitudinal impedances Zn and lateral admittance Gn . Its sound pressures pn in the nodes and velocities vln in the longitudinal elements, as well as vqn in the transversal elements, can be evaluated by iteration.

If the network is open (as shown), i.e. vl,N+1 = 0, one begins with an assumed value pN = 1. The backward recursion is vq,n = pn · Gn , v,n = vq,n + v,n+1 ,

(1)

pn−1 = pn + v,n · Zn . One iterates over n = N, N − 1, . . ., 1. The last result is p0 . All field quantities are proportional to pN . To replace this by p0 as the reference pressure, divide all (saved) quantities by the value of p0 . If parameters Zn , Gn are used and normalised with Z0 , the velocities are returned as Z0 vn . If the real network is terminated with a load impedance Zload , add 1/Zload to GN , so the network to be evaluated is open again. p0 p1 The input impedance of the network is Z= = Z1 + . (2) v,1 v,1

74

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The impedance of the part of the network behind the node n is

Yn =

pn ; n = 1, 2, . . . , N − 1. v,n+1

The load impedance at the node n is

Xn =

pn pn = ; n = 1, 2, . . . , N. (4) v,n vq,n + v,n+1

(3)

Suitable representations of a layer of material with thickness t, propagation constant a and wave impedance Za of the material are

Z G

Z = Za · sinh (a t) ; G =

Z

G

cosh (a t) − 1 ; Za · sinh (a t)

Z G

G=

sinh (a t) Za ; Z = (cosh (a t) − 1) . Za sinh (a t)

C.6 Partition Impedance of Orifices The method of equivalent networks was originally designed for a sequence of layers without constrictions. The impedance at a layer boundary is defined by the ratio of sound pressure p and normal velocity v, both averaged over the boundary surface S, Z = p S /v S. A layer with constrictions (e.g. a plate with the neck of a resonator) can be included in the equivalent network scheme if an orifice partition impedance (or simply: orifice impedance) ZM is added to the orifice of the constriction. This is a partition impedance of the type Zp = p S /v S with p = (pfront − pback ) the sound pressure drop across the plane of the orifice and v the particle velocity through the orifice (in the direction front→back). Assume the area of the orifice is s (e.g. s = the cross-section area of a neck) with the porosity = s/S (e.g. S = cross-section area of a single resonator). Let Zs = p s //v s be the impedance at the neck orifice (inside the neck) and ZK = p S /v S the impedance in the “chamber” (e.g. resonator volume) in the orifice plane (inside the chamber). The underlining will serve to remind the reader that ZK is a homogenised impedance. Then, from the conditions of continuity of volume flow and average pressure across the orifice plane follows for the orifice partition impedance ZM = Zs − ZK = p s /v s − ZK .

(1)

For its evaluation a field analysis must be made of the sound field in front of and behind the orifice for a plane wave incident on the orifice. Due to reciprocity it is sufficient to consider the case of sound incidence from the side of the narrow section (neck). The orifice partition impedance will be the same for sound incidence in the opposite direction.

Equivalent Networks

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75

For orifices radiating into free space or ending in an empty chamber,the orifice partition impedance is purely reactive with the sign of a mass reactance. Mostly it is represented in the literature by the end correction /a (with s = a2 ) by the general interrelation ZM = j k0a · /a,

(2)

i.e. /a represents the imaginary part of ZM . A number of end corrections are given in > Sect. F.2, End corrections, and in > Ch. H, “Compound Absorbers”, of this book. However, the orifice partition impedance becomes complex in some important configurations, and its dependence on the parameters of the configuration is not simple enough for a formulation of ZM as a regression polynomial of these parameters. Therefore, this section derives and presents the orifice partition impedance ZM for a number of configurations as explicit formulas in short, tabular form. The derivation of an explicit formula (i.e. no solutions of systems of equations) requires in some configurations the approximate assumption that only plane waves propagate in the narrow section (neck). At higher frequencies, for which higher neck modes are popagating, the conception of the equivalent networks no longer works. An important advantage of the orifice partition impedance ZM lies in the fact that it can be combined with the partition impedance ZF of (e.g.) a poro-elastic foil in the orifice by simple addition: Z = ZM + ZF . Below: • The use of ZM is demonstrated in the chain of formulas of an equivalent network for a Helmholtz resonator (as an example). • Then the typical procedure for deriving ZM is illustrated for this configuration. • Finally, the formulas for other configurations are collected in a table. All impedances and admittances are supposed to be normalised with Z0 . Use of ZM in the equivalent network for a Helmholtz resonator The sound field is subdivided into three zones: (I) in front of the resonator (II) in the neck (III) in the chamber (resonator volume)

σ

76

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Equivalent Networks

ZR ZMa , ZMi Zsv Zsh ZK Z1 , Z2 = s/S G v = · Gv Gv = 1/Zv Zv = ZMa + Zsv Zsv = Zi Zsv =

Zi · tanh(i d) + Zsh Zi + Zsh · tanh(i d)

j · tan(k0 d) + Zsh 1 + j · Zsh · tan(k0d)

Zsh = ZMi + ZK

tan (k0t) = Za tanh (a t)

ZK = · Zk = −j

= = ·Za ZMa → ZMa + ZFa ZMi → ZMi + ZFi

homogenised front-side impedance of the resonator array; outer and interior orifice partition impedances; entrance impedance of the neck (II); exit impedance of the neck (II); homogenised entrance impedance of the resonator chamber (III); impedances of the ¢-fourpole which represents the neck channel; porosity of the neck plate; homogenised front-side admittance of resonator; (3) Gv = front-side neck entrance admittance; (4) Zv = front-side neck entrance impedance; ZMa = front-side orifice partition impedance; (5) Zsv = front-side neck entrance impedance; front-side neck entrance impedance for (6a) a narrow neck, i , Zi characteristic capillary values; front-side neck entrance impedance for a medium-wide neck (plane waves only); exit impedance of neck; ZMi = interior orifice partition impedance; ZK = · ZK load impedance of neck; ZK = homogenised chamber entrance impedance;

(6b)

load impedance for an empty chamber of depth t;

(8)

chamber filled with porous material, char. values a , Za ; anechoic empty channel; channel filled with porous material; if orifice(s) is (are) covered with poro-elastic foil(s).

Field analysis for a circular neck orifice in a circular, empty chamber (example)

(7)

(9) (10) (11) (12)

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77

Field formulation in neck (I) for an incident plane wave with arbitrary amplitude B0 (e.g. B0 = 1 ) and higher radial modes with amplitudes Cm reflected at the orifice at x = 0: pI (x, r) = B0 e−j k0 x +

m≥0

2 Cm e+‰m x J0 (—m r) ; ‰m = —2m − k02

j ∂p ‰m Cm e+‰m x J0 (—m r) = B0 e−j k0 x + j k0 ∂x k0 m≥0

Z0 vIx (x, r) =

(13)

The radial eigenvalues follow from the condition of zero radial particle velocity of each mode at the neck wall at r = a and therefore are solutions of J1 (—m a) = 0; m = 0, 1, 2, . . .; with —0 a = 0, thus ‰0 = jk0. Field formulation in chamber (II) with unknown mode amplitudes Dn : pII (x, r) =

n≥0

Z0 vIIx (x, r) =

Dn cosh ‚n (x − t) J0 (†n r)

;

‚n2 = †n2 − k02

j ∂p ‚n =j Dn sinh ‚n (x − t) J0 (†nr) k0 ∂x k0 n≥0

(14)

The radial-mode eigenvalues †n b are solutions of J1 (†nb) = 0 (with n = 0, 1, . . .); †0 b = 0, i.e. †0 = 0 und ‚0 = jk0 . (II) The modes in each zone are orthogonal to each other and have mode norms N(I) m , Nn : ⎧ ⎪ ⎪ a ⎨ 0 ; m = ‹ J0 (—m r) J0 (—‹ r) r dr = a2 2 a2 ⎪ ⎪ J0 (—m a) −−−−→ ; m=‹ ⎩ N(I) m = 0 m=0 2 2 (15) ⎧ ⎪ b ⎨ 0 ; n = Œ J0 (†n r) J0 (†Œ r) r dr = 2 2 ⎪ ⎩ N(II) = b J2 († b) −−−→ b ; n=Œ 0 0 n n n=0 2 2

Mode-coupling integrals over the orifice area s= a2 between modes from both zones are a Sm,n = 0

= a2

J0 (—m r) J0 (†n r) r dr = a2

(†n a) J0 (—m a) J1 (†n a) −−−−−−−→ 0 ; n=0,m=0 (†na)2 − (—m a) 2

−−−→ a2 m=0

(—m a) J1 (—m a) J0 (†n a) − (†n a) J0 (—m a) J1 (†na) (—m a) 2 − (†n a)2

(16)

J1 (†na) a2 ; −−−−−−→ m,n=0,0 (†na) 2

Integrals for average values over the cross sections vanish for higher modes and in the averge over s are special cases of the mode-coupling integrals Sm,0 for n = 0. Therefore, in the evaluation of the average values < p >s and < Z0 vx >s for the impedances, only

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Equivalent Networks

the fundamental modes will contribute. This fact supports the approximation with only plane waves in the neck. ⎧ ⎪ ⎪ a ⎨ 0 ; m>0 J1 (—m a) 2 = Sm,0 = J0 (—m r) · r dr = a · ⎪ —m a ⎪ ⎩ a2/2 ; m = 0 0 b

J0 (†n r) · r dr = b2 ·

0

⎧ ⎪ ⎪ ⎨ 0

(17)

; J1 (†nb) = ⎪ †n b ⎪ ⎩ b2 /2

n>0 ;

n=0

Matching the axial particle velocity at x = 0 requires !

vIIx (0, r) =

⎧ ⎪ ⎨ 0 ; r > a ⎪ ⎩

with the field formulations

(18a)

vIx (0, r) ; r ≤ a

⎧ ⎪ ⎪ ⎨ 0 ; r > a

‚n ! −j Dn sinh ‚n t J0 (†n r) = ‰m ⎪ k0 ⎪ Cm J0 (—m r) ; r ≤ a. n≥0 ⎩ B0 + j k0 m≥0

(18b)

The range is 0 ≤ r ≤ b. Therefore, multiply and integrate over this range, i.e. b a left-hand side: . . . · J0 (†Œ r) r dr; right-hand side: . . . · J0 (†Œ r) r dr; Œ = 0, 1, 2, . . ., 0

giving −j DŒ

0

‚Œ ‰m sinh ‚Œ t N(II) = B0 S0,Œ + j Cm Sm,Œ ; Œ = 0, 1, 2, . . . . Œ k0 k0 m≥0

(19)

Matching the sound pressure at x = 0 requires !

pII (0, r) = pI (0, r) ; r ≤ a,

(20a)

Dn cosh ‚n t J0 (†n r) = B0 + Cm J0 (—m r).

n≥0

(20b)

m≥0

The range is 0 ≤ r ≤ a. Therefore, multiply and integrate over this range, i.e. a on both sides . . . · J0 (—‹ r) r dr ; ‹ = 0, 1, 2 . . .; giving 0

n≥0

Dn cosh ‚n t S‹,n = ƒ0,‹ B0 + C‹ N(I) ‹ ; ‹ = 0, 1, 2, . . .

(21)

Equivalent Networks

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79

(ƒm,n = Kronecker symbol). Equations (19) and (21) are two linear systems of equations for the unknown amplitudes Cm and Dn . One gets from (19) (changing Œ → n) j ‰m Cm Sm,n . Dn = B0 S0,n + j (22)

k0 (‚n k0) sinh ‚n t N(II) n m≥0 Inserting this into (21) leads to the following linear system of equations for the Cm :

‰m coth ‚n t (I) Cm Sm,n S‹,n + ƒm,‹ N‹ k0 n≥0 (‚n k0 ) N(II) m>0 n (23)

coth ‚n t (I) S S − ƒ0,‹ N‹ ; ‹ = 0, 1, 2, . . . = B0 j (II) 0,n ‹,n n≥0 (‚n k0 ) Nn With the solutions Cm inserted into (22) the Dn are are obtained; thus the sound field is known. The homogenised chamber entrance impedance ZK is

D0 cosh ‚0 t pII0 (0, r) b ZK = =j

= coth ‚0 t = −j cot(k0t). ‚0 Z0 vII0x (0, r) b D0 sinh ‚0 t k0

(24)

The neck exit impedance is Zsh =

pI (0, r) a B0 + C0 a B0 + C0 = = . Z0 vIx (0, r) a B0 − C0 a B0 − C0

(25)

Thus, when knowing the amplitude C0 of the reflected fundamental mode in the neck, the orifice partition impedance can be evaluated: ZMi = Zsh − · ZK = Zsh + j cot(k0t).

(26)

Because only the amplitude C0 of the fundamental mode (plane wave) in the mode sum (13) is used, an acceptable way to approximate medium-wide necks is to formulate the sound field in the neck only with plane waves. The computational advantage of this approximation lies in the fact that in such a case no system of equtions need be solved; the orifice partition impedance can be written in an explicit formula. The following tables present solutions for different arrangements. They cover the following variations:

80

Neck:

C

⎧ ⎨

Equivalent Networks

round

⎩

⎧ ⎨

wide (mode sum)

⎩

square medium-wide (only plane waves) ⎧ ⎧ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎫ "tube" ⎨ ⎨ round ⎨ empty ⎬ Chamber: ∞ length square ⎪ ⎩ ⎪ "duct" ⎪ ⎪ filled with absorber ⎪ ⎪ ⎭ ⎩ ⎩ rectangul. ⎧ ⎪ empty ⎪ ⎪ ⎪ ⎪ ⎨ filled with absorber "chamber" Chamber: finite length square ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ front absorber layer ⎪ ⎪ ⎪ ⎩ rectangul. ⎪ ⎩ ⎪ ⎩ rear absorber layer ⎧ ⎪ ⎪ ⎪ ⎨

⎧ ⎪ ⎪ ⎪ ⎨ round

The tables show: • A sketch of the arrangement and a short description; • Formulations of the sound fields in the zones; • Equations for the mode amplitudes; • Mode norm integrals; • Mode-coupling integrals; • Orifice partition impedance ZMi . The objects with the round chambers are suited as approximations for perforated panels with the perforations in a hexagonal arrangement; the objects with square or rectangular chambers are suited for similar arrangements of the perforations. Square necks are of interest in combination with square or rectangular chambers since their mode-coupling integrals are easier to evaluate.

Equivalent Networks

2b ø 2a ø

C

81

82

C

Equivalent Networks

Table 1 Wide round neck and round tube; full analysis A plane wave with amplitude B0 in a round neck with diameter 2a is incident on an orifice to an empty round tube with 2b diameter. Both zones have 1 lengths.A mode sum is reflected in the neck. = (a=b)2 .

Field formulation in (I)

pI (x; r) = B0 e−j k0 x + Z0 vIx (x; r) =

m0

2 = —2 − k 2 Cm e+‰m x J0 (—m r) ; ‰m m 0

‰m j @p Cm e+‰m x J0 (—m r) = B0 e−j k0 x +j k0 @x k0 m0

Eigenvalues —m a

J1 (—m a) = 0; m = 0; 1; 2; : : : ; with —0 a = 0

Field formulation in (II)

pII (x; r) =

n0

Z0 vIIx (x; r) =

Dn e−‚n x J0 (†n r) ; ‚n2 = †2n − k02 j @p ‚n −‚n x Dn e J0 (†n r) = −j k0 @x k0 n0

Eigenvalues †n b

System of equations for Cm

Equations for Dn

J1 (†n b) = 0; n = 0; 1; 2; : : :; with †0 b = 0 ⎡ ⎤ Sm;n S‹;n (I) ‰m ⎣ ⎦ Cm ƒ‹;m N‹ + (II) ‚n N m0 n n0 ; ‹ = 0; 1; 2; : : : S0;n (I) = B0 −ƒ0;‹ N‹ +j (II) n0 Nn ‚n =k0 Dn = j B0

S0;n N(II) n ‚n =k0

−

a Mode norms in (I)

N(I) m =

=

m0

‰m Sm;n ‚n N(II) n

a2 a2 2 J0 (—m a) −−−−! 2 m=0 2

J20 (†n r) r dr =

b2 b2 2 J0 (†n b) −−−! n=0 2 2

b N(II) n

Cm

J20 (—m r) r dr = 0

Mode norms in (II)

0

a Mode coupling (I)–(II)

J0 (—m r) J0 (†n r) r dr = a2

Sm;n = 0

−−−−−−−−! 0 ; −−−−−−−−! n=0; m6=0

Orifice partition impedance

n=0; m=0

ZMi = Zsh − ZK

;

hpI (0; r)is Zsh = = hZ0 vIx (0; r)is

(†n a) J0 (—m a) J1 (†n a) (†n a)2 − (—m a) 2

J1 (†n a) a2 ; −−−−! a2 2 m=0 (†n a)

ZK = 1

2 1+ Dn J1 (†n a) (†n a) D0 n>0

Equivalent Networks

C

83

Table 2 Medium-wide round neck and round tube A plane wave with amplitude B0 in a round neck with diameter 2a is incident on an orifice to an empty round tube with 2b diameter. Both zones have 1 lengths. A plane wave is reflected in the neck. = (a/b)2 .


pI (x; r) = B0 e−j k0 x + C0 ej k0 x Z0 vIx (x; r) =


pII (x; r) =

j @p = B0 e−j k0 x − C0 ej k0 x k0 @x

n0

Z0 vIIx (x; r) =

Dn e−‚n x J0 (†n r) ; ‚n2 = †2n − k02 ‚n −‚n x j @p = −j Dn e J0 (†n r) k0 @x k0 n0

Eigenvalues †n b

Equation for C0

J1 (†n b) = 0; n = 0; 1; 2; : : :; with †0 b = 0 ⎡ ⎤ S20;n 2j ⎦ C0 = −B0 ⎣1 − 2 (II) a n0 Nn ‚n =k0 ⎡ ⎤ −1 S20;n 2j ⎦ ⎣1 + 2 (II) a n0 Nn ‚n =k0

Equations for Dn

Dn = j

Mode norm in (I)

N(I) 0 =

S0;n N(II) n ‚n =k0

a r dr = 0

(B0 − C0 )

a2 2

b Mode norms in (II)

N(II) n =

J20 (†n r) r dr = 0

b2 2 b2 J0 (†n b) −−−! 2 n=0 2


J0 (†n r) r dr = a2

S0;n = 0

ZMi = Zsh − ZK Orifice partition impedance Zsh =

;

J1 (†n a) a2 ; −−−! (†n a) n=0 2

ZK = 1

hpI (0; r)is B + C0 = 0 hZ0 vIx (0; r)is B0 − C0

84

C

Equivalent Networks

Table 3 Narrow round neck and round tube A capillary wave with amplitude B0 in a round neck with diameter 2a is incident on an orifice to an empty round tube with 2b diameter. Both zones have 1 lengths. A fundamental capillary wave is reflected in the neck. i = capillary propagation constant; Zi = (normalised) capillary wave impedance (see > Sect. J.3). = (a=b)2 . Field formulation in (I)


pI (x; r) = [B0 e−ix + C0 eix ] J0 (—i r); —2i = i2 + k2 −1 @p 1 Z0 vIx (x; r) = [B0 e−ix − C0 ei x ] J0 (—i r) = i Zi @x Zi pII (x; r) = Dn e−‚n x J0 (†n r); ‚n2 = †2n − k02 n0

Z0 vIIx (x; r) =

j @p ‚n −‚n x Dn e J0 (†n r) = −j k0 @x k0 n0

Eigenvalues †n b

J1 (†n b) = 0; n = 0; 1; 2; : : :; with †0 b = 0

Capillary characteristic values

i k0

2 =−

eff Ceff ; 0 C0

Zi Z0

2 =

eff Ceff = 0 C0

1 C eff = ; eff = 1 + (‰ − 1) J1;0 (k0 a) 0 1 − J1;0 (kŒ a) C0 J1;0 (z) : = 2

J1 (z) – ; k 2 = −j ; z J0 (z) Œ Œ

‰– = ‰ Pr kŒ2 T0;n S0;n C0 = −B0 T0;0 − j (II) n0 Zi Nn ‚n k0 −1 T0;n S0;n T0;0 + j (II) n0 Zi Nn ‚n k0

2 = −j k0

Equation for C0

Equations for Dn

Dn = j

Mode norms in (II)

N(II) n =

T0;n (B0 − C0 ) Zi N(II) n ‚n k0 b J20 (†n r) r dr = 0

b2 2 b2 J0 (†n b) −−−! 2 n=0 2

a J0 (—i r) J0 (†n r) r dr −−−! a2

T0;n = Mode coupling (I)–(II)

n=0

0

= a2

J1 (—i a) —i a

—i a J1 (—i a) J0 (†n a) − †n a J0 (—i a) J1 (†n a) (—i a)2 − (†n a)2

Equivalent Networks

C

85

Table 3 continued a Modes in (II) – average over s=a2

J0 (†n r) r dr = a2

S0;n = 0

ZMi = Zsh − ZK Orifice partition impedance Zsh

;

J1 (†n a) a2 −−−! †n a n=0 2

ZK = 1

1 + C0 B0 hpI (0; r)is = = Zi hZ0 vIx (0; r)is 1 − C0 B0

Table 4 Medium-wide round neck and round tube with absorber A plane wave with amplitude B0 in a round neck with diameter 2a is incident on an orifice to a round tube with 2b diameter, filled with porous absorber material. Both zones have 1 lengths. A plane wave is reflected in the neck. Absorber characteristic values: i = a =k0 ; Zi = Za =Z0 . = (a=b)2 . Field formulation in (I)

pI (x; r) = B0 e−j k0 x + C0 ej k0 x j @p = B0 e−j k0 x − C0 ej k0 x Z0 vIx (x; r) = k0 @x


pII (x; r) =

n0

Z0 vIIx (x; r) =

Dn e−‚n x J0 (†n r) ; ‚n2 = †2n + a2 1 ‚n −‚n x −1 @p = Dn e J0 (†n r) a Zi @x i Zi k0 n0

Eigenvalues †n b

J1 (†n b) = 0; n = 0; 1; 2; : : :; with †0 b = 0 S0;0 − i Zi

Equation for C0

C0 = −B0 S0;0 + i Zi

Equations for Dn

Dn = i Zi

n0

n0

J20 (†n r) r dr =

Nn = 0

S20;n Nn ‚n k0

S0;n (B0 − C0 ) Nn ‚n k0


S20;n Nn ‚n k0

b2 2 b2 J (†n b) −−−! 2 0 n=0 2


J0 (†n r) r dr = a2

S0;n = 0


J1 (†n a) a2 ; −−−! (†n a) n=0 2

ZMi = Zsh − ZK = Zsh − Zi hpI (0; r)ia hB0 + C0 ia 1 + C0 B0 Zsh = = = hZ0 vIx (0; r)ia hB0 − C0 ia 1 − C0 B0

86

C

Equivalent Networks

Table 5 Wide round neck and round empty chamber; full analysis A round neck with 2a diameter ends in a round empty chamber with 2b diameter and depth t. A plane wave with amplitude B0 is incident in the neck; a mode sum is reflected. = (a=b)2 .


pI (x; r) = B0 e−j k0 x +

m0

Cm e+‰m x J0 (—m r) ;

2 = —2 − k 2 ‰m m 0 j @p Z0 vIx (x; r) = = B0 e−j k0 x k0 @x

+j

m0

Cm

‰m +‰m x e J0 (—m r) k0

Eigenvalues —m a

J1 (—m a) = 0; m = 0; 1; 2; : : :; with —0 a = 0


pII (x; r) =

n0

‚n2 = †2n − k02 Z0 vIIx (x; r) =

Dn cosh (‚n (x − t)) J0 (†n r) ;

j @p k0 @x

=j

n0

Eigenvalues †n b

System of equations for Cm ; ‹= 0,1,2,. . .

Dn

‚n sinh (‚n (x − t)) J0 (†n r) k0

J1 (†n b) = 0; n = 0; 1; 2; : : :; with †0 b = 0 ⎡ ⎤ coth (‚n t) ‰ m (I) Cm ⎣ S S + ƒm;‹ N‹ ⎦ (II) m;n ‹;n k0 m>0 n0 (‚n k0 ) Nn coth (‚n t) (I) S S − ƒ0;‹ N‹ = B0 j (II) 0;n ‹;n n0 (‚n k0 ) Nn Dn =

Equations for Dn

j (‚n k0 ) sinh (‚n t) N(II) n ‰m B0 S0;n + j Cm Sm;n k0 m0 a

Mode norms in (I)

N(I) m =

J20 (—m r) r dr =

a2 2 a2 J0 (—m a) −−−−! 2 m=0 2

J20 (†n r) r dr =

b2 b2 2 J0 (†n b) −−−! 2 n=0 2

0


N(II) n

= 0

Equivalent Networks

C

87

Table 5 continued a Sm;n = Mode coupling (I)–(II)

J0 (—m r) J0 (†n r) r dr 0

= a2

(†n a) J0 (—m a) J1 (†n a) (†n a)2 − (—m a) 2

−−−−−−−−! 0 ; −−−−−−−−! n=0; m6=0

n=0; m=0


;

a2 J1 (†n a) ; −−−−! a2 2 m=0 (†n a)

ZK = −j cot(k0 t)

hpI (0; r)ia hB0 + C0 ia 1 + C0 B0 = = = hZ0 vIx (0; r)ia hB0 − C0 ia 1 − C0 B0

Table 6 Medium-wide round neck and round empty chamber A round neck with 2a diameter ends in a round empty chamber with 2b diameter and depth t. A plane wave with amplitude B0 is incident in the neck; a plane wave is reflected. = (a=b)2 .



pI (x; r) = B0 e−j k0 x + C0 e+j k0 x j @p Z0 vIx (x; r) = = B0 e−j k0 x − C0 e+j k0 x k0 @x pII (x; r) = Dn cosh (‚n (x − t)) J0 (†n r) ; n0

‚n2 = †2n − k02

j @p k0 @x ‚n Dn sinh (‚n (x − t)) J0 (†n r) =j k0 n0

Z0 vIIx (x; r) =

Eigenvalues †n b

Equation for C0

Equations for Dn

J1 (†n b) = 0; n = 0; 1; 2; : : :; with †0 b = 0 S20;n coth (‚n t) C0 = B0 −N(I) + j 0 (II) n0 (‚n =k0 ) Nn −1 S20;n coth (‚n t) + j N(I) 0 (II) n0 (‚n =k0 ) Nn Dn = j (B0 − C0 )

S0;n (‚n =k0 ) sinh (‚n t) N(II) n

C

88

Equivalent Networks

Table 6 continued b N(II) n

Mode norms in (II)

J20 (†n r) r dr =

= 0

b2 2 b2 J0 (†n b) −−−! 2 n=0 2

a J0 (†n r) r dr = a2

S0;n =

Mode coupling (I)–(II)

0


;

J1 (†n a) a2 −−−! †n a n=0 2

ZK = −j cot(k0 t)

hpI (0; r)ia hB0 + C0 ia 1 + C0 B0 = = hZ0 vIx (0; r)ia hB0 − C0 ia 1 − C0 B0

|p(x/b,r/b,0)|, Kammer, a/b=0.3, b/lam=0.45, t/b=1.

2 1.5 1

1 0.5

0.5

0 -2

0

(y/b)

-1 -0.5 (x/b)

0 1 -1

Example of sound pressure matching at the orifice of a medium-wide neck and an empty chamber. a/b = 0.3 ; b/Š0 = 0.45; t/b = 1.0

Equivalent Networks

C

89

Table 7 Medium-wide round neck and round chamber with absorber A round neck with 2a diameter ends in a round chamber with 2b diameter and depth t, filled with porous absorber material. A plane wave with amplitude B0 is incident in the neck; a plane wave is reflected. Absorber characteristic values: i = a =k0 ; Zi = Za =Z0 . = (a=b)2 .



pI (x; r) = B0 e−j k0 x + C0 e+j k0 x j @p Z0 vIx (x; r) = = B0 e−j k0 x − C0 e+j k0 x k0 @x pII (x; r) = Dn cosh (‚n (x − t)) J0 (†n r) ; n0

‚n2 = †2n + a2 Z0 vIIx (x; r)

=

−1 @p a Zi @x

=

−1 ‚n Dn sinh (‚n (x − t)) J0 (†n r) i Z i k0 n0

Eigenvalues †n b

Equation for C0

J1 (†n b) = 0; n = 0; 1; 2; : : :; with †0 b = 0 ⎡ ⎤ 2 coth S t) (‚ Z n 0;n i i ⎦ C0 = B0 ⎣−1 + 2 2 (II) a n0 (‚n =k0 ) Nn ⎡ ⎤ −1 S20;n coth (‚n t) Z i i ⎦ ⎣1 + 2 2 a (‚n =k0 ) N(II) n n0

Equations for Dn

Dn = (B0 − C0 ) i Zi

Mode norms in (II)

N(II) n =

S0;n (‚n =k0 ) sinh (‚n t) N(II) n

b J20 (†n r) r dr = 0

b2 b2 2 J (†n b) −−−! 2 0 n=0 2


J0 (†n r) r dr = a2

S0;n = 0


;

J1 (†n a) a2 −−−! †na n=0 2

ZK = Zi coth (‚0 t)

hpI (0; r)ia hB + C0 ia 1 + C0 B0 = 0 = hZ0 vIx (0; r)ia hB0 − C0 ia 1 − C0 B0

Equation for C0

Field formulation in (III)

Eigenvalues †n b


n0

n0

n0

⎤ k0 S20;n ‚n +j i Zi ‰n tanh (‚n s) tanh (‰n(s−t)) 2 Z i i ⎦ C0 = B0 ⎣−1+ 2 ‚n Nn ‚n tanh (‚n s)+j i Zi ‰n tanh (‰n(s−t)) a n0 ⎤−1 ⎡ 2 S Z k ‚ +j Z ‰ tanh s) tanh (s−t)) 2 (‚ (‰ n n i i 0 0;n n i i n ⎦ ⎣ 1+ 2 a ‚n Nn ‚n tanh (‚n s)+j i Zi ‰n tanh (‰n (s−t))

⎡

Fn cosh (‰n (x − t)) J0 (†n r) ; ‰n2 = †2n − k02

‰n j @p =j Fn sinh (‰n (x − t)) J0 (†n r) k0 @x k0

n0

Z0 vIIIx (x; r) =

pIII (x; r) =

‚n2 = †2n + a2

1 ‚n

−1 @p = Dn e−‚n x − En e+‚n x J0 (†n r) a Zi @x i Zi k0

J1 (†n b) = 0; n = 0; 1; 2; : : :; with †0 b = 0

Z0 vIIx (x; r) =

n0

pI (x; r) = B0 e−j k0 x + C0 e+j k0 x j @p = B0 e−j k0 x − C0 e+j k0 x Z0 vIx (x; r) = k0 @x

pII (x; r) = Dn e−‚n x + En e+‚n x J0 (†n r) ;

C


A round neck with 2a diameter ends in a round chamber with 2b diameter and depth t, partially filled with porous absorber layer adjacent to the orifice. A plane wave with amplitude B0 is incident in the neck; a plane wave is reflected. Absorber characteristic values: i = a =k0 ; Zi = Za=Z0 . = (a=b)2 .

Table 8 Medium-wide neck and round chamber with front absorber layer

90 Equivalent Networks

S0;n =



Nn =

Mode norms in (II)

k0 S0;n −2‚n s ‚n − j i Zi ‰n tanh (‰n(s − t))

e ‚n Nn ‚n 1 − e−2‚n s + j i Zi ‰n 1 + e−2‚n s tanh (‰n (s − t))

k0 S0;n ‚n + j i Zi ‰n tanh (‰n (s − t))

‚n Nn ‚n 1 − e−2‚n s + j i Zi ‰n 1 + e−2‚n s tanh (‰n (s − t))

= =

=

ZMi

Zsh

ZK

0

a

0

2 hpI (0; r)ia hB + C0 ia B + C0 2i Zi k0 S0;n ‚n +j i Zi ‰n tanh (‚n s) tanh (‰n (s−t)) = 0 = 0 = 2 hZ0 vIx (0; r)ia hB0 − C0 ia B0 − C0 a ‚n Nn ‚n tanh (‚n s)+j i Zi ‰n tanh (‰n(s−t)) n0

1 + j Zi tanh i k0 s tan k0 (t − s) hpII (0; r)ib D0 + E0

= Zi = Zi hZ0 vIIx (0; r)ib D0 − E0 tanh i k0 s + j Zi tan k0 (t − s)

Zsh − ZK

J1 (†n a) a2 −−−! †na n=0 2

b2 b2 2 J0 (†n b) −−−! 2 n=0 2

J0 (†n r)r dr = a2

J20 (†n r) r dr =

Dn e−‚n s + En e+‚n s cosh (‰n (s − t)) b

Fn =

En = (B0 − C0 ) i Zi

Dn = (B0 − C0 ) i Zi

Equations for Fn

Equations for En

Equations for Dn

Table 8 continued

Equivalent Networks

C 91

-1

0

1 -1

-0.5

0 -2

0.5

1

(x/b)

-1 0 1 -1

-0.5

0 (y/b)

0.5

1

Example of sound pressure and axial particle velocity matching at the orifice of a medium-wide neck in a round chamber with absorber layer at its entrance. Parameters: a=b = 0:3 ; t=b = 1:0 ; s=t = 0:3 ; b=Š0 = 0:45 ; ¡b=Z0 = 20:0.

(x/b)

0.5

0.5 0 -2 0 (y/b)

1

1

C

1.5



Eigenvalues †n b



n0

n0

−1 @p −1 ‰n Fn sinh (‰n (x − t)) J0 (†n r) = a Zi @x i Zi k0

Fn cosh (‰n (x − t)) J0 (†n r) ;

=

n0

Z0 vIIIx (x; r)

pIII (x; r) =

‰n2 = †2n + a2

‚n

j @p Dn e−‚n x − En e+‚n x J0 (†n r) = −j k0 @x k0

J1 (†n b) = 0; n = 0; 1; 2; : : :; with †0 b = 0

Z0 vIIx (x; r) =

n0

j @p = B0 e−j k0 x − C0 e+j k0 x k0 @x

pII (x; r) = Dn e−‚n x + En e+‚n x J0 (†n r) ; ‚n2 = †2n − k02

Z0 vIx (x; r) =

pI (x; r) = B0 e−j k0 x + C0 e+j k0 x

A round neck with 2a diameter ends in a round chamber with 2b diameter and depth t, partially filled with porous absorber layer adjacent to the back side. A plane wave with amplitude B0 is incident in the neck; a plane wave is reflected. Absorber characteristic values: i = a =k0 ; Zi = Za=Z0 . = (a=b)2 .

Table 9 Medium-wide neck and round chamber with rear absorber layer

Equivalent Networks

C 93

n0

Fn =

Dn e−‚n (t−s) + En e+‚n (t−s) cosh (‰ns)

‚n i Zi − j ‰n tanh (‰ns) j k0 S0;n −2‚n (t−s)

e ‚n Nn ‚n i Zi 1 − e−2‚n (t−s) + j ‰n tanh (‰n s) 1 + e−2‚n (t−s)

En = (B0 − C0 )

Equations for En

Equations for Fn

‚n i Zi + j ‰n tanh (‰ns) j k0 S0;n

‚n Nn ‚n i Zi 1 − e−2‚n (t−s) + j ‰n tanh (‰ns) 1 + e−2‚n (t−s)

n0

⎤ −1 k0 S20;n ‚n i Zi + j ‰n tanh (‰ns) tanh (‚n (t − s)) 2j ⎦ ⎣1 + 2 ‚n Nn ‚n i Zi tanh (‚n (t − s)) + j ‰n tanh (‰n s) a ⎡

Dn = (B0 − C0 )

=

Equations for Dn

C0

⎤ k0 S20;n ‚n i Zi + j ‰n tanh (‰n s) tanh (‚n (t − s)) 2j ⎦ −B0 ⎣1 − 2 a ‚n Nn ‚n i Zi tanh (‚n (t − s)) + j ‰n tanh (‰ns) ⎡

C

Equation for C0

Table 9 continued



ZK

=

Zsh

=

=

=

0

ZMi

S0;n =


2 2j k0 S0;n ‚n i Zi + j ‰n tanh (‰n s) tanh (‚n (t − s)) 2 a ‚n Nn ‚n i Zi tanh (‚n (t − s)) + j ‰n tanh (‰n s) n0

hpII (0; r)ib D0 + E0 Zi + j tanh (a s) tan k0 (t − s)

= = hZ0 vIIx (0; r)ib D0 − E0 j Zi tan k0 (t − s) + tanh (a s)

hpI (0; r)ia hB0 + C0 ia B0 + C0 = = hZ0 vIx (0; r)ia hB0 − C0 ia B0 − C0

Zsh − ZK

a2 J1 (†n a) −−−! †na n=0 2

b2 b2 2 J (†n b) −−−! 2 0 n=0 2

J0 (†n r)r dr = a2

J20 (†n r) r dr = a

0

Nn =

b

Mode norms in (II)

Table 9 continued

Equivalent Networks

C 95

C

96

Equivalent Networks Im(ZMi), mat=1, Rb=5.0, t/b=1.6, s/t=0.4

Re(ZMi), mat=1, Rb=5.0, t/b=1.6, s/t=0.4

0.1 1 0.075

0.05 0.5 0.025 0.4 0

0.4 0

y=(b/lam)

0.2

0.2

0.4 x=(a/b)

y=(b/lam)

0.2

0.2

0.4

0.6 0.8

x=(a/b)

0.6 0.8

Example of variation of Re(ZMi ) and Im(ZMi ) over variables x= (a/b) and y = (b/Š0) for parameter values Rb = ¡b/Z0 = 5.0 ; t/b = 1.6; s/t = 0.4 (absorber = glass fibre, mat=1)

Equivalent Networks

C

97

Table 10 Medium-wide round neck and square empty duct A round neck with 2a diameter ends in an square empty duct with infinite length and 2b width.The incident wave in the neck with amplitude B0 and the reflected wave with amplitude C0 are plane waves. = =4(a=b)2 .

pI (x; r) = B0 e−j k0 x + C0 ej k0 x Field formulation in (I)

j @p = B0 e−j k0 x − C0 ej k0 x k0 @x pII (x; y; z) = Dn;Œ e−‚n;Œ x cos(†n y) cos(†Œ z) Z0 vIx (x; r) =


n;Œ0

j @p k0 @x ‚n;Œ −‚n;Œ x Dn;Œ e cos(†n y) cos(†Œ z) = −j k0 n;Œ0

Z0 vIIx (x; y; z) =

Eigenvalues †n b

Equation for C0

sin(†n b) = 0 ) †n b = n ; n = 0; 1; 2; : : : ; 2 = † 2 + †2 − k 2 ‚n;Œ n Œ 0 ⎡ ⎤ S2n;Œ j ⎦ C0 = B0 ⎣−1 + 2 (II) a n;Œ0 (‚n;Œ k0 ) Nn;Œ ⎡ ⎤ −1 S2n;Œ j ⎦ ⎣1 + 2 a (‚n;Œ k0 ) N(II) n;Œ n;Œ0

Equations for Dn;Œ

Dn;Œ = j

Mode norm in (I)

N(I) 0 =

Sn;Œ (B0 − C0 ) (‚n;Œ k0 ) N(II) n;Œ

a r dr = 0

a2 2

98

C

Equivalent Networks

Table 10 continued N(II) n;Œ

+b +b = cos(†n y) cos(†Œ z) −b −b

cos(†n0 y) cos(†Œ 0 z) dy dz ⎧ ⎪ ⎪ 0 ; n 6= n0 ⎪ ⎨

Mode norms in (II)

(y)

(y)

(z) 2 N(II) n;Œ = b Nn NŒ ; Nn =

⎧ ⎪ 0 ⎪ ⎪ ⎨0 ; Œ 6= Œ N(z) Œ = 1 ; Œ = Œ 0 = 6 0 ⎪ ⎪ ⎪ ⎩ 0 2 ; Œ = Œ = 0 Sn;Œ = Mode coupling (I)–(II)

in s= a2

1 ; n = n0 = 6 0 ; ⎪ ⎪ ⎪ ⎩ 2 ; n = n0 = 0

cos(†n y) cos(†Œ z) ds ! J1 a †2n + †2Œ 2 = 2 a −−−−−−−! a2 †n =†Œ =0 a †2n + †2Œ s


;

ZK = 1

hpI (0; r)is B0 + C0 = hZ0 vIx (0; r)is B0 − C0

Table 11 Medium-wide round neck and square duct filled with absorber A round neck with 2a diameter ends in a square duct with infinite length and 2b width, filled with porous absorber material. The incident wave in the neck with amplitude B0 and the reflected wave with amplitude C0 are plane waves. Absorber characteristic values: i = a =k0 ; Zi = Za =Z0 . = =4(a=b)2 .

pI (x; r) = B0 e−j k0 x + C0 ej k0 x Field formulation in (I) Z0 vIx (x; r) =

j @p = B0 e−j k0 x − C0 ej k0 x k0 @x

Equivalent Networks

C

Table 11 continued Field formulation in (II)

pII (x; y; z) =

n;Œ0

Dn;Œ e−‚n;Œ x cos(†n y) cos(†Œ z)

−1 @p 1 ‚n;Œ −‚n;Œ x Dn;Œ e = a Zi @x i Zi k0

Z0 vIIx (x; y; z) =

n;Œ0

cos(†n y) cos(†Œ z) sin(†n b) = 0 ) †n b = n ; n = 0; 1; 2; : : : ;

Eigenvalues †n b

2 = † 2 + †2 + 2 ‚n;Œ n Œ a ⎡ ⎤ S2n;Œ Z i i ⎦ C0 = B0 ⎣−1 + 2 (II) a n;Œ0 (‚n;Œ k0 ) Nn;Œ ⎡ ⎤ −1 S2n;Œ Zi i ⎦ ⎣1 + 2 a (‚n;Œ k0 ) N(II) n;Œ

Equation for C0

n;Œ0

Equations for Dn;Œ

Dn;Œ = i Zi

Mode norm in (I)

N(I) 0 =

Sn;Œ (B0 − C0 ) (‚n;Œ k0 ) N(II) n;Œ

a r dr = 0

N(II) n;Œ =

Mode norms in (II)

a2 2

+b +b cos(†n y) cos(†Œ z)

−b −b

N(II) n;Œ

N(z) Œ

=

⎧ ⎪ 0 ⎪ ⎪ ⎨ 0 ; Œ 6= Œ = 1 ; Œ = Œ 0 = 6 0 ⎪ ⎪ ⎪ ⎩ 0 2 ; Œ = Œ = 0

Sn;Œ = Mode coupling (I)–(II)

cos(†n0 y) cos(†Œ0 z) dy dz ⎧ ⎪ ⎪ 0 ; n 6= n0 ⎪ ⎨ (y) (y) b2 Nn N(z) Œ ; Nn = 1 ; n = n0 = 6 0 ; ⎪ ⎪ ⎪ ⎩ 2 ; n = n0 = 0

in s= a2

s

cos(†n y) cos(†Œ z) ds

= 2 a2

! J1 a †2n + †2Œ a †2n + †2Œ


;

−−−−−−−! a2 †n =†Œ =0

ZK = Zi

hpI (0; r)is B0 + C0 = hZ0 vIx (0; r)is B0 − C0

99

100

C

Equivalent Networks

Table 12 Medium-wide round neck and square empty chamber A medium-wide round neck with 2a diameter ends in a square empty chamber with 2b side length and depth t. The incident and the reflected waves in the neck are plane waves. = =4(a=b)2 .



pI (x; r) = B0 e−j k0 x + C0 e+j k0 x j @p = B0 e−j k0 x − C0 e+j k0 x Z0 vIx (x; r) = k0 @x

pII (x; y; z) = Dn;Œ cosh ‚n;Œ (x − t) n;Œ0

cos(†n y) cos(†Œ z)

‚n;Œ j @p =j Z0 vIIx (x; y; z) = Dn;Œ sinh ‚n;Œ (x − t) k0 @x k0 n;Œ0

cos(†n y) cos(†Œ z) Eigenvalues †n b

Equation for C0

Equations for Dn;Œ

sin(†n b) = 0 ) †n b = n ; n = 0; 1; 2; : : : ; 2 = † 2 + †2 − k 2 ‚n;Œ n Œ 0 ⎡

⎤ S2n;Œ coth ‚n;Œ t j ⎦ C0 =B0 = ⎣−1 + 2 a (‚n;Œ k0 ) N(II) n;Œ n;Œ0 ⎡

⎤ −1 S2n;Œ coth ‚n;Œ t j ⎦ ⎣1 + 2 (II) a n;Œ0 (‚n;Œ k0 ) Nn;Œ

Dn;Œ = j

Sn;Œ (B0 − C0 )

(‚n;Œ k0 ) sinh ‚n;Œ t N(II) n;Œ +b +b

N(II) n;Œ

cos(†n y) cos(†Œ z) cos(†n0 y) cos(†Œ0 z) dy dz

= −b −b

Mode norms in (II) (y)

(y)

(z) 2 N(II) n;Œ = b Nn NŒ ; Nn

N(z) Œ

⎧ ⎪ 0 ⎪ ⎪ ⎨ 0 ; Œ 6= Œ = 1 ; Œ = Œ 0 = 6 0 ⎪ ⎪ ⎪ ⎩ 0 2 ; Œ = Œ = 0

⎧ ⎪ 0 ⎪ ⎪ ⎨ 0 ; n 6= n = 1 ; n = n0 = 6 0 ; ⎪ ⎪ ⎪ ⎩ 2 ; n = n0 = 0

Equivalent Networks

C

101

Table 12 continued Sn;Œ = Mode coupling (I)–(II) in s= a2

s


= 2 a2

! J1 a †2n + †2Œ a †2n + †2Œ


;

−−−−−−−! a2 †n =†Œ =0

ZK = −j cot(k0 t)

hpI (0; r)ia hB0 + C0 ia 1 + C0 B0 = = hZ0 vIx (0; r)ia hB0 − C0 ia 1 − C0 B0

Table 13 Medium-wide round neck and square chamber with absorber A medium-wide round neck with 2a diameter ends in a square chamber with 2b side length and depth t, filled with porous absorber material. The incident and the reflected waves in the neck are plane waves. Absorber characteristic values: i = a =k0 ; Zi = Za =Z0 . = =4(a=b)2 .



pI (x; r) = B0 e−j k0 x + C0 ej k0 x j @p Z0 vIx (x; r) = = B0 e−j k0 x − C0 ej k0 x k0 @x

pII (x; y; z) = Dn;Œ cosh ‚n;Œ (x−t) cos(†n y) cos(†Œ z) n;Œ0

Z0 vIIx (x; y; z) =

−1 ‚n;Œ Dn;Œ sinh ‚n;Œ (x − t) i Z i k0 n;Œ0

cos(†n y) cos(†Œ z) Eigenvalues †n b

Equation for C0

sin(†n b) = 0 ) †n b = n ; n = 0; 1; 2; : : : ; 2 = † 2 + †2 + 2 ‚n;Œ n Œ a ⎡

⎤ S2n;Œ coth ‚n;Œ t Z i i ⎦ C0 =B0 = ⎣−1 + 2 (II) a n;Œ0 (‚n;Œ k0 ) Nn;Œ ⎡

⎤ −1 S2n;Œ coth ‚n;Œ t Z i i ⎦ ⎣1 + 2 a (‚n;Œ k0 ) N(II) n;Œ n;Œ0

Equations for Dn;Œ

Dn;Œ = i Zi

Sn;Œ (B0 − C0 )

(‚n;Œ k0 ) sinh ‚n;Œ t N(II) n;Œ

102

C

Equivalent Networks

Table 13 continued N(II) n;Œ

+b +b = cos(†n y) cos(†Œ z) −b −b

cos(†n0 y) cos(†Œ 0 z) dy dz ⎧ ⎪ 0 ⎪ ⎪ ⎨ 0 ; n 6= n (y) (y) (II) (z) 2 Nn;Œ = b Nn NŒ ; Nn = 1 ; n = n0 = 6 0 ; ⎪ ⎪ ⎪ ⎩ 2 ; n = n0 = 0 ⎧ ⎪ ⎪ ⎪ 0 ; Œ 6= Œ 0 ⎨

Mode norms in (II)

N(z) Œ =


in s= a2

1 ; Œ = Œ 0 = 6 0 ⎪ ⎪ ⎪ ⎩ 0 2 ; Œ = Œ = 0 s


= 2 a2

! J1 a †2n + †2Œ a †2n + †2Œ


;

−−−−−−−! a2 †n =†Œ =0

ZK = Zi coth(‚0 t)

1 + C0 B0 hpI (0; r)is = hZ0 vIx (0; r)is 1 − C0 B0

Table 14 Medium-wide round neck and rectangular empty duct A round neck with 2a diameter ends in a rectangular empty duct with 1 length and sides 2b, 2c.The incident and the reflected waves in the neck are plane waves. = a2 =(4bc).


pI (x; r) = B0 e−j k0 x + C0 ej k0 x Z0 vIx (x; r) =

j @p = B0 e−j k0 x − C0 ej k0 x k0 @x

Equivalent Networks

C


pII (x; y; z) =

n;Œ0

Z0 vIIx (x; y; z) =

Dn;Œ e−‚n;Œ x cos(†n y) cos(”Œ z)

‚n;Œ −‚n;Œ x j @p = −j Dn;Œ e k0 @x k0 n;Œ0

cos(†n y) cos(”Œ z) sin(†n b) = 0 ) †n b = n ; n = 0; 1; 2; : : : ; Eigenvalues †n b, ”Œ c

sin(”Œ c) = 0 ) ”Œ c = Œ ; Œ = 0; 1; 2; : : : ; 2 = †2 + ” 2 − k 2 ‚n;Œ n Œ 0 ⎡ ⎤ S2n;Œ j ⎦ C0 =B0 = ⎣−1 + 2 (II) a n;Œ0 (‚n;Œ =k0 ) Nn;Œ ⎡ ⎤ −1 S2n;Œ j ⎦ ⎣1 + 2 a (‚n;Œ =k0 ); N(II) n;Œ

Equation for C0

n;Œ0

Dn;Œ = j

Equations for Dn;Œ

N(II) n;Œ =

Sn;Œ (B0 − C0 ) (‚n;Œ k0 ) N(II) n;Œ

+c +b cos(†n y) cos(”Œ z) −c −b

cos(†n0 y) cos(”Œ 0 z) dy dz Mode norms in (II) (y) (y) (z) N(II) n;Œ = bc Nn NŒ ; Nn

⎧ ⎪ 0 ⎪ ⎪ ⎨ 0 ; Œ 6= Œ (z) NŒ = 1 ; Œ = Œ 0 6= 0 ⎪ ⎪ ⎪ ⎩ 2 ; Œ = Œ 0 = 0 Sn;Œ = Mode coupling (I)–(II)

in s= a2 =

s

cos(†n y) cos(”Œ z) ds

2 a2

! J1 a †2n + ”Œ2 a †2n + ”Œ2


⎧ ⎪ 0 ⎪ ⎪ ⎨ 0 ; n 6= n = 1 ; n = n0 6= 0 ; ⎪ ⎪ ⎪ ⎩ 2 ; n = n0 = 0

;

−−−−−−−! a2 †n =”Œ =0

ZK = 1


103

104

C

Equivalent Networks

Table 15 Medium-wide round neck and rectangular duct with absorber A round neck with 2a diameter ends in a rectangular duct with 1 length and sides 2b, 2c, filled with porous absorber material. The incident and the reflected waves in the neck are plane waves. Absorber characteristic values: i = a =k0 ; Zi = Za =Z0 . = a2 =(4bc).




pII (x; y; z) =

n;Œ0

Z0 vIIx (x; y; z) =


‚n;Œ −‚n;Œ x 1 Dn;Œ e cos(†n y) cos(”Œ z) i Z i k0 n;Œ0

sin(†n b) = 0 ) †n b = n ; n = 0; 1; 2; : : : ; Eigenvalues †n b, ”Œ c

sin(”Œ c) = 0 ) ”Œ c = Œ ; Œ = 0; 1; 2; : : : ; 2 = †2 + ” 2 + 2 ‚n;Œ n Œ a ⎡

C0 =B0

⎤ S2n;Œ Z i i ⎣−1 + ⎦ a2 (‚ k0 ) N(II) n;Œ n;Œ0 n;Œ ⎤ −1 ⎡ S2n;Œ Z i i ⎦ ⎣1 + 2 a (‚n;Œ k0 ) N(II) n;Œ

=

Equation for C0

n;Œ0

Equations for Dn;Œ

Dn;Œ = i Zi

N(II) n;Œ

Sn;Œ (B0 − C0 ) (‚n;Œ k0 ) N(II) n;Œ

+c +b = cos(†n y) cos(”Œ z) cos(†n0 y) cos(”Œ 0 z) dy dz −c −b

Mode norms in (II) (y) (y) (z) N(II) n;Œ = bc Nn NŒ ; Nn

N(z) Œ =

⎧ ⎪ 0 ⎪ ⎪ ⎨ 0 ; Œ 6= Œ 1 ; Œ = Œ 0 6= 0 ⎪ ⎪ ⎪ ⎩ 2 ; Œ = Œ 0 = 0

⎧ ⎪ 0 ⎪ ⎪ ⎨ 0 ; n 6= n = 1 ; n = n0 6= 0 ; ⎪ ⎪ ⎪ ⎩ 2 ; n = n0 = 0

Equivalent Networks

C

105

Table 15 continued Sn;Œ = Mode coupling (I)–(II) in s= a2

s


= 2 a2

! J1 a †2n + ”Œ2 a †2n + ”Œ2


;

−−−−−−−! a2 †n =”Œ =0

ZK = Zi

1 + C0 B0 hpI (0; r)is = = hZ0 vIx (0; r)is 1 − C0 B0

Table 16 Medium-wide round neck and rectangular empty chamber A round neck with 2a diameter ends in a rectangular empty chamber with depth t and sides 2b, 2c. The incident and the reflected waves in the neck are plane waves. = a2 =(4bc).




pII (x; y; z) =

n;Œ0

Z0 vIIx (x; y; z) = j

Dn;Œ cosh ‚n;Œ (x − t) cos(†n y) cos(”Œ z)

n;Œ0

Dn;Œ

‚n;Œ sinh ‚n;Œ (x − t) cos(†n y) cos(”Œ z) k0


Equation for C0

sin(”Œ c) = 0 ) ”Œ c = Œ ; Œ = 0; 1; 2; : : : ; 2 = †2 + ” 2 − k 2 ‚n;Œ n Œ 0 ⎡

⎤ S2n;Œ coth ‚n;Œ t j ⎦ C0 =B0 = ⎣−1 + 2 (II) a (‚ ) N k n;Œ 0 n;Œ n;Œ0 ⎡

⎤ −1 S2n;Œ coth ‚n;Œ t j ⎦ ⎣1 + 2 a (‚n;Œ k0 ) N(II) n;Œ n;Œ0

106

C

Equivalent Networks

Table 16 continued Dn;Œ = j

Equations for Dn;Œ

N(II) n;Œ

Sn;Œ (B0 − C0 )



Mode norms in (II) (y) (y) (z) N(II) n;Œ = bc Nn NŒ ; Nn

N(z) Œ

⎧ ⎪ 0 ⎪ ⎪ ⎨ 0 ; Œ 6= Œ = 1 ; Œ = Œ 0 6= 0 ⎪ ⎪ ⎪ ⎩ 2 ; Œ = Œ 0 = 0


in s= a2

s


= 2 a2

! J1 a †2n + ”Œ2 a †2n + ”Œ2


⎧ ⎪ 0 ⎪ ⎪ ⎨ 0 ; n 6= n = 1 ; n = n0 6= 0 ; ⎪ ⎪ ⎪ ⎩ 2 ; n = n0 = 0

;

−−−−−−−! a2 †n =”Œ =0

ZK = −j cot(k0 t)


Table 17 Medium-wide round neck and rectangular chamber with absorber A round neck with 2a diameter ends in a rectangular chamber with depth t and sides 2b, 2c, filled with porous absorber material. The incident and the reflected waves in the neck are plane waves. Absorber characteristic values: i = a =k0 ; Zi = Za =Z0 . = a2 =(4bc).



Equivalent Networks

C

107


pII (x; y; z) =

n;Œ0

Z0 vIIx (x; y; z) =

Dn;Œ cosh ‚n;Œ (x − t) cos(†n y) cos(”Œ z)

−1 ‚n;Œ Dn;Œ sinh ‚n;Œ (x − t) i Z i k0 n;Œ0


sin(”Œ c) = 0 ) ”Œ c = Œ ; Œ = 0; 1; 2; : : : ; 2 = †2 + ” 2 + 2 ‚n;Œ n Œ a ⎡

⎤ i Zi S2n;Œ coth ‚n;Œ t ⎦ ⎣ C0 =B0 = −1 + 2 (II) a n;Œ0 (‚n;Œ k0 ) Nn;Œ ⎡

⎤ −1 S2n;Œ coth ‚n;Œ t Z i i ⎦ ⎣1 + 2 a (‚n;Œ k0 ) N(II) n;Œ

Equation for C0

n;Œ0

Dn;Œ = i Zi

Equations for Dn;Œ

N(II) n;Œ

Sn;Œ (B0 − C0 )



Mode norms in (II) (y)

(y)

(z) N(II) n;Œ = bc Nn NŒ ; Nn =

N(z) Œ

⎧ ⎪ 0 ⎪ ⎪ ⎨ 0 ; Œ 6= Œ = 1 ; Œ = Œ 0 6= 0 ⎪ ⎪ ⎪ ⎩ 2 ; Œ = Œ 0 = 0


in s= a2 =

s

Zsh =

1 ; n = n0 6= 0 ; ⎪ ⎪ ⎪ ⎩ 2 ; n = n0 = 0


2 a2

! J1 a †2n + ”Œ2 a †2n + ”Œ2

ZMi = Zsh − ZK Orifice partition impedance

⎧ ⎪ 0 ⎪ ⎪ ⎨ 0 ; n 6= n

;

−−−−−−−! a2 †n =”Œ =0

ZK = Zi coth(a t)


Eigenvalues †n b, ”Œ c




n;Œ0

2 = †2 + ” 2 + 2 ‚n;Œ n Œ a

n;Œ0

Fn;Œ

sin(”Œ c) = 0 ) ”Œ c = Œ ; Œ = 0; 1; 2; : : : ;

2 = †2 + ” 2 − k 2 ‰n;Œ n Œ 0

‰n;Œ sinh ‰n;Œ (x − t) cos(†n y) cos(”Œ z) k0

Fn;Œ cosh ‰n;Œ (x − t) cos(†n y) cos(”Œ z)

sin(†n b) = 0 ) †n b = n ; n = 0; 1; 2; : : : ;

Z0 vIIIx (x; y; z) = j

pIII (x; y; z) =

sin(”Œ c) = 0 ) ”Œ c = Œ ; Œ = 0; 1; 2; : : : ;

sin(†n b) = 0 ) †n b = n ; n = 0; 1; 2; : : : ;

n;Œ0

1 ‚n;Œ

−1 @p = Dn;Œ e−‚n;Œ x − En;Œ e+‚n;Œ x cos(†n y) cos(”Œ z) a Zan @x an Zan k0

n;Œ0

Dn;Œ e−‚n;Œ x + En;Œ e+‚n;Œ x cos(†n y) cos(”Œ z)

Z0 vIIx (x; y; z) =

pII (x; y; z) =

pI (x; r) = B0 e−j k0 x + C0 e+j k0 x j @p Z0 vIx (x; r) = = B0 e−j k0 x − C0 e+j k0 x k0 @x

C


A round neck with diameter 2a ends in a rectangular chamber with sides 2b, 2c and depth t, with an absorber layer of thickness s adjacent to the orifice.The incident and the reflected waves in the neck are plane waves. Absorber characteristic values: i = a =k0 ; Zi = Za=Z0 . = a2 =(4bc).

Table 18 Medium-wide round neck and rectangular chamber with front absorber


Mode norms in (II)

Fn;Œ =

Equations for Fn;Œ

−c −b

⎧ ⎧ ⎪ ⎪ 0 0 ⎪ ⎪ 0 ; n = 6 n ⎪ ⎪ ⎨ ⎨ 0 ; Œ 6= Œ (z) = 1 ; n = n0 6= 0 ; NŒ = ⎪ 1 ; Œ = Œ 0 6= 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ 2 ; n = n0 = 0 2 ; Œ = Œ 0 = 0

+c +b cos(†n y) cos(”Œ z) cos(†n0 y) cos(”Œ 0 z) dy dz

(y) (y) (z) N(II) n;Œ = bc Nn NŒ ; Nn

N(II) n;Œ =

En;Œ

Equations for En;Œ Dn;Œ e−‚n;Œ s + En;Œ e+‚n;Œ s

cosh ‰n;Œ (s − t)

Dn;Œ = (B0 − C0 ) i Zi

‚n;Œ + j i Zi ‰n;Œ tanh ‰n;Œ (s − t) k0 Sn;Œ

−2‚n;Œ s + j Z ‰ −2‚n;Œ s tanh ‰ (s − t) ‚n;Œ N(II) n;Œ i i n;Œ 1 + e n;Œ ‚n;Œ 1 − e

‚n;Œ − j i Zi ‰n;Œ tanh ‰n;Œ (s − t) k0 Sn;Œ −2‚n;Œ s

= (B0 − C0 ) i Zi e ‚n;Œ N(II) ‚n;Œ 1 − e−2‚n;Œ s + j i Zi ‰n;Œ 1 + e−2‚n;Œ s tanh ‰n;Œ (s − t) n;Œ

n;Œ0

⎤ k0 S2n;Œ ‚n;Œ + j i Zi ‰n;Œ tanh ‚n;Œ s tanh ‰n;Œ (s − t) i Zi ⎣

⎦ C0 =B0 = −1 + 2 a ‚n;Œ N(II) n;Œ ‚n;Œ tanh ‚n;Œ s + j i Zi ‰n;Œ tanh ‰n;Œ (s − t) n;Œ0 ⎡

⎤ −1 k0 S2n;Œ ‚n;Œ + j i Zi ‰n;Œ tanh ‚n;Œ s tanh ‰n;Œ (s − t) Z i i

⎦ ⎣1+ 2 ‚n;Œ N(II) a n;Œ ‚n;Œ tanh ‚n;Œ s +j i Zi ‰n;Œ tanh ‰n;Œ (s−t)

⎡

Equations for Dn;Œ

Equation for C0

Table 18 continued

Equivalent Networks

C 109

s

cos(†n y) cos(”Œ z) ds = 2 a2 a †2n + ”Œ2

! J1 a †2n + ”Œ2 †n =”Œ =0

−−−−−−−! a2

A round neck with diameter 2a ends in a rectangular chamber with sides 2b, 2c and depth t, with an absorber layer of thickness s adjacent to the back side. The incident and the reflected waves in the neck are plane waves. Absorber characteristic values: i = a =0 ; Zi = Za =Z0 . = a2 =(4bc).

Table 19 Medium-wide round neck and rectangular chamber with rear absorber

Zsh =

hpI (0; r)ia hB0 + C0 ia B0 + C0 = = = hZ0 vIx (0; r)ia hB0 − C0 ia B0 − C0

i Zi k0 S2n;Œ ‚n;Œ + j i Zi ‰n;Œ tanh ‚n;Œ s tanh ‰n;Œ (s − t)

‚n;Œ N(II) a2 n;Œ ‚n;Œ tanh ‚n;Œ s + j i Zi ‰n;Œ tanh ‰n;Œ (s − t) n;Œ0

1 + j Zi tanh i k0 s tan k0 (t − s) D0;0 + E0;0 hpII (0; r)ib

ZK = = Zi = Zi hZ0 vIIx (0; r)ib D0;0 − E0;0 tanh i k0 s + j Zi tan k0 (t − s)

ZMi = Zsh − ZK

Sn;Œ =

C


Mode coupling (I)–(II) in s= a2

Table 18 continued


Equations for Dn;Œ

Equation for C0






Table 19 continued


2 = †2 + ” 2 − k 2 ‚n;Œ n Œ 0

2 = †2 + ” 2 + 2 ‰n;Œ n Œ a

sin(”Œ c) = 0 ) ”Œ c = Œ ; Œ = 0; 1; 2; : : : ; ⎡

⎤ k0 S2n;Œ j i Zi ‚n;Œ − ‰n;Œ tanh ‚n;Œ (t − s) tanh ‰n;Œ s 1

⎦ C0 B0 = ⎣−1 + 2 a ‚n;Œ N(II) n;Œ i Zi ‚n;Œ tanh ‚n;Œ (t − s) + j ‰n;Œ tanh ‰n;Œ s n;Œ0 ⎡

⎤ −1 k0 S2n;Œ j i Zi ‚n;Œ − ‰n;Œ tanh ‚n;Œ (t − s) tanh ‰n;Œ s 1

⎦ ⎣1 + 2 ‚n;Œ N(II) a n;Œ i Zi ‚n;Œ tanh ‚n;Œ (t − s) + j ‰n;Œ tanh ‰n;Œ s n;Œ0

j i Zi ‚n;Œ − ‰n;Œ tanh ‰n;Œ s k0 Sn;Œ

Dn;Œ = (B0 − C0 ) −2‚n;Œ (t−s) + j ‰ −2‚n;Œ (t−s) tanh ‰ s ‚n;Œ N(II) n;Œ 1 + e n;Œ n;Œ i Zi ‚n;Œ 1 − e

sin(†n b) = 0 ) †n b = n ; n = 0; 1; 2; : : : ;

n;Œ0

−1 ‰n;Œ Fn;Œ sinh ‰n;Œ (x − t) cos(†n y) cos(”Œ z) i Zi k0

n;Œ0

Z0 vIIIx (x; y; z) =

pIII (x; y; z) =

sin(”Œ c) = 0 ) ”Œ c = Œ ; Œ = 0; 1; 2; : : : ;

sin(†n b) = 0 ) †n b = n ; n = 0; 1; 2; : : : ;

n;Œ0

‚n;Œ

j @p Dn;Œ e−‚n;Œ x − En;Œ e+‚n;Œ x cos(†n y) cos(”Œ z) = −j k0 @x k0

n;Œ0

Dn;Œ e−‚n;Œ x + En;Œ e+‚n;Œ x cos(†n y) cos(”Œ z)

Z0 vIIx (x; y; z) =

pII (x; y; z) =

pI (x; r) = B0 e−j k0 x + C0 e+j k0 x j @p Z0 vIx (x; r) = = B0 e−j k0 x − C0 e+j k0 x k0 @x

Equivalent Networks

C 111


Mode coupling (I)–(II) in s= a2

Mode norms in (II)

Dn;Œ e−‚n;Œ (t−s) + En;Œ e+‚n;Œ (t−s)

cosh ‰n;Œ s

⎧ ⎪ 0 ⎪ ⎪ ⎨ 0 ; Œ 6= Œ

Zsh =

1 + C0 B0 1 k0 S2n;Œ j i Zi ‚n;Œ −‰n;Œ tanh ‚n;Œ (t−s) tanh ‰n;Œ s

= 2 1 − C0 B0 a n;Œ0 ‚n;Œ N(II) n;Œ i Zi ‚n;Œ tanh ‚n;Œ (t−s) + j ‰n;Œ tanh ‰n;Œ s

Zi + j tan k0 (t − s) tanh i k0 s hpII (0; r)ib D0;0 + E0;0

= = ZK = hZ0 vIIx (0; r)ib D0;0 − E0;0 Zi tan k0 (t − s) − j tanh i k0 s

ZMi = Zsh − ZK

Sn;Œ

⎧ ⎪ 0 ⎪ ⎪ ⎨ 0 ; n 6= n

(z) 1 ; n = n0 6= 0 ; NŒ = ⎪ 1 ; Œ = Œ 0 6= 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ 2 ; n = n0 = 0 2 ; Œ = Œ 0 = 0 ! J1 a †2n + ”Œ2 = cos(†n y) cos(”Œ z) ds = 2 a2 −−−−−−−! a2 †n =”Œ =0 s a †2n + ”Œ2

(y) (y) (z) N(II) n;Œ = bc Nn NŒ ; Nn =

−c −b

+c +b = cos(†n y) cos(”Œ z) cos(†n0 y) cos(”Œ 0 z) dy dz

Fn;Œ =

Equations for Fn;Œ

N(II) n;Œ

En;Œ

j i Zi ‚n;Œ + ‰n;Œ tanh ‰n;Œ s k0 Sn;Œ −2‚n;Œ (t−s)

= (B0 − C0 ) e ‚n;Œ N(II) i Zi ‚n;Œ 1 − e−2‚n;Œ (t−s) + j ‰n;Œ 1 + e−2‚n;Œ (t−s) tanh ‰n;Œ s n;Œ

C

Equations for En;Œ

Table 19 continued


Equivalent Networks

C

113

|vx(x/b,r/b;z/b)|, a/b=0.3, c/b=0.5, t/b=2.0, s/t=0.3, b/lam=1.2, Rb=5.0, z/b=0.

1 1 0.5 0.5 0 -2

0

(y/b)

-1 -0.5

0 (x/b)

1 2 -1

Example of axial particle velocity profiles. Parameters: a/b = 0.3; c/b = 0.5; t/b = 2.0; s/t = 0.3; b/Š0 = 1.2; Rb = ¡b/Z0 = 5.0

Table 20 Medium-wide square neck and rectangular duct; full analysis A square neck with 2a side length ends in a rectangular empty duct with 2b, 2c side lengths. A plane wave with amplitude B0 is incident in the neck on the orifice; a mode sum is reflected. = a2 =(bc).


pI (x; y; z) = B0 e−j k0 x + Cm;‹ e+‰m;‹ x cos(—m y) cos(—‹ z) m;‹0;0

Z0 vIx (x; y; z) = B0 e−j k0 x ‰m;‹ +‰m;‹ x +j Cm;‹ e cos(—m y) cos(—‹ z) k0 m;‹0;0 Eigenvalues —m a

sin(—m a) = 0 ) —m a = m ; m = 0; 1; 2; : : : ; 2 = — 2 + —2 − k 2 ‰m;‹ m ‹ 0

114

C

Equivalent Networks


pII (x; y; z) =

n;Œ0

Z0 vIIx (x; y; z) =


j @p ‚n;Œ −‚n;Œ x Dn;Œ e = −j k0 @x k0 n;Œ0

cos(†n y) cos(”Œ z) sin(†n b) = 0 ) †n b = n ; n = 0; 1; 2; : : : Eigenvalues †n b, ”Œ c

System of equations for Cm;‹ m0 , ‹0 = 0; 1; 2; : : :

sin(”Œ c) = 0 ) ”Œ c = Œ ; Œ = 0; 1; 2; : : : ; 2 = †2 + ” 2 − k 2 ‚n;Œ n Œ 0 ⎡ ⎤ ‹;Œ ‹0 ;Œ ‰m;‹ Sm;n Sm0 ;n (I) ⎦ Cm;‹ ⎣ƒm;m0 ƒ‹;‹0 Nm0 ;‹0 + (II) k0 m;‹0 n;Œ0 (‚n;Œ k0 ) Nn;Œ ⎡ ⎤ 0;Œ ‹0 ;Œ S0;n Sm0 ;n (I) ⎦ = B0 ⎣−ƒm0 ;0 ƒ‹0 ;0 N0;0 + j (II) n;Œ0 (‚n;Œ k0 ) Nn;Œ

⎛ Equations for Dn;Œ

Dn;Œ =

j

(‚n;Œ k0 ) N(II) n;Œ

⎝B0 S0;Œ + j 0;n

Cm;‹

m;‹0

⎞ ‰m;‹ ‹;Œ S ⎠ k0 m;n

+a N(I) y;m

cos(—m y) cos(—m0 y) dy

= −a +a

Mode norms in (I)

N(I) z;‹ =

cos(—‹ z) cos(—‹0 z) dz −a

N(I) m;‹ N(I) z;‹

=

N(I) y;m ⎧ ⎨

=

⎩

(I) N(I) z;‹ ; Ny;m

⎧ ⎨ =

Mode norms in (II) N(II) z;Œ =

⎩

2a ; m = m0 = 0

a ; ‹ = ‹0 6= 0 2a ; ‹ = ‹0 = 0

(II) (II) (II) N(II) n;Œ = Ny;n Nz;Œ ; Ny;n =

⎧ ⎨

⎩

a ; m = m0 6= 0

c ; Œ 6= 0 2c ; Œ = 0

⎧ ⎨ ⎩

b ; n 6= 0 2b ; n = 0

;

;

Equivalent Networks

C

115

Table 20 continued S‹;Œ m;n =

+a

+a cos(—m y) cos(†n y) dy

−a

cos(—‹ z) cos(”Œ z) dz −a

= Sm;n S‹;Œ



⎧ sin (a(—m − †n )) sin (a(—m + †n )) ⎪ ⎪ a + ; —m 6= †n ⎪ ⎪ ⎪ a(—m − †n ) a(—m + †n ) ⎪ ⎪ ⎪ ⎨ Sm;n = 2a ; —m = †n = 0 ⎪ ⎪ ⎪ ⎪ sin (a(—m + †n )) ⎪ ⎪ ; —m = †n 6= 0 a 1 + ⎪ ⎪ a(—m + †n ) ⎩ ⎧

sin a(—‹ − ”Œ ) ⎪ sin a(—‹ + ”Œ ) ⎪ ⎪a + ; —‹ 6= ”Œ ⎪ ⎪ a(—‹ − ”Œ ) a(—‹ + ”Œ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ S‹;Œ = 2a ; —‹ = ”Œ = 0 ⎪ ⎪ ⎪ ⎪

⎪ ⎪ sin a(—‹ + ”Œ ) ⎪ ⎪ ⎪ ; —‹ = ”Œ 6= 0 a 1 + ⎪ ⎪ a(—‹ + ”Œ ) ⎩

ZMi = Zsh − ZK ; ZK = 1 hpI (0; r)ia hB0 + C0;0 ia 1 + C0;0 B0 = = Zsh = hZ0 vIx (0; r)ia hB0 − C0;0 ia 1 − C0;0 B0

Table 21 Medium-wide square neck and rectangular duct A square neck with 2a side length ends in a rectangular empty duct with 2b, 2c side lengths. A plane wave with amplitude B0 is incident in the neck on the orifice; a plane wave with amplitude C0 is reflected. = a2 =(bc).

Cm;‹ ! C0 ; —m ; —‹ ! 0 ; ‰m;‹ ! ‰0;0 = j k0 ; Simplifications from Table 20 ‹;Œ 0;Œ (I) 2 Œ N(I) m;‹ ! N0;0 = 4a ; Sm;n ! S0;n =: Sn

pI (x; y; z) = B0 e−j k0 x + C0 e+j k0 x Field formulation in (I) Z0 vIx (x; y; z) =

j @p = B0 e−j k0 x − C0 e+j k0 x k0 @c

116

C

Equivalent Networks


pII (x; y; z) =

n;Œ0


j @p k0 @x ‚n;Œ −‚n;Œ x = −j Dn;Œ e cos(†n y) cos(”Œ z) k0 n;Œ0

Z0 vIIx (x; y; z) =


Equation for C0

sin(”Œ c) = 0 ) ”Œ c = Œ ; Œ = 0; 1; 2; : : : ; 2 = †2 + ” 2 − k 2 ‚n;Œ n Œ 0 ⎡ ⎤ (SŒn )2 j ⎣ ⎦ C0 =B0 = −1 + 2 (II) 4a (‚ ) N k n;Œ 0 n;Œ n;Œ0 ⎤ −1 ⎡ Œ )2 (S j ⎦ ⎣1 + 2 n 4a (‚n;Œ k0 ) N(II) n;Œ n;Œ0

Equations for Dn;Œ

Mode norms in (II)

j SŒ n (II) (B0 − C0 ) (‚n;Œ k0 ) Nn;Œ ⎧ ⎨ b ; n 6= 0 (II) (II) (II) N(II) ; n;Œ = Ny;n Nz;Œ ; Ny;n = ⎩ 2b ; n = 0 ⎧ ⎨ c ; Œ 6= 0 N(II) z;Œ = ⎩ 2c ; Œ = 0 Dn;Œ =

SŒn

+a +a =

cos(†n y) cos(”Œ z) dy dz −a −a

⎧ ⎨

Mode coupling (I)–(II) =

⎩

2a ; n = 0 a ; n 6= 0


;

⎫ ⎬ ⎭

ZK = 1

⎧ ⎨ ⎩

2a ; Œ = 0 a ; Œ 6= 0

⎫ ⎬ ⎭


Equivalent Networks

C

117

Table 22 Medium-wide square neck and rectangular duct with absorber A square neck with 2a side length ends in a rectangular duct with 2b, 2c side lengths, filled with porous absorber. A plane wave with amplitude B0 is incident in the neck on the orifice; a plane wave with amplitude C0 is reflected. Absorber characteristic values: i = a =k0 ; Zi = Za =Z0 . = a2 =(bc).

Changes rel. to Table 17



s = a2 ! 4a2 ; = a2 =(4bc) ! a2 =(bc) ; Sn;Œ ! SŒn pI (x; y; z) = B0 e−j k0 x + C0 e+j k0 x j @p = B0 e−j k0 x − C0 e+j k0 x Z0 vIx (x; y; z) = k0 @c pII (x; y; z) = Dn;Œ e−‚n;Œ x cos(†n y) cos(”Œ z) n;Œ0

Z0 vIIx (x; y; z) =

1 ‚n;Œ −‚n;Œ x Dn;Œ e i Z i k0 n;Œ0


sin(”Œ c) = 0 ) ”Œ c = Œ ; Œ = 0; 1; 2; : : : ; 2 = †2 + ” 2 + 2 ‚n;Œ n Œ a ⎡

C0 =B0

=

Equation for C0

⎤ Œ )2 Z (S i i n ⎣−1 + ⎦ (II) 4a2 n;Œ0 (‚n;Œ k0 ) Nn;Œ ⎡ ⎤ −1 i Zi (SŒn )2 ⎣ ⎦ 1+ 2 4a (‚n;Œ k0 ) N(II) n;Œ n;Œ0

Equations for Dn;Œ

Mode norms in (II)

Dn;Œ = i Zi

SŒn

(B0 − C0 ) (‚n;Œ k0 ) N(II) n;Œ

(II) (II) N(II) n;Œ = Ny;n Nz;Œ ;

N(II) z;Œ =

⎧ ⎨ ⎩

c ; Œ 6= 0 2c ; Œ = 0

N(II) y;n =

⎧ ⎪ ⎪ ⎨ b ; n 6= 0 2b ; n = 0 ⎪ ⎪ ⎩

;

118

C

Equivalent Networks

Table 22 continued SŒn =

+a +a cos(†n y) cos(”Œ z) dy dz −a −a

⎧ ⎨

Mode coupling (I)–(II) =

⎩

2a ; n = 0 a ; n 6= 0

⎫ ⎬ ⎭


;

⎧ ⎨ ⎩

2a ; Œ = 0 a ; Œ 6= 0

⎫ ⎬ ⎭

ZK = Zi


Table 23 Medium-wide square neck and rectangular empty chamber A square neck with 2a side length ends in a rectangular empty chamber with 2b, 2c side lengths and depth t. A plane wave with amplitude B0 is incident in the neck on the orifice; a plane wave with amplitude C0 is reflected. = a2 =(bc).




s = a2 ! 4a2 ; = a2 =(4bc) ! a2 =(bc) ; Sn;Œ ! SŒn pI (x) = B0 e−j k0 x + C0 ej k0 x j @p Z0 vIx (x) = = B0 e−j k0 x − C0 ej k0 x k0 @x

pII (x; y; z) = Dn;Œ cosh ‚n;Œ (x − t) cos(†n y) cos(”Œ z) n;Œ0

Z0 vIIx (x; y; z) = j

n;Œ0

Dn;Œ

‚n;Œ sinh ‚n;Œ (x − t) k0


sin(”Œ c) = 0 ) ”Œ c = Œ ; Œ = 0; 1; 2; : : : ; 2 = †2 + ” 2 − k 2 ‚n;Œ n Œ 0

Equivalent Networks

Table 23 continued

Equation for C0

C

119

⎡

⎤ (SŒn )2 coth ‚n;Œ t j ⎦ C0 =B0 = ⎣−1 + 2 (II) 4a n;Œ0 (‚n;Œ k0 ) Nn;Œ ⎡

⎤ −1 (SŒn )2 coth ‚n;Œ t j ⎦ ⎣1 + 2 4a (‚n;Œ k0 ) N(II) n;Œ n;Œ0

(B0 − C0 )

(‚n;Œ k0 ) sinh ‚n;Œ t N(II) n;Œ ⎧ ⎨ b ; n 6= 0 (II) (II) (II) N(II) ; n;Œ = Ny;n Nz;Œ ; Ny;n = ⎩ 2b ; n = 0 ⎧ ⎨ c ; Œ 6= 0 N(II) z;Œ = ⎩ 2c ; Œ = 0 Dn;Œ = j

Equations for Dn;Œ

Mode norms in (II)

SŒn = Mode coupling (I)–(II)

SŒn

+a +a cos(†n y) cos(”Œ z) dy dz −a −a

⎧ ⎨

in s= 4a2 =

⎩

2a ; n = 0 a ; n 6= 0


⎫ ⎬ ⎭ ;

⎧ ⎨ ⎩

2a ; Œ = 0 a ; Œ 6= 0

⎫ ⎬ ⎭

ZK = −j cot(k0 t)


Table 24 Medium-wide square neck and rectangular chamber with absorber A square neck with 2a side length ends in a rectangular chamber with 2b, 2c side lengths and depth t, filled with porous absorber material. A plane wave with amplitude B0 is incident in the neck on the orifice; a plane wave with amplitude C0 is reflected. Absorber characteristic values: i = a =k0 ; Zi = Za =Z0 . = a2 =(bc).


s = a2 ! 4a2 ; = a2 =(4bc) ! a2 =(bc) ; Sn;Œ ! SŒn

120

C

Equivalent Networks

Table 24 continued Field formulation in (I)

pI (x) = B0 e−j k0 x + C0 ej k0 x j @p Z0 vIx (x) = = B0 e−j k0 x − C0 ej k0 x k0 @x

pII (x; y; z) = Dn;Œ cosh ‚n;Œ (x − t) cos(†n y) cos(”Œ z)


n;Œ0

Z0 vIIx (x; y; z) =

‚n;Œ −1 Dn;Œ sinh ‚n;Œ (x − t) i Z i k0 n;Œ0


sin(”Œ c) = 0 ) ”Œ c = Œ ; Œ = 0; 1; 2; : : : ; 2 = †2 + ” 2 + 2 ‚n;Œ n Œ a ⎡

⎤ (SŒn )2 coth ‚n;Œ t Z i i ⎦ C0 =B0 = ⎣−1 + 2 (II) 4a n;Œ0 (‚n;Œ k0 ) Nn;Œ ⎡

⎤ −1 (SŒn )2 coth ‚n;Œ t Z i i ⎦ ⎣1 + 2 4a (‚n;Œ k0 ) N(II) n;Œ

Equation for C0

n;Œ0

SŒn (B0 − C0 )

(‚n;Œ k0 ) sinh ‚n;Œ t N(II) n;Œ ⎧ ⎨ b ; n 6= 0 (II) (II) (II) N(II) ; n;Œ = Ny;n Nz;Œ ; Ny;n = ⎩ 2b ; n = 0 ⎧ ⎨ c ; Œ 6= 0 N(II) z;Œ = ⎩ 2c ; Œ = 0 Dn;Œ = i Zi

Equations for Dn;Œ

Mode norms in (II)

SŒn Mode coupling (I)–(II) in s =

+a +a =

cos(†n y) cos(”Œ z) dy dz −a ⎧ −a

4a2

⎨

=

⎩

2a ; n = 0 a ; n 6= 0


⎫ ⎬ ⎭ ;

⎧ ⎨ ⎩

2a ; Œ = 0 a ; Œ 6= 0

⎫ ⎬ ⎭

ZK = Zi coth(at)

1 + C0 B0 hpI (0; r)is = = hZ0 vIx (0; r)is 1 − C0 B0







= a2 =(4bc) ! a2 =(bc) ;

Sn;Œ ! SŒn

n;Œ0

n;Œ0

n;Œ0

Fn;Œ

sin(”Œ c) = 0 ) ”Œ c = Œ ; Œ = 0; 1; 2; : : : ;

2 = †2 + ” 2 − k 2 ‰n;Œ n Œ 0

‰n;Œ sinh ‰n;Œ (x−t) cos(†n y) cos(”Œ z) k0


sin(†n b) = 0 ) †n b = n ; n = 0; 1; 2; : : : ;

Z0 vIIIx (x; y; z) = j

pIII (x; y; z) =

sin(”Œ c) = 0 ) ”Œ c = Œ ; Œ = 0; 1; 2; : : : ;

2 = †2 + ” 2 + 2 ‚n;Œ n Œ a

‚n;Œ

−1 @p 1 Dn;Œ e−‚n;Œ x − En;Œ e+‚n;Œ x cos(†n y) cos(”Œ z) = a Zan @x an Zan k0

sin(†n b) = 0 ) †n b = n ; n = 0; 1; 2; : : : ;

Z0 vIIx (x; y; z) =

n;Œ0

pI (x) = B0 e−j k0 x + C0 e+j k0 x j @p = B0 e−j k0 x − C0 e+j k0 x Z0 vIx (x) = k0 @x

Dn;Œ e−‚n;Œ x + En;Œ e+‚n;Œ x cos(†n y) cos(”Œ z) pII (x; y; z) =

s = a2 ! 4a2 ;

A square neck with 2a side length ends in a rectangular chamber with 2b, 2c side lengths and depth t, partially filled with a porous absorber layer of thickness s adjacent to the neck. A plane wave with amplitude B0 is incident in the neck on the orifice; a plane wave with amplitude C0 is reflected. Absorber characteristic values: i = a =k0 ; Zi = Za=Z0 . = a2 =(bc).

Table 25 Medium-wide square neck and rectangular chamber with front absorber

Equivalent Networks

C 121

Mode coupling (I)–(II) in s =

Mode norms in (II)

SŒn =

N(II) n;Œ

4a2

Fn;Œ =

Equations for Fn;Œ

−a −a

(II) N(II) z;Œ ; Ny;n

= ⎩

⎧ ⎨

⎩

⎧ ⎨

; N(II) z;Œ

=

a ; n 6= 0

2a ; n = 0

2b ; n = 0

b ; n 6= 0

cos(†n y) cos(”Œ z) dy dz =

N(II) y;n

+a +a

=

En;Œ

Equations for En;Œ Dn;Œ e−‚n;Œ s + En;Œ e+‚n;Œ s

cosh ‰n;Œ (s − t)

Dn;Œ = (B0 − C0 ) i Zi

Equations for Dn;Œ

⎭

⎩

⎧ ⎨

a ; Œ 6= 0

2a ; Œ = 0

2c ; Œ = 0

c ; Œ 6= 0 ⎫ ⎬

⎩

⎧ ⎨

⎭

⎫ ⎬

‚n;Œ + j i Zi ‰n;Œ tanh ‰n;Œ (s − t) k0 SŒn

−2‚n;Œ s + j Z ‰ −2‚n;Œ s tanh ‰ (s − t) ‚n;Œ N(II) n;Œ i i n;Œ 1 + e n;Œ ‚n;Œ 1 − e

‚n;Œ − j i Zi ‰n;Œ tanh ‰n;Œ (s − t) k0 SŒn −2‚n;Œ s

= (B0 − C0 ) i Zi e ‚n;Œ N(II) ‚n;Œ 1 − e−2‚n;Œ s + j i Zi ‰n;Œ 1 + e−2‚n;Œ s tanh ‰n;Œ (s − t) n;Œ

n;Œ0

⎡

⎤ k0 (SŒ )2 ‚n;Œ + j i Zi ‰n;Œ tanh ‚n;Œ s tanh ‰n;Œ (s − t) Z i i n

⎦ C0 B0 = ⎣−1 + 2 ‚n;Œ N(II) 4a n;Œ ‚n;Œ tanh ‚n;Œ s + j i Zi ‰n;Œ tanh ‰n;Œ (s − t) n;Œ0 ⎡

⎤ −1 k0 (SŒ )2 ‚n;Œ +j i Zi ‰n;Œ tanh ‚n;Œ s tanh ‰n;Œ (s−t) Z i i n

⎦ ⎣1 + 2 4a ‚n;Œ N(II) n;Œ ‚n;Œ tanh ‚n;Œ s +j i Zi ‰n;Œ tanh ‰n;Œ (s−t)

C

Equation for C0

Table 25 continued


hpI (0; r)ia hB0 + C0 ia B0 + C0 = = hZ0 vIx (0; r)ia hB0 − C0 ia B0 − C0

=

i Zi k0 (SŒn )2 ‚n;Œ + j i Zi ‰n;Œ tanh ‚n;Œ s tanh ‰n;Œ (s − t)

4a2 ‚n;Œ N(II) n;Œ ‚n;Œ tanh ‚n;Œ s + j i Zi ‰n;Œ tanh ‰n;Œ (s − t) n;Œ0

1 + j Zi tanh i k0 s tan k0 (t − s) D0;0 + E0;0 hpII (0; r)ib

ZK = = Zi = Zi hZ0 vIIx (0; r)ib D0;0 − E0;0 tanh i k0 s + j Zi tan k0 (t − s)

Zsh =

ZMi = Zsh − ZK



= a2 =(4bc) ! a2 =(bc);

pI (x) = B0 e−j k0 x + C0 e+j k0 x j @p = B0 e−j k0 x − C0 e+j k0 x Z0 vIx (x) = k0 @x

s = a2 ! 4a2 ;

Sn;Œ ! SŒn

A square neck with 2a side length ends in a rectangular chamber with 2b, 2c side lengths and depth t, partially filled with a porous absorber layer of thickness s at the back side. A plane wave with amplitude B0 is incident in the neck on the orifice; a plane wave with amplitude C0 is reflected. Absorber characteristic values: i = a =k0 ; Zi = Za=Z0 . = a2 =(bc).

Table 26 Medium-wide square neck and rectangular chamber with rear absorber


Table 25 continued

Equivalent Networks

C 123

Equations for Dn;Œ

Equation for C0



Eigenvalues †n b , ”Œ c


n;Œ0

2 = †2 + ” 2 + 2 ‰n;Œ n Œ a

(SŒn )2 j i Zi ‚n;Œ − ‰n;Œ tanh ‚n;Œ (t − s) tanh ‰n;Œ s

N(II) n;Œ i Zi ‚n;Œ tanh ‚n;Œ (t − s) + j ‰n;Œ tanh ‰n;Œ s

⎤ −1 (SŒn )2 j i Zi ‚n;Œ − ‰n;Œ tanh ‚n;Œ (t − s) tanh ‰n;Œ s ⎦

N(II) n;Œ i Zi ‚n;Œ tanh ‚n;Œ (t − s) + j ‰n;Œ tanh ‰n;Œ s

2; : : : ;

j i Zi ‚n;Œ − ‰n;Œ tanh ‰n;Œ s k0 SŒn

−2‚n;Œ (t−s) + j ‰ −2‚n;Œ (t−s) tanh ‰ s ‚n;Œ N(II) n;Œ 1 + e n;Œ n;Œ i Zi ‚n;Œ 1 − e

n;Œ0

sin(”Œ c) = 0 ) ”Œ c = Œ ; Œ = 0; 1; ⎡ 1 k0 C0 B0 = ⎣−1 + 2 ‚n;Œ 4a n;Œ0 ⎡ 1 k0 ⎣1 + 2 ‚n;Œ 4a

sin(†n b) = 0 ) †nb = n ; n = 0; 1; 2; : : : ;

Dn;Œ = (B0 − C0 )

2 = †2 + ” 2 − k 2 ‚n;Œ n Œ 0

‰n;Œ −1 Fn;Œ sinh ‰n;Œ (x − t) cos(†n y) cos(”Œ z) i Zi k0

n;Œ0

Z0 vIIIx (x; y; z) =

pIII (x; y; z) =

sin(”Œ c) = 0 ) ”Œ c = Œ ; Œ = 0; 1; 2; : : : ;

sin(†n b) = 0 ) †nb = n ; n = 0; 1; 2; : : : ;

n;Œ0

‚n;Œ

j @p = −j Dn;Œ e−‚n;Œ x − En;Œ e+‚n;Œ x cos(†n y) cos(”Œ z) k0 @x k0

n;Œ0

Dn;Œ e−‚n;Œ x +En;Œ e+‚n;Œ x cos(†n y) cos(”Œ z)

Z0 vIIx (x; y; z) =

pII (x; y; z) =

C


Table 26 continued



=

2b ; n = 0

b ; n 6= 0 ⎧ ⎨

; N(II) z;Œ = ⎩

⎧ ⎨ c ; Œ 6= 0

Zi + j tan k0 (t − s) tanh i k0 s hpII (0; r)ib D0;0 + E0;0

= == hZ0 vIIx (0; r)ib D0;0 − E0;0 Zi tan k0 (t − s) − j tanh i k0 s

1 + C0 B0 1 k0 (SŒn )2 j i Zi ‚n;Œ − ‰n;Œ tanh ‚n;Œ (t − s) tanh ‰n;Œ s

= = 1 − C0 B0 4a2 n;Œ0 ‚n;Œ N(II) n;Œ i Zi ‚n;Œ tanh ‚n;Œ (t − s) + j ‰n;Œ tanh ‰n;Œ s

ZK =

Zsh

⎩

⎧ ⎨

2c ; Œ = 0 ⎫ ⎧ ⎫ +a +a ⎬ ⎨ ⎬ 2a ; n = 0 2a ; Œ = 0 cos(†n y) cos(”Œ z) dy dz = ⎩ ⎭ ⎩ ⎭ a ; n 6= 0 a ; Œ 6= 0 −a −a

ZMi = Zsh − ZK

SŒn

(II) (II) (II) N(II) n;Œ = Ny;n Nz;Œ ; Ny;n =

Mode norms in (II)

Mode coupling (I)–(II) in s = 4a2

Fn;Œ =

Equations for Fn;Œ

Dn;Œ e−‚n;Œ (t−s) + En;Œ e+‚n;Œ (t−s)

cosh ‰n;Œ s

En;Œ

j i Zi ‚n;Œ + ‰n;Œ tanh ‰n;Œ s k0 SŒn −2‚n;Œ (t−s)

= (B0 − C0 ) e ‚n;Œ N(II) i Zi ‚n;Œ 1 − e−2‚n;Œ (t−s) + j ‰n;Œ 1 + e−2‚n;Œ (t−s) tanh ‰n;Œ s n;Œ

Equations for En;Œ

Table 26 continued

Equivalent Networks

C 125

126

C

Equivalent Networks

References Mechel, F.P.: Schallabsorber, Vol. II, Ch. 2: Equivalent networks. Hirzel, Stuttgart (1995)

D Reflection of Sound The limit between reflection and scattering of sound is not sharp. A generally applicable distinction could be that reflection sends sound back only into the half-space of incidence,whereas scattering sends sound also in the forward direction.This is the guideline for placing topics either in this chapter about reflection of sound or in the later chapter about scattering of sound. The general reference in this chapter is Mechel,“Schallabsorber” (Vol. I – III). Formulas for the input admittance and/or absorption coefficient can also be found in the chapter “Compound Absorbers”.

D.1 Plane Wave Reflection at a Locally Reacting Plane An absorber is said to be locally reacting if there is no sound propagation inside the absorber parallel to the absorber surface. A plane wave with amplitude A is incident in the plane (x,y) on an absorbent plane; the y axis is in the absorber surface; the x axis is normal to the absorber and directed into the absorber; the wave vector of the incident wave pi (x,y) forms a polar angle Ÿ with the normal to the surface.

Θ

Θ

The acoustic quality of the absorber is defined by the wall admittance:

Field formulation:

vx (0, y) j ∂p(0, y)/∂x = . p(0, y) k0 Z0 p(0, y)

(1)

p(x, y) = pi (x, y) + pr (x, y) , pi (x, y) = A · e−j k0 (x·cos Ÿ+y·sin Ÿ ) , pr (x, y) = r · A · e−j k0 (−x·cos Ÿ+y·sin Ÿ ) .

(2)

G=

D

128

Reflection of Sound

r=

Reflection factor:

pr (0, y) cos Ÿ − Z0 G = . pi (0, y) cos Ÿ + Z0 G

− −−−→ 1 G→0

;

(3)∗)

− −−−−−→ −1 , |G|→∞

(Ÿ) = 1 − |r(Ÿ)|2

Absorption coefficient with the

4z cos Ÿ (1 + z cos Ÿ)2 + (z cos Ÿ)2 4g cos Ÿ . = (g + cos Ÿ)2 + g2

normalised absorber

=

input impedance z=z +j· z = Z/Z0 or input admittance g = g +j · g = Z0 G.

(4)

α(Θ) % 10 0

5

10 20

8

z″⋅cos Θ

6 30

4 40 50 2

60 70 90

80

95

0 0

2

4

z′⋅cos Θ

6

8

10

Contour lines of (Ÿ) in per cent over z · cos Ÿ, z · cos Ÿ

The straight connecting line between the starting point at z , z, for Ÿ = 0, with the origin, for Ÿ = /2, mostly passes through higher absorption values at some finite angle Ÿ. ∗)

See Preface to the 2nd Edition.

Reflection of Sound

D

129

Sometimes the derivatives r(n) (Ÿ) = ∂ n r(Ÿ)/∂Ÿ n are needed (substitute Z0 G → g):

(5)∗)

−2g sin Ÿ , (g + cos Ÿ)2 −g 3 + 2g cos Ÿ − cos (2Ÿ) , r (Ÿ) = (g + cos Ÿ)3 −g 11 − 2 g2 + 8g cos Ÿ − sin Ÿ · cos (2Ÿ) (3) r (Ÿ) = , (g + cos Ÿ)4 r(4) (Ÿ) = r (Ÿ) =

−g 115 − 20 g2 + 2 g(47 − 4 g 2 ) cos Ÿ − 4 (19 − 11 g2 ) cos(2Ÿ) − 22 g cos(3Ÿ) + cos(4Ÿ) . 4 (g + cos Ÿ)5

″

Θ

αΘ

′

Θ

Contour lines of (Ÿ) in per cent over logarithmic z · cos Ÿ and linear z · cos Ÿ

D.2 Plane Wave Reflection at an Infinitely Thick Porous Layer “Porous layer” here stands for any homogeneous, isotropic material with characteristic propagation constant a and wave impedance Za . If the material is air, then a → jk0; ∗)


D

130

Reflection of Sound

Za → Z0 . Sound incidence is as in

> Sect. D.1.

Sound field above absorber: sound field in absorber:

p1 =pi +pr (as in > Sect. D.1) , p2 (x, y) = pt (x, y) = B · e−a (x cos Ÿa +y sin Ÿa ) .

(1)

Θ Θ Γ Θ The boundary conditions are: • Equal propagation constant in y direction on both sides; • Equal normal admittance component on both sides (is equivalent to matching sound pressure and normal particle velocity). Refracted angle Ÿ a (complex !):

j k0 sin Ÿa = . sin Ÿ a

Reflection factor r:

r=

Za / cos Ÿa − Z0 / cos Ÿ Zan − Z0n = . Za / cos Ÿa + Z0 / cos Ÿ Zan + Z0n

(2)

(3)

(in the second form Zan ,Z0n indicate normal components of the impedances). Absorption coefficient again is = 1 − |r|2 .

D.3 Plane Wave Reflection at a Porous Layer of Finite Thickness The absorber layer of thickness d is backed by a rigid wall. The input impedance of the layer is:

Z2 =

Reflection factor:

r=

Za · coth(a d · cos Ÿa ). cos Ÿa

Z2 / cos Ÿa − Z0 / cos Ÿ . Z2 / cos Ÿa + Z0 / cos Ÿ

(1)

(2)

With normalised characteristic values an = a /k0, Zan = Za /Z0 it is convenient to evaluate 2 + sin2 Ÿ = 2 + cos2 Ÿ . an · cos Ÿa = an 1 + an (3) a a (4) Ÿa follows from the law of refraction cos Ÿa = 1 + (sin Ÿ/an )2 in > Sect. D.2.

Reflection of Sound

D

131

Limit of layer thickness d above which the layer effectively behaves like an infinitely thick layer for normal sound incidence: The layer is locally reacting (either due to large R or by internal partitions): The limit follows

R ≥ 5.158/F0.5886 ;

from one of the relations

F ≥ 16.233/R1.699

for locally reacting layers:

2.699 f[Hz] · d[m]

·

F ≥ 2.81 · E0.629 ;

;

1.699 ¡[Pa·s/m 2]

(5)

6

≥ 2.274 · 10 .

Contour diagrams of (Ÿ) of a porous absorber layer with hard back, for Ÿ = 0 and Ÿ = 45◦ . α(0°)

Θ=0°

20. 0.2 0.3 0.4

0.5

0.97 0.6

0.7 0.8

0.9 0.95

0.99

10. R= Ξd/Z0 0.99 0.97 0.99 0.05 0.1 0.2

1.

0.2 0.01

0.1

1.

αΘ

F=fd/c 0 5.

Θ =45 ;̊ bulk reacting

20

0.97 0.2 0.3 0.4 0.5

0.7

0.6

0.8

0.9 0.95

0.99

10 R= Ξ d/Z0

0.05 0.1 0.2 1

0.2 0.01

0.1

1

5 F=fd/c0

132

D

Reflection of Sound

The layer is bulk reacting: The limit follows

R ≥ 3.209/F0.7245 ;

from one of the relations

F ≥ 5.00/R1.380

for bulk reacting layers:

2.380 f[Hz] · d[m]

with the non-dimensional quantities:

·

F ≥ 1.966 · E0.580 ;

;

1.380 ¡[Pa·s/m 2]

R = ¡ · d/Z0

;

(6)

6

≥ 0.70 · 10 ; F = f · d/c0 = d/Š0 ;

E = 0 f /¡ .

(7)

D.4 Plane Wave Reflection at a Multilayer Absorber The absorber consists of M layers of homogeneous porous material (or air); the layers are numbered m = 1, 2, . . . , M. The space in front of the layer (with characteristic values k0 , Z0) takes the index m = 0. Layer thicknesses:

dm ;

Characteristic values:

am , Zam ;

m = 1, 2, . . . , M m = 0, 1, . . . , M

(with a0 = j, Za0 = 1) Incidence and refracted angles:

Ÿm ;

m = 0, 1, . . ., M

Reflection factors

rm ;

m = 0, 1, . . . , M

Gm ;

m = 0, 1, . . . , M

Dm ;

m = 1, 2, . . ., M .

(r0 = reflection factor of the arrangement) Layer input admittances: (G0 = input admittance of the arrangement). Acoustic layer thicknesses:

One can apply the chain circuit algorithm of > Sect. C.5 for the evaluation of the input admittance and therewith of the reflection factor using the equivalent four poles of > Sect. C.2. Here will be given a more explicit scheme of iteration with the iteration of the reflection factors (rm is the reflection factor at the back side of layer m = 0, 1, 2, . . ., M):

rm−1

Wm 1 + rm · e−2 Dm − 1 − rm · e−2Dm W = m−1 . Wm 1 + rm · e−2Dm + 1 − rm · e−2Dm Wm−1

(1)

Reflection of Sound

D

133

Auxiliary quantities: Wm = Zam / cos Ÿm ; W0 = Z0 / cos Ÿ0 , cos Ÿm = 1 + (k0 /am )2 · (1 − cos2 Ÿ0 ) , Dm = am dm · cos Ÿm = k0dm 1 + (am /k0)2 − cos2 Ÿ0 .

(2)

If the arrangement has a rigid backing, start the iteration with rM =1. If the back side of the arrangement is in contact with free space (without a back cover of the last layer), start with rM =

Z0 / cos Ÿ0 − ZaM / cos ŸM . Z0 / cos Ÿ0 + ZaM / cos ŸM

(3)

Input admittance Gm of the mth layer (m = 1, . . ., M): Gm =

1 1 − rm . Wm 1 + rm

(4)

D.5 Diffuse Sound Reflection at a Locally Reacting Plane Generally, the absorption coefficient dif follows from the absorption coefficient (Ÿ) for oblique incidence by integration over the polar angle Ÿ. The integrals are in 2-dimensional space:

2−dif

/2 = (Ÿ) · cos Ÿ dŸ,

(1)

0

in 3-dimensional space:

3−dif

/2 = 2 (Ÿ) · cos Ÿ · sin Ÿ dŸ.

(2)

0

The integral in three dimensions has an analytical solution for a locally reacting plane with normalised input admittance Z0 G = g + j · g :

g 1 + 2 g g2 − g2 3−dif = 8 g 1 + · arctan − g · ln 1 + g g + g2 + g2 g2 + g2

1 + 2 g g2 (3) → 8 g − g · ln 1 + 1 + −−− − − − − − g =0 ; g =0 g + g2 g2 −−− −−−−−→ 0 g =0 ; g =0

or with the normalised input impedance Z/Z0 = z + j · z : z 3−dif = 8 2 z + z2

1 z2 − z2 z z 2 2 1 + 2 · arctan − · ln 1 + 2 z + z + z . z z + z2 1 + z z2 + z2

(4)

D

134

Reflection of Sound

Under condition z2 (1 + z )2 , the absorption coefficient 3−dif can be evaluated from the (measured or computed) absorption coefficient 0 for normal sound incidence: √ √ √

2 1 − 1 − 0 1 − 1 − 0 1 − 1 − 0 2 + 2 ln · − . (5) 3−dif = 8 √ √ 2 2 1 − 1 − 0 1 + 1 − 0 The maximum possible value of 3−dif for locally reacting absorbent planes is 3−dif = 0.951. The analytical solution for the integral of 2−dif follows from: /2 /2 2−dif = (Ÿ) · cos Ÿ dŸ = 0

1 = 0

1 = 0

and is

0

4 z cos2 Ÿ dŸ (1 + z cos Ÿ)2 + (z cos Ÿ)2

2

4z x dx √ (1 + z x)2 + z2 x2 1 − x2

(6)

4 z x2 dx √ 1 + 2 z x + (z2 + z2 ) x2 1 − x2 ⎡

2−dif = 2 z ⎣

z2

+ z2

⎫⎤ ⎧ √ ⎨ ln z + z2 − 1 ⎬ ⎦ . + 2 Im √ ⎩ z · z z2 − 1 ⎭

(7)

The difference 2−dif − 3−dif is small, in general, so that one absorption coefficient can be approximated by the other. This can be seen from the following diagram showing the difference over the complex plane of the normalised input impedance Z of a locally reacting plane. α2-diff − α3-diff

0.05

0

-0.05 0.01

100 0.1

10 Re{Z} 1

1 10

0.1 0.01 100

Im{Z}

D

Reflection of Sound

135

The limit of polar angle of incidence in the above formulas is assumed to be Ÿ = /2. In some of the literature it is recommended to integrate up to an angle Ÿ < /2 (values between 75◦ and 87◦ were proposed). Writing the normalised surface admittance as Z0 G = g1 + j · g2 and using the reflection factor R(‡) for the absorption coefficient (‡) for oblique incidence under a polar angle ‡: 4g1 cos ‡ , (cos ‡ + g1)2 + g22

(‡) = 1 − |R(‡)|2 = R(‡) =

cos ‡ − Z0 G , cos ‡ + Z0 G

(8)

Z0 G = g1 + j · g2 , one gets for the sound absorption coefficient dif = d¢a /d¢e of a plane wave with intensity I, with the absorbed sound power d¢a and the incident sound power d¢e on a surface element dS of the absorber: 2 d¢e = I dS

Ÿ dœ

cos ‡ sin ‡ d‡ = 2I dS

0

0

2

Ÿ

d¢e = I dS

dœ 0

Ÿ

cos ‡ sin ‡ d‡ = I dS sin2 Ÿ ;

0

(9)

Ÿ (‡) cos ‡ sin ‡ d‡ = 2I dS

0

(‡) cos ‡ sin ‡ d‡ . 0

The integral and its special values: dif

d¢a 2 = = d¢e sin2 Ÿ

cos ‡ → t

−−−−−−−−−−−−−−−→=

− sin ‡ d‡ = dt

Ÿ (‡) cos ‡ sin ‡ d‡ = 0

8g1 sin2 Ÿ

1 cos Ÿ

8g1 sin2 Ÿ

Ÿ 0

cos2 ‡ sin ‡ d‡ (cos ‡ + g1 )2 + g22

g2 − g22 t2 8g1 1 − cos Ÿ + 1 dt = 2 2 2 g2 (t + g1 ) + g2 sin Ÿ

g1 + cos Ÿ 1 + g1 − arctan arctan g2 g2

g2 + g22 + 2g1 cos Ÿ + cos2 Ÿ + g1 ln 1 1 + g12 + g22 + 2g1

g1 g12 + g22 g12 − g22 1 + g1 −arctan arctan +g1 ln −−−−−−−−→ 8g1 1+ g2 g2 g2 1 + g12 + g22 + 2g1 Ÿ = /2 −−−−−−→

8g1

g2 = 0 sin2 Ÿ

Ÿ = /2

1 − cos Ÿ +

−−−−−−−−−→ 8g1 1 + g1 −

g2 = 0

g12 g1 + cos Ÿ g12 − + 2g1 ln g1 + cos Ÿ 1 + g1 1 + g1

g12 g1 + 2g1 ln 1 + g1 1 + g1

.

(10)

D

136

Reflection of Sound

D.6 Diffuse Sound Reflection at a Bulk Reacting Porous Layer The integral for dif from > Sect. D.5 of bulk reacting absorbers generally must be evaluated numerically. The diagram below shows a contour plot of dif for a porous layer of thickness d with hard back over the non-dimensional parameters F = f · d/c0 and R = ¡ · d/Z0 . In the parameter range above and on the left-hand side of the dashdotted curve, the absorption coefficients of a locally reacting and a bulk reacting porous layer agree with each other. In the range above and on the right-hand side of the dashed straight line, the bulk reacting layer effectively has an infinite thickness. αdif

diffus, lateral

20. 0.3 0.4 0.5

0.6

0.7

0.8 0.85 0.9 0.925

0.95

10. 0.96

R= Ξd/Z 0

0.97 0.96 0.050.1 0.2

0.98

1.

0.2 0.01

0.1

1.

F=fd/c 0 5.

D.7 Sound Reflection and Scattering at Finite-Size Local Absorbers The wording“local”in this heading (and at other places in this book) is a shorter form of “locally reacting”; the corresponding abbreviation for “bulk reacting” will be “lateral”. If the side dimensions of the plane absorber are finite, scattering takes place at the borders between the absorber and the baffle wall. In fact, some theories determine the sound absorption of finite-size absorbers from the solution of the scattering problem. Let the absorber with area A be in the plane (x, y); the z co-ordinate shows into the space above the absorber. The sketch shows co-ordinates and angles used, as well as the incident plane wave pi and the specularly reflected wave pr . The field point is in P. Let s be a general co-ordinate.

Reflection of Sound

z ϑr

ϑi

pi

ϑ ϕi

D

137

pr ϕr

x

ϕ

A

r

–y

P(s)

Field composition: p(s) = pi (s) + pr (s) + ps (s) pi (s) = Pi · e−j (kx x+ky y−kz z) ,

(1)

pr (s) = ru · Pi · e−j (kx x+ky y+kz z) , with: pi (s) = incident plane wave; pr s) = specularly reflected plane wave; ps (s) = scattered wave with: ru = reflection factor of the baffle wall; F0 = normalised admittance of the baffle wall. Wave number components: kx = k0 · sin ˜i · cos œi kx2

+

ky2

+

kz2

=

;

ky = k0 · sin ˜i · sin œi

;

kz = k0 · cos ˜i ;

k02 .

Scattered wave:

∂ ∂ ps (s) = p(s0 ) − p(s0 ) · G (s|s0 ) ds0 , G (s|s0 ) · ∂n0 ∂n0

(2)

(3)

A

with Green’s function (in which s is a radius, see sketch) for field points at a large distance: G (s|s0 ) =

e−j k0 s j k0 z0 cos ˜ e + ru (˜ ) · e−j k0 z0 cos ˜ · ej k0 sin ˜ ·(x0 cos œ+y0 sin œ) . 4 s

(4)

The Green’s function corresponds to a superposition of the fields of point sources at Q and at the mirror-reflected point Q’. It satisfies the boundary condition at the baffle wall.

D

138

Reflection of Sound

Source point Q, mirror source point Q’ and field point P in the construction of Green’s function:

Q(x0,y0,z0) R

z

ϑi ϑ s0 s

R′

P(x,y,z)

ϕ

x

ϑ′

Q′(x0,y0,–z0) The integral equation above for ps holds also on the absorber surface A if the integral is multiplied with 1/2. It may be solved by iteration n = 0, 1, . . . for psn : • • •

Start with a suitable pso on A (e.g. with the value of pi + pr on A); Insert p = pi + pr + ps in the integrand; Evaluate the first approximation ps1 , and so on. The nth iteration gives

pn (s) = pi (s) + pr (s) + ps1 (s) + . . . + psn (s) (5)

and for ps (s):

ps (s) = ps1 (s) + . . . + psn (s) ∗ ¢s vns ps = Re · dS. Qs = Ii P∗i Pi /Z0 ∗ ¢a vn p Qa = = − Re · dS. Ii P∗i Pi /Z0

Scattering cross section of A: Absorption cross section of A: Extinction cross section of A:

Qe = Q a + Qs

with: ¢s ¢a Ii and

= = = the

scattered effective power; absorbed effective power; incident effective intensity, integrals over large hemispheres surrounding A.

(6) (7) (8)

Reflection of Sound

The absorption coefficient is:

(˜i , œi ) =

D

¢a ¢a Qa (˜i , œi ) = = . ¢i Ii · A · cos ˜i A · cos ˜i

139

(9)

Qa can be expressed with the far field angular distribution of ps (s) with the help of the extinction theorem: e−j k0 s . (10) ps (s) −−−−→ Pi · ¥s ˜i , œi |˜ , œ · The far field ps can be separated s→∞ s into an angular and a radial function: 4 (11) The extinction theorem: Qe = − · Im{¥s ˜i , œi |˜r , œr } , k0 (with ˜r , œr in the direction of the mirror-reflected wave,i.e.in our case ˜r = ˜i , œr = œi ). Finally, with Qa = Qe − Qs : 4 · Im{¥s ˜i , œi |˜r , œr } Qa = − k0

2 0

/2 dœ |¥s ˜i , œi |˜ , œ |2 · sin ˜ d˜ .

(12)

0

Thus one needs the angular distribution of the scattered far field. Example 1 The absorber area has the normalised admittance G, which possibly is a function of surface co-ordinates, G(x0, y0); then G is its average over A. The baffle wall has the constant normalised admittance F0 . An approximation to the angular far field distribution of the scattered field is −j k0 cos ˜ · cos ˜i (G − F0) e−j (‹x x0 +‹y y0 ) dx0 dy0 , ¥s = 2(cos ˜ + F0 ) (cos ˜i + G ) A (13) ‹x = k0 sin ˜i · cos œi − sin ˜ · cos œ , ‹y = k0 sin ˜i · sin œi − sin ˜ · sin œ . Because G − F0 = 0 outside A, the integral can be extended over the whole plane z = 0; then it just represents the two-dimensional Fourier integral of the admittance difference. In the special case F0 = 0, i.e. a hard baffle wall: −j k0 cos ˜i G e−j (‹x x0 +‹y y0 ) dx0 dy0 . ¥s = 2(cos ˜i + G )

(14)

A

Example 2 The absorber surface A = a · b is a rectangle, centred at the origin, with side length a in the x direction, side length b in the y direction, and G = const. The Fourier transform gives ¥s =

−j k0 ab (G − F0 ) cos ˜i · cos ˜ sin(a‹x /2) sin(b ‹y /2) , 2(cos ˜ + F0)(cos ˜i + G) a ‹x /2 b ‹y /2

(15)

D

140

Reflection of Sound

and for F0 = 0 ¥s =

−j k0 ab · G cos ˜i sin(a ‹x /2) sin(b ‹y /2) . 2(cos ˜i + G) a ‹x /2 b ‹y /2

(16)

Example 3 A circular absorber with radius a, centred at the origin, with a constant normalised admittance G [with J1 (z) the Bessel function of the first order]: ¥s =

−j k0 a2 (G − F0) cos ˜i · cos ˜ 2 J1 (‚ k0a) , 2(cos ˜ + F0 )(cos ˜i + G) ‚k0 a 2

(‚k0 ) =

‹x2

+ ‹y2

(17)

.

The diagram shows a directivity diagram of ¥sn over œ, ˜ of a square with k0a = 8; G = 1; œi = ˜i = 45◦ . Contour lines of ¥sn are displayed; the thick lines separate ranges with different signs of ¥sn .

ϑ

Φ

ϕ

The method of this section can also be applied for diffuse sound incidence. The scattered sound field for diffuse incidence is e−j k0 s ps,dif (s, ˜ , œ) = Pi s

2 0

/2 d œi ¥s (˜i , œi |˜ , œ) · sin ˜i d˜i . 0

(18)

Reflection of Sound

D

141

D.8 Uneven, Local Absorber Surface This section uses the method described in the previous > Sect. D.7. The “unevenness” may be modelled either with a variation of the normalised absorber admittance G(x,y) or with a variation of the co-ordinates of the surface. The first method can be applied to grooves and narrow valleys, for example; the second method is applicable for slow or random variations. If the absorber surface can be represented by a reference plane with a variable admittance G(x, y), then the admittance first is described by its Fourier series. The following example assumes in A a one-dimensional variation of the admittance F0 of the surrounding baffle wall G(x) = F0 + B · cos(2x/)

(1)

side length a in the x direction

G = F0 + B · si ( a/),

(2)

of a rectangular absorber with

A

= a · b,

(3)

with the function

si (z) = sin(z)/z .

(4)

in the form of a cosine modulation with the period length , and the average admittance over the absorber

The angular far field function of the scattered field is: ¥s =

−j k0 ab · B · cos ˜i cos ˜ 4(cos ˜ + F0 ) (cos ˜i + G ) · si a ‹x /2 − a/ + si a ‹x /2 + a/ · si b‹y /2 ,

‹x = k0 sin ˜i · cos œi − sin ˜ · cos œ , ‹y = k0 sin ˜i · sin œi − sin ˜ · sin œ .

(5)

(6)

Next, the geometrical profile of the absorber surface can be represented by z = …(x, y), and the normalised admittance G(x,y) is given at this surface. The co-ordinate s0 of the absorber surface in section D.7 now has a non-zero z component s0 = (x0 , y0 , …(x0 , y0)). The derivative normal to the surface becomes ∂ ∂ ∂… ∂ ∂… ∂ =− + · + · . ∂n0 ∂z0 ∂x0 ∂x0 ∂y0 ∂y0 If the variation of height is smaller than about half a wavelength, the angular far field distribution of the scattered field is ¥s =

−j cos ˜ cos ˜i 2(cos ˜ + F0 )(cos ˜i + G ) ∂… ∂… k0 (G − F0 ) + ‹x · e−j (‹x x0 +‹y y0 ) dx0 dy0 . + ‹y ∂x0 ∂y0 A

(7)

142

D

Reflection of Sound

In a special case (often encountered in reverberant room measurements) the profile …(x, y) is a constant height h of A over the surrounding baffle wall, with ∂… = h · [ƒ(x0 + a/2) − ƒ(x0 − a/2)] ∂x0 ∂… = h · ƒ(y0 + b/2) − ƒ(y0 − b/2) ∂y0

for

− b/2 < y0 < b/2 ,

for

− a/2 < x0 < a/2 ,

(8)

with the Dirac delta function ƒ(z). The contribution of the height step h to the far field angular distribution of the scattered field is ¥s… =

−j k0 ab · k0h · cos ˜i cos ˜ 2 ‚ · si a‹x /2 si b‹y /2 , (cos ˜ + F0 ) (cos ˜i + G )

(9)

with (‚k0)2 = ‹x2 + ‹y2 . The ratio of the contribution ¥s… to the contribution ¥sG which describes the difference of the absorber admittance G from the baffle wall admittance F0 is ¥s… k0 h 2 =2 ‚ . ¥sG G − F0

(10)

Next, the normalised absorber admittance G(s0 ) and/or the absorber surface contour …(s0 ) has random variations with correlation distances dG , d… , respectively, and correlation functions 2

KG (d) = (G − G )2 A · e−(d/dG ) /2 , 2

−(d/d… )2 /2

K… (d) = … A · e

(11)

.

With Gt (k) and …t (k) the Fourier transforms of G(s0 ) − G and …(s0 ), respectively, and using the relation between the far field effective intensity Is and angular distribution ¥s of the scattered field Is = |ps |2 /(2Z0) = Ii · |¥s |2 /s2, one gets for the far field contribution of the variations in G and/or … to the effective intensity: 2 cos ˜i cos ˜ A 2 Is,G,… = 4 · Ii · 2 s (cos ˜ + F0 ) (cos ˜i + G ) · k02 · |Gt (k0‚)|2 + k04 ‚ 4 · |…t (k0‚)|2 (12) 2 cos ˜i cos ˜ A = Ii · 2s2 (cos ˜ + F0 ) (cos ˜i + G ) 2 2 · (k0 dG )2 · (G − G )2 A · e−(k0 ‚ dG ) /2 + k04 ‚ 4 d…2 · …2 A · e−(k0 ‚ d… ) /2 .

D.9 Scattering at the Border of an Absorbent Half-Plane A hard half-plane and a locally reacting absorbent half-plane with the normalised surface admittance G have the y axis as common border line.A plane wave pi is incident from the

Reflection of Sound

D

143

side of the hard half-plane under the polar angle ˜i and azimuthal angle œi (measured in the x,y plane relative to the x axis).

pi

p rh Θi

z

ϑ

Θi

d s P

pr

x The problem becomes a two-dimensional one (in the x,z plane) by the substitutions k02 → k 2 = k02 1 − sin2 ˜i · sin2 œi ; kx = k · sin Ÿi ; kz = k · cos Ÿi ; sin Ÿi =

sin ˜i · cos œi

;

1 − sin2 ˜i · sin2 œi

cos Ÿi =

cos ˜i 1 − sin2 ˜i · sin2 œi

(1)

;

and a common factor e−j ky y to each field quantity. The sound field is composed of p(x, z) = pi (x, z) + prh (x, z) + ps (x, z) with • • •

pi = incident plane wave, prh = reflected wave with “hard reflection”, ps = scattered wave.

Combining pi + prh , the sound field is −jkx·sin Ÿi

p(x, z) = 2 Pi e

∞ · cos(kz · cos Ÿi ) − jkG

p(x0 , 0) · G(x, y|x0 , y0)dx0 ,

(2)

0

with the Green’s function j (2) G(x, y|x0 , y0) = − H(2) 0 (kR) + H0 (kR ) , 4 R2 = (x − x0 )2 + (z − z0 )2

;

(3)

R2 = (x − x0 )2 + (z + z0 )2 ,

containing Hankel functions of the second kind H(2) 0 (z). The far field of the sound pressure components prh (s) + ps (s) is prh+s (s) −−−−−→ ks→∞

⎧ ⎪ ⎪ ⎨ 1−

G (1 − C(u) − S(u)) + j (C(u) − S(u)) cos Ÿi + G Pi G cos Ÿi − G ⎪ ⎪ ⎩ + (1 − C(u) − S(u)) + j (C(u) − S(u)) cos Ÿi + G cos Ÿi + G

;

d0

, (4)

D

144

Reflection of Sound

with u from u2 =

1 d 1 1 kd = kd · sin(Ÿ − Ÿi ) = ks · sin2 (Ÿ − Ÿi ) 2 s 2 2

(5)

and C(u), S(u) the Fresnel’s integrals: C(u) =

2

u

2

cos(t )dt

;

S(u) =

0

2

u

sin(t2 )dt .

(6)

0

The sound pressure in the surface at z = 0 is

G 2 − U(−Ÿi , kx) + jV(−Ÿi , kx) , p(x, 0) = Pi e−jkx·sin Ÿi 2 − G + cos Ÿi

(7)

with the functions U(Ÿ, u), V(Ÿ, u) defined as the real and imaginary parts of u U(Ÿ, u) − jV(Ÿ, u) = 1 − cos Ÿ

J0 (w) − jY0 (w)

0 · cos(w · sin Ÿ) − j sin(w · sin Ÿ) dw ,

(8)

(J0 (z) and Y0 (z) are Bessel and Neumann functions, respectively). The evaluation of these integrals is described in Mechel, Vol. I, Ch. 8 (1989).

D.10 Absorbent Strip in a Hard Baffle Wall, with Far Field Distribution A locally reacting strip with normalised admittance G and width a, axial direction along the y axis, is placed in the x,y plane at (−a/2, +a/2). A plane sound wave pi is incident from the −x direction under the polar angle Ÿi .

p i Θi

z ϑ

x o·sin ϑ –a/2 See

> Sect. D.9

x0

P(x,z)

s R

x +a/2

for a possible component ky of the wave vector in the y direction.

The sound field is composed of p(x, z) = pi (x, z) + prh (x, z) + ps (x, z) with • • •

pi = incident plane wave, prh = reflected wave with “hard reflection”, ps = scattered wave.

Reflection of Sound

Combining pi + prh , the sound field is kG · cos(kz · cos Ÿi ) − 2

−jkx·sin Ÿi

p(x, z) = 2 Pi e

with R2 = (x − x0 )2 + z2 . In the far field,

∞

D

p(x0 , 0) · H(2) 0 (kR)dx0 ,

145

(1)

0

j e−jks √ · Vz (k sin ˜ ) , (2) 2k s where Vz (k sin ˜ ) is the Fourier transform of the particle velocity distribution (in the z direction) at the plane z = 0. From the equivalent form of the scattered field ps in the far field, e−jks j · ¥s (Ÿi |˜ ) · √ , ps (s) = −Pi (3) 2k s −jkx·sin Ÿi

p(x, z) = 2Pi e

· cos(kz · cos Ÿi ) +

follows +ka/2

¥s (Ÿi |˜ ) = −ka/2

p(x0 , 0) +jkx0 ·sin ˜ G· ·e d(kx0) = Pi

+∞ −∞

vz (x0 , 0) +jkx0 ·sin ˜ ·e d(kx0 ) , vi

(4)

and the absorption cross section Qa of the strip 2 1 Qa = Re{¥s (Ÿi |Ÿi )} − k 2k

+/2

|¥s (Ÿi |˜ )|2 d˜ .

(5)

−/2

The needed sound pressure distribution p(x0 , 0) has different possible approximations. For small ka and low values of G, p(x0 , 0) ≈ 2pi (x0 , 0) = 2Pi e−jkx0 ·sin Ÿi leading to ¥s (Ÿi |˜ ),

¥s (Ÿi |˜ ) = 2kaG · si (ka(sin ˜ − sin Ÿi )/2) , (6)

with si(z) = sin(z)/z,

and to Qa

Qa 2 = 4Re{G} − ka · |G|2 a

+/2

si 2 (ka(sin ˜ − sin Ÿi )/2) d˜

−/2

(7)

−−−−→ 4Re{G} − 2ka · |G|2 → 4Re{G} ka 1

Approximations for large ka

p(x0 , 0) ≈ Pi (1 + r) · e−jkx0 ·sin Ÿi = 2Pi

and the corresponding ¥s (Ÿi |˜ ) are:

¥s (Ÿi |˜ ) = 2

cos Ÿi · e−jkx0 ·sin Ÿi G + cos Ÿi

G cos Ÿi ka G + cos Ÿi

·si (ka(sin ˜ − sin Ÿi )/2) .

(8)

146

D

Reflection of Sound

The resulting Qa is +/2 Qa G cos Ÿi 2ka G cos Ÿi 2 = 4Re si2 (ka(sin ˜ − sin Ÿi )/2) d˜ . − a G + cos Ÿi G + cos Ÿi

(9)

−/2

D.11 Absorbent Strip in a Hard Baffle Wall, as a Variational Problem The geometry and field composition are as in

> Sect. D.10.

The variational principle is based on Helmholtz’s theorem of superposition, which requires that the power of the“cross intensity”p(x0 , 0)· vza(x0 , 0) of the desired field p(x, z) and the particle velocity vza (x, z) of the adjoint field be zero at the plane z = 0. The adjoint field is the solution for exchanged emission and immission points. The cross power is minimised by variation of the amplitude P of the estimate P · exp(−jkx · sin Ÿi ). The expression to be minimised is 1 4Pi ka G · P − kaG · P − k2 G2 P2 2 2

a/2 ·

+a/2

ejkx·sin Ÿi dx

−a/2

H(2) 0 (k|x

−j kx0 ·sin Ÿi

− x0 |) · e

(1) dx0 = Min(P) .

−a/2

The partial derivative ∂/∂P is zero for ⎫ ⎧ ⎛ ⎞ +a/2 ⎪ ⎪ a/2 ⎬ ⎨ kG ⎜ ⎟ −j kx0 ·sin Ÿi ej kx·sin Ÿi ⎝ H(2) (k |x − x |) · e dx dx . 1+ P = 2Pi 0 0⎠ 0 ⎪ ⎪ 2a ⎭ ⎩ −a/2

(2)

−a/2

Definition of auxiliary functions: 'u Z(Ÿ, u) = cosŸ J0 (|w|) − jY0(|w|) · e−jw·sin Ÿ dw 0 ⎧ ⎪ −1 + U(Ÿ, u) − jV(Ÿ, u) ; u < 0 ⎨ = , ⎪ ⎩ 1 − U(Ÿ, u) + jV(Ÿ, u) ; u < 0 1 W(Ÿ, ka) = ka

(3)

ka Z(Ÿ, u)du 0

[the functions U(Ÿ, u), V(Ÿ, u) are defined in

> Sect. D.9].

The amplitude factor P becomes P=

2Pi · cos Ÿi , cos Ÿi + G/2 · (W(Ÿi , ka) + W(−Ÿi , ka))

(4)

Reflection of Sound

D

and the sound pressure at z = 0 is

G Z(Ÿi , ka/2 − kx) + Z(−Ÿi , ka/2 + kx) p(x, 0) = 2Pi 1 − · e−jkx·sin Ÿi 2 cos Ÿi + G/2 · (W(Ÿi , ka) + W(−Ÿi , ka)) ⎧ −jkx·sin Ÿi ; |x| a ⎪ ⎨ 2Pi · e → − cos Ÿi ⎪ ⎩ 2Pi · e−j kx·sin Ÿi ; |kx| |ka| 1 . G + cos Ÿi

147

(5)

The angular far field distribution of the scattered field is ¥s (Ÿi |˜ ) =

cos Ÿi +

G 2

2kaG cos Ÿi · si (ka (sin Ÿ − sin Ÿi )/2) , (W(Ÿi , ka) + W(−Ÿi , ka))

(6)

with si(z) = sin(z)/z. The absorption cross section Qa of the strip follows with this from the previous Sections. Numerical examples for sound pressure distributions in the plane z = 0: (equivalencies for the parameters in the plot labels: “Theta” ∼ Ÿi ; “F” ∼ G; “k0a” ∼ k0a). The curve dashes become shorter for later entries in the parameter lists {. . .}. Sound incidence is normal to the strip axis (ky = 0). Θ

Sound pressure distributions for different k0a

D

148

Reflection of Sound

Θ

k0 a = 8 F = 1 + 0.5 · j

Sound pressure distributions for different angles of incidence Ÿi

2

k0 a = 8 Ÿi = 60◦

F = 0.5 + 1j

1.5

1

F = 0.5 – 1j 0.5 -10

-5

0 k0x

5

10

15

Sound pressure distributions for two normalised admittances F G

D.12 Absorbent Strip in a Hard Baffle Wall, with Mathieu Functions

See also: Mechel (1997), for notations and relations of Mathieu functions.

The sound field around a locally reacting strip with normalised admittance G in a hard baffle wall can be formulated as a boundary value problem with exact solutions

Reflection of Sound

D

149

in elliptic-hyperbolic cylinder co-ordinates (, ˜ ). The co-ordinate curves are confocal ellipses and orthogonal confocal hyperbolic branches. The radial and azimuthal eigenfunctions in these co-ordinates are Mathieu functions. Transformation between Cartesian

x = x1 = c · cosh · cos ˜ ,

and elliptic-hyperbolic co-ordinates:

y = x2 = c · sinh · sin ˜ ,

The common foci are at x = ±c.

z = x3 = z .

(1)

The boundary surface of the absorbent strip is at = 0, the focus distance is c = a/2, and the boundaries of the baffle wall are at ˜ = 0 and ˜ = . The Helmholtz differential equation (wave equation) ( + k02) u = 0 in elliptic-hyperbolic co-ordinates is: 2 ∂ 2u ∂ 2u 2 2 2∂ u 2 + + cosh − cos ˜ · c + (k c) u =0. (2) 0 ∂2 ∂˜ 2 ∂z2 4 3

ρ=

2.0

2

0.7π

0.8π 1 y/c 0 -1

0.6π

ϑ= π/2

0.4π

0.2π

1.5

0.9π

1.0

π-0.1 ϑ=±π -π+0.1

0.1π 0.5

0.1 ϑ=0 -0.1

ρ=0

-0.1 π

-0.9 π

-0.2 π

-0.8 π

-2

-0.3 π

-0.7 π -0.6 π

-3 -4 -4

0.3π

-3

-2

-1

ϑ=- π/2

-0.4 π

0

1

x/c

2

3

Co-ordinate lines in elliptic-hyperbolic cylinder co-ordinates

4

150

D

Reflection of Sound

d2 R() − Š − 2q · cosh(2) · R() = 0 , 2 d

For separated field functions u(, ˜ ) = T(˜ ) · R(). This is equivalent to the pair of Mathieu differential equations:

d2 T(˜ ) + Š − 2q · cos(2˜ ) · T(˜ ) = 0 . d˜ 2

(3)

The parameter q is determined by q = (k0c)2 /4. The parameter Š stands for characteristic values of the Mathieu functions, for which the Mathieu differential equations have finite and periodic (in ˜ ) solutions; they will be called Š = acm for cos-like (symmetrical in ˜ ) azimuthal Mathieu functions T(˜ ) = cem (˜ , q) and Š = bcm for sin-like (antisymmetrical in ˜ ) azimuthal Mathieu functions T(˜ ) = sem (˜ , q).The azimuthal functions are associated,respectively,with radial Mathieu functions of the Bessel type, Jcm (), Jsm (), of the Neumann type, Ycm (), Ysm (), and of the Hankel-type for outward propagating waves Hc(2) m () = Jcm () − j · Ycm ()

or

Hs(2) m () = Jsm () − j · Ysm (). A plane wave incident at an angle Ÿ against the major axis of the ellipses, i.e. against the plane of the strip and the baffle wall, is in Cartesian co-ordinates pi (x, y) = e−jk0 (x cos Ÿ+y sin Ÿ) = e−2jw

√

q

= pi (, ˜ ) ,

w = cos cos ˜ cos Ÿ + sin sin ˜ sin Ÿ .

(4)

Its expansion in Mathieu functions is pi (, ˜ ) = 2

∞ (

(−j)m cem (Ÿ; q) · Jcm (; q) · cem (˜ ; q)

m=0 ∞ )

+2

(5) (−j)m sem (Ÿ; q) · Jsm (; q) · sem (˜ ; q) .

m=1

The sum of the incident wave pi and of the reflected wave pr with reflection at a hard plane containing the major axis of the ellipses (i.e. the baffle wall) is pi (, ˜ ) + pr (, ˜ ) = 4

∞ )

(−j)m cem (Ÿ; q) · Jcm (; q) · cem (˜ ; q) .

(6)

m=0

A scattered field ps (, ˜ ) is added to these field components; it is formulated as a sum of terms as in pi + pr , but with yet undetermined term amplitudes am , and the Mathieu-Bessel functions Jcm (; q) (which represent radial standing waves) replaced with Mathieu-Hankel functions Hc(2) m (; q); so it satisfies the boundary condition at the baffle wall. Thus: p(, ˜ ) = pi (, ˜ ) + pr (, ˜ ) + ps (, ˜ ) =4

∞ ( m=0

(−j)m cem (Ÿ; q) · cem (˜ ; q) · Jcm (; q) + am · Hc(2) m (; q) .

(7)

Reflection of Sound

D

151

The gradient in elliptic-hyperbolic co-ordinates is e e˜ ∂u ∂u ∂u grad u = + + ez · , ∂ ∂˜ ∂z c sinh2 + sin2 ˜ c sinh2 + sin2 ˜ and therefore the particle velocity v v in the direction of the hyperbolic -lines:

=

∂p/∂ j j . grad p = k0 Z0 k0 cZ0 sinh2 + sin2 ˜

−−−→

This is used for the boundary condition at the strip: Z0 v (0, ˜ ) = −G · p(0, ˜ ).

(8)

=0

j ∂p/∂ , k0c Z0 | sin ˜ |

−−−−−−→ ˜ =0 ˜ =

(9)

j ∂p/∂ . k0 cZ0 sinh

The boundary condition gives (a prime at the Mathieu functions indicates the derivative in ) ∞ )

(−j)m cem (Ÿ; q) · cem (˜ ; q) · Jcm (0; q) + am · Hc(2) m (0; q)

m=0

= jk0 cG · | sin ˜ |

∞ )

m

(−j) cem (Ÿ; q) · cem (˜ ; q) · Jcm (0; q) + am ·

Hc(2) m (0; q)

(10) .

m=0 The functions cem (˜ ; q) are orthog- cem (˜ ; q) · cen (˜ ; q)d˜ = ƒm,n · Nm . onal in ˜ over (0, ) with the norms Nm : 0

(11)

Application of the orthogonality in tegral on both sides of the bound- Tm,n = cem (˜ ; q) · cen (˜ ; q) · | sin ˜ |d˜ . ary condition gives, with the mode0 coupling coefficients:

(12)

the linear, inhomogeneous system of equations for the amplitudes am : ∞ ) m=0

=−

(2) am · (−j)m cem (Ÿ; q) · j k0c G · Tm,n · Hc(2) m (0; q) − ƒm,n Nn · Hcn (0; q) ∞ )

(−j) cem (Ÿ; q) · j k0 c G · Tm,n · Jcm (0; q) − ƒm,n Nn · Jcn (0; q) .

(13)

m

m=0

The mode norms are Nm = /2; the coupling coefficients Tm,n can be expressed in terms of the Fourier coefficients of the Fourier series representation of cem (˜ ) (see Mechel (1997), > Sect. 19.5). After the solution of this (truncated) system of equations, the sound field is known.

152

D

Reflection of Sound

Numerical examples: The horizontal dashed lines in the plots below are the squared sound pressure magnitudes at an absorber of infinite extend. Θ

Distribution of sound pressure magnitude squared at the surface of an absorbent strip with mass-type reactance Θ

Distribution of sound pressure magnitude squared at the surface of an absorbent strip with spring-type reactance The acoustic corner effect: Due to scattering at the borders of a finite-size absorber, its absorption in general is different from the absorption of an infinite, but otherwise equal, absorber. The quantitative corner effect is defined as the ratio of the effective power ¢ absorbed by the strip

Reflection of Sound

˜ k0 a mhi

= = =

D

153

45◦ 12 6

The corner effect may be positive or negative, depending on the sign of the reactance to the power ¢∞ absorbed by an area of the same size of an infinite, but otherwise equal, absorber: 1 ¢ G 2 = CE(˜ ) = · 1 + ¢∞ 4 k0a · Re {G} sin ˜ (14) ) * + m ∗ 2 · 4cem (˜ ; q) · Re (−j) am + | am | . m0

D.13 Absorption of Finite-Size Absorbers, as a Problem of Radiation


The surface impedance ZA of an infinite absorber is generally easily evaluated. The problem with finite-size absorbers in a baffle wall is the influence of border scattering. This influence can be taken into account by a simple equivalent network. |Pi |2 4 Re{ZA } . 2 Z0 |ZA + Zs | 2

The absorbed effective power is:

¢a

=

A·

The normalised absorption cross section is:

Qa A

=

4 Re{ZA } . |ZA + Zs |2

(1) (2)

D

154

A Pi ZA Zs

= = = =

Reflection of Sound

area of the absorber; amplitude of the incident plane wave; surface impedance of the infinite absorber; radiation impedance of a radiator with the size and shape of the absorber, when its surface oscillation pattern agrees with that of the exciting wave at the absorber surface

This can be represented in an equivalent network: The pressure source has an amplitude 2Pi ; The radiation impedance Zs is the internal source impedance; The impedance ZA of the infinite absorber is the load impedance; The power in ZA is the absorbed power ¢a .

• • • •

2P i

~

Zs

ZA

Thus the determination of the absorption by finite-size absorbers is reduced to the determination of their radiation impedance.

D.14 A Monopole Line Source Above an Infinite, Plane Absorber; Integration Method

See also: Mechel, A line source above a plane absorber (2000)

A monopole line source placed at Q is parallel to the absorber, with a normalised surface admittance G. S is the mirror-reflected point to Q. P is a field point.

Q ξ η h

θs F

h

r∙cos ϑ ϑ r

x

y S

Rp r

'

θs

r∙sinϑ P

G

2h-r∙sinϑ =r ∙cos θs '

r'∙s i n θs The absorber may be locally or bulk reacting; it will be mentioned if results are valid for locally reacting absorbers only. In what follows, k is the wave vector component in the plane containing Q, S, P. The field is set up as p = pQ + pr ; pQ = source free field; pr = reflected field.

Reflection of Sound

D

pQ (r) = H(2) 0 (kr).

Source free field (with unit amplitude)

155

(1)

Field of plane wave incident under polar angle ‡ [with reflection factor R(‡) for this angle of incidence, and Ÿ = ‡ + ˜ ]: pe + per = e−j kr·sin Ÿ + R(Ÿ − ˜ ) · e−2j kh·cos(Ÿ−˜ ) · e−jkr·sin(Ÿ−2˜ ) . After application of the integral operation: 1 pe + per dŸ

(2)

(3)

C(Ÿ)

the first term is the integral representation of the Hankel function 1 H(2) (kr) = e−j kr·sin Ÿ dŸ ; Re{kr} > 0 . 0

(4)

C(Ÿ)

whith path C(Ÿ): −j∞ → 0 → → + j∞. Thus the second term yields 1 R(Ÿ − ˜ ) · e−j kr·(sin(Ÿ−2˜ )+2h/r·cos(Ÿ−˜ )) dŸ, pr =

(5)

C(Ÿ)

and after a horizontal shift of the path, with œ = ‡ − ‡s (see sketch for ‡s and r ): 1 cos(œ + ‡s ) − G . R(œ + ‡s ) · e−j kr ·cos œ dœ ; R(œ + ‡s ) = pr = cos(œ + ‡s ) + G

(6)

C(œ)

This is an exact representation for pr ; the path C(œ) is shown in the diagram, together with the shaded range, where poles of R(œ + ‡s ) are possible (only for Im{G} > 0), and the “path of steepest descent” (pass way) Pw. If during the deformation C(œ) → Pw a pole is crossed, a “pole contribution” must be added to the integral of steepest descent; it has the form of a surface wave.

Im (ϕ) Pole range with Im{G} >0 -π/2 C (ϕ) Pw

π/2

R e (ϕ)

D

156

Reflection of Sound

For direct numerical integration, use 1 pr = +

(−/2)

(−/2)

2j

∞ 0

cos(œ + ‡s ) − G −j kr ·cos œ ·e dœ cos(œ + ‡s ) + G (7) 2

1 + G2 − cos2 ‡s + sinh œ · e−kr ·sinh œ dœ . 1 − G2 − cos2 ‡s + sinh2 œ + 2jG cos ‡s · sinh œ

The first integrand oscillates strongly for kr 1. Therefore use the method of integration along the steepest descent (also “saddle point integration” or “pass integration”). Some cases must be distinguished. Saddle point at œs = 0; on the pass way is œ(s) = ± arccos(1 − j · s2 ); s ≷ 0, s being a running parameter on the pass way from −∞ to +∞; the saddle point is at s = 0; the slope of the pass way in the saddle point is dœ(0)/ds = 1 + j. No pole crossing, and no pole near the saddle point: pr =

1 · 3 · 5 · . . . (2n − 1) (2n) 1 1 −j kr e ¥ (0) + . . . + ¥ (0) , ¥ (0) + kr 4 kr (2n)!(2 kr )n

(8)

(primes at ¥ indicate derivatives with respect to s), with: 2j cos(œ(s) + ‡s ) − G 2j ¥ (s) = R(œ(s) + ‡s ) · = . · 2 cos(œ(s) + ‡s ) + G 2j+ s 2 j + s2

(9)

With some derivatives performed [leaving R(n) (‡s ) unevaluated] we have

pr =

2j −j kr j 9 75 j 3675 e · 1 + − − + · R(‡s ) kr 8 kr 128(kr)2 1024(kr)3 32768(kr)4 (2) R (‡s ) j 5 259 j 3229 · + − − + 2 16 kr 768(kr )2 6144(kr )3 kr (4) 35 j 329 1 R (‡s ) + − − · 2 8 192 kr 1024(kr ) (kr )2 (6)

7 R (‡s ) 1 R(8) (‡s ) j − · · . − + 48 128 kr (kr )3 384 (kr)3

(10)

This form is valid for both locally and bulk reacting absorbers. The first terms in the parentheses give the geometrical acoustic approximation (or mirror source approximation): pr −−−−−→ R(‡s ) · H(2) 0 (kr ). kr 1

(11)

Reflection of Sound

D

157

For locally reacting absorbers the derivatives R(n) (‡s ) can be evaluated in advance (for bulk reacting absorbers they depend on the internal structure of the absorber); G = normalised admittance: cos ‡s − G R(‡s ) = , cos ‡s + G R(2) (‡s ) = −2G

2 − cos2 ‡s + G · cos ‡s , (cos ‡s + G)3

R(4) (‡s ) = 2G

(G − 5 cos ‡s )(cos ‡s + G)2 cos ‡s + 4(cos ‡s + G)(2G − 7 cos ‡s ) sin2 ‡s − 24 sin4 ‡s , (cos ‡s + G)5

R(6) (‡s ) = 2 G

(cos ‡s + G)3(28 G cos ‡s − G2 − 61 cos2 ‡s ) cos ‡s + . . . (cos ‡s + G)7

(12)

. . . + 2(cos ‡s + G)2(193 G cos ‡s − 16 G2 − 331 cos2 ‡s ) sin ‡s + . . . ... . . . + 120(cos ‡s + G)(4 G − 11 cos ‡s ) sin4 ‡s − 720 sin6 ‡s , ... R(8) (‡s ) = 2 G

(cos ‡s + G)4(G3 − 123 G2 cos ‡s + 1011 G cos2 ‡s − 1385 cos3 ‡s ) × . . . (cos ‡s + G)9

. . . × cos ‡s + 8(cos ‡s + G)3 (16 G3 − 519 G2 cos ‡s + 2694 G cos2 ‡s − . . . ... . . . − 3071 cos3 ‡s ) sin2 ‡s + 1008(cos ‡s + G)2 (−8 G2 + 59 G cos ‡s − . . . ... . . . − 83 cos2 ‡s ) sin4 ‡s + 20160(cos ‡s + G)(2 G − 5 cos ‡s ) sin6 ‡s − . . . ... . . . − 40320 sin8 ‡s . ... Special case ‡s = 0, i.e. field point P on line through S and Q: j 9 75 j 3675 2 j −j kr e · 1+ − − + pr (r , 0) = kr 8 kr 128 (kr )2 1024 (kr )3 32768 (kr)4 259 j 3229 2G 5 1−G j − + · − − × 2 3 1+G 2 16 kr 768 (kr ) 6144 (kr ) kr (1 + G)2 35 j 7 2 G (5 − G) 329 1 j + − · + − + 8 192 kr 1024 (kr )2 (kr )2 (1 + G)3 48 128 kr

1 2 G(1385 − 1011 G + 123 G2 − G3) 2 G(61 − 28 G + G2 ) · . − × (kr )3 (1 + G)4 384 (kr )4 (1 + G)5

(13)

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D

Reflection of Sound

Special case ‡s = /2, i.e. P and Q are on the absorber: 2 j −j kr pr (r , /2) = e kr j 9 75 j 3675 · − 1+ − − + 8 kr 128 (kr )2 1024 (kr )3 32768 (kr)4 4 5 259 j 3229 j · 2+ − − − + (14) 2 3 2 16 kr 768 (kr ) 6144 (kr ) kr G 48 − 16 G2 35 j 7 329 j 1 · + − − + + 8 192 kr 1024 (kr)2 (kr )2G4 48 128 kr , 2(720 − 480 G2 + 32 G4 ) 1 2 40320(1 − G2 ) + 8064 G4 − 128 G6 × . · − (kr )3 G6 384 (kr)4 G8 A pole is crossed which is not near the saddle point: The pole is circumvented by an indentation of Pw; the first-order pole of R gives a “pole contribution”, which must be added to the above result. Im (ϕ)

pole

-π/2 C (ϕ)

R e (ϕ) π/2

Pw

The pole contribution is 1 prp (r , ‡s ) = R(œ + ‡s ) · e−j kr ·cos œ dœ œp = 2j · Res R(œ + ‡s ) · e−j kr ·cos œ

œ=œp

(15) ,

where œp is the position of the pole in the complex œ and Res(f (z)) is the residue of f (z) at a pole of f (z). For locally reacting absorbers 4j G prp (r , ‡s ) = √ · e−j kr ·cos œp 2 1−G √ 4j G 2 =√ · e+j kG·r ·cos ‡s · e−j k 1−G ·r ·sin ‡s 2 1−G √ 4j G 2 =√ · e+j kG·h · e+j kG·|y| · e−j k 1−G ·x . 2 1−G

(16)

Reflection of Sound

D

159

Condition for pole crossing (if ≤ holds): G ≤

1 (G + cos ‡s )(G · cos ‡s + 1) 1 (G + cos ‡s )(G · cos ‡s + 1) = , √ sin ‡s sin ‡s 1 + 2G cos ‡s + G2 (G + cos ‡s )2 + sin2 ‡s

−−−−−−→ ≤ √

1 + G2

‡s →/2

−−−−→ ≤ ‡s →0

G

(17)

−−− −→ 0 , G →0

1 → ∞. sin ‡s

10 θs=

G'' 8

10° 15° 20°

6

30° 40°

4 2 0

50°

60° 75°

0

0.5

1

1.5

2

2.5

G'

90° 3

The diagram shows limits for pole crossing in the complex plane of G = G + j · G with ‡s as parameter for a locally reacting absorber. Pole contributions are below the curves (their magnitude, however, may make them negligible) A uniform pass integration: It can be used also if the pole is near the saddle point; however,the simple pass integration above is preferable if the pole is not near the saddle point. Let the integral to be computed along the pass way Pw be of the following form, with real x > 1: I = ex·f (œ) · F(œ)dœ . (18) Pw

It can be evaluated by

√ x·f (œs ) ·T ; Im{b} ≷ 0 , I=e · ±ja · W ±b x + x

√ 2 Im{b} = ex·f (œs ) · ja · W b x + · T − ja · e−x·b ; Re{b} = x · T ; Im{b} = 0 , ex·f (œs ) · x

0 0

,

(19)

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D

Reflection of Sound

with the following definitions (œs = value of œ at saddle point): dD , a =: lim (œ − œp ) · F(œ) = N(œp ) œ→œp dœ œ=œp b =: f (œs ) − f (œp ) ‚ =: −2/f (œs ) T =: ‚ · F(œs ) +

;

;

œp − œs , œp →œs ‚ arg(‚) = arg(dœ) œs ; b −−−−−→

œ

along

Pw ,

(20)

a , b

2

W(u) =: e−u · erfc(−j u)

;

2 erfc(z) =: √

∞

2

e−y dy .

z

It is supposed that F(œ) = N(œ)/D(œ) can be written as the quotient of a numerator and denominator; thus a is the residue of F(œ). The quantity b distinguishes cases of the relative position œp of the pole to the pass way Pw or to the saddle point œs . For Im{b} > 0 the pole is still outside Pw; for Im{b} < 0 it has been crossed by C(œ) → Pw, and Im{b} = 0 describes the situation where œp is on Pw.The addendum in the definition of b defines the sign of the root in b. The addendum in the definition of ‚ also serves to select the sign of the root; it demands that the argument of ‚ should agree with the argument of a step dœ from the saddle point œs in the direction of the pass way. The function W(u) is based on the complementary error function erfc(z). The correspondences to the present integral along the pass way Pw as defined above are x ⇒ kr

;

f (œ) ⇒ −j cos œ

;

F(œ) ⇒ R(œ + ‡s ) =

cos(œ + ‡s ) − G , cos(œ + ‡s ) + G

(21)

and the required quantities are œs = 0

;

œp = arccos(−G) − ‡s ,

cos(œp + ‡s ) = −G , cos œp = −G · cos ‡s +

(22) √ 1 − G2 · sin ‡s

;

/ .√ Im 1 − G2 ≤ 0 .

Reflection of Sound

D

161

Other quantities in the above definitions are for a locally reacting absorber: cos(œp + ‡s ) − G 2G =√ , sin(œp + ‡s ) − G 1 − G2 1−j œs b = ± j cos œp − 1 −−−−−−−→ ∓ œp →œs =0 2 √ = ±(j)3/2 1 + G · cos ‡s − 1 − G2 · sin ‡s , ‚ = 2j , 2 jG 1 . T = 2 j · R(‡s ) ∓ √ 2 1 − G 1 + G · cos ‡ − √1 − G2 · sin ‡ s s a =−

(23)

The sign convention for ‚ is satisfied; the sign convention in b requires the lower signs in b and T if the last root in b, T is evaluated with a positive real part. The desired field pr = I/ can be evaluated by insertion. The diagrams below compare in 3D plots (as “wire graphics”) the magnitude of the sound pressure |p| over kx, ky from numerical integration of the exact integral (thick lines) with results from approximate methods (to which the pass integration belongs).

kh=0 G=1 + 0.5∙j 1 |p| 0.8 0.6 0.4 10

0.2

7.5

0 0

5

k|y|

2.5 2.5

5 7.5

kx

10 0

Comparison of numerical integration of exact integral (thick lines)with the mirror source approximation (thin lines)

D

162

Reflection of Sound

kh=0 G=1 + 0.5·j 1 |p| 0.8 0.6 0.4 10

0.2

7.5

0 0

5

k|y|

2.5 2.5

5 7.5

kx

10 0

Comparison of numerical integration of exact integral (thick lines) with the simple pass integration (thin lines)

D.15 A Monopole Line Source Above an Infinite, Plane Absorber; with Principle of Superposition See also: Mechel, Modified Mirror and Corner Sources with a Principle of Superposition (2000)

A monopole line source with volume flow q (per unit length) is placed at Q with a height h above a locally reacting plane with normalised surface admittance G. S is the mirror-reflected point to Q, and P is a field point.

θ ϑ ϑ

ϕ ψ θ

′ ϑ ″

θ

Reflection of Sound

D

163

The sound field is formulated as pa (r) = A · ph (r) + B · ps (r), with ph the field above a hard plane and ps the field above a soft plane, both satisfying individually the source condition. The principle of hard-soft superposition (third principle of superposition in B.10) gives pa (r) =

1 · ph (r) + G · X(s) · ps (r) , 1 + G · X(s)

>

Sect.

(1)

with s the projection of the field point P (along co-ordinate lines) on the plane, and the “cross impedance” X(s) =

ph (s) jk0 · ph (s) =− . Z0 vsn (s) gradn ps (s)

In the present task is ph = pQ + pSh

(2) ;

pw = pQ + pSw ,

with pQ the free source field: pQ = P0 · H(2) 0 (k0 r ) =

k0 Z0 · q pQ (k0 h) · H(2) · H(2) 0 (k0 r ) = 0 (k0 r ) , 4 H(2) (k h) 0 0

(3)

(the second form replaces the amplitude P0 by the source volume flow q; the third form describes the source strength by the free field sound pressure pQ (h) at the origin), and ph = pQ + pSh ; ps = pQ + pSs , where pSh , pSs are the fields from the mirror sources in the case of a hard or soft plane, respectively, which for “ideal” reflection are exact forms of the scattered field: pSh = P0 · H(2) 0 (k0 r );

pSs = −P0 · H(2) 0 (k0 r ) .

(4)

One gets for the cross impedance X(y) 0 1 k0 rq (k r ) 1 −Z0 vQsx H(2) 0 = · 1 + 2(2) = −j · X(y) pQh 2 H0 (k0r ) 0 1 (k0 y)2 + (k0 rq )2 H(2) k 0 rq 2 · 1 + (2) = −j · . 2 H0 (k0 y)2 + (k0 rq )2

(5)

With pSw = −pSh one can simplify to pa (x, y) = pQ (x, y) +

1 − G · X(y) · pSh (x, y) 1 + G · X(y)

(6)

1/X(y) − G · pSh (x, y) . = pQ (x, y) + 1/X(y) + G Numerical comparison with saddle point integration (see

> Sect. D.14):

164

D

Reflection of Sound

Sound pressure magnitude from a line source above an absorbing plane (on x axis), evaluated using the principle of superposition

This diagram compares the above diagram (in a 3D wire plot) with results from the method of saddle point integration

Reflection of Sound

D

165

D.16 A Monopole Point Source Above a Bulk Reacting Plane, Exact Forms


A monopole point source is placed at a point Q at height h above an absorbent plane; a field point is at P. The plane may be bulk reacting. See > Sect. D.17 for a locally reacting plane. See Mechel (1989) for references to the extensive literature about this problem.As an exception, the time factor in this section is e−i –t in order to facilitate the comparison with the literature, where this sign convention mostly is used. The free field of the point source is

pQ (r1 ) = P0

ei k0 r1 , k 0 r1

(1)

the field p above the absorber is

p ei k0 r1 pr = + , P0 k 0 r1 P0

(2)

with r1 = dist(Q, P) and pr the reflected field. The task is to find pr . An exact integral expression for pr is pr =i P0

/2−i∞

J0 (k0 r · sin Ÿ) · ei k0 (z+h)·cos Ÿ · R(Ÿ) · sin Ÿ dŸ ,

(3)

0

where R(Ÿ) = (cosŸ − G)/(cosŸ + G) is the reflection factor of a plane wave incident under a polar angle Ÿ, z is the co-ordinate normal to the plane directed into the halfspace above the plane, r is the radius of P from the foot point of Q on the plane, and J0 (z) is the Bessel function of zero order. The path of integration in the complex Ÿ plane is 0 → /2 → /2 − i · ∞ . If the absorber is a half-space (indicated with index ß = 2, in contrast to index ß = 1 for half-space above the absorber) of a homogeneous, isotropic material, the characteristic wave numbers and wave impedances in both half-spaces are kß , Zß , respectively, and the ratios k = k2/k1, Z = Z2 /Z1. An exact formulation (Sommerfeld) of the field in the upper half-space ß = 1 is √2 ∞ y · J0 (y k1r) · e−k1 z y −1 p1 (r, z) = (1 + kZ) dy . (4) P0 y 2 − k 2 + kZ y 2 − 1 0

This integral is used for numerical integration (as a reference for approximations), for which the interval of integration is subdivided into (0,∞) = (0,1)+(1,2)+(2,yhi)+(yhi ,∞), and the precision and convergence are checked separately in each subinterval. In the case of two half-spaces, the integral above for pr /p0 can be transformed into (a form, which is suited for saddle point integration) pr (r, ˜ ) i i k1 H·cos ˜ = H(1) · R(˜ ) · sin ˜ d˜ , (5) 0 (k1 r · sin ˜ ) · e P0 2 C

166

D

Reflection of Sound

with the reflection factor kZ cos ˜ − k 2 − sin2 ˜ R(˜ ) = . kZ cos ˜ + k 2 − sin2 ˜

(6)

This form can be applied also for bulk reacting layers of finite thickness if a corresponding reflection factor is used. The path of integration is C = −/2 + i · ∞

→ −/2 → +/2 → +/2 − i · ∞ .

The cross-over from the positive bank of Re{˜ } to the negative bank is at Re{˜ } = 0. Further exact forms (Butov) for the reflected field above a homogeneous half-space and the field in the lower half-space are i pr = P0 2 k1 p2 i = P0 2 k1

∞ ∞ −∞ −∞ ∞ ∞

−∞ −∞

k2z − kZ · k1z i (kx x+ky y) ei k1z |z+h| ·e dkx dky ; k2z + kZ · k1z k1z 2kZ · ei (kx x+ky y) ei k1z h e−i k2z z dkx dky ; k2z + kZ · k1z

(7)

with wave number components kßx, kß y, kß z of kß . The first line can be transformed into ∞ ) pr =i (−1)n (4n + 1) · V2n · h(1) 2n (k1 r2 ) · P2n (cos ˜ ) , P0 n=0

(8)

with r2 = dist (mirror point of Q, P); P2n (z) = Legendre polynomial; h(1) 2n (z) = spherical Hankel function of the first kind; and V2n =

1 2

1 V(x) · P2n (x) dx −1

;

V(x) =

√ k 2 − 1 + x2 . √ kz · x + k 2 − 1 + x2

kz · x −

(9)

Although this form is elegant, it is not suited for numerical evaluations because of problems of convergence caused by the spherical Hankel functions. Another exact form for p1 (Brekhovskikh) above a homogeneous absorber half-space is

Q h h

r1 r2

Θ0 Q′

r

P z

Reflection of Sound

p1 (r, z; h) ei k1 r1 ei k1 r2 p1 (r, z + h; 0) = − + , P0 k1r1 k1 r 2 P0 √2 ∞ y · e−k1 H y −1 p1 (r, z + h; 0) = 2 (1 + kZ) J0 (y · k1 r) dy , P0 y 2 − k 2 + kZ y 2 − 1

D

167

(10)

0

or with ‚ = y · k1 : p1 (r, z + h; 0) 2 (1 + kZ) = P0 k1

∞ 0

‚ · e−H

√

‚ 2 −k12

J0 (‚r) d‚ , ‚ 2 − k22 + kZ ‚ 2 − k12

(11)

with r1 = dist(Q, p); r2 = dist (mirror point of Q, P); H = h + z = sum of heights of P and Q. The inclusion of the source height h in p1 (r, z; h) indicates Brekhovskikh’s rule: if one subtracts from the source-free field the mirror source field, the remaining scattering term depends only on the sum of source and receiver heights. The second form can be further modified to a form which is suited for saddle point integration: √2 2 ∞ p1 (r, H; 0) 2 (1 + kZ) ‚ · e−H ‚ −k1 +i‚ r −‚ r = · H(1) d‚ . (12) 0 (‚r) · e 2 2 2 2 P0 k1 ‚ − k2 + kZ ‚ − k1 −∞

The path of integration is parallel to Re{‚} with a small distance above this axis for Re{‚} < 0 and a small distance below it for Re{‚} > 0. The exact form of Van der Pol for two half-spaces is ∂ 2 ei k2 r1 ei k1 r2 1 p1 (r, z) ei k1 r1 ei k1 r2 = + − · r0 dr0 dz dœ , P0 k 1 r1 k1 r2 k1 ∂z2 r1 r2

(13)

V2

with integration over the half-space V2 below the plane which contains the mirrorreflected point to Q, and r0 , œ determined from r12 = r02 + z2

;

r22 = r2 − 2 r r0 cos œ + r02 + (H + kZ · z)2 .

(14)

D.17 A Monopole Point Source Above a Locally Reacting Plane, Exact Forms

See also: Mechel, Vol. I, Ch. 13 (1989); Ochmann (2004)

A monopole point source is placed at a point Q at height h above an absorbent plane; a field point is at P. The plane is locally reacting with a normalised admittance G = 1/Z. See > Sect. D.16 for a bulk reacting plane; some of the forms for the field above the absorbent plane in that section can be used also for a locally reacting plane, if such forms apply the reflection factor R of a plane wave at the plane. See Mechel (1989) for references to the extensive literature about this problem.As an exception, the time factor in this section is e−i–t in order to facilitate the comparison with the literature, where this sign convention mostly is used.

D

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The free field of the point source is

pQ (r1 ) = P0

ei k0 r1 , k 0 r1

(1)

the field p above the absorber is

ei k0 r1 pr p = + , P0 k 0 r1 P0

(2)

with r1 =dist (Q,P) and pr the reflected field. The task is to find pr . The reflection factor of a plane wave incident under a polar angle Ÿ is (with Z=1/G ) R(Ÿ) = (cos Ÿ − G)/(cos Ÿ + G) = (Z · cos Ÿ − 1)/(Z · cosŸ + 1) .

(3)

An exact form of pr is: /2−i∞

pr =i P0

J0 (k0r · sin ˜ ) · ei k0 H·cos ˜ · R(˜ ) · sin ˜ d˜ .

(4)

0

H = z + h, z is the co-ordinate normal to the plane directed into the half-space above the plane, r is the radius to P from the foot point of Q on the plane, and J0 (z) is the Bessel function of zero order. The path of integration in the complex ˜ plane is: 0 → /2 → /2 − i · ∞. Decomposition into real and imaginary parts with ˜ = ˜ + i · ˜ of sin ˜ = sin ˜ · cosh ˜ + i · cos ˜ · sinh ˜ , cos ˜ = cos ˜ · cosh ˜ − i · sin ˜ · sinh ˜ ,

(5)

gives pr =i P0

1

Z·y −i dy J0 k0 r · 1 − y 2 · ei k0 H·y Z·y +i

0

∞ Z·y +i + J0 k0 r · 1 + y 2 · e−k0 H·y dy . Z·y −i

(6)

0

Replacement of the Bessel function with the Hankel function yields i pr i k0 H·cos ˜ = H(1) · R(˜ ) · sin ˜ d˜ , 0 (k0 r · sin ˜ ) · e P0 2

(7)

C

with the path of integration C = −/2 + i · ∞ → −/2 → +/2 → +/2 − i · ∞. A different exact form is pr = P0

∞

J0(k0 r · y) · e 0

√

− k0 H·

y2 − 1 + i y dy , 2 2 Z· y −1−i y −1

y 2 −1 Z

·

with a negative imaginary root in 0 y < 1.

(8)

Reflection of Sound

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169

With a field composition p(r, z; h) ei k0 r1 ei k0 r2 p1 (r, z + h; 0) = − + P0 k0 r1 k0r2 P0

(9)

one gets (H = z + h) p1 (r, H; 0) = 2Z P0

Q h

r1

∞ 0

Θ0

(10)

P z

r2

h

√2 y · e−k0 H y −1 J0 (y · k0 r) dy . Z y2 − 1 − i

r

Q′ Butov’s form for bulk reacting absorbers can be transformed so that it can be applied to locally reacting absorbers: ∞ ) pr =i (−1)n (4n + 1) · V2n · h(1) 2n (k1 r2 ) · P2n (cos ˜ ) , P0 n=0

(11)

with h(1) 2n (z) spherical Hankel functions of the first kind,P2n (z) are Legendre polynomials, r2 = dist (mirror point of Q, P), and V2n=0

1 = 2Z

Z −Z

1 1+Z y −1 dy = 1 − ln , y +1 Z 1−Z

(12)

−2 Q2n (1/Z) Z ⎤ ⎡ (13) [n−1/2] ) 1+Z 2 (n − m) − 1 −1 ⎣P2n (1/Z) · ln −4 P2(n−m)−1 (1/Z)⎦ , = Z 1−Z (2m + 1) (2n − m) m=0

V2n>0 =

with Q2n (z) Legendre polynomials of the second kind and [n − 1/2] the highest integer ≤ (n − 1/2). All integrals of the exact forms have oscillating integrands, and the interval of integration extends to infinity. If numerical integration is applied, the convergence must be improved. This is done according to the scheme ∞ I=

∞ f (x) dx =

y

∞ f∞ (x) dx +

y

(f (x) − f∞(x)) dx , y

(14)

170

D

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where f∞ (x) is an asymptotic approximation to f (x), and the analytical integral over f∞ (x) is known (it is dangerous to apply an approximation to that integral). In (f (x) −f∞ (x) the oscillations at large x are reduced. An integral solution with favourable numerical behaviour of the numerical integral (due to a fast convergence of the integrand) was recently published by Ochmann (2004). The structure of the solution is [cf. (9)] p(r, z; h) ei k0 r1 ei k0 r2 ps (r, z + h; 0) = + − , P0 k0 r1 k0r2 P0 √ √ 2 2 2 2 ei 2 (r/Š0 ) +(z/Š0−h/Š0 ) ei 2 (r/Š0 ) +(z/Š0 +h/Š0 ) p(r, z; h) = + P0 2 (r/Š0 )2 + (z/Š0 − h/Š0 )2 2 (r/Š0 )2 + (z/Š0 + h/Š0 )2 √ ∞ i 2 (r/Š0 )2 +((z/Š0 )+(h/Š0 )+i )2 e −2G e−2G d . (r/Š0)2 + ((z/Š0) + (h/Š0 ) + i )2 0

(15)

(16)

This solution does not need explicit additional surface wave terms for spring-type reactive surfaces (the contributions of the surface wave are contained implicitly in the integral).

D.18 A Monopole Point Source Above a Locally Reacting Plane, Exact Saddle Point Integration


The method of saddle point integration mostly is considered as an approximate method to evaluate an integral which satisfies some criteria. In the present task, the saddle point integration can be applied so that it is an exact transformation of the integral, which makes it suited to numerical integration with high precision. If an integral as it appears in the present problem can be cast exactly as an integral over the path of steepest descent, it has the best possible form for a precise numerical evaluation. ϑ

π π

ϑ

ϑ

Reflection of Sound

Suppose the integral to be evaluated is of the form I = ea·f (˜ ) · F(˜ ) d˜ .

D

171

(1)

C

This integral is suited for saddle point integration if a 1 is a large real number and the path C goes to infinity on both sides. The saddle point integration in most of its applications is an approximation because f (˜ ) is approximated as f (˜ ) ≈ + · s2 with a real variable s on the pass way and, more seriously, F(˜ ) is expanded as a power series. If a pole is near the saddle point, the radius of convergence becomes small and the precision goes down. The start integral I0 in our problem comes from the third integral of multiplication and division of the integrand by exp(±ik0r · sin ˜ ): Q G

h

z

r1

pr i = P0 2

Q' C

P

Θ0

h Θ0

> Sect. D.17, after

r2 z r

x,r

i −i k0 r·sin ˜ · R(˜ ) · sin ˜ d˜ = · I0 , (2) ei k0 r2 ·cos(˜ −Ÿo ) · H(1) 0 (k0 r · sin ˜ ) · e 2

with the geometrical quantities as in the sketch and making use of k0 ((h + z) cos ˜ + r sin ˜ ) = k0r2 · cos(˜ − Ÿ0 ) .

(3)

The integration path C(œ) and the path of steepest descent (pass way Pw) are shown above in the sketch in the complex plane of œ = ˜ − Ÿ0 . If during the deformation C(œ) → Pw a pole of the reflection factor R(˜ ) is crossed, it is encircled as shown. This extra circle will give a “pole contribution”. The oscillations of the term in brackets in the integral go to zero for large argument values because the Hankel function oscillations are compensated by the exponential factor. Comparing I0 with the general integral I, correspondences are a → k0r2 ; f (˜ ) → i · cos œ. The saddle point ˜s with the maximum exponential factor (outside the brackets) follows from df (˜ )/d˜ = 0, which in our case is œs = 0, i.e. ˜s = Ÿ0 . The parameter form of the pass way equation is (with ˜ = ˜ + i · ˜ ; values of ˜ on Pw are called ˜Pw ) cos(˜Pw − Ÿ0 ) · cosh ˜Pw =1, − Ÿ0 ) = cos(˜Pw

1 , cosh ˜Pw

− Ÿ0 ) = − tanh ˜Pw , sin(˜Pw

(4)

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172

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or, equivalently, = sin ˜Pw

cos ˜Pw

sin Ÿ0 − cos Ÿ0 · sinh ˜Pw , cosh ˜Pw

(5)

cos Ÿ0 + sin Ÿ0 · sinh ˜Pw = , cosh ˜Pw

and therefore the function f (˜ ) in the exponent can be expressed as follows: 2 f (˜Pw ) = − tanh ˜Pw · sinh ˜Pw + i −−−−−−→ −(˜Pw ) +i.

(6)

|˜Pw | 1

All factors in the integrand of I0 can be expressed as functions of ˜Pw , especially

= R(˜Pw ) = R ˜Pw

Z · cos ˜Pw

−1 , Z · cos ˜Pw + 1

(7)

with the definitions: cos ˜Pw

= cos Ÿ0 + sin Ÿ0 · sinh ˜Pw , −i · tanh ˜Pw · sin Ÿ0 − cos Ÿ0 · sinh ˜Pw

(8)

= sin Ÿ0 − cos Ÿ0 · sinh ˜Pw sin ˜Pw . +i · tanh ˜Pw · cos Ÿ0 + sin Ÿ0 · sinh ˜Pw With the transition ˜ → ˜Pw the general integral I is transformed into ⎧ g(˜Pw ) = i − tanh ˜Pw · sinh ˜Pw , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −∞ d˜ ⎨ G(˜ ) = F ˜ (˜ ) a·g(˜Pw ) Pw Pw , e · G(˜Pw ) d˜Pw ; I= d˜Pw ⎪ ⎪ ⎪ +∞ ⎪ dg(˜Pw ⎪ )/d˜Pw ⎪ ⎩ . ) = F ˜ (˜Pw df (˜ )/d˜

(9)

The last fraction becomes dg(˜Pw )/d˜Pw · (2 − tanh 2 ˜Pw ) sinh ˜Pw 2 − tanh2 ˜Pw =− =− . df (˜ )/d˜ tanh ˜Pw + i · sinh ˜Pw i + 1/ cosh ˜Pw

(10)

The desired integral I0 finally is (substitute for ease of writing ˜Pw → t) i k0 r2

I0 = e

∞

e−k0 r2 ·tanh t·sinh t

2 − tanh2 t i + 1/ cosh t

0 −i k0 r sin t · R t · sin t · H(1) 0 (k0 r sin t ) · e −i k0 r sin −t dt . + R −t · sin −t · H(1) r sin −t ) · e (k 0 0 ∗)


∗)

(11)

Reflection of Sound

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173

In the special case h = z = 0, i.e. Ÿ0 = /2, one gets cos ˜Pw

= sinh ˜Pw − i · tanh ˜Pw = − cos −˜Pw

,

= 1 + i · tanh ˜Pw · sinh ˜Pw = sin −˜Pw

, sin ˜Pw

− i · tanh ˜Pw Z · sinh ˜Pw −1 R ˜Pw = = 1/R −˜Pw

, Z · sinh ˜Pw − i · tanh ˜Pw + 1

(12)

and therewith ∞ I0 =

e−k0 r2 ·tanh t·sinh t

0

2 − tanh 2 t · (1 + i · tanh t · sinh t) i + 1/ cosh t

(13)

· H(1) 0 (k0 r(1 + i · tanh t · sinh t)) · (R t + 1/R t ) dt . The integrand in I0 decreases quickly with increasing t. This is paid for with a complex argument of the Hankel function. The scattered field is pr /P0 = i/2 · I0 . If during the deformation C(œ) → Pw a pole of the reflection factor R(˜ ) is crossed, the pole contribution prp must be added to pr : prp −2 (1) H0 (k0 r 1 − 1/Z2) · e−i k0 H/Z = P0 Z

;

Re{ 1 − 1/Z2 } > 0 .

(14)

D.19 A Monopole Point Source Above a Locally Reacting Plane, Approximations


See > Sect. D.17 for exact integral formulations of the solution, and see Mechel (1989) for a discussion of the approximations and their precision. A monopole point source is placed at a point Q at height h above an absorbent plane; a field point is at P with height z above the plane and horizontal distance r from Q. Q is the mirror-reflected point to Q. The plane is locally reacting with a normalised admittance G = 1/Z.

Q h h

r1 r2

Θ0

P z

r

Q′ See Mechel (1989) for a detailed discussion of the approximations. As an exception, the time factor in this section is e−i–t in order to facilitate comparison with the literature, where this sign convention mostly is used. Radii used are r1 = dist(Q, P) and r2 = dist(Q , P), and the angle Ÿ0 = ∠ ((Q, Q), (Q , P)).

D

174

Reflection of Sound

The start equation for a first approximation to the reflected field pr is pr i = P0 2

i (1) ei k0 r2 ·cos (˜ −Ÿo ) · H0 (k0 r · sin ˜ ) · e−i k0 r·sin ˜ · R(˜ ) · sin ˜ d˜ = · I0 . (1) 2

C

See > Sect. D.17 for definitions of r, r2, R t . An approximate saddle point integration is applied to I0 [condition: a pole at cos˜p = −1/Z of the reflection factor R(˜ ) is not near the saddle point ˜s = Ÿ0; see sketch in > Sect. D.18 for Ÿ0 ]. The first-order approximation is

2 3 prp i pr (r, z) ei k0 r2 = R(Ÿ0 ) − . R (Ÿ0 ) cot Ÿ0 + R (Ÿ0 ) + P0 k 0 r2 2 k0 r 2 P0

(2)

The term prp /P0 indicates a possible pole contribution prp (for more, see below). In these equations,r = horizontal distance between source Q and field point P; z = height of field point P; r1 = dist(Q, P); r2 = dist(Q , P); Ÿ0 = ∠ ((Q, Q), (Q , P)); see > Sect. D.1 for R(Ÿ0 ) [there r(Ÿ)] and derivatives. A higher approximation (condition: pole not near the saddle point) is pr (r, z) ei k0 r2 =− P0 k0r2

2 3 prp 1 1 , + i · F(Ÿ0) + F (Ÿ0 ) + 8 k0 r 2 2 k0 r 2 P0

(3)

with F(Ÿ0 ) = R(Ÿ0 ) 1 − 4

2

sin Ÿ0

−

i sin Ÿ0

;

=

9 128 (k0r)2

;

=

1 ; 8 k0 r ,

1 6 + 11 cos2 Ÿ0 2 + 3 cos2 Ÿ0 1 − 3 sin2 Ÿ0 R(Ÿ0 ) − − i 4 sin2 Ÿ0 sin4 Ÿ0 sin3 Ÿ0

3 i 1 + + + R (Ÿ0 ) cos Ÿ0 sin Ÿ0 sin3 Ÿ0 sin2 Ÿ0

i − . +R (Ÿ0) 1 − sin2 Ÿ0 sin Ÿ0 F (Ÿ0 ) =

(4)

If the pole of the reflection factor is near the saddle point, the integral to be evaluated, and which can be transformed into the general form I=e

a·f (˜s )

∞

e−a·f (˜ ) · ¥ (s) ds ,

(5)

−∞

is modified further by separation of the simple pole in ¥ (s), i.e. by setting ¥ (s) =

+ T(s), s−

(6)

Reflection of Sound

D

175

with

√ 2 (1) 2 H0 (k0r 1 − 1/Z2 ) · e−i k0 r 1−1/Z ; Re{ 1 − 1/Z2 } > 0 ; Z = i 1 + cos Ÿ0/Z − sin Ÿ0 1 − 1/Z2 ;

=

(7)

⎧ ⎨ > 0 ; pole above pass way Im{} = 0 ; pole on pass way ⎩ < 0 ; pole below pass way The cases for Im{} correspond to Im{} 0 ⇔ G

−1 (cos Ÿ0 + G ) (1 + cos Ÿ0 · G ) √ . sin Ÿ0 1 + 2 cos Ÿ0 G + G2

(8)

One gets the approximation pr i · ei k0 r2 = P0 2 ⎧ √ ⎨ ±i · W(± k0r2 ) + /(k0r2 ) · T(0) ; Im{} ≷ 0 · √ ⎩ 2 i · W( k0r2 ) + /(k0r2 ) · T(0) − i · e−k0 r2 · ; Im{} = 0

(9)

with cosŸ0 − G −i k0 r sin Ÿ0 · sin Ÿ0 + (1 − i) H(1) 0 (k0 r sin Ÿ0 ) · e cosŸ0 + G ∞ 2 2 2 W(u) = e−u · erfc(−iu) ; erfc(z) = √ e−x dx .

T(0) =

(10)

z

where erfc(z) is the complementary error function. Van Moorhem’s approximation: ei k0 r2 pr = [R(Ÿ0 ) + (1 − R(Ÿ0 )) · F(Ÿ0 , k0r2 )] , P0 k 0 r2

(11)

with F(Ÿ0 , k0r2 ) =

i 1 + G · cos Ÿ0 1 (3 G2 − 1) cos2 Ÿ0 + 4 G cos Ÿ0 − 3 − G2 + 2 2 k0 r2 (G + cos Ÿ0 ) (k0 r2 ) (G + cos Ÿ0)4

−

3i (5 G3 − 3G) cos3 Ÿ0 + (9 G2 − 3) cos2 Ÿ0 + (9 G − 3G2) cos Ÿ0 + 5 − 3G2 (k0r2 )3 (G + cos Ÿ0 )6

−

3 (3 − 30 G2 + 35 G4 ) cos4 Ÿ0 + (80 G3 − 48 G) cos3 Ÿ0 − (30 − 108 G2 + 30 G4) (k0r2 )4 (G + cos Ÿ0 )8 . . . · cos2 Ÿ0 + (80 G − 48 G3 ) cos Ÿ0 + 3G4 − 30G2 + 35 . ...

(12)

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Lawhead / Rudnick’s approximation (valid for Im{G} < 0, i.e. spring-type reactance): pr ei k0 r2 = [R(Ÿ0 ) + (1 − R(Ÿ0 )) · F(u)] , P0 k 0 r2 F(u) = 1 + with

u = (1 − i)

√

(13)

2

· u · eu · erfc (−u) k0 r2 G + cos Ÿ0 2 sin Ÿ0

;

Im{G} < 0

(14)

Ing˚ard’s approximation is like Rudnick’s approximation; however, the function F now is √ √ F = 1 − · e · 1 − ¥ ( ) i k0 r2 (G + cos Ÿ0 )2 = 2 1 + G cos Ÿ0

;

2 ¥ (x) = √

x

2

e−t dt

(15)

0

Approximation by Chien / Soroka: (valid for |G| 1 and k0 r 1) with pr /P0 as above, but with the function F(u): √ 2 F(u) = 1 + i · u · e−u · erfc (−i u) (16) u = i k0 r2 /2 · (G + cos Ÿ0 ) These authors also derived an approximation with a wider range of applicability: pr ei k0 r2 ps pp = + + , P0 k 0 r2 P0 P0 with

(17)

6 1 + G cos Ÿ0 1 √ + −1 + √ 1 + 2 sin Ÿ0 1 − G2 i sin Ÿ0 √ + − 1 − G2 √ 1 + G cos Ÿ 0 k0 r2 (G + cos Ÿ0 )2 8 2 ,7 1 + G cos Ÿ0 3/2 1 + G cos Ÿ0 · 1+ √ √ 3+ ; sin Ÿ0 1 − G2 sin Ÿ0 1 − G2

2G ps ei k0 r2 = P0 G + cos Ÿ0 k0 r2

5

√ √ pp (1) = − G erfc (−i x0 / 2) · H0 (k0 r 1 − G2 ) · e−i k0 (h+z)·G ; P0 √ x02 = i k0r2 1 + G cos Ÿ0 − sin Ÿ0 1 − G2 . 2 This approximation was also derived by Attenborough et al.

(18)

(19)

Reflection of Sound

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177

Thomasson presented the approximation with good precision: √ ei k0 r2 pr = R(Ÿ0 ) + (1 − R(Ÿ0 )) · U(±i x0 / 2) P0 k 0 r2 √ √ 2 U(±u) = 1 ∓ i · e−x0 /2 · erfc (±i x0 / 2) ,

;

Im{x0 } ≷ 0 ;

(20)

with x0 as above. A further approximation by Thomasson is based on √ pr ei k0 r2 −i k0 (z+h)· G 2 = − (1 − C) · G · H(1) − 2 G · ei k0 r2 0 (k0 r 1 − G ) · e P0 k 0 r2

∞ 0

e−t dt,(21) √ V(t)

with

V(t) = A2 + t B2 − t ; A = ei k0 r2 (‚0 − 1) ; B = ei k0 r2 (1 − ‚1 ) ; √ ‚0,1 = −G cos Ÿ0 ± sin Ÿ0 1 − G2 √ and the sign rules Re{ V(0)} > 0 and 5 /4 < arg(A) < /2 , Re{ V(t)} < 0 if t > Im{A2 B2 }/Im{A2 − B2 } √ Re{ V(0)} > 0 else 5 C is a “switch function”:

C=

+1 ; −/2 ≤ arg(A) ≤ /4 −1 ; /4 < arg(A) < /2 .

(22)

(23)

(24)

The signs of other square roots are selected so that their real parts are positive. This form is well suited for numerical integration; the results coincide with those of numerical integrations of other exact forms. It can be applied also for h = z = 0, i.e. source and receiver on the plane. An approximation which is derived from the last form of pr /P0 is for k0r2 1 and |B|2 |A|2 and |B|2 k0r2 : 4 , ∞ C) pr ei k0 r2 (2m) ! = · Im 1 − 2 k0r2 G P0 k0r2 B m=0 (m !)2(4B2 )m (25) √ (1) − (1 − C) G · H0 k0 r 1 − G2 · e−i k0 (h+z)·G , with iterative evaluation of the Im : I0 =

√ 2 · eA · erfc (A)

;

I1 = A +

1 − A2 · I0 ; 2

1 2 Im = m − − A · Im−1 + (m − 1) A2 · Im−2 . 2

(26)

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This approximation computes very precisely in the mentioned range of conditions. An approximation by Nobile: ∞

) pr ei k0 r2 4iG·B = − · ei k0 r2 (e0 · En + Kn ) · Tn , P0 k 0 r2 G + cos Ÿ0 n=0

(27)

with

√ √ B = −i 1 + G cos Ÿ0 − sin Ÿ0 1 − G2 ; Re{ . . .} > 0 , √ √ C = 1 + G cos Ÿ0 − sin Ÿ0 1 − G2 ; Re{ 1 − G2 } ≥ 0 , 0 1 n−m [n/2] 1 ) n−m −4 B2 · an−m · Tn = , (2 B)n m=0 C m 1 −n 1 2 2 · an−1 ; e0 = a0 = 1 ; an = · e−Š · erfc (−i Š) , n 2 i k 0 r2 √ √ Š = i k0r2 1 + G cos Ÿ0 + sin Ÿ0 1 − G2 ; all Re{ . . .} > 0 ,

(28)

and iterative evaluation of E0 = 1 ;

E1 = −B

K0 = 0 ;

K1 = −

;

En = −B · En−1 − i

i 2 k0 r 2

;

n−1 · En−2 ; 2 k0 r 2

Kn = −B · Kn−1 − i

n−1 · Kn−2 . 2 k0r2

(29)

Another approximation by Nobile, in which is a “reflection factor for a spherical wave” (see > Sect. D.20) reads as follows: ei k0 r2 pr = P0 k 0 r2

;

=1+

∞ ) 2G (e1 · E¯ n + K¯ n ) · T¯ n ; G + cos Ÿ0 n=0

with auxiliary quantities from above, except for the newly defined quantities: 0 1 n−m [n/2] ) n−m −4 B2 · an−m · ; e1 = −2 i B k0 r2 · e0 ; T¯ n = C m m=0 E¯ 0 = 1 ; K¯ 0 = 0 ;

1 2 1 K¯ 1 = − 2 E¯ 1 = −

; ;

1 n−1 E¯ n = − · E¯ n−1 − i · E¯ n−2 ; 2 8 k0r2 B2 1 n−1 K¯ n = − · K¯ n−1 − i · K¯ n−2 . 2 8 k0r2 B2

(30)

(31)

(32)

In the finite sum the upper limits is [x], the highest integer ≤ x. Finally, the approximation obtained with the principle of superposition applied in > Sect. D.15 is ei k0 r2 pr = P0 k 0 r2

;

=

1/X(r) − G , 1/X(r) + G

(33)

Reflection of Sound

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179

with the normalised cross impedance X(r) of the plane z = 0 defined and given by 1 −Z0 vsz (r, z = 0) i ∂ pQ (r, 0) − pQ (r, 0) /∂z = = , (34) X(r) ph (r, z = 0) k0 pQ (r, 0) + pQ (r, 0) where pQ (r1 ) is the source free field and pQ (r2 ) the free field of a point source (of same strength) in the mirror-reflected point Q to Q · ph is the field, for which the plane z = 0 is hard; ps is the field for which that plane is soft. It is ei k0 r1 ei k0 r2 ; pQ = ; k 0 r1 k 0 r2 r1 = (h − z)2 + r2 ; r2 = (h + z)2 + r2 ;

pQ =

and therefore √ 2 + r2 h i + k h k 0 0 1 = X(r) k02 (h2 + r2 )

√ i − G (k0h + r/h · k0r) + k0 h2 + r2 i + k0 h (1 − G) = √ −−→ i + G (k0h + r/h · k0r) + k0 h2 + r2 r=0 i + k0 h (1 + G)

(35)

(36)

with the limit → −1 for r → ∞. can be considered as a “reflection factor for spherical waves”.

Q h h

r1 r2

Θ0

P z

r

Q′

D.20 A Monopole Point Source Above a Bulk Reacting Plane, Approximations


See > Sect. D.16 for conventions used, and see Mechel (1989) for a discussion of the approximations and their precision. The object is a half-space with homogeneous,isotropic material having the characteristic wave number k2 and wave impedance Z2 . The point source is in the upper half-space with k1 , Z1 as characteristic wave number and wave impedance; it is at the source point Q with a height h above the plane. The field point P has a horizontal distance r of Q and a height z. The ratios k = k2 /k1 and Z = Z2 /Z1 are used; further H = h + z.

D

180

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The approximation by Delany / Bazley starts from Van der Pol’s exact form; it is pr ei k0 r2 = +2ik P0 k 0 r2

∞ 0

ei (k1 r3 +ky) dy ; k1r3

k1 r3 = (k1 r)2 + (k1 H + kZ · y)2

(1) ;

√ Im{ . . .} > 0 .

The exponential function in the integrand decays exponentially, therefore this form is suited for numerical integration. Norton / Rudnick propose a correction to this approximation: √ 2 kZ → k Z k 2 − sin2 Ÿ0 ; Re{ . . .} > 0.

(2)

Soomerfeld’s approximation for the total field in the upper space: p1 ei k1 r1 pr = + P0 k1r1 P0 = 2C ·

H(1) 0 (‚p r)

with

6

−k1 z

·e

k2 − 1 (kZ)2 − 1

√

(‚p /k1 )2 −1

√ C1 (z) i k1 r C2 i k1 r·k−k1 z k2 −1 + · e + · e , (k1 r)2 (k1 r)2

kz kZ ; C1 (z) = −2 i (1 + kZ) +√ 2 1−k 1 − k2 6 Z2 − 1 k /k = k C2 = −2 i (1 + kZ) ; ‚ . p 1 (kZ)2 (k 2 − 1) (kZ)2 − 1

kZ C = 1 − kZ

The approximation by Paul uses the notations ƒ = 1/(kZ) ei k0 r2 pr =− + V(H, r) ; P0 k 0 r2

;

V(H, r) = V1 (H, r) + V2(H, r) ,

(3)

(4)

k1 H = k1 (h + z): (5)

with

1+ƒ F1 (H) F2 (H) ei k1 r V1 (H, r) = −2 i 2 F0(H) − i − ƒ (1 − k 2 ) (k1 r)2 2 k1 r 8 (k1r)2 √ F0 (H) = 1 + k1 H · ƒ · 1 − k 2

(6)

Reflection of Sound

F1 (H) =

D

181

√ 1 . 4 − 3(k1 H)2 + k2 2 + 3 (k1H)2 + k1 H 1 − k 2 2 1−k · ƒ · 1 − k1H + k 2(2 + k1H) − 6/ƒ / √ +k1 H 1 − k 2 ƒ · 1 − k1 H + k 2 (2 + k1H) − 6/ƒ + 6/ƒ2 ;

F2 (H) =

. √ 1 · k1H ƒ 1 − k 2 g + 2(k1H)2 + (k1 H)4 + k 2 2 2 (1 − k ) · 36 − 14(k1H)2 − (k1 H)4 + 12 k4(k1 H)2 2

4

2

2

4

+ 48 − 24(k1H) + 5(k1H) + k 72 − 12(k1H) − 5(k1H)

(7)

k1 H √ 1 − k 2 108 − 20(k1H)2 + k 2 72 + 20(k1H)2 ƒ 1 k 1H √ 120 − 168 − 60(k1H)2 + k 2 72 + 60(k1H)2 + 120 3 1 − k 2 + 4 ; ƒ ƒ ƒ √

2 G2 (H) ei (k1 r·k+k1 H 1−k ) G1(H) V2 (H, r) = −2 i k(1 + ƒ) − G0(H) − , (k1r)2 2 k1r · k 8(k1r)2 · k 2 + 36 k 4 (k1H)2 −

G0 (H) =

ƒ 1 − k2

√ ƒ 2 2 2 2 2 2 + 4 k − 6 ƒ k + 3 i k H · k 1 − k 1 (1 − k 2)2 √ k2 ƒ 72 + 48 k 2 − i k1H 36 + 39 k 2 1 − k2 − 15 (k1H)2 G2 (H) = − 2 2 (1 − k ) √ ·k 2(1 − k 2) − ƒ2 72 + 168 k 2 − 60 i k1H · k 2 1 − k 2 + 120 k2ƒ4 . G1 (H) = −

An approximation given by Attenborough/Hayek/Lawther is

pr cos Ÿ0 iF ei k1 r2 −1 + 2(1 + kZ) 1+ , √ = P0 k 1 r2 k 1 r2 kZ · cos Ÿ0 + k 2 − sin Ÿ0 √ with Im{ k 2 − sin Ÿ0 } > 0 and 0 1 √ cos Ÿ0 + kZ k 2 − sin Ÿ0 sin2 Ÿ0 F =1− √ k 2 − sin2 Ÿ0 kZ · cos Ÿ0 + k 2 − sin Ÿ0

(8)

(9)

(10)

kZ · sin2 Ÿ0 √ cos Ÿ0 · (kz · cos Ÿ0 + k 2 − sin Ÿ0 ) 1,7 4 0 5 cos Ÿ0 1 3 cos2 Ÿ0 · sin2 Ÿ0 2 2 cos Ÿ0 + √ cos Ÿ0 − sin Ÿ0 + · 1− 2 2(k 2 − sin Ÿ0 ) sin2 Ÿ0 kZ k2 − sin Ÿ0 +

182

D

Reflection of Sound

The denominators create problems when they go to zero, i.e. for large flow resistivity values of a porous material in the lower half-space, and at the same time Ÿ0 → /2, i.e. source and receiver on the plane. An approximation obtained by saddle point integration is

pr ei k1 r2 −i 1 1 = ·√ + i · F(Ÿ0 ) + F (Ÿ0 ) , P0 k 1 r2 8 k1 r 2 2 k1 r2 sin Ÿ0 with

F(Ÿ0 ) = R(Ÿ0 ) sin Ÿ0 1 −

iß − 2 sin Ÿ0 sin Ÿ0

,

(11)

(12)

4 , R(Ÿ0 ) 6 + 11 cos2 Ÿ0 2 + 3 cos2 Ÿ0 1 − 3 sin2 Ÿ0 sin Ÿ0 − −iß F (Ÿ0 ) = 4 sin2 Ÿ0 sin4 Ÿ0 sin3 Ÿ0

3 1 1 + 3 +iß 2 + R (Ÿ0 ) sin Ÿ0 cos Ÿ0 sin Ÿ0 sin Ÿ0 sin Ÿ0

iß − + R (Ÿ0 ) sin Ÿ0 1 − 2 sin Ÿ0 sin Ÿ0

and the reflection factor and its derivatives: R(Ÿ0 ) =

kZ · cos Ÿ0 − w , kZ · cos Ÿ0 + w

kZ sin Ÿ0 · 1 + sin2 Ÿ0 − 2k 2 , w · (kZ · cos Ÿ0 + w)2 (13) kZ 2 2 2 R (Ÿ0 ) = Ÿ − 2k Ÿ · cos Ÿ cos Ÿ 1 + sin + 2 sin 0 0 0 0 w · (kZ · cos Ÿ0 + w)2 1, 0 sin2 Ÿ0 · cos Ÿ0 2 sin2 Ÿ0 kZ · w + cos Ÿ0 2 2 + 1 + sin Ÿ0 − 2 k · + ; w2 w kZ · cos Ÿ0 + w

R (Ÿ0 ) =

using 9 = 128 (k1r)2

;

1 ß= 8 k1 r

;

w=

k 2 − sin2 Ÿ0 ; Im{w} > 0 .

For k1 r2 1 this approximation can be simplified to

pr ei k1 r2 i R(Ÿ0 ) − R (Ÿ0 ) · cot Ÿ0 + R (Ÿ0 ) . = P0 k 1 r2 2 k1r2

(14)

(15)

Reflection of Sound

D

183

References to Part D Mechel, F.P.: Schallabsorber. Vol. I–III, Hirzel, Stuttgart (1989, 1995, 1998) Mechel, F.P.: Schallabsorber. Vol. I, Ch. 8: “Plane absorbers with finite lateral dimensions”, Hirzel, Stuttgart (1989) Mechel, F.P.: Mathieu Functions; Formulas, Generation, Use. Hirzel, Stuttgart (1997) Mechel, F.P.: A line source above a plane absorber. Acta Acustica 86, 203–215 (2000)

Mechel, F.P.: Modified mirror and corner sources with a principle of superposition.Acta Acustica 86, 759–768 (2000) Mechel, F.P.: Schallabsorber.Vol. I, Ch. 13: Spherical waves over a flat absorber. Hirzel, Stuttgart (1989) Ochmann, M.: The complex equivalent source method for sound propagation over an impedance plane. J. Acoust. Soc. Am. 116, 3304–3311 (2004)

E Scattering of Sound E.1 Plane Wave Scattering at Cylinders


See

> Sect. E.2

for a survey of formulas for cylinders and spheres.

The cylinder with diameter 2a is either bulk reacting, i.e. it consists of a homogeneous material with characteristic propagation constant a and wave impedance Za , or it consists of a similar material with same characteristic values, but locally reacting either in the axial direction or locally reacting in all directions. Local reaction is obtained either by a high flow resistivity ¡ of the porous material or by thin partitions at mutual distances smaller than about Š0 /4. Sound incidence of the plane wave with unit amplitude is in the x,z plane. y r 2aø

Θ

P ϕ x

pi

Γa , Za

bulk reacting

axially local

omnidirectional local

The field formulation will be given below for the bulk reacting cylinder; the fields for the other cylinders follow from that by simplifications. Notations: • • • • •

an = a /k0; Zan = Za /Z0 normalised characteristic values; pi = incident plane wave; ps = scattered wave; p = pi + ps = total exterior field; pa = interior field in the absorbing cylinder.

186

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Scattering of Sound

Expansion of the incident plane wave in Bessel functions: pi (r, œ, z) = e−j k0 z·sin Ÿ ƒm (−j)m m≥0 1; m=0 · cos (mœ) · Jm (k0r · cos Ÿ); ƒm = 2; m>0 Formulation of the scattered field: ps (r, œ, z) = e−j k0 z·sin Ÿ Dm · ƒm (−j)m · cos (mœ) · H(2) m (k0 r · cos Ÿ).

(1)

(2)

m≥0

Formulation of the interior field: Em · ƒm · cos (mœ) pa (r, œ, z) = e−j k0 z·sin Ÿ m≥0

·Im (a r · cos Ÿ1 );

Im (z) = (−j)m Jm (jz),

(3)

with Bessel functions Jm (z),Hankel functions of the second kind H(2) m (z),modified Bessel functions Im (z). From the boundary conditions of matching pressure and radial particle velocity: 2 − cos2 Ÿ j k0 · sin Ÿ = a · sin Ÿ1 ; an · cos Ÿ1 = 1 + an (Ÿ1 is the refracted angle), and

(4)

cos Ÿ m + · Jm ()−cos Ÿ · Jm+1 () Jm () + j Wm · Jm () j Wm k0a Dm = − (2) = − (2) cos Ÿ m (2) Hm ()+j Wm · Hm () + · H(2) m ()−cos Ÿ · Hm+1 () j Wm k0a Jm ()+Dm · H(2) m () Em = Jm (ß)

(5)

with the abbreviations

(6)

= k0a · cos Ÿ;

= j k0 a · an · cos Ÿ1

and the modal normalised surface impedances Wm = Zan

cos Ÿ Im (a a · cos Ÿ1) cos Ÿ Jm (y) = −j Zan cos Ÿ1 Im (a a · cos Ÿ1) cos Ÿ1 Jm (y)

an · cos Ÿ1 cos Ÿ =− j Wm an Zan

Jm+1 (y) m − . Jm (y) y

For the cylinder which is locally reacting in the axial direction: • Set Ÿ1 = 0. For the cylinder which is locally reacting in all directions: • Retain in pa only the term m = 0; • Set Ÿ1 = 0;

(7)

Scattering of Sound

E

187

• Replace everywhere Wm → W0 , in which case: cos Ÿ cos Ÿ 1 J1 (j k0 a · an ) → =− . j Wm j W0 Zan J0 (j k0 a · an )

(8)

The result then describes the scattering of a cylinder consisting of a (porous) material and made locally reacting. For a cylinder which is locally reacting in all directions and described by a normalised surface admittance G: • Neglect pa ; • Replace everywhere Wm → W0 = 1/G, in which case: cos Ÿ cos Ÿ → = −j G · cos Ÿ. j Wm j W0

(9)

The incident plane wave is temporarily assumed to have an amplitude p0 (which above

was set at p0 = 1). The integrals below are taken at the cylinder surface (r = a).A star ∗ indicates the complex conjugate. Scattering cross section(ratio of scattered power to incident intensity): v∗ ps · rs dS −−−→ Re{ps · Z0 vrs∗ } dS, Qs = Re p0 =1 p0 p0 /Z0 Qs 2 = ƒm · |Dm |2. 2a k0 a m≥0

(10) (11)

Absorption cross section (ratio of absorbed power to incident intensity): Qa = − Re{p · Z0 vr∗ } dS,

−2 Qa = ƒm · Re{Dm } + |Dm |2 . 2a k0a m≥0 Extinction cross section: Qe = Qs + Qa = − Re{p∗i · vr + p · vri∗ } dS, Qe −2 ƒm · Re{Dm }. = 2a k0a m≥0

(12) (13)

(14) (15)

With the always possible separation of the scattered far field into an angular and a radial factor: ps (r, ˜ , œ) −−−−−→ ¥ (˜ , œ) · k0 r1

e−j k0 r + O(r−2 ) r

(16)

188

E

Scattering of Sound

the extinction cross section is (extinction theorem) Qe = −

4 Im{¥ (˜0 , œ0 )}, k0

(17)

where a radius with the angles ˜0 , œ0 points in the forward direction of the incident wave. Backscattering cross section (measures the strength of the backscattering to the source): Qr = 2r

| ps (r, ˜0 + , œ0 + )|2 p20

;

k0r 1.

(18)

Absorption cross section for diffuse sound incidence: There exist several definitions in the literature (differing from each other in the reference intensity). ¢a = absorbed power; Ii = intensity of an incident plane wave Qa1 = ¢a Ii , First definition:

Qa1

/2 = 4 Qa (Ÿ) · cos Ÿ dŸ .

(19)

0

Qa2 = ¢a Ii,dif Second definition:

Qa2

Qa1 =4 =

;

Ii,dif = · Ii ,

/2 Qa (Ÿ) · cos Ÿ dŸ .

(20)

0

Third definition: with ¢i,dif = incident power in a diffuse field on a cylinder of unit length and diameter Qa3 = ¢a ¢i,dif ; ¢i,dif = 4 · Ii Qa3

Qa1 Qa2 = = = 4 4

/2 Qa (Ÿ) · cos Ÿ dŸ

(21)

0

E.2 Plane Wave Scattering at Cylinders and Spheres


See previous > Sect. E.1 for an oblique plane wave incident on a cylinder. This section briefly gives the fundamental relations for a plane wave incident on a sphere and then collects equations for both spheres and cylinders (with normal incidence on the cylinder axis). In the case of a bulk reacting sphere, it consists of a homogeneous material with characteristic propagation constant a and wave impedance Za .

Scattering of Sound

z

E

189

P r ϑ x

pi

z

ϑ

P

r ϕ

x

pi

Diameter 2a; incident plane wave pi; field point P Notations: • • • • •

an = a /k0; Zan = Za /Z0 normalised characteristic values; pi = incident plane wave (with amplitude p0 = 1); ps = scattered wave; p = pi + ps = total exterior field; pa = interior field in the absorbing cylinder or sphere.

Sound field formulations for a sphere: Incident plane wave: pi (r, ˜ ) = e−j k0 r·cos ˜ =

(2m + 1) (−j)m · Pm (cos ˜ ) · jm (k0r).

(1)

m≥0

Scattered wave: Dm · (2m + 1) (−j)m · Pm (cos ˜ ) · h(2) ps (r, ˜ ) = m (k0 r).

(2)

m≥0

Total exterior field: p = pi + ps , where Pm (z) = Legendre polynomial, jm(z) = spherical Bessel function, h(2) m (z) = spherical Hankel function of the second kind. The following table contains corresponding quantities for cylinders and spheres, of diameter 2a, which are locally reacting with a normalised surface admittance G. Hankel functions of the second kind are written as Hm (z). The argument k0 a of Bessel and Hankel functions is dropped. In some equations W = 1/G = R + j · X will be used.

190

E

Scattering of Sound

The amplitude factors Cm are Cm = 2Dm − 1. Quantity

Symbol

Cylinder

Factors

Dm =

−

Sphere

−j G + m k0 a Jm −j G + m k0 a Hm

−j G + m k0 a Hm − −j G + m k0 a Hm

Cm =

cos (m˜)

ƒm =

⎧ ⎨ 1; m=0 ⎩ 2; m>0

Cross section

S=

2a

Incident wave

pi (r; ˜) = m0

ps (r; ˜) ! !

p(r; ˜) =

Scattering cross section

Qs =

2m+1 a2

ƒm (−j)m Tm Jm (k0 r)

m0

Dm ƒm (−j)m Tm Jm (k0 r)

m0

e−j k0 r Dm ƒm Tm k0 r

2j e−j k0 r ¥(˜) p k0 r

e−j k0 r ¥(˜) k0 r

m0

m0

m0

ƒm (−j)m Tm

m0

Jm (k0 r) + Dm Hm (k0 r)

ƒm (−j)m Tm

jm (k0 r) + Dm hm (k0 r)

g dS Refps vrs

2 ƒm jDm j2 k0 a

4 ƒm jDm j2 (k0 a)2

2 ƒm j1 − Cm j2 k0 a

4 ƒm j1 − Cm j2 (k0 a)2

m0

=

Dm ƒm (−j)m Tm jm (k0 r)

j

Qs =S =

ƒm (−j)m Tm jm (k0 r)

2j e−j k0 r p Dm ƒm Tm k0 r

Total field

− Hm+1 − Hm+1

e−j k0 rcos ˜ m0

Scattered far field

− Hm+1

−j G + m k0 a jm − jm+1 − −j G + m k0 a hm − hm+1

−j G + m k0 a hm − hm+1 − −j G + m k0 a hm − hm+1 Pm (cos ˜)

Tm =

Scattered wave ps (r; ˜) =

− Jm+1

m0

m0

m0

Scattering of Sound

E

191

Table continued: Quantity

Symbol

Cylinder

Sphere

Scattering cross section; approximations k0 a 1; R = 0 or = 1

Qs =S =

2

2

k0 a1; G=0

Qs =S =

3 2 (k0 a)3 8

7 (k0 a)4 9

k0 a1; jGj!1

Qs =S =

k0 a1; G = else

Qs /S =

Absorption cross section

Qa =

2

4

2 k0 a ln2 (1=k0 a)

4 (k0 a)2 jGj2

5 k0 ajGj2

− Refp vr g dS

Qa =S =

−2 ƒm RefDm g + jDm j2 k0 a m0

=

1 ƒm 1 − jCm j2 2 k0 a m0

−4 2 RefD ƒ g + jD j m m m (k0 a)2 m 1 2 1 − jC ƒ j m m (k0 a)2 m0

Absorption cross section; approximations k0 a 1; jXj R

Qa =S =

R

4 R

k0 a1; R<jXj/2

Qa =S =

R X2

4R X2

Extinction cross section

Qe =

with scattered far field

Qe =S =

− Refpi vr + p vri g dS −2 ƒm Tm (0) RefDm g k0 a m0

ps ! Qe =S =

p

e−j k0 r 2j= ¥(˜) p k0 r

−2 Ref¥(0)g k0 a

−4 ƒm Tm (0) RefDm g (k0 a)2 m0

j ¥(˜)

e−j k0 r k0 r

−4 Ref¥(0)g (k0 a)2

192

E

Scattering of Sound

Table continued: Quantity

Symbol

Cylinder

Sphere

Backscatter cross section

Qr =

2r jps (r; )j2

4r 2 jps (r; )j2

Backscatter cross-section; approximations k0 a 1; jGj ! 0 or 1

Qr =S =

=2

1

k0 a 1; jGj ! 1

Qr =S =

22 k0 a 2 + 4 ln2 (1:123=k0 a)

4

k0 a < first resonance; G else

Qr =S =

2 k0 ajGj2 2

4 k0 ajGj2

Reactance at m-th resonance k0 a < 1 = 1:123

X0 =

−k0 a ln

ß k0 a

1 (k0 a)2 ln 2 1 1 − (k0 a)2 ln 2 1−

X1 =

−k0 a

−k0 a 1 + (k0 a)2 ß k0 a ß k0 a

ß k0 a ß 1 − (k0 a)2 ln k0 a

1 + 2:14(k0 a)2 ln

(k0 a)2 k0 a 4(m − 1) − m (m − 2)(k0 a)2 1+ 4m (m − 1) 2 k0 a

−k0 a

1+

Xm =

Qs at lowest resonance; R = 0; X = X0

Qs =S =

Frequency of lowest backscatter minimum

k0 a =

2 jXj 2 + 3 jXj2 p jXjmax = 2=3 p (k0 a)max = 1= 6

−

1 + 2(k0 a)2 1 + (k0 a)2

k0 a m

1 (k0 a)2 1− 2m 2

4 (k0 a)2

6 jXj 3 + 5 jXj2 jXjmax = (k0 a)max =

p

3=10

Scattering of Sound

E

193

The next table contains corresponding values for cylinders and spheres, with diameter 2a,consisting of a homogeneous (bulk reacting) material with characteristic propagation constant a and wave impedance Za . Hankel functions of the second kind are written as Hm (z) and spherical Hankel functions of the second kind as hm (z); the arguments k0a of Bessel and Hankel functions are dropped. The abbreviation z = j · a a will be used, and an = a /k0, Zan = Za /Z0 . A prime at functions indicates the derivative; a prime or double prime at an or Zan indicates the real or imaginary part, respectively. An asterisk indicates the complex conjugate. Quantity

Symbol

Factors

Dm = Cm = Tm = ƒm =

Cylinder

−j Gm + m k0 a Jm − Jm+1 − −j Gm + m k0 a Hm − Hm+1

−j Gm + m k0 a Hm − Hm+1 − −j Gm + m k0 a Hm − Hm+1

Sphere

−j Gm + m k0 a jm − jm+1 − −j Gm + m k0 a hm − hm+1

−j Gm + m k0 a hm − hm+1 − −j Gm + m k0 a hm − hm+1

cos (m˜) ⎧ ⎨ 1; m=0 ⎩ 2; m>0

Pm (cos ˜)

2a

a2

Cross section

S=

Modal admittance

Z0 J0m (z) Gm = j Za Jm (z) (z = j a a)

Incident wave

pi (r,˜) =

m0

Scattered wave ps (r,˜) =

Scattered far field

ps (r,˜)!

m0

p(r,˜) =

j

Dm ƒm (−j)m Tm Hm (k0 r)

m0

ƒm (−j)m Tm

Jm (k0 r) + Dm Hm (k0 r)

Z0 j0m (z) Za jm (z)

e−j k0 rcos ˜

2 j −j k0 r − Cm ƒm Tm e k0 r m0

Total ext. field

ƒm (−j)m Tm Jm (k0 r)

2m+1

m0

m0

ƒm (−j)m Tm jm (k0 r) Dm ƒm (−j)m Tm hm (k0 r)

j −j k0 r Cm ƒm Tm e k0 r m0

=: m0

e−j k0 r ¥(˜) k0 r ƒm (−j)m Tm

jm (k0 r) + Dm hm (k0 r)

194

E

Scattering of Sound

Table continued:

Quantity Scattering cross section Absorption cross section

Symbol

Cylinder

Sphere

Qs = S

1 ƒm j1 − Cm j2 2 k0 a m

1 ƒm j1 − Cm j2 (k0 a)2 m

Qa = S

1 ƒm 1 − jCm j2 2 k0 a m

1 ƒm 1 − jCm j2 2 (k0 a) m

Approximations: k0 a 1 Modal admittance

Scattering cross section

G0 =

Qs = S

k0 a an 2 Zan

Absorption cross section, sphere

1 ƒm j1 − C0 j2 (k0 a)2 m an 2 4(k0 a)4 1 + j = 9 Zan

1 ƒm j1 − C0 j2 2 k0 a m =

Absorption cross section, cylinder

k0 a an 3 Zan

2 (k0 a)3 16 1+

an Zan

0 −

an Zan

1 + j an Zan 2 +3 1 − 2j an Zan

00 2

an Zan

0

Qa = S

(1 − jC0 j2 ) k0 a = 2k0 a 2

Qa = S

4 k0 a 1 + j an Zan an (1 − jC0 j2 ) = − + 3 Im 1 + j (k0 a)2 3 Zan 1 − 2j an Zan

1 + k0 a ln 0:890 k0 a 2

(k0 a)2 an 00 1 + k0 a ln 0:890 k0 a 1 + 2 Zan

Scattering of Sound

E

Table continued:

Quantity

Symbol

Cylinder

Sphere

Approximations: k0 a 1 Total ext. field at surface, sphere

p(a; ˜) =

1 − j k0 a

Radial particle velocity at surface, sphere

vr (a,˜) =

−k0 a an 3j + cos (˜) 3 Zan j + 2 an Zan

Scattered field directivity in the far field, cylinder

¥(˜) =

Scattered field directivity ¥(˜) = in the far field, cylinder

3 an Zan cos (˜) j + 2 an Zan

) an an Zan − j −j 3 +2 (k0 a) − 1 + j cos (˜) 8 k0 Zan an Zan + j

) k02 a3 an Zan − j an +2 − 1+j cos (˜) 3 Zan an Zan + j

Some numerical examples will illustrate the quantities and relations given above.

∗)


195

E

196

Scattering of Sound

20 lg|pg|

0

-5 dB -10 4

-15 2 -20 0

-4

y/a

-2 -2

0 x/a

2 4

-4

Total sound pressure level around a cylinder of homogeneous, bulk reacting, porous glass fibre material. Sound incidence from the left-hand side. f = 1000 [Hz]; a = 0.25 [m]; Ÿ = 45◦ ; ¡ = 10000 [Pa s/m2 ]; mhi = 32 (upper summation limit) 20 lg|ps|

0

dB

-10

4

-20 2 0

-4

y/a

-2 -2

0 x/a

2 4

-4

Scattered sound pressure level around a cylinder of homogeneous, bulk reacting, porous glass fibre material. Sound incidence from the left-hand side. f = 1000 [Hz]; a = 0.25 [m]; Ÿ = 45◦ ; ¡ = 10000 [Pa s/m2 ]; mhi = 32 (upper summation limit)

Scattering of Sound

E

197

Qa/2a 2 1 0.5

0.2 0.1 0.05

0.02 0.01 0.1

0.5

1

5 k0a

10

50

100

Normalised absorption cross section Qa/2a of a cylinder of homogeneous,bulk reacting, porous glass fibre material. Ÿ = 0◦ ; mhi = 32 (upper sum limit); parameter values of curves: R = ¡ · a/Z0 : R={0.2, 0.5, 1., 2.} (dashes are shorter in that order)

Qa/2a 2 1 0.5

0.2 0.1 0.05

0.02 0.01 0.1

0.5

1

5 k0a

10

50

100

Normalised absorption cross section Qa /2a of a cylinder of homogeneous, porous glass fibre material, made fully locally reacting Ÿ = 0◦ ; mhi = 32 (upper sum limit); parameter values of curves: R = ¡ · a/Z0: R={0.2, 0.5, 1., 2.} (dashes are shorter in that order)

E

198

Scattering of Sound

Qs/2a, fully loc.cyl., G=var.

10

1

0.1

0.01 0.01

0.05 0.1

k0a

0.5

1

5

10

Normalised scattering cross section of a locally reacting cylinder with given values of the normalised surface adimittance G={0, 0.5j, 1j, 2j, 4j} (curves from low to high in that order).The graph illustrates the scattering resonances (the exterior vibrating mass resonates with the resilience of the surface)

E.3 Multiple Scattering at Cylinders and Spheres


Consider an “artificial medium” consisting of an arrangement (preferably random) of hard scatterers (cylindrical or spherical) with a root mean square average radius a and mutual distances such that the “massivity” ‹ of the arrangement (fraction of the space occupied by the scatterers) holds. A sound wave propagates through that medium with an effective (complex) wave number keff and wave impedance Zeff given by 2 keff eff Ceff = · 2 0 C0 k0

;

Z2eff eff Ceff = / , 2 0 C0 Z0

(1)

with the effective density eff and compressibility Ceff 8‹ eff = 1+j 0 (k0 a)2

n=1,3,5...

Dn

;

Ceff 8‹ = 1+j · 0.5 D0 + Dn , (2) C0 (k0a)2 n=2,4,6...

where the coefficients Dn are taken from the table in

> Sect. E.2, and C0= 1/(0 c20 ).

(3)

Scattering of Sound

E

199

E.4 Cylindrical Wave Scattering at Cylinders A line source at Q is parallel to the axis of a locally reacting cylinder with radius a and (normalised) surface admittance G. The field point is at P. The source distance rq defines two radial zones (a),(b). The sound field is composed of the sum of the source free field pQ and the scattered field ps : p(r, ˜ ) = pQ (r) + ps (r, ˜ ).

′

(1)

ϑ

The source free field pQ is transformed with the addition theorem for Hankel functions to the co-ordinates (r,˜ ): ⎧ (2) ⎪ ⎪ ⎨ m≥0 ƒm · Jm (k0 r) · Hm (k0 rq ) · cos (m˜ ); pQ (r) = P0 · H(2) 0 (k0 r ) = P0 · ⎪ ⎪ ƒm · Jm (k0 rq ) · H(2) ⎩ m (k0 r) · cos(m˜ ); m≥0

with

ƒm =

1 ; m = 0, 2 ; m > 0.

in (a), in (b),

(2)

(3)

Formulation of the scattered field: ps (r, ˜ ) = P0 ·

m≥0

am · ƒm · Jm (k0 rq ) · H(2) m (k0 r) · cos (m ˜ ).

(4)

E

200

Scattering of Sound

! The boundary condition −Z0 vQr + vsr = G · pQ + ps gives the amplitudes am = −

j · Jm (k0 a) + G · Jm (k0a) j · Hm (2) (k0 a) + G · H(2) m (k0 a)

− −−−→ − G→0

Jm (k0a) · H(2) m (k0 rq ) (2)

Jm (k0rq ) · Hm (k0 a)

− −−−−−→ − |G|→∞

·

H(2) m (k0 rq ) Jm (k0 rq )

= ahm

Jm (k0a) · H(2) m (k0 rq ) Jm (k0rq ) · H(2) m (k0 a)

(5)

= asm

with the special cases G→0 (hard) and |G| → ∞ (soft). In a different notation, the scattered field is (2) ƒm · cm · H(2) ps (r, ˜ ) = −P0 · m (k0 rq ) · Hm (k0 r) · cos (m˜ ), m≥0

m · Jm (k0a) − j · Jm+1 (k0a) k0 a . cm = m (2) · H(2) G+ m (k0 a) − j · Hm+1 (k0 a) k0 a

G+

(6)

|pg/P0|; exact

1

0.75

0.5 0.25 - 10

0 10

-5 0

5 0 k0x

5 -5 - 10

k 0y

10

Sound pressure magnitude from a line source around a locally absorbing cylinder. G = 0.5 − 2 · j; k0 a = 2 ; k0 rq = 6.1; upper summation limit mhi = 8

Scattering of Sound

E

201

E.5 Cylindrical or Plane Wave Scattering at a Corner Surrounded by a Cylinder

See also: Mechel, Improvement of Corner Shielding by an Absorbing Cylinder (1999)

The apex line of a corner with hard flanks at ˜ = 0 and ˜ = ˜0 ≤ 2 is surrounded by a locally reacting cylinder of radius a and (normalised) surface admittance G. The line source at Q has the co-ordinates (rq , ˜q ). The sound field is formulated as a mode sum: R† (kr) · T(†˜ ). p(r, ˜ , z) = Z(kz z)

(1)

†

The factor Z(kz z) may be any of the functions e±jkz z , cos(kz z), sin(kz z) or a linear combination thereof. If kz = 0, set k 2 = k02 − kz2 . Below it will be supposed (for simplicity) that kz = 0,

Z(kz z) = 1.

The azimuthal functions are T(†˜ ) = cos(†˜ ), and the azimuthal wave numbers satisfy the characteristic equation (†n ˜0 ) · tan (†n˜0 ) = 0

(2)

with the solutions †n ˜0 = n · ; n = 0, 1, 2, . . .. Field formulations in the two radial zones (a),(b): (1) (2) An · H(2) a ≤ r ≤ rq pa (r, ˜ ) = †n (krq ) · H†n (kr) + rn · H†n (kr) · cos (†n ˜ ); n≥0

pb (r, ˜ ) =

n≥0

(2) (2) An · H(1) †n (krq ) + rn · H†n (krq ) · H†n (kr) · cos (†n ˜ );

rq ≤ r < ∞

(3)

202

E

Scattering of Sound

with the modal reflection factors at the cylinder: k (1) H (ka) k0 †n rn = − . k (2) G · H(2) (ka) + j H (ka) †n k0 †n G · H(1) †n (ka) + j

(4)

The mode amplitudes An follow from the source condition: An =

Z0 q pQ (0) cos (†n ˜q ) = cos (†n ˜q ). k 0 rq 4 ˜0 Nn ˜0 Nn H(2) 0 (krq )

(5)

In the second form the source strength is given by the free field source pressure pQ (0) in the corner apex line. The terms Nn are the mode norms: 1 Nn = ˜0

˜0 0

sin (2†n˜0 ) 1 1 ; 1+ cos (†n ˜ ) d˜ = = 2 2†n˜0 1/2; 2

n=0 . n>0

(6)

An alternative formulation, showing separately the contributions of the corner alone and of the cylinder with radius a, is pa (r, ˜ ) = pa,Corner + pa,Cyl =

2 pQ (0) cos (†n ˜q ) · H(2) †n (krq ) · J†n (kr) · cos (†n ˜ ) ˜0 H(2) N n 0 (krq ) n≥0 −

(7)

cos (†n ˜q ) 2 pQ (0) (2) Cn · · H(2) †n (krq ) · H†n (kr) · cos (†n ˜ ) , (2) ˜0 H0 (krq ) n≥0 Nn

pb (r, ˜ ) = pb, Corner + pb, Cyl =

2 pQ (0) cos (†n ˜q ) · J†n (krq ) · H(2) †n (kr) · cos (†n ˜ ) ˜0 H(2) N n (kr ) q n≥0 0 −

(8)

cos (†n˜q ) 2 pQ (0) (2) Cn · · H(2) †n (krq ) · H†n (kr) · cos (†n ˜ ) . ˜0 H(2) N n (kr ) q n≥0 0

with the following coefficients: k J (ka) k0 †n Cn = ; k (2) G · H(2) H†n (ka) †n (ka) + j k0 J†n (ka) J†n (ka) ; Cn −−−−−−→ (2) ; Cn −−−−→ (2) G→0 H |G|→∞ H (ka) †n (ka) †n G · J†n (ka) + j

(9) Cn −−−−→ 0. ka→0

Level reduction in the zones i = a, b by the cylinder: Li (r, ˜ ) = 20 · lg 1 + pi,Cyl pi,Corner ; i = a, b.

(10)

Scattering of Sound

E

203

Plane wave incidence from the direction ˜q : p(r, ˜ ) = pCorner + pCyl =

ej †n /2 2 pQ (0) · J†n (kr) · cos (†n ˜q ) · cos (†n ˜ ) ˜0 Nn n≥0 −

(11)

ej †n /2 2 pQ (0) Cn · · H(2) †n (kr) · cos (†n ˜q ) · cos (†n ˜ ). ˜0 N n n≥0

A special case of a thin screen is obtained with ˜0 = 2 and †n = n/2; n = 0, 1, 2, . . . ϑ0=270° ; ϑq=225° ; ka=5 ; kr=50

3 -16 2 Im{G}

-18

1 0 -4 -8

-10

0

-12 ΔL(r,0)= -14

-15

-1

-2 -16 -3

0

1

2

3

4

5

6

Re{G}

Contour lines in the complex plane of G of the sound pressure level reduction L(r, 0) by an absorbent cylinder, at the shadowed flank ˜ = 0 at a distance kr = 50, for a plane wave incident under ˜q = 225◦ , on a rectangular convex corner (˜0 = 270◦ ), with a cylinder radius given by ka = 5 Below it is assumed that no cylinder exists at the corner line. The sound field in the shadow zone of a corner decays monotonously with the distance to the corner and on approaching the shadowed flank.If the distance between two consecutive convex corners

204

E

Scattering of Sound

is not too small, the sound field at the later corner can approximately be assumed to be generated by a line source situated at the earlier corner. Thus the sound shielding by buildings can be evaluated by iteration over the surrounded corners. The required intermediate values L(kr, ˜ = 0) = 20 · lg|p(kr, 0)/pQ(0)| can be evaluated by regression over kr for corner parameters ˜0 (r = distance between earlier and later corner): L(kr, 0) = a0 + a1 · x + a2 · x2 + . . .

ai =

⎧ ai (˜0 , krq , ˜q ) ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ai (˜0 , ˜q )

;

i=

;

x = lg (kr)

⎧ 0, 1, 2 ; ⎪ ⎪ ⎨

line source

⎪ ⎪ ⎩ 0, 1, 2, 3;

(12)

plane wave

In the case of a line source at (rq , ˜q ), the coefficients ai are expanded as follows: For ˜0 = 2: ai ( krq , ˜q ) = b0,0 + b1,0 · z + b2,0 · z2 + b3,0 · z3 + b0,1 · y + b0,2 · y 2 + b1,1 · z · y + b1,2 · z · y 2 + b2,1 · z2 · y + b2,2 · z2 · y 2 z = lg (krq );

y = (˜0 − ˜q )rad ;

bm,n = bm,n (˜0 , i);

i = 0, 1, 2 ;

(13)

line source

˜0 = 2

For ˜0 = 2 (note change in sign of y): ai ( krq , ˜q ) = b0,0 + b1,0 · z + b2,0 · z2 + b3,0 · z3 + b0,1 · y + b0,2 · y 2 + b0,3 · y 3

(14)

+ b1,1 · z · y + b1,2 · z · y 2 + b1,3 · z · y 3 + b2,1 · z2 · y + b2,2 · z2 · y 2 + b2,3 · z2 · y 3

z = lg (krq )

;

y = (˜q − ˜0 )rad

bm,n = bm,n (˜0 , i) ;

;

i = 0, 1, 2 ;

line source

˜0 = 2

The following diagrams show the sound pressure level at the shadowed flank (˜ = 0), evaluated with mode sums (points) and from the above regressions (lines).

Scattering of Sound

ϑ0=270° ; krq=1.122

20lg|p(kr,0)/pQ(0)| 0 dB -5

ϑq= 180°

-10 225° -15 270° -20 -25

1

2

5

10

20

50

100

kr

A line source at a small distance krq = 1.122 ϑ0=270° ; krq=8.913

20lg|p(kr,0)/pQ(0)| 0 dB

ϑq= 180°

-5 -10

225°

-15 270° -20 -25

1

2

5

10

20

kr

50

100

A line source at a medium distance krq = 8.913 ϑ0=270°

20lg|p(kr,0)/pQ(0)| 0

ϑq=

dB -5

180° 195° 210°

-10 225° 240°

-15

255° 270° -20 -25

1

2

5

10

20

kr

50

Plane wave incidence from different directions ˜q

100

E

205

206

E

Scattering of Sound

In the case of a plane wave from ˜q , the coefficients ai are expanded similarly. The coefficients are: Line source; ˜0 = 270◦ . a0 = − 3.508914089 + 2.522196950· z − 1.883348105·z2 + 0.4967954203 · z 3 + 0.02544989020 · y + 0.8345544874·y 2 + 0.3679533261 · z · y + 0.3777085214 · z · y 2 − 0.09024391927 · z2 · y − 0.1947245060 · z2 · y 2 a1 = − 8.048512868 − 0.009541528038·z + 0.5769454995·z2 − 0.3051673769·z3 + 0.3821548481 · y + 0.2651097458 · y 2 − 0.6159740550 · z · y + 3.292101514 · z · y 2 + 0.2958236606 · z2 · y − 0.8724895618 · z2 · y 2

(15)

a2 = − 0.6310612470 − 0.1423972091·z + 0.01678228228·z2+ 0.04756468413·z3 − 0.1407494337 · y − 0.08464913202 · y 2 + 0.5043646313 · z · y − 1.107819931 · z · y 2 − 0.7181971540 · z2 · y + 0.8256508092 · z2 · y 2 Line source; ˜0 = 225◦ : a0 = − 0.09540693872 + 3.1825983742·z − 2.211437627·z2 + 0.5797754874·z3 − 1.521470673 · y + 2.965063818 · y 2 + 3.512579747 · z · y − 4.438049225 · z · y 2 − 1.739392722 · z2 · y + 2.146307680 · z2 · y 2 a1 = −7.323833505 + 3.020507185·z − 0.08553799890·z2 − 0.4849823570·z3 + 3.284316693 · y − 4.259941774 · y 2 − 7.601695001 · z · y + 16.521364584 · z · y 2 + 4.276159321 · z2 · y − 7.413341455 · z2 · y 2 a2 = −0.8765745279 − 1.203362984·z + 0.3513458955·z2 + 0.03401430131·z3 − 1.529012091 · y + 1.973388860 · y 2 + 4.011726713 · z · y − 7.347839244 · z · y 2 − 2.352498508 · z2 · y + 4.809651759 · z2 · y 2

(16)

Scattering of Sound

E

207

Line source; ˜0 = 360◦ (y = (˜q − ˜0 )rad ): a0 = −9.074649329 + 1.352053882·z − 1.152752755·z2 + 0.3306858082·z3 − 0.1455294858 · y + 0.5960196681 · y 2 − 0.008012063348 · y 3 + 0.6291707040 · z · y + 0.8532798218 · z · y 2 + 0.1325783868 · z · y 3 − 0.2481813528 · z2 · y − 0.3078462069 · z2 · y 2 − 0.04647002117 · z2 · y 3 a1 = −8.796878642 − 0.4605118745·z + 0.2689865167· z2 − 0.03586357874·z3 + 0.6092864775 · y + 0.8416378006 · y 2 + 0.1356184578 · y 3 − 1.530126829 · z · y − 1.589422436 · z · y 2 − 0.6606164056 · z · y 3 + 0.7158176859 · z2 · y + 0.6233222389 · z2 · y 2 + 0.2175905849 · z2 · y 3

(17)

a2 = −0.4550505284 + 0.1300965967·z − 0.06995829336·z2 − 0.03107882535·z3 − 0.2438201115 · y − 0.3079336093 · y 2 − 0.05023497076 · y 3 + 0.7849193467 · z · y + 0.7181240230 · z · y 2 + 0.2441835343 · z · y 3 − 0.7782697727 · z2 · y − 0.7604596554 · z2 · y 2 − 0.2139967639 · z2 · y 3 The corresponding expansions for plane wave incidence are as follows: Plane wave; ˜0 = 270◦ : a0 = −2.389774901 − 0.1395262072·y + 2.414652403·y 2 − 2.082638349·y 3 + 1.727712526 · y 4 − 0.5024423058 · y 5 a1 = −6.387181695 + 1.429899038·y − 6.294135235·y 2 + 22.1143696537·y 3 −19.7690393085 · y 4 + 5.565221750 · y 5 a2 = −2.552124597 − 2.342005059·y + 14.7867785184·y 2 − 38.0273605653·y 3 + 36.1731866937 · y 4 − 10.7284415031 · y 5 a3 = 0.5883191145 + 0.7953788225 · y − 5.524362056 · y 2 + 13.7842093181 · y 3 − 13.6837211307 · y 4 + 4.349290260 · y 5

(18)

Plane wave; ˜0 = 225◦ : a0 =

1.568078212 + 0.04605330206·y + 0.9901780451·y 2 + 1.888227817·y 3 −1.204873300 · y 4 − 0.5572513587 · y 5

a1

−3.081659731 − 0.5413285417·y + 8.6690503927·y 2 − 24.2944777516·y 3 + 19.7879485353 · y 4 + 0.7720284647 · y 5

=

a2 = a3 =

−3.729226627 + 1.080888001·y − 4.076862638·y 2 + 52.4234791286·y 3 − 59.9341011899 · y 4 + 11.3285552296 · y 5 0.6710694247 − 0.5457149106·y + 3.322227192·y 2 − 28.5329448742 · y 3 + 41.1343130200 · y 4 − 13.8714159690 · y 5

(19)

E

208

Scattering of Sound

Plane wave; ˜0 = 360◦ (y = ˜0 − ˜q rad): a0 = −8.619029147 − 0.04914242523·y + 0.98893551678·y 2 − 0.09677973083·y 3 +0.04314542296 · y 4 − 0.007092745198 · y 5 a1 = −8.495414675 + 0.6349841469·y − 0.9414382840·y 2 + 0.9074414634·y 3 −0.03120088534 · y 4 − 0.03111306942 · y 5 a2 = −1.051533957 − 2.397034644·y + 6.111232667·y 2 − 5.903508575·y 3 + 2.008747402 · y 4 − 0.2088099622·y 5 a3 = 0.2143116375 + 1.396801316·y − 3.902386481·y 2 + 3.949226446·y 3 −1.549939902·y 4 + 0.2033883155·y 5

(20)

E.6 Plane Wave Scattering at a Hard Screen The hard screen is a special case with ˜0 = 2 of

> Sect. E.5; > Sect. E.7.

A plane wave is incident in the direction ˜q . Its sound pressure at the screen corner is pQ (0). The radial wave number component is k (see > Sect. E.5).

r P

ϑ

ϑq pi

ϑ0

The sound pressure around the screen is √ ˜ − ˜q 1 + j j kr cos (˜ −˜q ) 1 − j e +C 2kr cos p(r, ˜ ) = pQ (0) 2 2 2 √ ˜ − ˜q −jS 2kr cos 2 √ ˜ + ˜q 1−j +C 2kr cos + ej kr cos (˜ +˜q ) 2 2 √ ˜ + ˜q 2kr cos −jS 2

(1)

Scattering of Sound

E

209

with the Fresnel integrals defined by S(x) =

2

x

2

sin(t ) dt

;

C(x) =

0

2

x

cos(t2) dt.

(2)

0

E.7 Cylindrical or Plane Wave Scattering at a Screen with an Elliptical Cylinder Atop

See also: Mechel (1997) for notation, formulas and evaluation of Mathieu functions.

A hard, thin screen of height h has a locally absorbing, elliptical cylinder at its top; the surface admittance of the cylinder is G; its long and short axes are 2a, 2b. The eccentricity of the ellipse is c. A line source parallel to the axis of the ellipse is at Qu with the elliptical co-ordinates (q , œq ). For q → ∞; a plane wave incidence prevails. First,the height h is taken as h → ∞; then the arrangement is mirror-reflected at y = −h, and the scattering of the field from the mirror-reflected source is evaluated, then both fields are superposed.

Transformation between Cartesian (x, y) and elliptic-hyperbolic (, ˜ ) co-ordinates: x = c · cosh · cos ˜ ; y = c · sinh · sin ˜ x ± jy = c · cosh( ± j˜ ).

;

z = z;

(1)

Also used are the co-ordinates (, œ) with œ = ˜ + /2

;

cos ˜ = sin œ

;

sin ˜ = − cos œ.

(2)

210

E

Scattering of Sound

Geometrical parameters (with c on the elliptical cylinder): a = cosh c c

b = sinh c c

;

;

b = tanh c a

;

c=

a . cosh c

(3)

With a separation p(, ˜ , z) = R() · T(˜ ) · Z(z) and an axial factor Z(z) proportional to either one or a linear combination of the functions Z(z) = e±jkz z ; cos(kz ); sin(kz z) given by a wave with a wave number kz leading to the wave number k in the plane normal to the axis with k 2 = k02 − kz2 , the axial factor Z(z) can be dropped; only p(, ˜ ) will be given. Sound field from a line source The line source is placed at (q , ˜q ) or (q , œq ). Its polar distance to the origin is rq with (4) rq2 = xq2 + yq2 = c2 · cosh2 q · sin2 œq + sinh2 q · cos2 œq . When it has a volume flow q (per unit length), then it will produce the sound pressure in free space at the position of the origin: 1 Z0 k0 rq · q · H(2) (5) 0 (krq ). 4 General field formulations in the two zones with c ≤ < q and q < < ∞, respectively, separated from each other by the elliptic radius q of the line source position, are (integer summation index m) (2) p1 (, œ) = am · Hcm/2 (2) (q ) · Hc(1) m/2 () + rm · Hcm/2 () · cem/2 (œ), pQ (0) =

m≥0

p2 (, œ) =

m≥0

(1) (2) am · Hc(2) m/2 () · Hcm/2 (q ) + rm · Hcm/2 (q ) · cem/2 (œ).

(6)

The term amplitudes am follow from the source condition; they are am =

Z0 q · k0 c cen/2(œq ) · Icm,n (q ) 2 n≥0

(7)

with the integrals (about which see below) 1 Icm,n () : = 2

2 sinh2 + cos2 œ · cem/2(œ) · cen/2 (œ) dœ.

(8)

0

The modal reflection factors rm (at the cylinder surface) are obtained from the boundary condition at the cylinder with the form bm = rm · am from the system of equations (with explicitly known am ): j (2) (2) (2) Hc (c ) bm · Hcm/2 (q ) · Gc · Hcm/2 (c ) · Icm,n (c ) + ƒm,n · 2k0c m/2 m≥0 (9) j (2) (1) (1) Hc (c ) am · Hcm/2 (q ) · Gc · Hcm/2(c ) · Icm,n (c ) + ƒm,n · =− 2k0c m/2 m≥0 (ƒm,n the Kronecker symbol).

Scattering of Sound

E

211

First special case: In the first special case, the cylinder is supposed to be rigid, Gc = 0. The system of equations for the rm simplifies to bm = rm · am = −am ·

Hc(1) m/2 (c ) (2)

Hcm/2(c )

.

(10)

Second special case: In the second special case with c = 0, in which the cylinder degenerates to an absorbing strip of width 2c,the equations formally remain unchanged, but the integrals Icm,n (c ) simplify drastically, as will be seen below. Third special case: The third special case is a combination of the first and second: Gc = 0 and c = 0; the cylinder changes to a rigid strip. Then (1)

bm = rm · am = −am ·

Hcm/2(0) Hc(2) m/2 (0)

.

(11)

There still remain the integrals Icm,n () to be evaluated. They can be expressed in terms of the Fourier coefficients AŒ of the even azimuthal Mathieu functions ce‹ (œ), BŒ of the odd azimuthal Mathieu functions se‹ (œ), which are needed at any rate for the evaluation of such Mathieu functions. When ‹ and Œ are integers and both even, then: Icm.n () =

(−1)(‹+Œ)/2 (−1)s+ A2s (‹) · A2 (Œ) · I|s−| () + Is+ () 4 s,≥0

(12)

with 2 Ii () =

cos(2iœ) ·

sinh2 + cos2 œ dœ

0

/2 = 4 cos(2iœ) · sinh2 + cos2 œ dœ

(13) ;

2i = even

0

=0

;

2i = odd

and the values Ii () = 2(−1)i+1 · cosh i≥1

;

>0 ;

(2(i + k) − 3)!! · (2(i + k) − 1)!! 1 ; k! · (2i + k)! (2 cosh )2(i+k) k≥0

(0)!! = (−1)!! = 1 ;

(14)

(2n + 1)!! = 1 · 3 · 5 · . . . · (2n + 1)

with the special value I0 () = 4 cosh · E(1/ cosh ), (E(z) the exponential integral), and for = 0: ⎧ ⎪ ; 2i + 1 = 0; ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ sin(2i − 1)/2 sin(2i + 1)/2 (15) Ii (0) = 2 + ; 2i = even; ⎪ ⎪ 2i − 1 2i + 1 ⎪ ⎪ ⎪ ⎪ ⎩ 0 ; 2i = odd = 1.

212

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Scattering of Sound

When ‹ and Œ are integers and both odd, then Icm,n () =

(−1)(‹+Œ−2)/2 (−1)s+ B2s+1 (‹) · B2+1(Œ) · I|s−| () + Is++1 () . 4 s,≥0

(16)

When ‹ and Œ are integers, one even the other odd, then Icm,n () = 0. When ‹ is half-valued, Œ is an integer, or inversely, then Icm,n () = 0. When ‹, Œ are both half-valued, with ‹ = ‹ + 1/2, Œ = Œ + 1/2, and both ‹ , Œ even or odd: +∞ 1 (−1)s+ C2s (‹) · C2 (Œ) · I|(‹ −Œ )/2+s−| (). Icm,n () = 4 s,=−∞

(17)

When ‹, Œ are both half-valued, with ‹ even, Œ odd, or inversely: Icm,n () =

+∞ 1 (−1)s+ C2s (‹) · C2 (Œ) · I|(‹ +Œ +1)/2+s+| (). 4 s,=−∞

One gets in the limit → ∞ with

(18)

sinh2 + cos2 œ = cosh2 − sin2 œ → cosh : (19)

⎧ ⎪ 2 cosh ; i = 0; ⎪ ⎪ ⎪ ⎪ /2 ⎨ Ii () → 4 cosh cos(2iœ) dœ = 0; i=

0, ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎩ 2 cosh · sin(i)/(i); else.

integer;

(20)

This gives in the above cases of non-zero Icm,n () 1 (‹+Œ)/2 cosh · A2s (‹) · A2 (Œ) + A0(‹) · A0(Œ) , Icm,n () → (−1) 2 s,≥0;s= 1 B2s+1 (‹) · B2+1(Œ), Icm,n () → (−1)(‹+Œ)/2−1 cosh · 2 s,≥0;s= Icm,n () = Icm,n () =

∞

1 cosh · 2 1 cosh · 2

(−1)s+ C2s (‹) · C2 (Œ),

(21)

s,=−∞

(‹ −Œ )/2+s−=0 ∞

(−1)s+ C2s (‹) · C2 (Œ).

s,=−∞

(‹ +Œ +1)/2+s+=0

Sound field from a plane wave This case is treated as the limit q → ∞. The polar radius rq of the source position √ approaches rq → c · cosh q , whence 2 q cosh q → krq . One further replaces Z0 q =

4 pQ (0) (2)

k0rq · H0 (krq )

.

(22)

Scattering of Sound

E

213

Finally, one replaces above am · Hc(2) −−−−→ 2pQ (0) · ejm/4 · m/2 (q ) − q →∞

and in

cen/2 (œq ) · Icm,n ,

(23)

n≥0

(2) bm · Hc(2) m/2 (q ) = rm · am · Hcm/2 (q ).

(24)

Rigid screen with a mushroom-like hat The cylindrical body atop the screen has a semielliptical shape; its surface is curved and rigid on the upper side and flat and absorbing on its lower side with the admittance Gc . The boundary condition at the cylinder is

Gc =

c =

⎧ ⎪ ⎨ 0;

/2 ≤ œ ≤ 3/2 ,

⎪ ⎩ G ; else , c ⎧ ⎪ ⎨ c ; /2 ≤ œ ≤ 3/2 , ⎪ ⎩ 0;

(25)

else ,

i.e.: ⎧ ⎪ ⎨ 0

∂p1 = ⎪ ∂ 2 ⎩ −G · p k0 c sinh + cos2 œ c 1

j

; = c

; /2 ≤ œ ≤ 3/2,

; =0

; else.

(26)

E

214

Scattering of Sound

The term amplitudes am remain as above; the system of equations for the bm = rm · am changes to m≥0

bm · Hc(2) m/2 (q )

1 3/2 − cem/2 (œ) · cen/2(œ) dœ · 2 /2 1 − j k0c · Gc · Hc(2) (0) | cos œ| · cem/2(œ) · cen/2 (œ) dœ m/2 2 else (2) + ƒm,nNm Hcm/2 (0) am · Hc(2) =− m/2 (q ) Hc(2) m/2 (c )

Hc(2) m/2 (0)

(27)

m≥0

3/2 1 · − cem/2 (œ) · cen/2(œ) dœ 2 /2 1 (1) | cos œ| · cem/2(œ) · cen/2 (œ) dœ − j k0c · Gc · Hcm/2(0) 2 else (1) + ƒm,nNm Hcm/2 (0) .

Hc(1) m/2 (c )

Hc(1) m/2 (0)

The Nm are the azimuthal mode norms: 1 Nm = 2

2

ce2m/2 (œ) dœ.

(28)

0

E.8 Uniform Scattering at Screens and Dams See also: Mechel, A Uniform Theory of Sound Screens and Dams (1997), see Mechel, Mathieu Functions (1997) for notation, formulas and evaluation of Mathieu functions

This section describes the plane wave scattering • at a “high” absorbent dam, with the limit case of • a thin absorbent screen; • at a “flat” absorbent dam, with the limit case of • a flat absorbent strip in a rigid baffle wall. A semicircular absorbent dam could also be treated as a limiting case of this uniform theory (it will, however, be discussed separately in > Sect. 10). All objects are situated on a hard ground. All absorbent objects are locally reacting with a normalised surface admittance G. The distinction between “high” and “flat” dam is necessary because of the orientation of the axes and co-ordinates of the elliptical cylinder, with which the objects are modelled.

Scattering of Sound

E

215

A high dam with its elliptical-hyperbolic co-ordinate system. The plane x = 0 is the hard ground. p+e is the incident plane wave; p−e is the mirror-reflected wave. The height of the dam is h; its width at ground level is 2b

A flat dam with its elliptical-hyperbolic co-ordinate system. The plane y = 0 is the hard ground. p+e is the incident plane wave; p−e is the mirror-reflected wave.The height of the dam is b; its width at ground level is 2h

216

E

Scattering of Sound

Limit cases are:

A thin, absorbent screen with its elliptical-hyperbolic co-ordinate system. The plane x = 0 is the hard ground. p+e is the incident plane wave; p−e is the mirror-reflected wave. The height of the screen is h

An absorbent strip in a hard baffle with its elliptical-hyperbolic co-ordinate system. The plane y = 0 is the hard baffle wall. p+e is the incident plane wave; p−e is the mirrorreflected wave. The width of the strip is 2h

Scattering of Sound

E

217

The co-ordinate transformation between Cartesian (x,y) and elliptic-hyperbolic coordinates (˜ ) is, with the eccentricity of the ellipses c, ⎫ x = c · cosh · cos ˜ ⎪ ⎬ ; 0 ≤ < ∞ ; − ≤ ˜ ≤ + (1) ⎪ y = c · sinh · sin ˜ ⎭ and in the backward direction: + j˜ = area cosh( + j†) = ln + jn ±

( +

j†)2

−1 ,

(2)

with = x/c, † = y/c. Geometrical parameters are with the elliptical radius c on the object: # h h c = cosh c . (3) ; b h = tanh c ; c = cosh c b c = sinh c The geometrical shadow limit for plane wave incidence with an angle Ÿ 0 is given by h/c − tgŸ0 · sinh · sin ˜ = cosh p · cos ˜ .

(4)

90° 80 3.2

ϑ

2.0 60

1.6 1.2 1.0 0.8

40

0.6 0.5 0.4 0.3

20

0.2 0.1

0 -90° -75

-50

-25

0

Geometrical shadow limits for h/c = 1

25

50

ρ=0.05

Θ0

75

90°

218

E

Scattering of Sound

With the special cases = 0:

˜ = arccos (h/c) −−−−−→ 0, h/c→1

Ÿ0 = 0: ˜ = arccos 1:

h/c , cosh

˜ ≈ − arctan

(5)

h/c . tan Ÿ0 tanh

+ − Let p± e be the incident wave and the mirror-reflected wave at the ground. pe = pe + pe + − then is the “exciting” wave with hard ground, and pe = pe + r · pe the exciting wave with absorbing ground having a reflection factor r. The total field is composed of the sum p = pe + prs of the exciting wave pe and a “reflected plus scattered” wave prs .

The following description uses the second principle of superposition from > Sect. B.10, i.e. the task is subdivided in two subtasks (ß) = (h), (w), in which the plane of symmetry through the scattering object (where it is sound transmissive) is considered first as hard (h), second as soft (w). The reflected plus scattered wave is marked and () () () () decomposed in both subtasks as prs = pr + ps , where pr is the reflected wave at the plane of symmetry, with hard reflection for (ß) = (h) and soft reflection for (ß) = (w), () (h) respectively, i.e. p(w) r (y) = −pr (y). The component ps is the “truly” scattered wave. At the high dam, the co-ordinate normal to the plane of symmetry is y → ˜ . According to the principle of superposition the sound field on the front side (side of sound incidence) is 1 (h) pfront (˜ < 0) = pe (˜ ) + ps (˜ ) + p(w) s (˜ ) 2 (6)

1 (h) (h) (w) (w) pe (˜ ) + pr (˜ ) + ps (˜ ) + pe (˜ ) + pr (˜ ) + ps (˜ ) = 2 and the transmitted sound field on the back side is: 1 (h) p (−˜ ) − p(w) pback (˜ > 0) = pe (˜ ) + s (−˜ ) 2 s 1 = pe (−˜ ) + p(h) r (−˜ ) 2

(w) (w) + p(h) s (−˜ ) − pe (−˜ ) + pr (−˜ ) − ps (−˜ ) .

(7)

The basis for the field analysis is the decomposition of a plane wave in Mathieu functions: u(x, y) = e−j k0 (x cos +y sin ) =2

∞ m=0

+2

(−j)m cem () · cem (˜ ) · Jcm ()

∞ m=1

(−j)m sem () · sem (˜ ) · Jsm ()

(8)

Scattering of Sound

E

219

( is the angle between the wave number vector and the positive x axis), and the decomposition of the Hankel function of the second kind in Mathieu functions: ⎧ ⎡ Jcm (0 )Hc(2) > 0 ⎪ m () ; ⎪ ⎨ ⎢ (2) H0 (k0R) = 2 ⎢ ⎣m≥0 cem (˜0 ) · cem (˜ ) · ⎪ Jc ()Hc(2) ( ) ; < 0 0 ⎪ m ⎩ m

+

m≥1

sem (˜0 ) · sem (˜ ) ·

⎧ Jsm (0 )Hs(2) ⎪ m () ⎪ ⎨

;

(2) ⎪ ⎪ ⎩ Jsm ()Hsm (0 )

;

> 0

⎤

(9)

⎥ ⎥ < 0 ⎦

with the source of the Hankel function in the elliptical co-ordinates (0 , ˜0 ). The parameter of the Mathieu differential equation is q = (k0c)2 /4; cem (˜ ), sem (˜ ) are even and odd azimuthal Mathieu functions; Jcm (), Ycm (), Hc(2) m () = Jcm () − j · Ycm () and Jsm (), Ysm (), Hs(2) () = Js () − j · Ys () are the associated radial Mathieu-Bessel, m m m Mathieu-Neumann and Mathieu-Hankel functions. High dam Let the incident wave be a plane wave with ± = /2 ± Ÿ0 . The exciting wave on hard ground is: pe = p+e + p−e = Pe e−jky y (e+jkx x + e−jkx x ) = 2Pe cos kx x · e−jky y ,

(10)

and with absorbing ground: pe = p+e + r · p−e = Pe e−jky y (e+jkx x + r · e−jkx x )

(11)

with

(12)

kx = k0 sin Ÿ0

;

ky = k0 cos Ÿ0 ,

With hard ground, the exciting and reflected waves are in both subtasks h, w: (−j)2r ce2r (+ ) · ce2r (˜ ) · Jc2r (), pe + p(h) r = 8Pe

(13)

r≥0

pe + p(w) = 8Pe r

(−j)2r+1 se2r+1 (+ ) · se2r+1 (˜ ) · Js2r+1 ().

(14)

r≥0

The scattered waves are formulated as: (2) Cr · ce2r (˜ ) · Hc2r (), p(h) s = Pe

(15)

r≥0

p(w) = Pe s

Sr · se2r+1 (˜ ) · Hs(2) 2r+1 ()

(16)

r≥0

with still undetermined term amplitudes Cr ,Sr .They follow from the boundary condition at the elliptic cylinder:

! (ß) (ß) (ß) + vs = G · pe + p(ß) (17) −Z0 ve + vr r + ps =c =c

220

E

Scattering of Sound

using the integrals Icm,‹ (u) : =

cem (t) · ce‹ (t) ·

*

u 2 + sin2 t dt

;

u 2 = sinh2

0

Ism,‹ (u) : =

sem (t) · se‹ (t) ·

*

(18) u 2 + sin2 t dt

0

and the orthogonality relation:

cem (t) · ce‹ (t) dt =

0

sem (t) · se‹ (t) dt = ƒm,‹ · /2.

(19)

0

One gets the following systems of equations for the term amplitudes: . . . · ce2s (t) ·

For (ß) = (h) by:

* u 2 + sin2 t dt

;

s≥0

0

j (2) (2) Hc (c ) + G · Hc2r (c ) · Ic2r,2s Cr · ƒr,s 2 k0c 2r r≥0 j 2r + Jc (c ) + G · Jc2r (c ) · Ic2r,2s ; (−j) ce2r ( ) · ƒr,s = −8 2 k0c 2r r≥0

For (ß) = (w) by:

. . . · se2s+1 (t) ·

* u2 + sin2 t dt

;

(20) s ≥ 0.

s≥0

0

j (2) (2) Sr · ƒr,s Hs2r+1 (c ) + G · Hs2r+1 (c ) · Is2r+1,2s+1 = −8 (−j)2r se2r+1 (+ ) 2 k0 c r≥0 r≥0

j · ƒr,s Js2r+1 (c ) + G · Js2r+1 (c ) · Ic2r+1,2s+1 ; 2 k0 c

(21)

s ≥ 0.

With the Fourier coefficients AŒ (‹) of the ce‹ (z) and BŒ (‹) of the se‹ (z), the required integrals are Icm,‹ =

Ism,‹ =

n,v≥0

n,v≥1

An (m) · AŒ (‹) ·

* cos(n˜ ) cos(Œ˜ ) u2 + sin2 ˜ d˜

0

Bn (m) · BŒ (‹) · 0

*

sin(n˜ ) sin(Œ˜ ) u 2 + sin2 ˜ d˜ ,

(22)

Scattering of Sound

E

221

and 1 An (m) · AŒ (‹) · [I(n−Œ)/2 + I(n+Œ)/2 ], 2 n,Œ≥0 1 Ism,‹ = Bn (m) · BŒ (‹) · [I(n−Œ)/2 − I(n+Œ)/2 ], 2 n,Œ≥0 Icm,‹ =

(23)

where Ii = Ii (u): =

* cos(2i˜ ) u2 + sin2 ˜ d˜ .

(24)

0

Special case of a hard high dam, i.e. G = 0: Has the explicit solutions: Cr = −8(−1)2r ce2r (+ )

Jc2r (W ) Hc(2) 2r (W )

;

Sr = −8(−1)2r+1 se2r+1 (+ )

Js2r+1 (W ) Hs(2) 2r+1 (W )

.

(25)

Special case of a thin screen, i.e. c → 0: Has the special values: Jcm (0) = 0

;

Jsm (0) = 0

;

Hcm (2) (0) = −j Ycm (0)

;

Hs(2) m (0) = −j Ysm (0) (26)

and if further G = 0, i.e. the screen is hard: Cr = 0. With absorbent ground, having the reflection factor r: Exciting and reflected wave (± = /2 ± Ÿ0 ): (−j)m cem (+ )(1 + r · (−1)m ) · cem (˜ ) · Jcm (), pe + p(h) r = 4Pe m≥0

pe + p(w) r = 4Pe

m≥1

(−j)m sem (+ )(1 − r · (−1)m ) · sem (˜ ) · Jsm ().

(27)

Formulation of the scattered waves for both subtasks h, w: Cm · cem (˜ ) · Hc(2) p(h) s = Pe m (), m≥0

p(w) s = Pe

m≥1

Sm · sem (˜ ) · Hs(2) m ().

(28)

The systems of equations for the term amplitudes Cm , Sm become j (2) (2) Hc (c ) + Z0 G · Hcm (c ) · Icm,‹ Cm · ƒm,‹ 2 k0 c m m≥0 j m + m = −4 Jcm (c ) + Z0 G · Jcm (c ) · Icm,‹ ; (29) (−j) cem ( ) (1 + r · (−1) ) · ƒm,‹ 2 k c 0 m≥0 ‹≥0

222

E

Scattering of Sound

j (2) (2) Hs (c ) + Z0 G · Hsm (c ) · Ism,‹ Sm · ƒm,‹ 2 k0 c m m≥1 j =− 4 Jsm (c )+Z0 G · Jsm (c ) · Ism,‹ .; . (30) (−j)m sem (+ ) (1 − r·(−1)m ) · ƒm,‹ 2 k0 c m≥1

‹≥1 The orders m,‹ in the integrals Icm,‹ , Ism,‹ have the same parity. High dam and cylindrical incident wave The original source (1) is at (0 , ˜0 ). Some mirror sources (2). . . (4) are used. The original free field is p+e = p1 = Pe · H(2) 0 (k0 R1 ).

(31)

The exciting wave with hard ground is: (2) pe = p+e + p−e = Pe · [H(2) 0 (k0 R1 ) + H0 (k0 R2 )],

(32)

and with absorbent ground: (2) pe = p+e + r · p−e = Pe · [H(2) 0 (k0 R1 ) + r · H0 (k0 R2 )].

(33)

In the range < 0 , and especially = c one has with a hard ground: (2) ce2r (˜0 ) · ce2r (˜ ) · Jc2r () · Hc2r (0 ), pe + p(h) r = 8Pe

(34)

r≥0

= 8Pe pe + p(w) r

r≥0

se2r+1 (˜0 ) · se2r+1 (˜ ) · Js2r+1 () · Hs(2) 2r+1 (0 ).

(35)

Scattering of Sound

E

223

The formulations for the scattered waves of the subtasks (ß) = (h), (w) remain as above. The systems of equations for the term amplitudes are obtained from those above by the following substitutions (± = /2 ± Ÿ0 ): ⎧ m + (2) ⎪ ⎨ (h): (−j) · cem ( ) → cem (˜0 ) · Hcm (0 ), (36) () = ⎪ ⎩ (w): (−j)m · se (+ ) → se (˜ ) · Hs(2) ( ). m m 0 0 m An absorbent ground is introduced as above. 20lg|p(x/h, y/h)/pnorm| h=4 [m] ; b/h=0.5 ; f=500 [Hz] ; 2c/λ0=10.091 ; G=0.25 - j1 ; ρ0=1.5 ; Θ0=-87° ; Δr=8 ; Δϑ=6° ; Δρ=0.25

10

0 dB -20

-40 8 -60 8

6 x/h

6

4 4

2 2

y/h

0

Sound pressure level on the shadow side behind a high dam; the line source is near the ground at 0 = 1.5; Ÿ0 = −87◦

E.9 Scattering at a Flat Dam Scattering at a flat dam is contained in a separate section, because the formulas are different from those for a high dam. See the previous > Sect. E.8 for the distinction between flat and high dams.

224

E

Scattering of Sound

The sound field is again evaluated with the principle of superposition (see previous > Sect. E.8). The exciting wave pe , with the incident plane wave p+ e and the plane wave reflected at ground p−e (reflection factor r of the ground), is:

pe = p+e + r · p−e = Pe · ejkx x · ejky y + re−jky y , (1) kx = k0 cos Ÿ0 ; ky = k0 sin Ÿ0 .

A flat dam with its elliptical-hyperbolic co-ordinate system. The plane y = 0 is the hard ground. p+e is the incident plane wave; p−e is the mirror-reflected wave. The height of the dam is b; its width at ground level is 2h. The angle Ÿ0 of sound incidence is measured with respect to the ground. The dam is a semi-ellipse with eccentricity c. See the previous > Sect. E.8 for relations with other geometrical parameters The sound fields in front of (side of incidence) and behind the dam are: 1 (w) pfront (x > 0, y) = pe (x, y) + p(h) s (x, y) + ps (x, y) , 2 (2) 1 (h) (w) pback (x < 0, y) = pe (x, y) + ps (−x, y) − ps (−x, y) 2 (ß) with ps (x, y) the scattered fields for the subtask (ß) = (h) with hard plane x = 0 and the subtask (ß) = (w) with soft plane x = 0. The field in x ≥ 0 in the two subtasks is (ß) (h) pe + p(ß) r + ps , where pr is the exciting wave after hard reflection at the plane x = 0, (w) and pr is the exciting wave after soft reflection at the plane x = 0. The sums pe + p(ß) r are (± = /2 ± Ÿ0 ): (−j)2r ce2r (+ )ce2r (˜ )Jc2r () pe + p(h) r = 4Pe (1 + r) r≥0

+ 4Pe (1 − r)

r≥1

(−j)2r se2r (+ )se2r (˜ )Js2r (),

(3)

Scattering of Sound

pe + p(w) r = 4Pe (1 + r)

r≥0

+ 4Pe (1 − r)

E

225

(−j)2r+1 ce2r+1 (+ )ce2r+1 (˜ )Jc2r+1 ()

r≥0

(4)

(−j)2r+1 se2r+1 (+ )se2r+1 (˜ )Js2r+1 ().

The formulations for the scattered fields are, with still unknown term amplitudes C(ß) r , S(ß) r : (2) (2) (h) (h) (h) Cr ce2r (˜ )Hc2r () + Sr se2r (˜ )Hs2r () , (5) ps = Pe r≥0

p(w) = Pe s

r≥1 (2) C(w) r ce2r+1 (˜ )Hc2r+1 ()

r≥0

+

(2) S(w) r se2r+1 (˜ )Hs2r+1 ()

.

(6)

r≥0

(ß) (ß) + vs The boundary condition −Z0 ve + vr

=c

! (ß) = G · pe + p(ß) at the r + ps =c

dam surface gives for (ß) = (h) the following two systems of equations (ƒr,s = Kronecker symbol): (h) j Hc2r (2) (c ) + G · Hc(2) Cr ƒr,s ( ) · Ic c 2r,2s 2r 2k0c r≥0 (7) j 2r + Jc (c ) + G · Jc2r (c ) · Ic2r,2s ; s ≥ 0, = −4 (1 + r) (−j) ce2r ( ) ƒr,s 2k0c 2r r≥0 j (2) (2) ƒ Hs S(h) ( ) + G · Hs ( ) · Is r,s c c 2r,2s 2r r 2k0 c 2r r≥1 j 2r + Js (c ) + G · Js2r (c ) · Ic2r,2s ; = −4 (1 − r) (−j) se2r ( ) ƒr,s 2k0 c 2r r≥1

and for (ß) = (w) two more systems of equations: (w) j (2) (2) Hc Cr ƒr,s (c ) + G · Hc2r+1 (c ) · Ic2r+1,2s+1 2k0c 2r+1 r≥0 = −4 (1 + r) (−j)2r+1 ce2r+1 (+ )

r≥0

j ƒr,s Jc (c ) + G · Jc2r+1 (c ) · Ic2r+1,2s+1 ; 2k0c 2r+1

s≥0

(8) s ≥ 1,

(9)

E

226

Scattering of Sound

j (2) (2) ƒr,s Hs (c ) + G · Hs2r+1 (c ) · Is2r+1,2s+1 2k0c 2r+1 r≥0 = −4 (1 − r) (−j)2r+1 se2r+1 (+ )

S(w) r

r≥0

ƒr,s

j Js (c ) + G · Js2r+1 (c ) · Ic2r+1,2s+1 ; 2k0c 2r+1

The integrals Icm,n , Ism,n are described in (ß) solving for the C(ß) r , Sr .

(10)

>

s≥0

Sect. E.8. The sound field is known after

E.10 Scattering at a Semicircular Absorbing Dam on Absorbing Ground


A semicircular, locally reacting dam, with radius a and normalised surface admittance G, sits on an absorbent ground plane with reflection factor r. A plane or cylindrical wave p+e is incident at an angle Ÿ0 with the ground plane. The ground plane produces the mirror-reflected wave p−e . A field point is at the cylindrical co-ordinates (, ˜ ). The second diagram (b) shows the co-ordinates as they are generally used in scattering problems at cylinders, such as in > Sects. E.1 and E.2.

Scattering of Sound

E

227

20lglp( ρ,ϑ)/2PeI k0h=36.60 ; h=4 [m] ; f=500 [Hz] ; d=0.15 [m] ; Ξ=40 [kPa s/m 2] ; G=0.272+j . 0.232 ; Θ0=3 °

10

0

dB -20

6 -40 4 -50 4

x/h 3

2 2 y/h

1 0

A semicircular absorbent dam on a hard ground plane. The absorption of the dam corresponds to that of a d = 0.15 [m] thick glass fibre layer with a flow resistivity of ¡ = 40[kPa · s/m2 ]. A plane wave is incident under an angle of elevation of Ÿ0 = 3◦ . The diagram shows the sound pressure level in the shadow area The exciting wave is pe = p+e + r · p−e .

(1)

Incident plane wave The exciting wave expanded in Bessel functions:

pe = Pe · e−jkx x e+jky y + r · e−jky y = Pe

m≥0

ƒm(−j)m [cos(m(ƒ + Ÿ0)) + r · cos(m(˜ − Ÿ0 ))] · Jm (k0)

(2)

E

228

with

Scattering of Sound

ƒm =

1; m = 0 ; 2; m = 0

kx = k0 cos Ÿ0 ;

ky = k0 sin Ÿ0 .

Formulation of the scattered field with as yet undetermined term amplitudes Dm : ƒm (−j)m · Dm · [cos(m(˜ + Ÿ0 )) + r · cos(m(˜ − Ÿ0 ))] · H(2) ps = Pe m (k0 ).

(3)

(4)

m≥0

The boundary condition at the cylinder gives for the term amplitudes: Dm = −

Jm (k0h) − j Z0 G · Jm (k0 h) (2) Hm (k0 h) − j Z0 G · H(2) m (k0 h)

m k0h − j Z0 G · Jm (k0h) − Jm+1 (k0 h) = − . (2) m k0h − j Z0 G · H(2) m (k0 h) − Hm+1 (k0 h)

(5)

20lg|p(ρ,ϑ)/2Pe| h=4 [m] ; f=500 [Hz] ; k0h=36.60 ; d=0.15 [m] ; Ξ=40 [kPa s/m2] ; G=0.272+j . 0 .232

10

0 dB -20

6 -40 4 -50 4

x/h

3

2 2 y/h

1 0

As above, but with a fully absorbent ground plane (r = 0)

Scattering of Sound

E

229

Incident cylindrical wave Let the line source Q of the cylindrical wave be at a distance 0 from the dam axis under an elevation angle Ÿ0 with the ground plane. The ground plane with the reflection factor r produces the mirror-reflected wave p−e to the original incident wave p+e . The exciting wave is in the radial range < 0 : (2) (k R ) + r · H (k R ) pe = Pe H(2) 0 1 0 2 0 0 = Pe

m≥0

ƒm(−1)m · H(2) m (k0 0 ) · Jm (k0 ) · [cos (m(˜ + Ÿ0 ))

(6)

+ r · cos (m(˜ − Ÿ0 ))] .

The total field is p = pe + ps with the scattered field: ps = Pe

m≥0

(2) ƒm (−1)m · Dm · H(2) m (k0 0 ) · Hm (k0 ) · [cos (m(˜ + Ÿ0 ))

(7)

+ r · cos (m(˜ − Ÿ0 ))]. The term amplitudes Dm follow from the boundary condition at the dam surface = a as:

m k0 a − j G · Jm (k0a) − Jm+1 (k0a) Jm (k0a) − j G · Jm (k0a) Dm = − (2) = − . (8) (2) Hm (k0a) − j G · H(2) m k0 a − j G · H(2) m (k0 a) m (k0 a) − Hm+1 (k0 a) (2) The component form pe = Pe H(2) 0 (k0 R1 ) + r · H0 (k0 R2 ) is valid for all ≥ a, like ps . The radii R1 , R2 are given by * k0 R1 = k0 ( cos ˜ + 0 cos Ÿ0 )2 + ( sin ˜ − 0 sin Ÿ0 )2 , * k0 R2 = k0 ( cos ˜ + 0 cos Ÿ0 )2 + ( sin ˜ + 0 sin Ÿ0 )2 .

(9)

E

230

Scattering of Sound

20lg|p(ρ,ϑ)/pnorm| h=4 [m] ; f=500 [Hz] ; k0h=36.60 ; G=0.272+j. 0.232 ; ρ0/h=4 ; Θ0=3°

10

0 dB -20

-40

6

-50 6

4 4 2 y/h

x/h

2 0

Sound pressure level in the shadow zone of a semicircular absorbing dam on a hard ground plane for a cylindrical incident wave from the source position 0 /h = 4; Ÿ0 = 3◦

E.11 Scattering in Random Media, General


This section presents general distinctions and concepts for the scattering of sound in random media. The composite medium consists of a fluid with randomly distributed scatterers. The fluid

• May have no losses, • May have viscous and thermal losses.

Scattering of Sound

E

231

The scatterers

• May be different with respect to their shape, e.g. below: • Spheres, • Cylinders, • May be different with respect to their consistency: • Rigid or soft, • Fluid, with or without losses, • Elastic, • May be different with respect to their dynamical behaviour: • Not moving (though oscillating at their surface), • Moving as a total under the influence of acoustical forces. The composite medium • May be disperse, i.e. multiple scattering negligible, • May be dense, i.e. multiple scattering not negligible. The scattering • May retain the wave type (monotype scattering), • May change the wave type into the triple of density, thermal, viscous waves (triple type scattering).

Table 1 on the next page gives a survey of some of the different scattering processes. The upper rows belong to monotype scattering: both the exciting wave ¥e and the scattered wave ¥s are of the same type. The lower rows describe triple-type scattering: the exciting wave ¥e generates the triple of density (), thermal () and viscous (Œ) waves as scattered waves. The propagation of sound through the composite medium is composed of elementary scattering processes at single scatterers, which is indicated in the first column. In disperse media (second column) the multiple scattering (scattering of scattered fields) is neglected, i.e. the scattered waves propagate freely through the medium. In dense media (third column) multiple scattering must be taken into account. The exciting wave then is an “effective” wave ¥E . The scatterers will have a uniform random distribution in the composite medium. If there are inhomogeneities, e.g. holes or clusters, they are supposed to be randomly distributed as well and to form a subsystem of scatterers.

2

1

Scattered wave

Exciting wave

Triple-type scattering

ρ

ρ

f¥ (k r); ¥ (k r); ¦Œ (kŒ r)g

¥s (k rj )

f¥ (k r); ¥ (k r); ¦Œ (kŒ r)g

;j6=i

Ψν ν Φα α Φρ ρ

¥e (k x) = e−jk x +

Φ

¥s (k rj )

ρ

¥s (k r) =

Φρ ρ

Φα α

j6=i

Φ

¥s (k r) =

¥e (k x) = e−jkx

Φ

Ψν ν

ρ

¥s (k r)

Φ

¥s (k r)

ρ

Scattered wave

Φ

¥e (k ) = e−jk x +

ρ

¥e (k x) = e−jkx

Φ

Disperse medium

Exciting wave

Monotype scattering

Single scatterer

2

ρ

ρ

¥s (k rj )

Ψν ν Φα α Φρ ρ

¥s (k rj )

;j6=i

j6=i

Φ

f¥ (k r); ¥ (k r); ¦Œ (kŒ r)g

¥s (k r) =

¥E = e− x +

Φ

¥s (k r)

¥E = e− x +

Φ Γ

Dense medium

3

E

Type of scattering

1

Table 1 Survey of scattering in random media

232 Scattering of Sound

Scattering of Sound

E

233

Reiche’s experiment: A simple experiment is the background of many theories for the determination of the characteristic propagation constant and wave impedance Zi inside a composite medium with random scatterers: A plane wave ¥e is incident on a layer of thickness D of the investigated material. A receiver on the front side “collects” the backscattered wave components from inside the layer as reflected wave ¥r ; a receiver on the back side“collects”the forward-scattered wave components from inside the layer as transmitted wave ¥t .The characteristic values of the material are determined from the reflection and transmission factors re , te on the front side and ra , ta on the back side (reflection factors are underlined for distinction with the later used symbol r for radius and/or general co-ordinate). The forward and backward waves inside the layer are ¥+ , ¥− , respectively. The fluid inside the layer (between the scatterers) equals the fluid outside the layer.

Γ , Zi

k0 , Z0

k0 , Z0

Φe Φt Φr

x Φ+

Φ

re

ra ta

te D

Monotype scattering:

¥e (x) = e−j k0 x

;

¥r (x) = re · e+j k0 x

re = ¥r (0)/¥e(0)

;

ra = ¥− (D)/¥+ (D)

te = ¥+ (0)/¥e(0) ; t = ¥t (D)/¥e (0)

ta = ¥t (D)/¥+ (D)

;

¥t (x) = t · e−j k0 (x−D)

(1)

234

E

Scattering of Sound

For transmission with y = D: ta = 1 + ra =

2 1 + Zi /Z0

1 − r2a t = · e−y 1 − r2a · e−2y

;

te =

1 − ra 1 − r2a · e−2y

(2)

−y

;

t = te ta e

For reflection with y= D: ra =

1 − Zi /Z0 1 + Zi /Z0

;

r e = −ra ·

1 − e−2y . 1 − r 2a · e−2y

(3)

Field inside the layer:

¥I (x) = ¥+ (x) + ¥− (x) = te · e− x + ra · e+ (x−2D) .

(4)

Field at a field point r (which may be well outside the layer): us (r − rs ), ¥ (r) = ¥e (r) + U(r) ; U(r) =

(5)

s

where us is the scattered field of a single scatterer with running index s having its position at rs . This often-used elementary decomposition implicitly assumes that the exciting wave can reach the scatterers (even deeply inside the layer) without attenuation; this form therefore is restricted to disperse media. The single scatterer functions us are cylindrical scatterers or sums over Hankel functions of the second kind H(2) n (k0 r) for * (2) spherical Hankel functions of the second kind h(2) (k r) = /(2k 0 0 r) Hn+1/2 (k0 r) for n spherical scatterers. It is a further assumption in Reiche’s experiment that receiving points for the transmitted and reflected fields are chosen at large distances,so that k0 |x| 1.Then the scattered far field can be written as a product of a factor (k0 r) with only a radial variation, and an angular factor g(o, i) which contains the angle between the direction o to the field point with the direction i of incidence on the scatterer. Replacing the summation over the index s of the scatterers by an integration and taking into account the symmetry of the problem around the axis of propagation, one can write for the scattered fields outside the layer (at large distances of it): D U> (x) = ¥e (k0x) · C 0

ejk0 · G( , i) d

D

U< (x) = ¥e (−k0 x) · C

;

x>D, (6)

e−jk0 · G( , −i) d

;

x (x) = (1 − QQ ) e−2jkE D F · e−jk0 x

;

x ≥ D.

(25)

Thus the characteristic values of the composite material and the sound fields are known using this method (and with its restrictive assumptions) if the scattered far field functions g> = g(i, i) and g< = g(–i, i) are known from single scatterer evaluations, which are the scattered far fields in forward and backward directions.With them the characteristic values can be given other forms, using R = C g< and kE = k0

/

;

T = C g> − jk0

eff Ceff · 0 C0

;

Zi = Z0

C eff = 1 + j (g> − g< ) 0 k0

;

;

/

eff 0

Q=

0

C g< C g> − j (k0 + kE )

Ceff , C0

Ceff C = 1 + j (g> + g 2 k0 k0

;

Z2i C ≈ 1 + 2j · g< . 2 k0 Z0

(29)

In this approximation the wave number only depends on the forward scattering and the wave impedance only on the backward scattering. With the extinction cross section Qe of a single scatterer ⎧ ⎨ 2/k Cylinder , 0 Qe = −2C · Re{g> } ; C = (30) ⎩ 2/k02 Sphere , one finds the plausible result for the attenuation (in the present approximation): −2 Im{kE } ≈ N · Qe ,

(31)

i.e. the attenuation is proportional to the number density of the scatterers and to their extinction cross section.

E.12 Function Tables for Monotype Scattering This section gives tables of functions for monotype scattering to be applied in the general scheme of the previous > Sect. E.11, i.e. incident and scattered waves are of the same type, to say, density waves. It is not necessary to use potential functions ¥ for the field with monotype scattering; therefore the field function here will be the sound pressure p. The time factor is, as usual, ej –t . The incident plane wave has a unit amplitude. The first table compiles radial functions Rn (r) and azimuthal functions Tn (˜ ) for cylindrical and spherical scatterers. Zn (z) stands for a Bessel, Neumann or Hankel function; Kn (z) stands for a spherical Bessel, Neumann or Hankel function. Because Hankel functions only are of the second kind,the upper index (2) will be dropped (for ease of writing). The second table gives formulations of the incident plane wave and of the scattered wave for both geometries of the scatterer. The third table collects modal amplitudes and mode admittances (normalised with Z0 ) for both locally reacting and bulk reacting scatterers. Bulk reacting scatterers are supposed to be of an isotropic, homogeneous material having the characteristic propagation constant and wave impedance Z ; thus this type of scatterer can represent also fluids with losses. Hard and soft cylinders and spheres are treated as special cases of locally reacting scatterers. a is the radius of the scatterer; N is the number density of scatterers (number of scatterers per unit volume); ‹ is the massivity, i.e. the ratio of space occupied by the scatterers. The argument in radial functions is dropped if it is k0 a . Some of the contents of these tables may be found also in > Sects. E.1 and E.2, but the tables here have been completed with terms required in the previous > Sect. E.11.

Scattering of Sound

E

Table 1 Radial and polar functions for cylindrical and spherical co-ordinates Quantity 1

Zn(z)

R0n (z)

=

Hn (z)

=

Z0n(z)

= =

Jn (z) −−−−−−! 3

Sphere Kn (z)

=

fJn (z); Yn (z); Hn (z)g

Kn (z)

=

fjn (z); yn (z); hn (z)g p =(2z) Zn+1=2 (z)

Jn (z) − j Yn (z)

hn (z)

=

jn (z) − j yn (z)

n Zn(z) − Zn+1 (z) z n Zn−1 (z) − Zn(z) z

Kn0 (z)

=

Radial functions Rn (z)

2

Cylinder

=

n Kn (z) − Kn+1 (z) z n Kn−1 (z) − Kn (z) z

jn (z) −−−−−−!

(z=2)n n!

zn 1 2 3 : : : (2n + 1)

Jn (z) −−−−−−!

1 cos z − n z 2

jzj 1

jzj 1

4

jzj 1

jn (z) −−−−−−!

2 cos z − n − z 2 4

jzj 1

Yn (z) −−−−−−! 5

jzj 1

yn (z) −−−−−−! jzj 1

Yn (z) −−−−−−! 6

jzj 1

2 ln z (n − 1)! ; n>0 Yn (z) ! − (z=2)n Y0 (z) !

yn (z) −−−−−−!

2 sin z − n − z 2 4

−

1 3 5 : : : (2n − 1) zn+1

1 sin z − n z 2

jzj 1

7

H0 (z) ! −

hn (z) −−−−−−!

j(n − 1)! ; n>0 Hn (z) ! (z=2)n 2 e−j(z−n=2−=4) z

jzj 1

jzj 1

Hn (z) −−−−−−! 8

2j ln z

Hn (z) −−−−−−!

jzj 1

j

1 3 5 : : : (2n − 1) zn+1

j −j(z−n=2) e z e−jz = jn (−jz)

hn (z) −−−−−−!

jn e−jz = p p =2 −jz

9

Jn+1 (z)Yn (z) –Yn+1 (z)Jn (z)

2/(z)

1/z2

10

Polar functions Tn (˜)

cos (n˜)

Pn (cos ˜)

11

T0 (˜)

1

1

12

Tn (0)

1

1

13

Tn ()

(−1)n

(−1)n

jzj 1

239

240

E

Scattering of Sound

Table 2 Incident and scattered waves Quantity 1

Cylinder

Sphere pe (r; ˜) = e−jk0 rcos ˜

Incident wave 1

pe (r; ˜) =

n=0

ƒn (−j)n Tn (˜) Ren (r)

2

ƒn =

⎧ ⎨ 1; n= 0 ⎩ 2; n>0

3

Tn (˜) =

cos (n˜)

Pn (cos ˜)

4

Ren (r) =

Jn (k0 r)

jn (k0 r)

5

Scattered wave ps (r; ˜) =

6

Rsn (r) =

7

ps (r; ˜) −−−−−−!

8

Sect. E.12 the mode n = 0 with mode amplitude D0 dominates for small k0 a. The static compressibility of the composite model medium is reduced by 1 → 1 − ‹; therefore one corrects D0 → D0 /(1 − ‹). In eff /0 the mode n = 1 with the mode amplitude D1 dominates. eff is corrected with the ratio of the oscillating mass of a free disk to that of a sphere. Thus 2 1+ ‹ 1 . (5) D0 → D0 · ; D1 → D1 · 3 1−‹ 1+ ‹ 2 Higher-order mode amplitudes Dn remain unchanged. The curves in the diagram below show that the transmission loss for a parameter set a, ‹, D can be evaluated from the transmission loss for a parameter set a0 , ‹0 , D0 by the transformation a0 ‹D a L(f , a, ‹, D) = · L f , a0, ‹0, D0 . (6) a ‹0 D0 a0 D = 100 [m] ; n max = 16

10.

2a = 20 ΔL

30 40 [cm]

dB

2a = 4

3

2 [cm]

1.

Cylinder μ=0.002

0.1 100.

Leaves μ=0.02

1k

f [Hz]

10 k

Transmission loss through a model forest, D = 100 [m] wide. The left-hand curves only consider cylindrical, hard trunks; the right-hand curves only consider the leaves, which are modelled as scattering spheres

244

E

Scattering of Sound

E.14 Mixed Monotype Scattering in Random Media The fundamentals of this section are presented in > Sect. E.11. The difference in this section with respect to > Sects. E.11–E.13, all dealing with monotype scattering, lies in the fact that there not only is the exciting wave of the same type as the scattered wave (density wave), but also their free field wave numbers are supposed to agree with each other. For not too low scatterer densities N, however, the wave which excites a reference scatterer deep in the layer of the composite material will have characteristic values different from those of the wave in free space. This section still makes the assumption that nearby neighbouring scatterers (to the reference scatterer) placed in the forward direction are not shadowed by the reference scatterer with respect to nearby neighbouring scatterers in the backward direction (in front of the scatterer). This condition implies that ⎧ 2‹ ⎪ ⎪ N Qs ⎨ k0a = ⎪ k0 ⎪ ⎩ 3‹ 4 k0 a

Qs S Qs S

⎧ 3 2 ⎪ ⎪ ⎨ 4 ‹ (k0 a) ! 1 ≈ ⎪ ⎪ ⎩ 1 ‹ (k0a)3 3

;

⎧ ⎪ ⎪ ⎨ Cylinder, ⎪ ⎪ ⎩ Sphere.

(1)

At the theoretically possible upper limit ‹ = 1 (which, however, would be in conflict with conditions for the application of monotype scattering), the limits above give k0a 0.65 for the cylinder and k0a 1.44 for the sphere. N = number density of scatters; ‹ = massivity of composite material; D = material layer thickness QS = scattering cross section; a

= radius of scatterer;

S = cross section of scatterer

The effective propagation constant and wave number will be symbolised with = jkE , the effective wave impedance with ZE . As in > Sect. E.11, the scattered far field angular distribution g(o, i) will be used (o = outward direction of the scattered field, i = inward direction of the exciting wave); but the different wave numbers k0 , kE in both directions will also be indicated; and because only the forward and backward directions (parallel and antiparallel to the incident wave) will be relevant, one changes: g(o, i) → g(±k0 , ±kE ). The sound field in the material layer is

¥ (x) = A e−jkE x + B e+jkE x

(2)

Scattering of Sound

E

245

with the relations between the amplitudes as follows ( > Sect. E.11): ⎡ ⎤ x A e−jkE x = e−jk0 x · ⎣1 + ejk0 · (A · S+ e−jkE + B · R+ e+jkE ) d ⎦ , 0

B e+jkE x = e+jk0 x ·

D

(3)

e−jk0 · (A · R− e−jkE + B · S− e+jkE ) d ,

x

where the following abbreviations are used: S+ = C g(k0, kE )

;

R+ = C g(k0, −kE )

;

S− = C g(−k0, −kE ),

(4)

R− = C g(−k0, kE ).

The constant factor C can be taken from Table 5 in following system of equations: R− S+ − = −j kE − k0 kE + k 0 A · S+ B · R− − = −j kE − k0 kE + k 0

> Sect. E.12. Integration

;

S− R+ − = −j kE − k 0 k E + k 0

;

B · S− · e+jkE D A · R− · e−jkE D − =0 kE − k0 kE + k 0

yields the

(5)

and the solutions of the last two equations: A = (1 − Q) · F where (as in

;

B = (1 − Q ) Q e−2jkE D · F,

(6)

> Sect. E.11) the following auxiliary quantities are used:

F = [1 − Q Q e−2jkE D ]−1

;

Q=

R− kE − k0 S+ kE + k0

Q =

;

R+ kE − k0 . S− kE + k0

(7)

For scatterers with front-to-back symmetry (in the statistical average) simplifications are g(k0 , kE ) = g(−k0, −kE ) S+ = S− = S F=

;

;

R+ = R− = R

g(−k0 , kE ) = g(k0, −kE ) , ;

Q = Q ,

(8)

1 . 1 − Q2 e−2jkE D

For such scatterers the field inside the layer is: ¥I (x) =

1−Q · e−j kE x − Q e+j kE (x−2D) 2 −2j k D E 1−Q e

;

0 ≤ x ≤ D;

(9)

the reflected field in front of the layer is: ¥r (x) = −Q (1 − e−2j kE D ) F · e+j k0 x

;

x ≤ 0;

(10)

246

E

Scattering of Sound

the transmitted field behind the layer is: ¥t (x) = (1 − Q2 ) e−2j kE D F · e−j k0 x

;

x ≥ D.

(11)

The analogy with a homogeneous layer gives the correspondences Q ↔ ra ; j kE ↔ , where ra is the reflection factor of the internal plane wave at the back side of the material layer ( > Sect. E.11). Despite the close analogy to the results of > Sect. E.11 (with pure monotype scattering) there are differences in the g(±k0, ±kE ) [as compared to g(±o, ±i); see below for values] and in the definition of Q. For ease of writing (and in close analogy to > Sect. E.11) the values of the scattered far field angular distribution are defined (for symmetrical scatterers) as g> (k0, kE ) = g(+k0 , +kE ) = g(−k0 , −kE ),

(12)

g< (k0, kE ) = g(−k0 , +kE ) = g(+k0 , −kE ), with which the wave impedance of the effective wave is, kE (g> − g< ) + (g> + g< ) ZE 1 − Q k = 0 = , kE Z0 1+Q (g> + g< ) + (g> − g< ) k0 and a square equation holds for the wave number: C C kE2 kE − · j (g − g ) − 1 + j (g + g ) = 0. > < > < k0 k02 k0 k0

(13)

(14)

Since both g> and g< contain kE , ZE , the equation for kE must be solved numerically, in general. The still sought g> and g< follow from the solution of the scattering task at a single scatterer. The scatterer shall consist of a homogeneous material with a characteristic propagation constant = jk and a characteristic wave impedance Z . The scattered field is formulated as in > Sect. E.11, i.e. with the scattered mode amplitudes Dn , except for the substitution k0 → kE . The interior field is formulated as p (r, ˜ ) =

∞

ƒn (−j)n En Tn (˜ ) Rn (r)

(15)

n=0

with ƒn and the azimuthal functions Tn (˜ ) taken from Table 2 of radial functions ⎧ ⎪ ⎪ ⎨ Jn (k r) ; Cylinder, Rn (r) = ⎪ ⎪ ⎩ j (k r) ; Sphere. n

> Sect. E.12, and

the

(16)

Scattering of Sound

E

247

The boundary conditions at the surface give for the scattered mode amplitudes Dn and the interior mode amplitudes En (below for a cylinder, similarly for a sphere): 1 1 1 Dn = Jn (kE a) Jn (k a) − Jn (k a) Jn (kE a) , X Z /Z0 ZE /Z0 (17) 1 1 (2) (2) Jn (kE a) Hn (k0 a) − H (k0 a) Jn (kE a) En = X ZE /Z0 n with the abbreviation X = Jn (k a) H(2) n (k0 a) −

1 H(2) (k0a) Jn (k a) Z /Z0 n

(18)

(a prime indicates the derivative with respect to the argument). With the modal (normalised) admittances Gn the scattered mode amplitudes can be written as: j Jn (j a) , Z /Z0 Jn (j a) ZE n −j Gn · Jn (kE a) − Jn+1 (kE a) −1 k a Z0 E Dn = . n ZE /Z0 (2) − j Gn · H(2) (k a) − H (k a) n 0 n+1 0 k0 a Gn =

(19)

A locally reacting scatterer with a (normalised) surface admittance G is obtained by the substitution Gn → G. The usual representation of the characteristic values kE , ZE of the composite medium / / eff Ceff eff Ceff kE ZE = · ; = / (20) k0 0 C0 Z0 0 C0 with the effective density eff and compressibility Ceff is possible with eff = 0

1 k0 C 1− ·j (g> − g< ) kE k0

;

Ceff C =1+j (g> + g< ). C0 k0

(21)

In the special case of a composite medium consisting of hard, parallel cylinders with radius a, and the plane wave incident normally on the cylinders (a forest of trunks, see > Sect. E.13), the expressions needed in the equations for the characteristic values are g> (k0, kE ) + g< (k0, kE ) = D0 (k0 , kE ) + 2 g> (k0, kE ) − g< (k0, kE ) = 2

n max n=1, 3, 5,...

n max n=2, 4,...

ƒn Dn (k0, kE ), (22)

ƒn Dn (k0, kE ).

The diagram shows the attenuation coefficient Re{ E /k0 } in such a medium where the scatterers have a massivity ‹ = 0.02 (cf. > Sect. E.13).

E

248

Scattering of Sound

μ=0.02

10 -2 Re{ΓΕ/ k0 }

10 -3

10 -4

10 -5 0.01

0.1

1.

k0 a

10.

Attenuation coefficient in a composite medium of parallel, hard cylinders of radius a, forming a massivity ‹ = 0.02,evaluated with the method of mixed monotype scattering

E.15 Multiple Triple-Type Scattering in Random Media See > Sect. E.11 for general distinctions and notations. The sections about sound in capillaries in the chapter“Duct Acoustics”contain fundamentals about sound fields with thermal and viscous losses. A sound wave with a scalar potential function ¥e is incident on a scatterer in the composite medium. If the mutual distances of the scatterers are larger than the thickness of the shear boundary layer at a scatterer, it will be a density wave; otherwise it will be an “effective” wave ¥E (i.e. influenced by the three wave types). The scatterer produces a scattered density wave ¥ , a temperature wave ¥ and a viscous wave ¦ . ¥ß ; ß = e, , are scalar potentials with particle velocities vß = −grad ¥ß , and ¦ is a vector potential, so that the total particle velocity is v = −

grad ¥ß + rot ¦ .

(1)

ß

The component fields obey the wave equations ( + kß2 ) ¥ß = 0

;

( + kŒ2 ) ¦ = 0

(2)

Scattering of Sound

with characteristic wave numbers (given as squares) 2 – – 2 2 k ≈ k0 = ; kŒ2 = −j ; c0 Œ ‰– 2 = ‰ Pr ·kŒ2 . k2 ≈ k0 = −j

E

249

(3)

c0 = adiabatic sound velocity; 0 = air density; C0 = air compressibility; Œ = †/ = kinematic viscosity; † = dynamic viscosity; = /(0 cp ) = temperature conductivity; = heat conductivity; cp = specific heat at constant pressure; Pr = Œ/ = Prandtl number; – = angular frequency; p0 = atmospheric pressure The sound pressure p in the scattered field is p = p0 ¢ · ¥ + p0 ¢ · ¥ .

(4)

The coefficients ¢ß are given in the mentioned sections about capillaries. The ratio ¢E /¢ for an effective wave ¥E can be expressed by the effective density eff of the composite medium: p0 ¢E = jkE ZE = j– eff

;

p0 ¢E kE Zi eff = = . p0 ¢ k0 Z0 0

(5)

If the composite medium is statistically homogeneous, the scattered vector potentials ¦ compensate each other in the forward and backward directions of scattering; thus the scattered far field angular distributions g(o, i) do not contain the viscous wave in those directions. A similar compensation for the thermal wave ¥ does not exist; it can be neglected in the propagating wave ¥e only if the immediate neighbours of a reference scatterer are outside the boundary layer. The scatterer is assumed below to be either a cylinder or a sphere of a fluid with thermal and viscous losses (hard, soft or locally reacting scatterers can be treated as special cases of this general assumption). Field quantities and material parameters inside the scatterer are marked with a prime.Only Hankel functions of the second kind will appear; the upper index (2) therefore will be dropped (for ease of writing). The exciting wave ¥e is supposed to have unit amplitude: ¥e = e−jke x = e−jke r

cos ˜

.

(6)

E

250

Scattering of Sound

The particle velocities are:

→

v = −grad

ß=e,,

→

v = −grad

ß = ,

¥ß + rot ¦Œ

;

outside,

¥ß + rot ¦Œ ;

inside.

(7)

z

r ϑ

Φe

x 2aŒ r ϑ

Φe

x

ϕ 2aŒ

The vector potential of the viscous wave has the components ⎧ ⎪ ⎪ ⎨ {0, 0, ¦ßz } ; Cylinder ¦ß = ; ß = Œ, Œ . ⎪ ⎪ ⎩ {0, 0, ¦ } ; Sphere

(8)

ßœ

The boundary conditions are: (a):

vr = vr

;

(b):

v˜ = v˜ ;

(c):

T = T

;

(d):

· ∂T/∂r = · ∂T /∂r ;

(e):

prr = prr

;

(f ):

pr˜ = pr˜ .

(9)

(a),(b) fit the radial and tangential particle velocities; (c),(d) fit the (alternating) temperature and heat flow; (e), (f) fit the radial and tangential tensions, which are (i, j standing for co-ordinates; † = dynamic viscosity): ∂vj ∂vi ∂vi ; i = j. (10) ; pij = † · + pii = p + 2† · ∂xi ∂xi ∂xj

Scattering of Sound

The pressure field is (p0 = atmospheric pressure): ⎧ ⎪ ⎪ ⎨ e, , outside, p = p0 · ¢ß ¥ ß ; ß= ⎪ ⎪ ß ⎩ , inside.

E

251

(11)

The following tables give: • Strain velocities in cylindrical and spherical co-ordinates; • Vector components of grad and rot in both systems; • Field formulations of the incident wave, the scattered wave, and the wave inside the scatterer; • Terms appearing in the boundary conditions. Table 3 contains • Pn (z) Legendre polynomials, • P1n (z) associate Legendre functions, with the useful relations: P1n (z)

=

P1n (cos ˜ ) = − sin ˜ Pn (cos ˜ )

=

d 1 (sin ˜ · P1n (cos ˜ )) sin ˜ d˜

= =

√ √ dPn (z) = − 1 − z2 Pn (z) , − 1 − z2 dz d Pn (cos ˜ ) , d˜ sin2 ˜ · Pn (cos ˜ ) − 2 cos ˜ · Pn (cos ˜ )

(12)

(13)

−n(n + 1) · Pn (cos ˜ )

and the recursive evaluation Pn+1 (z) =

1 [(2n + 1) z Pn (z) − n Pn−1 (z)] n+1

;

from which can be evaluated: n P1n (cos ˜ ) = [cos ˜ Pn (cos ˜ ) − Pn−1 (cos ˜ )] . sin ˜

P0 (z) = 1

;

P1 (z) = z,

(14)

(15)

252

E

Scattering of Sound

Table 1 Components of the strain velocity in cylindrical and spherical co-ordinates Cylinder

Sphere

1

@vr s˙ rr = @r

s˙ rr =

2

s˙ ˜˜ =

3

s˙ zz =

4

s˙ r˜ = s˙ ˜r

1 @v˜ vr + r @˜ @vz @z

s˙ z˜

1 @v˜ vr + r @˜ 1 @vœ s˙ œœ = vr sin ˜ + v˜ cos ˜ + sin ˜ @œ s˙ ˜˜ =

s˙ r˜ = s˙ ˜r

1 @v˜ v˜ 1 @vr − + 2 @r r r @˜ 1 1 @vz @v˜ = s˙ ˜z = + 2 r @˜ @z

= 5

=

s˙ zr = s˙ rz =

1 @vr @vz + 2 @z @r

@v˜ 1 @vr r − v˜ + 2r @r @˜

s˙ rœ = s˙ œr =

6

@vr @r

@vœ 1 @vr +r sin ˜ − vœ sin ˜ 2r sin ˜ @œ @r

s˙ ˜œ = s˙ œ˜ =

@vœ 1 @v˜ sin ˜ − vœ cos ˜ + 2r sin ˜ @˜ @œ

Table 2 Components of grad and rot in cylindrical and spherical co-ordinates Component

Cylinder

1

Sphere grad U

2

r

@U @r

@U @r

3

˜

1 @U r @˜

1 @U r @˜

4

z, œ

@U @z

1 @U r sin œ @œ

5

rot V

6

r

7

˜

8

z, œ

@V˜ @ (sin ˜ Vœ ) − @˜ @œ @Vr 1 @ 1 (rVœ ) − r sin ˜ @œ r @r

1 @Vz @V˜ − r @˜ @z

1 r sin ˜

@Vr @Vz − @z @r 1 @(rV˜ ) @Vr − r @r @˜

1 @ 1 @Vr (rV˜ ) − r @r r @˜

Scattering of Sound

E

253

Table 3 Field formulations of the incident wave, the scattered wave and the interior wave Quantity 1

Cylinder

Sphere

Incident wave

¥e (r; ˜) = e−jkercos ˜ 1

¥e (r; ˜) =

n=0

2

ƒn =

⎧ ⎪ ⎨ 1; n=0 ⎪ ⎩ 2; n>0

3

Tn (˜) =

cos (n˜)

4

Ren (r) =

Jn (ke r)

5

Scattered wave

1

¥ß (r; ˜) =

n=0

6 7

Tn (˜) =

jn (ke r)

Pn (cos ˜) 1

¦Œz(œ) (r; ˜) =

n=0

9

Ran (r) =

Hn (kß r) ;

12

ƒn (−j)n AŒn Tn1 (˜) Ran (r) ;

ß = ; ; Œ

Interior wave

1

¥ß0 (r; ˜) =

n=0

hn (kß r) ;

ß = ; ; Œ

ƒn (−j)n Aß0 n Tn (˜) Rin (r) Pn (cos ˜)

cos (n˜) 1

¦Œ 0 z(œ) (r; ˜) =

AŒ0 = 0

P1n (cos ˜)

sin (nœ)

Tn (˜) =

ƒn (−j)n Aßn Tn (˜) Ran (r)

cos (n˜)

Tn1 (˜) =

11

2n + 1

Pn (cos ˜)

8

10

ƒn (−j)n Tn (˜) Ren (r)

n=0

13

Tn1 (˜) =

sin (nœ)

14

Rin (r) =

Jn (kß0 r) ;

ƒn (−j)n AŒ 0 n Tn1 (˜) Rin (r) ;

AŒ 0 0 = 0

P1n (cos ˜) ß0 = 0 ; 0 ; Œ 0

jn (kß0 r) ;

ß0 = 0 ; 0 ; Œ 0

254

E

Scattering of Sound

Table 4 Terms in the boundary conditions Quantity 1

vr =

Cylinder @ 1 @¦Z − ¥ + @r r @˜

2

v˜ =

−

1 @ @¦z ¥ − r @˜ @r

Sphere @ 1 @ − ¥ + (sin ˜¦œ ) @r r sin ˜ @˜

−

3

T = T0

4

@ T = @r T0

5

Prr =

p0

6

Pr˜ =

Ÿ ¥ Ÿ

¦œ @¦z 1 @ − ¥ − r @˜ r @r

@¥ @r

¢ ¥

p0

@2 ¥ + 2† − 2 @r ¦z 1 @¦z @ − 2 + + @˜ r @r r @ 1 @¥ ¥ † −2 − 2 r @˜ r @r 2 1 @2 ¦z @ ¦z 1 @¦z + − − r @r @r 2 r 2 @˜2

Ÿ¥ Ÿ

@¥ @r ¢ ¥

@2 ¥ + 2† − 2 @r ¦œ 1 @¦œ 1 @ sin ˜ − 2 + + sin˜ @˜ r @r r @ 1 @¥ ¥ † −2 − 2 r @˜ r @r 2 @ ¦œ 2 − − 2 ¦œ @r 2 r 1 @ 1 @ + 2 (sin ˜¦œ ) r @˜ sin ˜ @˜

The following Table 5 contains the equations of the boundary conditions with the above field formulations. Use was made of the derivatives of the radial functions: z · Rn (z) = n · Rn (z) − z · Rn+1 (z)

(16)

and for the second derivatives of the cylindrical radial functions Zn (z) and spherical radial functions Kn (z): z2 · Zn (z) = (n2 − n − z2) · Zn (z) + z · Zn+1 (z), z2 · Kn (z) = (n2 − n − z2) · Kn (z) + 2z · Kn+1 (z).

(17)

The following terms, appearing in the table, can be simplified as: k2 1 − (k /k0)2 p0 a2 ¢ = −(kŒ a)2 2 ≈ −(kŒ a)2 , † k0 1 − ‰(k /k0)2 p0 a2 ¢E 4 p0 a2 ¢ eff ≈ −(kŒ a)2 1 − ‰ Pr ; = −(kŒ a)2 . † 3 † 0

(18)

Scattering of Sound

E

Table 5a Equations for cylinder Bound cond.

Equations for cylinder nJn (ke a) − ke aJn+1 (ke a) +

vr =

vr0

=

0 =0 ;0

nJn (ke a) + v˜ = v˜0 T T0 = T0 T0

=

0 =0 ;0

An[nHn (k a) − kaHn+1 (k a)] − AvnnHn (kv a)

=;

AnnHn (k a) − Avn [nHn (kv a) − kv aHn+1 (kv a)]

A0 n nJn (kß0 a) − Av 0n [nJn (kv 0 a) − kv 0 aJn+1 (kv0 a)] =;

AnŸ Hn (k a) =

0 =0 ;0

Ÿe [n Jn (ke a) − ke a Jn+1 (ke a)] +

@ T = @r T0 @ T0 0 @r T0

=

†

0

ß0 =0 ; 0

A0 n Ÿ0 Jn (k0 a)

ß=;

Aßn Ÿß [n Hn (kß a) − kß a Hn+1 (kß a)]

Aß0 n Ÿß0 [n Jn (kß0 a) − kß0 a Jn+1 (kß0 a)]

p0 a2 ¢e + 2(n − n2 + (ke a)2 ) Jn (ke a) − 2ke aJn+1 (ke a) †

p a2 0 ¢ + 2(n − n2 + (k a)2 ) Hn (k a) − 2kaHn+1 (k a) † =; + AŒn 2n[(n − 1)Hn (kv a) − kv aHn+1 (kv a)]

+ prr = p0rr

=;

A0 n [nJn (kß0 a) − kß0 aJn+1 (kß0 a)] − Av0 n nJn (kv0 a)

Ÿe Jn (ke a) +

An

= †0 AŒ 0 n 2n[(n − 1)Jn (kv0 a) − kv 0 aJn+1 (kv 0 a)] +

0 =0 ;0

A 0 n

p a2 0 2 2 0 0 0 0 0 ¢ + 2(n − n + (k a) ) J (k a) − 2k aJ (k a n n+1 ß †0

† 2n[(n − 1)Jn (ke a) − ke aJn+1 (ke a)] + 2n

pr˜ =

p0r˜

=;

An [(n − 1)Hn (k a) − kaHn+1 (k a)]

− Avn [(2n(n − 1) − (kv a)2 )Hn (kv a) + 2kv aHn+1 (kv a)] = †0 2n A0 n [(n − 1)Jn (k0 a) − k0 aJn+1 (kß0 a)] 0 =0 ;0

− Av 0 n [(2n(n − 1) − (kv 0 a)2 )Jn (kv 0 a) + 2kv0 aJn+1 (kv 0 a)]

255

E

256

Scattering of Sound

Table 5b Equations for sphere Bound. cond.

Equations for Sphere njn (ke a) − ke a jn+1 (ke a) +

vr =

vr0

=

0 =0 ;0

jn (ke a) + v˜ =

v˜0

T T0 = T0 T0 @ T = @r T0 @ T0 0 @r T0

=

=;

An[nhn (k a) + k a hn+1 (k a)] + Avn n(n + 1)hn (kv a)

Anhn (k a) + Avn [(n + 1)hn(kv a) − kv a hn+1 (kv a)]

A0 n jn (kß0 a) + Av 0 n [(n + 1)jn (kv0 a) − kv 0 ajn+1 (kv 0 a)]

Ÿe jn(ke a) +

=;

A0 n [njn (kß0 a) − kß0 a jn+1 (kß0 a)] + Av 0 n n(n + 1)jn (kv 0 a)

0 =0 ;0

=;

An Ÿ hn (k a) =

0 =0 ;0

Ÿe [n jn (ke a) − ke a jn+1 (ke a)] +

=

0

ß0 =0 ; 0

A0 n Ÿ0 jn (k0 a)

ß=;

Aßn Ÿß [n hn (kß a) − kß a hn+1 (kß a)]

Aß0 n Ÿß0 [n jn (kß0 a) − kß0 a jn+1 (kß0 a)]

p0 a2 ¢e + 2(n − n2 + (ke a)2 ) jn (ke a) − 4ke a jn+1 (ke a) † p 0 a2 ¢ + 2(n − n2 + (k a)2 ) hn(k a) − 4k a hn+1 (k a) + An † =; − AŒn 2n(n + 1)[(n − 1)hn(kv a) − kv ahn+1 (kv a)]

†

prr = p0rr

= †0 −AŒ 0 n 2n(n + 1)[(n − 1)jn(kv 0 a) − kv0 ajn+1 (kv0 a)] p a2 0 A0 n [ 0 ¢0 + 2(n − n2 + (k0 a)2 )]jn (k0 a) − 4k0 ajn+1 (k0 a) † 0 =0 ;0 † 2[(n − 1)jn(ke a) − ke a jn+1 (ke a)]

+

+2

pr˜ =

p0r˜

=;

An[(n − 1)hn (k a) − ka hn+1 (k a)]

+ Avn [(2(n2 − 1) − (kv a)2 )hn (kv a) + 2kv a hn+1 (kv a)] = †0 2 A0 n [(n − 1)jn(k0 a) − k0 a jn+1 (k0 a)] 0 =0 ;0

+ Av0 n [(2(n2 − 1) − (kv0 a)2 )jn (kv 0 a) + 2kv 0 a jn+1 (kv 0 a)]

Scattering of Sound

E

257

Further, the coefficients ŸE , Ÿ , Ÿ can be written as: ‰−1 ‰ kE2 ‰(‰ − 1) kE2 ≈ , 2 2 2 2 k0 1 − ‰ · kE /k0 k0 ⎧ 2 ⎪ ‰(‰ − 1) k ⎪ ⎨ ‰ kß2 ‰−1 2 k0 ≈ Ÿß = 2 2 ⎪ −‰ k0 1 − ‰ · kß2 /k0 ⎪ ⎩

ŸE =

Ÿß ≈ −(‰−1) Ÿ

kß2 2 k0

;

ß = , E

;

;

ß = ,

;

ß = ,

⎧ 2 ⎪ ⎨ 0 cp (‰ − 1) kß 2 k0 Ÿß ≈ ⎪ ⎩ −0 cp

(19)

;

ß = , E,

;

ß = .

(20)

The six boundary equations in the above tables (for each shape of the scatterer) originally contained common factors on both sides; they have been divided out. If these boundary equations are to be used for other types of scatterers besides fluid cylinders or spheres, the factors must be included again before the modification of the equations. These factors are contained in the following Table 6, the first column of which indicates the number of the corresponding row in the above tables and the index letter used above for the boundary conditions. Table 6 Common factors on both sides of the boundary equations Quantity

Factors Cylinder

1 (a)

vr

cos(n˜) −(−j)n ƒn a

2 (b)

v˜

(−j)n ƒn

T1 T0

(−j)n ƒn cos(n˜)

3 (c)

sin(n˜) a

Sphere Pn (cos ˜) −(−j)n ƒn a −(−j)n ƒn

dPn (cos ˜)=d˜ a

(−j)n ƒn Pn (cos ˜)

(−j)n ƒn

cos(n˜) a

(−j)n ƒn

Pn (cos ˜) a

prr

(−j)n ƒn

cos(n˜) a2

(−j)n ƒn

Pn (cos ˜) a2

pr˜

(−j)n ƒn

sin(n˜) a2

−(−j)n ƒn

4 (d)

5 (e) 6 (f )

@ T1 @r T0

dPn (cos ˜)=d˜ a2

258

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Scattering of Sound

Special types of scatterers Elastic scatterers: Replace the dynamic viscosity † with the shear modulus G or the Lamé constant ‹ and the sound velocity c0 with the compression modulus K or the second Lamé constant Š according to † → ‹ /j– = G /j–, c0

→ K 0

(21)

2 K = Š + ‹ . 3

;

Rigid scatterer at rest: Set vr = v˜ = 0

;

k = k → 0

;

Rn>0 (kß a) = 0

;

R0 (kß a) = 1.

(22)

Delete the boundary conditions (e), (f); in (a)–(d) delete on the right-hand sides all interior wave terms except the temperature wave term. In the special case of an isothermal surface delete (d) and set the right-hand side in (c) to zero. In this special case only a set of amplitudes Aßn with ß = , , Œ must be determined. Porous scatterers: Mostly |k a|2, |kE a|2 1 and |k /kß|2 1 ; ß = , Œ. Then the radial functions Rn (k a) can be approximated with the first term of their power series, and terms with Ÿ /Ÿ and ¢ /¢ can be neglected. Isothermal, freely oscillating hard scatterer: This case will be fully formulated here. The oscillation is in the x direction (ƒ1 = 2 is retained from the general formulations): ¥x = (−j)1 ƒ1 · Ax

x r = (−j)1 ƒ1 · Ax cos ˜ , a a

Ax vx = jƒ1 a

cos ˜ vr = jƒ1 Ax a

;

;

sin ˜ v˜ = −jƒ1 Ax . a

(23)

On the right-hand sides of the boundary conditions (a), (b) all terms with n = 1 vanish; and for n = 1 there will appear Ax . The right-hand side of the boundary condition (c) disappears, and also the complete boundary condition (d). The two last boundary conditions (e), (f) are replaced by an equation for the balance of force Kx (which is the integral of the stresses in the x direction over the scatterer surface): Ax ! (24) Kx = (prr cos ˜ − pr˜ sin ˜ ) · dA = j–M · vx = j–M · jƒ1 , a

Scattering of Sound

E

Table 7 Boundary conditions for an isothermal, hard, freely movable scatterer Bound. cond.

Equations Cylinder

vr = vx cos ˜

=;

= − [nJn (ke a) − ke aJn+1 (ke a)]

v˜ = −vx sin ˜ T =0 T0

=;

An nHn (k a) − Avn [nHn (kv a) − kv aHn+1 (kv a)] − Ax ƒ1;n

= −nJn(ke a) =;

=;

Kx = j–Mvx

An [nHn (k a) − k a Hn+1 (k a)] − AvnnHn (kv a) − Ax ƒ1;n

An ŸB Hn (k a) = −Ÿe Jn (ke a) A1

p0 a2 ¢ + 2(k a)2 H1 (k a) †

− Av1(kŒ a)2 H1 (kv a) − Ax (kv a)2

00 0

p0 a2 ¢e + 2(ke a)2 J1 (ke a) =− † Sphere

vr = vx cos ˜

=;

= −[njn(ke a) − ke a jn+1 (ke a)]

v˜ = −vx sin ˜ T =0 T0

=;

An hn (k a) + Avn [(n + 1)hn (kv a) − kv a hn+1 (kv a)] − Ax ƒ1;n

= − jn (ke a) =;

=;

Kx = j–Mvx

An [nhn (k a) − ka hn+1 (k a)] + Avnn(n + 1)hn (kv a) − Ax ƒ1;n

An ŸB hn (k a) = −Ÿe jn (ke a) A1

p0 a2 ¢ + 2(k a)2 h1 (k a) †

+ Av12(kŒ a)2 h1 (kv a) − Ax (kv a)2

00 0

p0 a2 =− ¢e + 2(ke a)2 j1 (ke a) †

259

E

260

Scattering of Sound

where M = 0 V0 is the scatterer’s mass (per unit length of the cylinder). The integrals over the terms with prr , pr˜ lead respectively to Irr =

+1 Pn (cos ˜ ) cos ˜ sin ˜ d˜ =

0

Ir˜ = 0

y · Pn (y) dy = −1

2 ƒ1,n, 3 (25)

∂ 4 (Pn (cos ˜ )) sin2 ˜ d˜ = −2 Irr = − ƒ1,n ∂˜ 3

with the Kronecker symbol ƒm,n . The boundary equations for an isothermal, movable, hard scatterer are given in Table 7.

E.16 Plane Wave Scattering at Elastic Cylindrical Shell

See also: Paniklenko/Rybak (1984)

A plane wave pe is incident (under an angle ‡ with the radius) on a cylindrical shell with radius R and thickness h. The exterior sound field is written as p = pe + pr + ps with pr the scattered field from a hard cylinder and ps additional scattering due to elasticity. z h θ

r

ϕ x

pe

2R±

Parameters of the surrounding medium: 0 , c0 , k0, Z0 = density, sound speed, free field wave number, free field wave impedance. Parameters of the shell: , ,E † = density, Poisson ratio, Young’s modulus, loss factor; cD , kD = speed and wave number of the dilatational wave in a plate of thickness h; E = E · (1 + j · †) = complex Young’s modulus.

Scattering of Sound

E

261

Abbreviations: §0 = k0R;

§ = kD R;

= k0 r · cos ‡;

ƒ0 = 1;

ƒn>0 = 2.

(1)

Field component formulations: pe (r, œ) = e−j (k0 r·cos ‡·cos œ+k0 z·sin ‡) , pr (r, œ) = ps (r, œ) =

Jn ( ) n≥0

,

H(2) n ( )

(2)

ƒn (−j)n H(2) ( ) · cos (nœ) −2 Z0 n e−j k0 z·sin ‡ 2 §0 cos ‡ (2) n≥0 Hn (§0 cos ‡) (Zmn + Zsn )

with radiation impedance Zsn of the n-th mode of the shell: Zsn =

−j Z0 H(2) n (§0 cos ‡) cos ‡ Hn (2) (§0 cos ‡)

(3)

and the mechanical impedance Zmn of the n-th shell mode: Zmn =

−j h D §2 D1

;

D = Det {Aik } ;

D1 = A11 · A22 − A12 · A21 ;

i, k = 1, . . . , 3 (4)

with matrix coefficients A11 = §2 − (§0 sin ‡)2 − (1 − ) n2/2 ; A12 = −A21 = −j (1 + ) §0 sin (n‡/2) , A13 = −A31 = − j §0 sin ‡ , A22 A33

(§0 sin ‡)2 − n2 ; A23 = A32 = −n , = §2 − (1 − ) 2

= §2 − h2 (§0 sin ‡)2 +n2 12 − 1 .

Asymptotic form of ps (for k0 r 1): / −2j −j (k0 +k0 z·sin ‡) e ps ≈ · ¥s (§0 , œ, ‡) , −2 Z0 ƒn · cos (nœ) . ¥n = ¥s (§0 , œ, ‡) = 2 §0 cos ‡ n≥0 n≥0 H(2) (Zmn + Zsn ) n (§0 cos ‡)

(5)

(6)

Shell resonances without shell losses: Resonance condition:

Im{Zmn + Zsn } = 0.

(7)

For n ≥ 2 the resonances with fluid load are about at the resonances of the shell without fluid load.

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Approximation for low frequencies §0 2n + 1: Z0 §0 cos ‡ §0 cos ‡ 2n+1 4 Zsn ≈ +j . cos ‡ (n !)2 2 n

(8)

Far field angular distribution of radiating mode in resonance: ¥nres (œ) ≈ −ƒn cos (nœ).

(9)

Scattered far field of the n-th mode in resonance: / −2j −j (k0 +k0 z·sin ‡) e · ƒn cos (nœ). ps ≈ −

(10)

Scattering cross section in resonance: 2 ƒ2n Qs = k0 cos ‡

2

cos2 (nœ) dœ =

0

8 . k0 cos ‡

(11)

Quality factor qn of the resonance of the n-th mode (without shell losses): Im{Zsn } + h (n !)2 2 2n h qn = = n . 1+ Re{Zsn } 2 n §0 0 R Shell resonances with shell losses:

(12)

c2D → c2D (1 + j†).

(13)

With

E → E · (1 + j†);

Mechanical shell mode impedance: 2 cD 2 1 2 2 h . Zmn = j –h 1 − (n − 1) 12 R2 c0 §20

(14)

Resonances at

n Bn h cD 2 h2 ; Bn : = (n2 − 1)2 . h 0 R c0 12 R2 1+n 0 R For n ≥ 2, with losses Zmn,† , without losses Zmn,0 : §20,res (n) =

(15)

†Z0 Bn . §0 Ratio of radiation loss to internal loss: 1 §0 2n+1 8 Re{Zsn } . = Re{Zmn } † (n !)2 Bn 2 Far field angular distribution in resonance with losses and ‡ = 0 for §0,res 1: −2 ƒn cos œ ¥nres ≈ . (n ! 2)2(2 §0 )2(n+1) † Bn

(17)

Relation to far field angular distribution ¥nh of a hard cylinder: 2 ¥nres . = h ¥nh † 1+n 0 R

(19)

Zmn,† ≈ Zmn,0 +

(16)

(18)

Scattering of Sound

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263

E.17 Plane Wave Backscattering by a Liquid Sphere

See also: Johnson (1977)

Consider a fluid sphere with radius a and 1 ,c1 ,k1 for density,sound speed,free field wave number, respectively, of the sphere fluid in an outer medium with 0 , c0 , k0 , respectively. Ratios of densities: g = 1 /0 ; of sound velocities ‚ = c1/c0 . The backscattering cross section for an incident plane wave is: 2 (−1)m (2m + 1) = , a2 k0 a 1 + j · Cm m≥0

Cm

m ym (k0 a) ßm − g‚ m jm (k1a) m , = m jm (k0 a) − g‚ m jm (k1 a)

(1)

m = m · jm (k0 a)−(m+1) · jm+1 (k0a) ; m = m · jm (k1 a)−(m+1) · jm+1 (k1a) , ßm = m · ym (k0a)−(m+1) · ym+1 (k0a) ; ßm = m · ym (k1 a)−(m+1) · ym+1 (k1 a) with jm (z) spherical Bessel functions, ym(z) spherical Neumann functions. 2 2 1−g 4 1 − g‚ ≈ 4 (k a) + . Approximation for k0 a 1: 0 a2 3 g‚ 2 1 + 2g

(2)

Special case: air bubble in water: 12 3 2 2 2 ≈ 4 (f − 1 + ƒ f ) 0 a2

;

f0 =

1 3 ‰P 0 2a

(3)

with f 0 the first bubble resonance frequency, ‰ = adiabatic exponent of air, P = static pressure, ƒ ≈ 1/5 an attenuation exponent of the bubble oscillation. 2 1 0.5 0.2 0.1 0.05 0.02 0.2

0.5

1 k0 a

2

5

10

Normalised backscatter cross section /(a2) for an air bubble in water (exact form)

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E.18 Spherical Wave Scattering at a Perfectly Absorbing Wedge

See also: Rawlins (1975)

Attention: The time factor here is e−i –t . A point source at Q = {rq , ¥q , zq } sends a spherical wave onto a wedge with half wedge angle §. The object treated is an idealised model for an absorbing wedge; the scattered field is half the sum of the fields for a hard and a soft wedge. Thus the wedge here is perfectly absorbing for all directions of incident sound.

Ω

Ω

z Q R P zq rq

x

z r ρ

y

ρq

Φq Φ

Field composition: with incident wave

p(, ¥ , z) = pi (, ¥ , z) + ps (, ¥ , z) ei k0 R . pi = k0 R

(1)

General solution: p R

− − ¥ − ¥q 1 ei k0 R() · cot d, 2 i Œ k0R() 2Œ C1 +C2 2 = + q2 − 2 q cos (¥ − ¥q ) + (z − zq )2 , =

R() = Œ

(2)

2 + q2 + 2 q cos + (z − zq )2 ,

= 2( − §)/.

The path of integration circumvents the branch points at ± ic with R c = 2 cosh−1 √ . 2 q

(3)

Scattering of Sound

E

265

Im{α}

π+ic C1 π

0

π

2π

Re{α}

C2 πic

Near field: n (¥ − ¥q ) 1 · Sn/Œ ; ƒ0 = 1 ƒn · cos Œ n≥0 Œ ⎧ * ⎪ +∞ 2 2 ⎨ J‘ k − t i 0 ei t(z−zq ) · S‘ = * dt ; ⎪ 2k0 ⎩ H(1) k02 − t2 ‘ −∞ p =

;

ƒn>0 = 2,

⎧ ⎨ < *k 2 − t2 0 ⎩ > *k 2 − t 2

(4)

0

with J‘ (z) = Bessel function; H(1) ‘ (z) = Hankel function of the first kind. Approximation for k0 1: i p ≈ h(1) k0 2 + q2 + (z − zq )2 0 Œ (1) 2 2 2 1/Œ h1/Œ k0 + q + (z − zq ) ¥ − ¥q 2i + k0 q /2 · 1/(2Œ) · cos (1/Œ) Œ 2 + q2 + (z − zq )2

+ O (k0)min(2/Œ,2)

(5)

with h(1) n (z) = spherical Hankel function of the first kind; (z) = Gamma function. Far field, k0q /R1 1: p≈

ei k0 R(nm ) 4 5 + V(− − ¥ + ¥q ) − V( − ¥ + ¥q ) k R(nm ) n,m 0

(6)

with summation over all n, m with |¥ − ¥q + 2nmŒ| < , and nm = − ¥ + ¥q − 2nmŒ 1 V(ß) = 2Œ

∞ 0

;

R1 =

* ( + q )2 + (z − zq )2 ,

ei k0 R(it) sin (ß/Œ) dt. k0R(it) cosh (t/Œ) − cos (ß/Œ)

(7)

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Approximation for the scattered far field: 1 1 ei (k0 R1 +/4) sin (/Œ) ps ≈ √ . √ 2k0R1 k0 q Œ cos (/Œ) − cos ((¥ − ¥q )/Œ)

(8)

E.19 Impulsive Spherical Wave Scattering at a Hard Wedge

See also: Biot/Tolstoy (1957), Ouis (1997)

> Sects. E.5 and E.6. This section will give exact solutions in the time domain for an impulsive point source and approximations for a point source with harmonic signal.

A hard wedge has its apex line on the z axis of a cylindrical co-ordinate system (r, ˜ , z) and its flanks at ˜ = 0,˜ = Ÿ0 .The wedge may be convex (Ÿ0 > ) or concave (Ÿ0 < ). A point source Q with volume flow q is at (rq , ˜q , 0); the observer point P is in (r, ˜ , z). z ϑ Q

rq

r

ϑq

z

P

Θ0

ϑ=0

ϑ=Θ 0

0 q · ƒ(t − r c0 ), 4r where t is time; r the distance from Q; and ƒ(z) is the Dirac delta function. Composition of the field: p = pq (Rq ) + ps (Rs ) + pd , The point source sends a delta pulse

pq (r) =

s

(1)

(2)

where pq = direct source contribution; ps = mirror source contribution; pd = diffracted wave. Some or all of the contributions may vanish, depending on the time interval and the geometrical situation. * In t< t0 ; t0 = R0 /c0; R0 = (r − rq )2 + z2 , no signal is received, and p = 0. * In t0 < t < ‘0; ‘0 = Ra /c0 ; Ra = (r + rq )2 + z2 the shortest distance between Q and P passing the apex line, only pq (Rq ) and (possibly) mirror source contributions ps (Rs ) are received. pq (Rq ) is obtained by the substitution r → Rq = r2 + rq2 + z2 − 2r rq cos (˜ − ˜q ) in pq (r), and ps (Rs ) is obtained by a similar substitution in pq (r ) with rq → rs ; ˜q → ˜s , where rs , ˜s are the co-ordinates of the mirror source.

Scattering of Sound

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267

Mirror sources (of different orders) represent specular reflections at the wedge flanks. The original source Q and a mirror source S produce a new mirror source at one of the flanks only if they are on the “field side” of that flank (which for that decision is extended to infinity). The black dots represent possible image sources, the open circles are excluded. In general there are conditions in which no image source contribution exists.

For a contribution ps (Rs ) the condition arccos

rs2 + rq2 + z2 − (c0 t)2 2rs rq

≤ must hold.

(3)

The diffracted wave pd in time: The diffracted wave is received for t > ‘0. It vanishes if the wedge angle is an integer fraction of : Ÿ0 = /m. Its time function is pd (t) =

y {ß}

−q Z0 e−y/Ÿ0 · {ß} · , 4Ÿ0 r rq · sinh (y)

= arccos h =

(c0 t)2 − (r 2 + rq2 + z2 ) 2r rq

(4)

,

sin ( ± ˜ ± ˜q )/Ÿ0

, 1 − 2e−y/Ÿ0 sin ( ± ˜ ± ˜q )/Ÿ0 + e−2y/Ÿ0

where {ß} is the sum of terms with the four possible combinations of signs.

(5)

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E.20 Spherical Wave Scattering at a Hard Screen

See also: Biot/Tolstoy (1957), Ouis (1997)

The hard screen is the special case Ÿ0 = 2 of the previous > Sect. E.19. The diffracted wave pd (t) in this case can be given an alternative form, valid for z = 0 (see sketch in E.19): / ##

cos (˜ ± ˜q )/2 −q 0 t+2 − t−2

· pd (t) = 2 ; 4 c0 t2 − t+2 t2 − t+2 + (t+2 − t−2 ) cos2 (˜ ± ˜q )/2 (1) t± = (r ± rq )/c0, where {{. . . }} is the abbreviation for the sum of two terms corresponding to different signs in the argument of the trigonometric function. This form is suited for the (approximate) Fourier transformation: ∞ pd (–) =

pd (t) · ej – t dt

‘0

;

‘0 =

r + rq . c0

(2)

Sound field for a harmonic point source: The sound field for a harmonic point source with angular frequency – = 2f is obtained by a Fourier transformation. The contributionspq , ps are the values of the spherical wave p(R) =

j k02 qZ0 e−j k0 R , 4 k0 R

(3)

where q is the volume flow amplitude and R → Rq ; R → Rs ,respectively (see > Sect. E.19).

Approximations of different orders pdi (f) will be given below for the diffracted field pd (f) in the frequency range. First order: development of pd (‘) for ‘ = t − ‘0 ‘0 ## 1 1 −q 0 1 + j j –‘0

· √ ·e . pd1 (–) = * 42 c0 2t+ (t+2 − t−2 ) cos (˜ ± ˜q )/2 2 f Second order: in the range of the first order, but improved: / −q 0 t+2 − t−2 ej –‘0 pd2 (–) = 2 4 c0 2t+ 2t+ ##

* cos (˜ ± ˜q )/2 · e−j – a± · · erfc −j – a± √ a±

= (t+2 − t−2 ) cos2 (˜ ± ˜q )/2 (2t+ ) ≥ 0 a± with the complementary error function erfc(z).

(4)

(5)

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Scattering of Sound

Third order: ‘2 + 2t+ ‘ (t+2 − t−2 ) cos2 (˜ ± ˜q )/2 ## −q 0 ej –(‘0 −t+ ) 1

pd3 (–) = * · · K0 (−j –t+ ) 42 c0 t+2 − t−2 cos (˜ ± ˜q )/2

(6) (7)

with K0 (z) the modified Bessel function of the second kind and order zero. √ √ Fourth order: ‘(‘ + 2t+ ) ≈ 2t+ ‘ / −q 0 t+2 − t−2 ej –‘0 pd4 (–) = 2 4 c0 2t+ 2 ##

* cos (˜ ± ˜q )/2 e−j –‘1,2 √ erfc −j –‘1,2 · √ ‘1,2 ±

± = t+2 − t+2 − t−2 cos2 (˜ ± ˜q )/2 ≥ 0

;

‘1,2 = t+ ∓

269

(8)

(9)

* ± ≥ 0,

(10)

where [[. . . ]] denotes the difference of the term with index 1 minus term with index 2. Fifth order: after expansion of ⎛ ⎞ 1 1 ‘ n ⎝ −1/2 ⎠ √ = √ 2t+ ‘ + 2t+ 2t+ n≥0 n

;

‘ ≤ 2t+

(11)

and using three series terms: /

cos (˜ ± ˜q )/2 −q 0 t+2 − t−2 ej –‘0 √ pd5 (–) = 2 · 4 c0 2t+ 2 ± √ 3/2

‘1,2 3 ‘1,2 1 1 −j –‘1,2 · + erfc −j –‘1,2 − * + ·e √ ‘1,2 4 t+ 4t+ −j– 64 t+2 1 · K0 (−j –‘1,2 /2)−K1 (−j –‘1,2 /2) 1+ j –‘1,2 with more terms: /

cos (˜ ± ˜q )/2 −q 0 t+2 − t−2 ej –‘0 · pd5 (–) = 2 √ 4 c0 2t+ 2 ± ⎡⎡ ⎛ ⎞ −j –‘1,2

e −1/2 ⎝ ⎠ 1 · ⎣⎣ √ · erfc −j –‘1,2 + ‘1,2 (2t+ )n n≥1 n * n−1/2 · (−1)n ‘1,2 e−j –‘1,2 · erfc −j –‘1,2 +2

1−n

n−1 2 (−2j –)1/2−n (2n − 2m − 3) !! · (−2j –‘1,2 )m m=0

(12)

(13)

## .

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Sixth order: with development of the fraction in {{. . . }} of pd (t) and U(a , b ; z) = 1/za · 2 F0 (a , 1 + a − b , −1/z) the Tricomi function: −q 0 ej –‘0 1 1 pd6 (–) = 2 * (−2t+ )n (n + ) · U(n + , n + 1 ; −2j –t+ ) 4 c0 t+2 − t−2 n≥0 2 2 (14) ## n 1 1

· . n−m cos (˜ ± ˜q )/2 m=0 ‘m 1 · ‘2

E.21 Spherical Wave Scattering at a Cone The scattering object is a circular cone, infinitely long, with a hard or soft surface. The cone may be tipped with a hard, soft, or absorbing sphere. The incident sound field comes from a point monopole source Q. Plane wave excitation is obtained by letting Q go an infinite distance, combined with Bessel function asymptotics.

ϑ

ϑ

ϑ

ϕ

ϕ

ϑ

Cone angle Ÿ = − ˜0 . Tipping sphere radius a (≥ 0). Time factor e+j–t . Cartesian and spherical co-ordinates of field point P = (x, y, z) = (r, œ, ˜ ). Cone tip in origin. Source co-ordinates Q = (rq > a, œq , ˜q ≤ ˜0 ). The sound field is represented by a series of azimuthal and polar modes (eigenfunctions of œ, ˜ ; Carslaw derives an integral representation of the field). Modes of order ‹ in azimuth œ and order Œ in polar angle ˜ are of the form: ‹

p‹,Œ (r, ˜ , œ) = RŒ (k0 r) · ŸŒ (˜ ) · ¥‹ (œ)

(1)

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Scattering of Sound

271

The radial factor RŒ (k0r) satisfies the Bessel differential equation and is of the form: d2 RŒ 2 dRŒ Œ(1 + Œ) 2 (2) RŒ = 0 ⇒ RŒ (r) = A · h(1) + − + k 0 Œ (k0 r) + B · hŒ (k0 r) (2) dr2 r dr r2 (k0r) = jŒ (k0r) with spherical Hankel functions of the first and second kind: h(1,2) Œ ‹ ±j · yŒ (k0r). The polar factor ŸŒ (˜ ) satisfies the Legendre differential equation and is of the form: ‹ ‹ ‹2 dŸŒ d2 ŸŒ ‹ ‹ ‹ + Œ(1 + Œ) − + cot ˜ (3) ŸŒ = 0 ⇒ ŸŒ (˜ ) = PŒ (cos ˜ ) d˜ 2 d˜ sin2 ˜ ‹

with the associated Legendre functions of the first kind PŒ (cos ˜ ) [the associated Leg‹ endre functions of the second kind QŒ (cos ˜ ) do not appear because they are singular ‹ ‹ at ˜ = 0]. The identity PŒ (x) = P−(Œ+1) (x) should be noticed. The azimuthal factor ¥‹ (œ) obeys the differential equation and is of the form: d2 ¥ + ‹ 2¥ = 0 ⇒ ¥ (œ) = A · cos (‹œ) + B · sin (‹œ) dœ2

(4)

= cos (m(œ − œq )) ; m = 0, ±1, ±2, . . . . The last form takes into account the field symmetry with respect to œ = œq . The mode orders ‹ = m are an integer because of the period in œ with œ = 2. The boundary condition at the cone surface for the hard cone is zero polar particle velocity v˜ at ˜ = ˜0, i.e. with Z0 v˜ =

RŒ (k0r) ∂Pm j j ∂p Œ (cos ˜ ) =j grad˜ p = k0 k0r ∂˜ k0 r ∂˜

(5)

RŒ (k0 r) ∂Pm Œ (cos ˜ ) = −j sin ˜ k0 r ∂(cos ˜ ) it leads to the characteristic equation (mode eigenvalue equation) for eigenvalues Œ: !

PŒ m (cos ˜0 ) −−−−−−−−→ PŒ m (x0 ) = 0.

(6)

cos ˜0 →x0

The special cases ˜0 = 0 and ˜0 = here are irrelevant; a prime indicates the derivative. In the case of the soft cone the condition of zero sound pressure at the cone surface ˜ = ˜0 leads to the following characteristic equation for Œ: !

Pm −−−−−−−→ Pm Œ (cos ˜0 ) − Œ (x0 ) = 0.

(7)

cos ˜0 →x0

Because of P00 (x) = 1 ; Pm>0 (x) ≡ 0, the value Œ = 0 is the trivial solution for the hard 0 ∧

cone; it is a forbidden value for the soft cone. Because of the equivalence Œ = −(Œ + 1) it is sufficient to consider eigenvalues Œ > 0. The conditions of regularity at r = 0, the Sommerfeld far field condition for large r, and the source condition at r = rq suggest a radial subdivision of the field formulation by

E

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Scattering of Sound

two zones (1) and (2) with their common limit at r = rq and the field formulations in zone (1): (1) (2) AŒm h(2) p1 (r, ˜ , œ) = Œ (k0 rq ) hŒ (k0 r) + rŒ · hŒ (k0 r) |m| ≥ 0

(8)

Œ>0

·Pm Œ (cos ˜ ) cos (m(œ − œq )),

and in zone (2): p2 (r, ˜ , œ) =

(1) (2) AŒm h(2) Œ (k0 r) hŒ (k0 rq ) + rŒ · hŒ (k0 rq )

|m| ≥ 0

(9)

Œ>0

· Pm Œ (cos ˜ ) cos (m(œ − œq )). The summation index |m| > 0 indicates that both signs of m = ±1, ±2, . . . must be considered although the sign has no influence on the azimuthal factor, but it has an important influence on the polar eigenvalues Œ(±m), and therefore on P±m Œ(±m) (x) (see below). The factors rŒ in (8) and (9) are the modal reflection factors of the sphere around the cone tip. They are for a sphere surface with admittance G: rŒ = −

(1) j Z0 G h(1) Œ (k0 a) − hŒ (k0 a)

(2) j Z0 G h(2) Œ (k0 a) − hŒ (k0 a) (1) j Z0 G − (Œ/k0a) hŒ (k0a) + h(1) Œ+1 (k0 a) = − . (2) j Z0 G − (Œ/k0a) hŒ (k0a) + h(2) Œ+1 (k0 a)

(10)

The special cases of a hard sphere, G = 0, and of a soft sphere, |G| = ∞, lead to: rŒ −−−−→ − G→0

hŒ (1) (k0a) h(1) Œ (k0 a) ; r → − − − − − − − −−−−−→ 1. Œ |G|→∞ k a→0 hŒ (2) (k0a) h(2) Œ (k0 a) 0

(11)

The modal amplitudes AŒm in (8) and (9) are determined from the source condition, which fits the step of the radial volume flow at r = rq to the volume flow q of the point source: !

vr2 (rq + 0) − vr1 (rq − 0) = q ·

ƒ(˜ − ˜q ) ƒ(œ − œq ) ƒ(˜ − ˜q ) ƒ(œ − œq ) · =q · · (12) h2 h3 rq rq · sin ˜q

with the Dirac delta functions ƒ(. . . ) and the scale factors h2 = rq ; h3 = rq sin ˜q of the transformation betwen the Cartesian and the spherical co-ordinates. The factors on the right-hand side will be expanded in polar and in azimuthal modes, respectively:

ƒ(œ − œq ) ƒ(˜ − ˜q ) = cm cos m (œ − œq ) ; = bŒ Pm Œ (cos ˜ ). rq · sin ˜q m≥0 rq Œ>0

(13)

Scattering of Sound

The coefficients are:

cm =

1 Mm rq sin ˜q

E

273

(14)

with the azimuthal mode norms 2 Mm = 0

⎧ ⎨ 1 ; m = 0 2 cos2 (m(œ − œq )) dœ = ; ƒm = ⎩ 2 ; |m| > 0 ƒm

(15)

from the orthogonal integrals 2

⎧ ⎨ 0 ; m = m integer cos (m(œ − œq )) · cos (m (œ − œq )) dœ = ⎩ M ; m = m integer

(16)

m

0

bŒ =

and

sin ˜q m PŒ (cos ˜q ) Nm Œ rq

(17)

with the polar mode norms Nm Œ from the integrals ˜0

m Pm Œ (cos ˜ ) · PŒ (cos ˜ ) sin ˜ d˜

0 x=cos ˜

x0

−−−−−−−−−−−−−−→ − dx = − sin ˜ d˜ x0 = cos ˜0

1

⎧ ⎨ 0 ; Œ = Œ m Pm . Œ (x) · PŒ (x) dx = ⎩ Nm ; Œ = Œ

(18)

Œ

The source condition (12), when applied term-wise, gives: 5 4 (2) (1) (2) j AŒm h(1) Œ (k0 rq ) hŒ (k0 rq ) − hŒ (k0 rq ) hŒ (k0 rq ) =

Z0 q Pm (cos ˜q ), (19) 2 Œ Nm Œ · Mm rq

where the brackets {. . . } contain the Wronski determinant of the spherical Hankel functions:

(2) W h(1) Œ (k0 rq ) , hŒ (k0 rq ) =

−2j . (k0 rq )2

(20)

Thus, the amplitudes AŒm in (8) and (9) will become: AŒm =

k02 Z0 q k02Z0 q ƒm m m P (cos ˜ ) = PŒ (cos ˜q ). q Œ 2 Nm 4 Nm Œ · Mm Œ

(21)

A useful modification of field formulations (8) and (9) is obtained by the identical substitutions of rŒ with CŒ : rŒ = 1 + (rŒ − 1) = 1 − 2CŒ ⇒ 2CŒ = 1 − rŒ ,

(22)

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together with the relations between spherical Hankel and Bessel and Neumann func2) (z) = jŒ (z) ± j · yŒ (z), giving the intermediate result: tions: h(1, Œ 2CŒ =

(1) j Z0 G hŒ (2) (k0 a) − hŒ (2) (k0a) + j Z0 G h(1) Œ (k0 a) − hŒ (k0 a) (2) (2) j Z0 G hŒ (k0 a) − hŒ (k0a)

=2

j Z0 G jŒ (k0 a) − jŒ (k0a) (2)

j Z0 G hŒ (k0 a) − hŒ (2) (k0a)

(23)

,

from which follow the special cases: CŒ −−−−→ G→0

jŒ (1) (k0a) j(1) Œ (k0 a) ; CŒ −−−−−→ 0. ; C → − − − − − − Œ (2) (2) |G|→∞ k0 a→0 hŒ (k0 a) hŒ (k0a)

(24)

With these amplitudes and rŒ = 1 − 2CŒ , the sound fields in (1) and (2) finally are k 2 Z0 q ƒm (2) hŒ (k0rq ) jŒ (k0r) − CŒ · h(2) p1 (r, ˜ , œ) = 0 Œ (k0 r) m 2 NŒ |m| ≥ 0 Œ>0

(25)

m · Pm Œ (cos ˜q ) PŒ (cos ˜ ) cos (m(œ − œq )) = p1, cone + p1, sphere ;

k02 Z0 q 2

p2 (r, ˜ , œ) =

|m| ≥ 0 Œ>0

ƒm (2) hŒ (k0r) jŒ (k0 rq ) − CŒ · h(2) Œ (k0 rq ) m NŒ

(26)

m · Pm Œ (cos ˜q ) PŒ (cos ˜ ) cos (m(œ − œq )) = p2, cone + p2, sphere .

The first terms in the brackets [. . . ] describe the sound fields in each zone including the scattered field from the cone. The second terms describe the additional scattering by the tip sphere; they vanish when the sphere disappears, k0 a → 0 [see (24)]. The amplitude factor in front of the sum supposes a point source with a volume flow q. Results (25) and (26) and their derivation so far are quite normal. The problems begin with the numerical application by difficulties in the determination of the polar eigenvalues Œ (because modes of associated Legendre functions have a steady transition to useless trivial solutions for positive integer azimuthal mode numbers m) and continue in the evaluation of the polar mode norms Nm Œ by numerical integrations of (18) because of the extremely large variation with m of the order of magnitude of the oscillating associated Legendre functions Pm Œ (x). An analytical description of the polar mode norm Nm Œ

cos ˜0

=−

2 Pm Œ (x) dx

(27)

1

follows from the orthogonal integral:

‹ PŒ (x)

·

‹ P (x) dx

=

‹ ‹ (1 − x2 ) PŒ (x) · P ‹ (x) − PŒ ‹ (x) · P (x) (Œ − )(1 + Œ + )

,

(28)

Scattering of Sound

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275

which with mode solutions obeying the boundary conditions of eq. (6) or (7) vanishes for different mode numbers Œ, . For equal mode numbers Œ = in the norm integral it is evaluated as a limit (x0 = cos ˜0 , a prime for the derivative):

2 Nm Œ = −(1 − x0 ) lim

→Œ

m m m Pm Œ (x0 ) · P (x0 ) − PŒ (x0 ) · P (x0 ) (Œ − )(1 + Œ + )

(29)

or, with the recursion for the derivative:

‹

‹

−(1 − x 2 )PŒ (x) = ‹ x PŒ (x) +

√

‹+1

1 − x2 PŒ (x),

(30)

one gets a limit representation: m+1 m+1 Pm (x0 ) · Pm Œ (x0 ) · P (x0 ) − PŒ (x0 ) 2 Nm . = 1 − x lim Œ 0 →Œ (Œ − )(1 + Œ + )

(31)

Numerical application of (29) and (31) requires the availability of precisely computing programs for associated Legendre functions.

E.22 Polar Mode Numbers at a Soft Cone The characteristic equation for polar mode numbers Œ at a cone with soft surface at polar angle ˜ = ˜0 (see > Sect. E.21) is:

!

Pm −−−−−−−→ Pm Œ (cos ˜0 ) − Œ (x0 ) = 0 cos ˜0 →x0

with integer m = 0, ±1, ±2, . . .

(1)

The difficulties with the evaluation of polar mode numbers Œ may be illuminated by the fact that for positive integers m and n the associated Legendre functions are identically zero, Pm n (x) ≡ 0 for m > n, and they are constant for m = n. When m > n, eq. (1) holds, but Pm n (x) then does not represent a mode since it is a trivial solution. Because there are no other solutions Œ for m > 0 and Œ < m, the transition between modes and trivial solutions is steady.

276

E

Scattering of Sound

-lg|P(nu,mu,cos(th))|, theta=165.

4

2

0

-2

-4 -2

-4 10

0 8

mu

2

6 4 nu

2

4

‹ − lg PŒ (cos ˜ ) over Œ and ‹, for ˜ = 165◦. Equivalences in the plot labels: ‹ nu → Œ; mu → ‹; theta → ˜ ; P(nu, mu, cos(th)) → PŒ (cos ˜ ) ‹

For getting an overall view of the space ‹, Œ of mode solutions the magnitude |PŒ (cos ˜ )| is 3D-plotted over ‹, Œ for constant values of ˜ (indeed, the negative common logarithm ‹ − lg |PŒ (cos ˜ )| is plotted, so that zeros are visible as crests). The mesh points of the crests (i.e. the solutions) are collected in the roof plane of the enclosing cube of the 3D-plot. The diagram above is an example for ˜ = 165◦ . The large variation of the order of magnitude with ‹ and the change of the structure of the function for Œ > ‹ are clearly visible. The next plot combines points of mode solutions Œ(‹) from the level plot with straight approximation lines suited for the evaluation of approximations as starters to Muller’s numerical solution method for the characteristic equation (see > Sect. J.4 “Lined ducts, general”).

Scattering of Sound

E

277

nu(mu), soft; theta=165.

10 8 6 nu

4 2 0 -4

-2

0 mu

2

4

Points of solutions (‹, Œ) from the 3D-level plot for ˜ = 165◦ , and straight lines of approximation (see below). The thick points mark the values for which the solutions approach Œ(‹) ≈ ‹; the thick inclined line through the origin represents Œ = ‹ The solution point curves will be enumerated with k = 0, 1, 2, . . . from low to high; k = 0 belongs to the lowest curve ending near (‹, Œ) = (0, 0). The approximations are composed of three line sections in the ranges ‹ ≤ 0, 0 < ‹ ≤ ‹2, ‹ > ‹2, respectively, where ‹2 belongs to the “thick points” in the plot. The approximations are evaluated according to the following table: ˜0

‹≤0

0 < ‹ ≤ ‹2

‹ > ‹2

90◦ < ˜0 ≤ 145◦

Œap (‹)

Œap (‹)

Œ(‹) = k

Œap (‹) + Œap1 (‹) /2

Œap (‹)

Œ(‹) = k

Œap1 (‹)

Œap (‹)

Œ(‹) = k

145◦ < ˜0 ≤ 170◦ 170◦ ≤ ˜0

with the endpoints (‹1, Œ1 ) on the axis ‹ = 0, and (‹2, Œ2 ) marking the transition to constant values Œ ≈ k (“thick points”): • Endpoint on the axis ‹ = 0: ⎧ ⎨ ‹ =0 1 −−−−→ Œap1 (‹); ⎩ Œ = k · /˜ − ‹ + 2(1 − ˜ /) ‹1 →‹ 1 0 1 0 • Transition point to Œap2 (‹) = k = const.; k = 0, 1, 2, . . . ⎧ ⎨ ‹ = 2(1 − ˜ /) (1 + k) 2 0 −−−−→ Œap2 (‹) = k; ‹>‹2 ⎩ Œ =k 2

(2)

(3)

E

278

Scattering of Sound

• Connection of the points (‹1 , Œ1) and (‹2 , Œ2): Œap (‹) = Œ2 + (Œ1 − Œ2) · (‹ − ‹2 )/(‹1 − ‹2 ).

(4)

These approximations are safe starters to Muller’s procedure for a set of mode solutions (in contrast to approximations published in the literature or produced by widely distributed mathematical computer applications) if the (small) increments applied to them for the second and third Muller starters are taken in directions for partial compensation for the deviations to solution points for which the plot above is typical. Analytical approximations to mode solutions for a soft cone, usable as start solutions in Muller’s procedure, are obtained from the series representation of the associated Legendre functions in (1) for ‹ → m, integer: Pm Œ (x) =

(−1)m (Œ + m + 1) (1 − x2 )m/2 2 F1 (m + Œ + 1 , m − Œ ; m + 1 ; (1 − x)/2) (5) 2m m! (Œ − m + 1)

with the hypergeometric function

2 F1 (a1 , a2 ; b1 ; z)

=

∞ (a ) · (a ) 1 k 2 k k z k! (b ) 1 k k=0

(6)

containing the recursively evaluated Pochhammer symbols (k, n = integers; a = real): (a)0 = 1 : (a)k = a · (a + 1) · . . . · (a + k − 1) = (a)k−1 · (a + k − 1) = (a + k)/ (a) ; k ≥ 1, (n)k =

(7)

(n + k − 1)! −−−→ (1 + k)!. (n − 1)! n→2

For negative integer orders ‹ → −m holds: m P−m Œ (cos ˜ ) = (−1)

(Œ − m + 1) m P (cos ˜ ) = 0, (Œ + m + 1) Œ

(8)

and therefore, with (5), the characteristic equation for eigenvalues Œ of soft cones then is: P−m Œ (cos ˜ ) =

(1 − x2 )m/2 2 F1 (m + Œ + 1 , m − Œ ; m + 1 ; (1 − x)/2) . 2m m!

(9)

Truncation of the series returns a polynomial in Œ whose solutions may be used as approximations to mode numbers Œ. Some of the polynomial solutions may appear several times, while others may be complex; they should be excluded from application as starters of Muller’s procedure. An important decision is the limit ‹ up to which azimuthal mode numbers ‹ = m > 0 can be used in a modal field synthesis so that trivial solutions are avoided in the field series. From numerical investigations one can derive the recommendation for a soft cone: ‹ ≤ Min(k, ‹2) = Min(k, 2(1 + k)(1 − ˜0 /)).

(10)

Scattering of Sound

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279

E.23 Polar Mode Numbers at a Hard Cone The characteristic equation for mode eigenvalues Œ is: PŒ m (cos ˜0 ) = 0 ; Œ > 0 ; m = 0, ±1, ±2, . . . .

(1)

The derivative may be evaluated with the recursion ‹

d PŒ (cos ˜ ) ‹+1 ‹ = PŒ (cos ˜ ) + ‹ cot ˜ PŒ (cos ˜ ), d˜

(2a)

‹

x 1 d PŒ (x) ‹+1 ‹ PŒ (x) − ‹ PŒ (x). = −√ dx 1 − x2 1 − x2

(2b)

Reliable starters solver procedure can be obtained from 3D-survey ‹for Muller’s equation plots of − lg ∂PŒ (cos ˜ )/∂˜ over Œ and ‹ for a fixed parameter ˜ . Solutions therein are marked by maxima which arrange in crests. These solutions (‹, Œ) are collected as points in the roof plane of the enclosing cube of the 3D-plot. The following example is for ˜ = 120◦ . Equivalences in the plot labels: nu → Œ; mu → ‹; theta → ˜ ; ‹ P(nu, mu, cos(th)) → PŒ (cos ˜ ). -lg|dP(nu,mu,cos(th))/dth|, theta=120.

4

2

0 -5

-2 0

-4 10

mu

8 6 nu

5

4 2

‹ − lg ∂PŒ (cos ˜ )/∂˜ over Œ and ‹ for ˜ = 120◦ The next plot combines points of mode solutions Œ(‹) from the level plot with straight approximation lines suited for the evaluation of approximations as starters to Muller’s

E

280

Scattering of Sound

numerical solution method for the characteristic equation (see general”).

> Sect. J.4,“Lined ducts,

nu(mu), hard; theta=120.

10 8 6 nu

4 2 0 -4

-2

0

mu

2

4

6

8

Points of solution (‹, Œ) from the 3D-plot for ˜ = 120◦ of a hard cone, and straight lines of approximation The structure of the approximations is shown in the next graph. The sequences (curves) of the solution points are enumerated from low to high with k = 0, 1, 2, . . .. The approximation lines are composed of line sections (a), (b), (c), (d) through endpoints (0), (1), (2). ν

μ

Construction of the straight approximations to the moden solutions Œ(‹) for a hard cone Points: (0): Value of Œap1 at ‹ = 0 : Œap1(0) = k · /˜

(2): ‹2 , Œ2 = (2k(1 − ˜ /), k)

:

‹0 , Œ0 = (0 , k · /˜ ) (3)

Scattering of Sound

E

281

(1): Extension of the line (‹2 , Œ2) → (‹0, Œ0) to ‹ = −‹2 :

‹1 , Œ1 = −‹2 , k + ‹2/2 · /˜ Sections: → −‹ ; k = 0 ; ‹ < 0 (a): Œ(‹) = Œap1 = k · /˜ − ‹ − −− k=0

(b): Œ(‹) = Œap2 = k;

k = 1, 2, 3, . . . ; ‹ ≥ ‹2

(c): Connection between (1) & (2); ‹1 ≤ ‹ ≤ ‹2 : Œ(‹) = Œap = Œ0 + (Œ2 − Œ0 )

‹ − ‹0 = k − ‹/2 · /˜ ‹ 2 − ‹0

(d): Straight line through (1) with slope of Œap1 ;

Œ(‹) = Œap3 = Œ1 − ‹ − ‹1

(4)

‹ < ‹1 :

Special case ‹ = m = 0 for a hard cone: Then the characteristic equation: d PŒ (x0 ) dx

=

1 (Œ + 1) Œ 2 F1 (2 + Œ , 1 − Œ ; 2 ; (1 − x0 )/2) 2

=

(2 + Œ)k · (1 − Œ)k 1 ! (Œ + 1) Œ (1 − x0 )k = 0 2 2k k! (1 + k)!

∞

(5)

k=0

can be formulated with the hypergeometric function 2 F1

(a1 , a2 ; b1 ; z) =

∞ (a1 )k · (a2 )k k z k! (b1)k

(6a)

k=0

containing Pochhammer’s symbols: (a)0 = 1 : (a)k = a·(a+1)·. . . ·(a+k −1) = (a)k−1 ·(a+k −1) = (a+k)/ (a) ; k ≥ 1.(6b) Truncation of the series will return a polynomial equation in Œ. The factor (Œ + 1)Œ in front of the polynomial produces the solution Œ = 0 and the equivalent solution −(1 + Œ) = −1. Expansion up to the fourth degree gives the following approximations (of moderate precision; x0 = cos ˜0 ): * (1 − x0 ) ± (1 − x0 )2 + 4(1 − x0 ) 10 − 4x0 ± 2 4 − 8x0 + x02 Œ≈− . (7) 2 (1 − x0 ) As the upper limit ‹ = m > 0 up to which azimuthal mode numbers m can be used in a modal field synthesis so that trivial solutions are avoided in the field series one can recommend for a hard cone: ‹ ≤ Min(k, ‹2) = Min(k, 2k(1 − ˜0 /)).

(8)

E

282

Scattering of Sound

The next 3D-plot shows œ-orbits (thick) for the sound pressure magnitude |p(r, ˜ , œ)/pQ (0)| with fixed ˜ = ˜0 = 145◦ , i.e. immediately on the hard cone surface, for some values of k0r. The sound pressure is plotted as radial distance from the orbit circle (thin). The point source Q (thick point) is placed at the height of the cone tip (˜q = 90◦ ) at a radial distance k0 rq = 12. (Naming equivalences in the plot label: |p(phi)/pQ(0)| → |p(r, ˜ , œ)/pQ (0)|; tha0 → ˜0 ; k0rq → k0 rq ; thaq → ˜q ). |p(phi)/pQ(0)|; tha0=145., k0rq=12., thaq=90.

0 z

-2 -4 -6

5

-5

2.5 0

0 5

y+|p|

-2.5

x+|p| 10

-5

|p(œ)/pQ(0)| on œ-orbit for point source Q at (k0 rq , ˜q , œq = 0) near a hard cone with ˜0 = 145◦ ; field point P = (k0r, ˜ , œ) parameter values: ˜0 = 145.◦ ; k0 r = 2. & 4. & 6. & 8.; ˜ = 145.◦ ; k0 rq = 12.; ˜q = 90.◦ ; œq = 0.; khi = 6; ‹lo = −8; œ = 15.◦

E.24 Scattering at a Cone with Axial Sound Incidence This section treats a special case with axial sound incidence, ˜q = 0, œq = 0, of the preceding sections on sound scattering at a cone (see those sections for definitions of symbols). This special case avoids some analytical and numerical difficulties of the more general task. The associated Legendre functions go over to the Legendre functions, ‹ PŒ (x) −−−→ PŒ (x); PŒ (1) = 1. ‹=0

Mode series for the sound field in zones (1) and (2) with common boundary at the radius r = rq of the point source with its strength defined by volume flow q (k0 = –/c0; Z0 = 0 c0 ): k02Z0 q 1 (2) hŒ (k0rq ) jŒ (k0r) − CŒ · h(2) Œ (k0 r) · PŒ (cos ˜ ), 2 Œ>0 NŒ 2 k Z0 q 1 (2) hŒ (k0r) jŒ (k0 rq ) − CŒ · h(2) p2 (r, ˜ , œ) = 0 Œ (k0 rq ) · PŒ (cos ˜ ), 2 Œ>0 NŒ p1 (r, ˜ , œ) =

(1a)

Scattering of Sound

ϑ

E

283

ϑ ϕ

ϑ

and with reference to the source free field pressure pQ (0) at the origin: 1 2 p1 (r, ˜ , œ) (2) = (2) h(2) Œ (k0 rq ) jŒ (k0 r)−CŒ · hŒ (k0 r) · PŒ (cos ˜ ), pQ (0) h0 (k0rq ) Œ>0 NŒ 1 2 p2 (r, ˜ , œ) (2) = (2) h(2) Œ (k0 r) jŒ (k0 rq )−CŒ · hŒ (k0 rq ) · PŒ (cos ˜ ). pQ (0) N h0 (k0rq ) Œ>0 Œ

(1b)

Eigenvalues (mode numbers) Œ of the polar modes are solutions of the characteristic equations PŒ (x0 ) = 0 with a soft cone, and dPŒ (x0 )/dx = 0 with a hard cone; x0 = cos ˜0 for a cone half-angle ˜0 . The terms with the factors CŒ in the brackets represent the scattered field generated additionally by a sphere of radius a and surface admittance G around the cone tip; they will be dropped if there is no sphere. CŒ =

j Z0 G jŒ (k0a) − jŒ (k0 a)

(2)

(2) j Z0 G h(2) Œ (k0 a) − hŒ (k0 a)

with special values: CŒ −−−−→ G→0

jŒ (1) (k0a) j(1) Œ (k0 a) ; C → ; CŒ −−−−−→ 0. − − − − − − Œ |G|→∞ h(2) (k a) k0 a→0 hŒ (2) (k0 a) 0 Œ

(3)

The NŒ are the mode norms Nm Œ for m = 0. One gets by partial integration: x0 NŒ = −

(PŒ (t)) 2 dt =

1

+

x0 · (PŒ (x0 ))2 PŒ−1 (x0 ) · PŒ (x0 ) + 1 + 2Œ 1 + 2Œ

Œ PŒ (x0 ) ∂Py (x0 )/∂y 1 + 2Œ

y=Œ−1

− PŒ−1 (x0 ) · ∂Py (x0 )/∂y

y=Œ

(4) .

This simplifies in the case of a soft cone, because of PŒ (x0 ) = 0, to: x0 NŒ = − 1

(PŒ (t)) 2 dt =

−Œ PŒ−1 (x0 ) · ∂Py (x0 )/∂y y=Œ . 1 + 2Œ

(5)

284

E

Scattering of Sound

Alternatively, evaluation of the mode norm by integration of the product of the series for the Legendre function: PŒ (x) = 2 F1 (−Œ , 1 + Œ ; 1 ; (1 − x)/2) =

(−Œ)k (1 + Œ)k k≥0

2k (k!)2

(1 − x)k

(6a)

will return: x0 NŒ = −

(PŒ (x))2 dx =

(−Œ)k (1 + Œ)k (−Œ)‰ (1 + Œ)‰ (1 − x0 )1+k+‰ . 2k (k!)2 2‰ (‰!)2 1+k+‰

(6b)

k, ‰≥0

1

The upper summation limits for k and ‰ must be definitely higher than Œ. The convergence becomes slow for ˜0 ≈ . Analytical approximations to mode numbers (eigenvalues) Œ for a soft cone can be obtained by truncation of the series in the characteristic equation: PŒ (x0 ) = P0Œ (x0 ) = 2 F1 (−Œ , 1 + Œ ; 1 ; (1 − x0 )/2) =

(−Œ)k (1 + Œ)k k≥0

2k (k!)2

!

(1 − x0 )k = 0,(7)

and for a hard cone by truncation of: d PŒ (x0 ) 1 = (Œ + 1) Œ 2 F1 (2 + Œ , 1 − Œ ; 2 ; (1 − x0 )/2) dx 2 ∞

(2 + Œ)k · (1 − Œ)k 1 ! = (Œ + 1) Œ (1 − x0 )k = 0. k 2 2 k! (1 + k)!

(8)

k=0

The summation over k must go rather high, because a number of the polynomial solutions Œ must be excluded from the application as starters in Muller’s procedure, due to multiple solutions, negative values, complex values, or too large magnitudes of the characteristic equations at the polynomial solutions. A “more economic” construction of starters makes use of the construction rules given in > Sects. 22 or 23. The sound field for axial incidence of a plane wave is obtained by starting with the field in zone (1) from eq. (1b) in the limit rq → ∞ with asymptotic approximations for the Hankel functions with argument z = k0 rq → ∞: H(2) (z) ∼

√ 2/z e−j (z−/2−/4) ;

(2) h(2) Œ (k0 rq )/h0 (k0 rq )

h(2) Œ (z) =

√ 1 −j (z−Œ/2−/2) /2z H(2) Œ+1/2 (z) ∼ e z

(9)

∼ e+j Œ/2

leading to [pQ (0) is the plane wave pressure in the origin]: e+j Œ/2 p(r, ˜ , œ) =2 jŒ (k0 r) − CŒ · h(2) Œ (k0 r) · PŒ (cos ˜ ). pQ (0) NŒ Œ>0

(10)

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E

285

References Biot, M.A.; Tolstoy, I.: Formulation of wave propagation in infinite media by normal coordinates with an application to diffraction. J.Acoust.Soc. Am. 29 381–391 (1957) Carslaw, H.S.: The scattering of sound waves by a cone. Math. Annalen 75, 133–147 (1914) Johnson, R.K.: J. Acoust. Soc. Amer. 61 375–377 (1977) Mechel, F.P.: Schallabsorber, Vol. I, Ch. 6: Cylindrical sound absorbers Hirzel, Stuttgart (1989) Mechel, F.P. Schallabsorber,Vol. II, Ch. 14:“Characteristic values of composite media”Hirzel,Stuttgart (1995) Mechel, F.P.: A uniform theory of sound screens and dams. Acta Acustica 83, 260–283 (1997)

Mechel, F.P.: Mathieu Functions; Formulas, Generation, Use Hirzel, Stuttgart (1997) Mechel, F.P.: Schallabsorber, Vol. III, Ch. 22: Semicircular absorbing dam on absorbing ground Hirzel, Stuttgart (1998) Mechel, F.P.: Improvement of corner shielding by an absorbing cylinder. J. Sound Vibr. 219, 559–579 (1999) Ouis, D.: Report TVBA-3094, Lund Inst.of Technology Theory and Experiment of the Diffraction by a Hard Half Plane (1997) Paniklenko, A.P.; Rybak, S.A.: Sov. Phys. Acoust. 30, 148–151 (1984) Rawlins: J.Sound Vibr. 41, 391–393 (1975)

F Radiation of Sound Radiation of sound takes place, not only if a surface is driven by an internal force, but also if the surface is set in vibration by an incident sound wave. Then radiation is the back reaction of the surface to the incident sound in the process of reflection and/or scattering. Part of the power which the vibrating surface produces with the exciting sound pressure is radiated as effective power to infinity; this gives rise to the radiation loss of the surface. Part of the reaction is contained in non-radiating near fields; they will influence the tuning of resonating surfaces by the inertia of their oscillating mass. This oscillating mass can be represented as the mass contained in a prism with the cross section of the vibrating surface (e.g. an orifice) and the length of an end correction. The advantage of the concept of the oscillating mass and of the end correction is the possibility to include them as members in equivalent networks (they are determined just so that this is possible). Recall the distinction between “mechanical impedance”, “impedance”, and “flow impedance” from > Sect. A.3 conventions.

F.1

Definition of Radiation Impedance and End Corrections


Let vn (s) be the oscillating velocity in a surface A with the co-ordinate s in A,and directed normal to the surface towards the side, on which a sound pressure p(s) exists. The time average sound power produced is: 1 ∗ ¢ = ¢ +j·¢ = p(s) · vn (s) dA = In (s) dA (1) 2 A

A

with the normal time average sound intensity In (s). The radiation impedance Zr = Zr + j · Zr is defined by: 1 ¢: = Zr · |vn (s)|2 dA. (2) 2 A

The mechanical radiation impedance Zmr (which is suitable for a small surface A and/or conphase excitation) is defined by: 1 ¢: = Zmr · |vn (s)|2 A , 2 where . . .A stands for the average over A. It is evident that: Zmr =A·Zr .

(3)

F

288

Radiation of Sound

A normal component ZFn (s) of a field impedance can be defined by: p(s) = ZFn (s) · vn (s) on A. Then 1 ¢= ZFn (s) · |vn (s)|2 dA. (4) 2 A

Special case:

ZFn (s) = const(s):

Special case:

|vn(s)| = const(s):

Zr = ZFn . 1 1 dA Zr = ZFn (s) dA = , A A Gn A

(5) (6)

A

where the field admittance component Gn = 1/ZFn . Special case:

vn (s) = const(s) in magnitude and phase: Zr =

Special case:

p(s) = const(s):

1 ¢= p 2

vn∗ (s) dA =

A

p(s)A . vn

1 p · q∗ , 2

(7) (8)

where q = volume flow of the surface A. Related quantities: The radiation efficiency is defined as the ratio of the real (effective) power radiated by A to the effective power, which a section of size A of an infinite surface with constant surface velocity vn would radiate: 1 Z =¢ |vn (s)|2 dA = r . 2 Z0

(9)

A

The oscillating mass Mr is given by Zmr = j – · Mr or a mass surface density mr given by Zr = j – · mr with Mr = A · mr . The end correction is the height of a prism of cross section A containing the oscillating mass Mr : =

mr Z Z Z Mr = = mr = r = r 0 A 0 –0 A –0 k0Z0

;

Zr = . a k0a · Z0

(10)

The non-dimensional form /a may contain any meaningful length a,mostly the radius of surface A. Also used is the radiation factor S, which is the ratio of the power ¢ of A to the power ¢0 of a small spherical radiator with the same square average of the volume flow density |q|2 A as the considered surface A: ¢ = S · ¢0

;

¢0 =

Z0 k02 |q|2 A 2 4

;

Zr = Z0

k02 A · S. 4

(11)

Radiation of Sound

F.2

F

289

Some Methods to Evaluate the Radiation Impedance

The simplest radiators are piston radiators and “breathing” radiators with constant normal particle velocity over the radiator surface A: vn(s) = const. According to > Sect. F.1, only the average sound pressure p(s)A at the surface must be evaluated. Also simple are radiators with a surface A which is on a co-ordinate surface of a coordinate system in which the wave equation is separable (e.g. spheres, cylinders, ellipsoids, etc.) and if the vibration pattern agrees with an eigenfunction (mode) in that system, because then the modal field impedance of the vibration is constant over A, so it agrees with the radiation impedance ( > Sect. F.1). P z

r

y ϑ

ϕ x A

P z

r

R

ϑ dA

x

An important family of radiators are plane surfaces A in a surrounding plane baffle wall. Let the normal particle velocity at points (x0 , y0) of A be v(x0 , y0). The sound pressure at a field point P(x, y, z) is then: p(x, y, z) =

j k0 Z0 2

j k0 Z0 = 2

v(x0 , y0) A

e−j k0 R dx0 dy0 R

v(x0 , y0) · G(x, y, z|x0, y0, 0) dx0 dy0 A

with Green’s function G(x, y, z|x0 , y0, 0).

(1)

290

F

Radiation of Sound

One gets with the Fourier transform of v(x0 , y0) (in the hard baffle wall z = 0): +∞

V(k1 , k2) =

v(x0 , y0) · e−j (k1 x0 +k2 y0 ) dx0 dy0,

(2)

−∞

for the complex power k0 Z0 ¢ = ¢ +j·¢ = 8 2

+∞

−∞

|V(k1, k2)|2 dk1 dk2 , k02 − k12 − k22

(3)

and therefore for the radiation impedance: +∞

|V(k1, k2)|2 dk1 dk2 k02 − k12 − k22

Zr = k0Z0 −∞

+∞

|V(k1 , k2)|2 dk1 dk2 .

(4)

−∞

The sound pressure in the far field is given by: k0 Z0 p(x, y, z) = 4 2

+∞

−∞

V(k1, k2) k02

−

k12

− k22

· e−j (k1 x+k2 y+z

√

k02 −k12 −k22 )

dk1 dk2 .

(5)

Special case: Surface A is a strip with the strip axis on the y axis and v(x0 , y0) = const(y):

p(x, z) =

k0 Z0 2

k0 Z0 ¢ = 4

+k0 −k0

+∞ −∞

√2 2 V(k1) · e−j (k1 x+z k0 −k1 ) dk1 , k02 − k12

|V(k1)|2 dk1 , k02 − k12

k0 Z0 Zr = 2 A|vn |2 A

+∞ −∞

|V(k1)|2 dk1 k02 − k12

(¢ and Zr per unit strip length; A = strip width). Special case: Plane surface A and the velocity v(r) have a radial symmetry.

(6)

(7)

(8)

Radiation of Sound

F

291

The role of the Fourier transform of v(r) is taken over by a Hankel transform: ∞ V(kr ) = 2

v(r0) · J0 (kr r0 ) · r0 dr0 .

(9)

0

One gets for the sound pressure far field: p(r, ˜ ) =

j k0 Z0 e−j k0 r · V(k0 sin ˜ ), 2 r

(10)

and for the effective sound power ¢ and the radiation impedance Zr : k 2 Z0 ¢ = 0 4

Zr =

/2 k0 |V(kr )|2 k0Z0 2 |V(k0 sin ˜ )| · sin ˜ d˜ = · kr dkr , 4 k02 − kr2 0

(11)

0

k0 Z0 2 A|vn |2 A

+∞ −∞

|V(kr )|2 · kr dkr . k02 − kr2

(12)

Bouwkamp (1945/46), evaluates the radiation impedance of a plane piston radiator with particle velocity distribution v(x, y) = const as: Z0 k02A Zr = 42

2

/2+j∞

|D(˜ , œ)|2 · sin ˜ d˜ ,

dœ 0

(13)

0

where D(˜ , œ) is the far field directivity function of the radiated sound (directivity pattern with unit value in the maximum). The integration over ˜ = 0 → ˜ = /2 returns the real part of Zr ; the integration ˜ = /2 + j · 0 → ˜ = /2 + j · ∞ returns the imaginary part of Zr .

F.3

Spherical Radiators


Let v(˜ , œ) be the pattern of the normal (outward) particle velocity on the sphere with radius a. The pattern is synthesised with spherical modes: v(˜ , œ) =

n ∞ n=0 m=0

Vm,n · Pm n (cos ˜ ) · cos (mœ)

(1)

292

F

Radiation of Sound

z y

ϑ

r

P ϕ x

2aŒ

with associate Legendre functions dm Pn (x) ; m≥1 dxm

2 m/2 Pm n (x) = (1 − x )

;

Pn (x) = P0n (x) =

1 dn 2 (x − 1)n 2n n! dxn

(2)

defined via the Legendre polynomials Pn (x). Some special values: P0 (x) = 1 2

P2 (x) = (3x − 1)/2

;

P1 (x) = x ;

;

P3 (x) = (5x3 − 3x)/2.

(3)

The modal velocity amplitudes are: Vm,n =

2

1 Nm,n

dœ

0

v(˜ , œ) · Pm n (cos ˜ ) · cos (mœ) · sin ˜ d˜

(4)

0

with the mode norms 2 Nm,n =

cos2 (mœ) dœ

0

1 −1

2 2 2 (n + m)! 1; m = 0, ; ƒ (x) dx = = Pm m n 2; m > 0. ƒm 2n + 1 (n − m)!

(5)

The sound pressure at the surface of the sphere is: p(a, ˜ , œ) =

∞ n

Zn · Vm,n · Pm n (cos ˜ ) · cos (mœ),

(6)

n=0 m=0

where Zn are the modal impedances at the sphere surface (directed inward): Zn = −j 0 c0

h(2) n (k0 a)

hn(2) (k0a)

with the spherical Hankel functions of the second kind h(2) n (z).

(7)

Radiation of Sound

F

293

If the sphere oscillates in a single mode,the modal impedance is the radiation impedance ( > Sect. F.1). Special case: The vibration pattern v(˜ , œ) = const(œ), i.e. the oscillation is symmetrical around the z axis. v(˜ ) =

∞

Vn · Pn (cos ˜ ),

n=0

1 Vn = n + 2

(8)

v(˜ ) · Pn (cos ˜ ) · sin ˜ d˜ , 0

p(r, ˜ ) = −j 0 c0

∞

Vn · Pn (cos ˜ )

n=0

h(2) n (k0 r) (2)

hn (k0a)

−−−→ r→a

∞

Zn Vn · Pn (cos ˜ ).

(9)

n=0

Special case: Breathing sphere: Vn>0 = 0; V0 = v(˜ , œ) = const(˜ , œ).

(10)

Radiation impedance (= zero mode impedance): Zr0 = 0 c0

j k0 a (k0a)2 + j k0a . = 0 c0 1 + j k0 a 1 + (k0 a)2

(11)

Oscillating mass: Mr0 = A ·

Zr0 0 · 4a3 = −−−−−→ 0 · 4a3 = 0 · 3 Vol. – 1 + (k0a)2 k0 a 1

(12)

Special case: Oscillating rigid sphere: Vn=1 = 0; v(˜ ) = V1 · cos ˜ .

(13)

Radiation impedance (= first-order mode impedance): Zr1 =

(k0a)4 + j k0 a 2 + (k0a)2 = 0 c0 4 + (k0a)4 h(2) 0 (k0 a)

j 0 c0

2 − k0 a h(2) 1 (k0 a) −−−−−→ 0 c0 k0 a 1

(k0 a)4 + j –0 a/2, 4

Mr1 −−−−−→ 0 · 32 Vol. k0 a 1

(14)

F

294

Radiation of Sound

In general, for the n-th mode oscillation (n > 0): Zrn = 0 c0

(k0a)2n+2 (n + 1)2[1 · 3 · . . . · (2n − 1)]2

= 0 c0 Zrn = 0 c0

(k0 a)2 |2n − 1|

;

(15)

2

;

k0a n + 1,

k0 a n+1

= 0 c0 /k0a Mr n −−−−−−→ 0 · k0 a Sect. F.1) the modal wave impedance Zm,n (a) is the radiation impedance Zr : Zr = Zm,n (a) =

−j k0 Z0 H(2) n (krm a) . (2) krm Hn (krma)

(5)

For an axially conphase oscillation (km = 0): Zr n = Z0,n (a) = −j 0 c0 For thin cylinders (k0a 1) and n > 0:

For n = 0:

H(2) n (k0 a)

Hn(2) (k0a)

.

(6)

Zr n (k0a)2n ≈ k0 a ; 0 c0 (n !)2 · 22n−1 Zr n k0 a . ≈ 0 c0 n

(7)

Zr 0 k0 a − j k0a · ln (k0 a). ≈ 0 c0 2

(8)

Special case: 2 > k02 . km

A slow mode in the axial direction:

The modal radiation impedances then are:

⎤ ⎡ 2 − k2 K a k n+1 m 0 n k0 ⎣ ⎦

− Zm,n (a) = j 0 c0 2 − k2 2 − k2 2 − k2 km a k a K k m 0 0 n m 0

(9)

with Kn (z) modified Bessel functions of the second kind. Correspondence in the graphs below:“Zsn”→ Zrn /Z0 ;“k0a”→ k0a.They are for km = 0. The curves are ordered from left to right as in the parameter list {n} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. 2

Re{Zsn}, {n}={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

1.75 1.5 1.25 1 0.75 0.5 0.25 2

4

k0a

6

8

10

Radiation of Sound

2

F

297

Im{Zsn}, {n}={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

1.75 1.5 1.25 1 0.75 0.5 0.25 2

F.5

4

k0a

6

8

10

Piston Radiator on a Sphere


This case corresponds to the classical Helmholtz resonator. A hollow hard sphere with radius a has a circular hole which subtends an angle ˜0 with the z axis. z

ϑ0 x 2aØ

Let the particle velocity be constant in the hole: ; 0 ≤ ˜ < ˜0 v v(˜ ) = 0 0 ; ˜0 < ˜ ≤ . Modal velocity amplitudes at r = a: 1 Vn = (n + 1/2) · v0 cos ˜0

Pn (x) dx =

v0 [Pn−1 (cos ˜0 ) − Pn+1 (cos ˜0 )] 2

with Pn (z) being Legendre polynomials and P−1 (z) = 1.

(1)

298

F

Radiation of Sound

Radial particle velocity and sound pressure at r = a: v(a, ˜ ) =

∞

Vn · Pn (cos ˜ )

;

p(a, ˜ ) =

n=0

∞

Zn (a) · Vn · Pn (cos ˜ )

(2)

n=0

using the modal (radial) impedances: Zn (a) = −j 0 c0

h(2) n (k0 a)

(3)

hn(2) (k0a)

with the spherical Hankel functions of the second kind h(2) n (z). Because v(˜ ) = const over the hole, its radiation impedance is given by the average sound pressure and the particle velocity ( > Sect. F.1) with the radiator surface:

A = 2a

2

˜0

sin ˜ d˜ = 2a2 (1 − cos ˜0 ),

(4)

0

p(a, ˜ )A = v0

∞ a2 Zn (a) [Pn−1 (cos ˜0 ) − Pn+1 (cos ˜0 )]2 . A n=0 2n + 1

(5)

This gives the radiation impedance: ∞

Zr =

1

Zn (a) 1 [Pn−1 (cos ˜0 ) − Pn+1 (cos ˜0 )]2. 2 (1 − cos ˜0 ) n=0 2n + 1 Re{Zs/Z0}, {theta0}={5., 10., 15., 20., 40., 60.}

0.8 0.6 0.4 0.2

2

4

k0a

6

8

10

(6)

Radiation of Sound

F

299

Im{Zs/Z0}, {theta0}={5., 10., 15., 20., 40., 60.}

1 0.8 0.6 0.4 0.2

2

4

k0a

6

8

10

In the limit of low frequencies: 1 + cos ˜0 Zr0 1 + cos ˜0 j k0a Zr . ≈ = 0 c0 2 0 c0 2 1 + j k0 a

(7)

Correspondence in the diagrams above: “Zs/Z0” → Zr /0 c0 ; “theta0” → ˜0 ; “k0a” → k0 a. The dashes become shorter for higher list entries of ˜0 ; The curves are ordered from right to left in the sequence of parameter values in the parameter list {˜0 }.

F.6

Strip-Shaped Radiator on Cylinder


A hard cylinder with radius a has a vibrating strip on its surface, which subtends an angle œ0 with the x axis. z

2ϕ0

y

x 2aŒ

F

300

Radiation of Sound

The radial particle velocity be constant in the azimuthal direction and may have a propagating or standing wave pattern in the axial direction: v(a, œ, z) =

v0 · g(km z)

;

−œ0 ≤ œ ≤ œ0

0

;

œ0 < œ < 2 − œ0 .

(1)

The modal particle velocity amplitudes are: Vm,n

sin (nœ0 ) ƒm v0 œ0 = nœ0

;

ƒm =

1; m=0 2; m>0

;

2 kr2 = k02 − km .

(2)

The radiation impedance is evaluated as: Zr =

∗

p · v dA A

|v|2dA =

A

−j œ0 k0a = 0 c0 kr a

∞ n=0

ƒn

H(2) n (kr a) (2)

Hn (kr a)

∞ œ0 sin (nœ0 ) 2 ƒn Zm,n (a) n=0 nœ0

sin (nœ0 ) nœ0

2

(3)

.

At high frequencies, Zr → 0 c0 · k0/kr .

(4)

Correspondence in the diagrams below:“Zs/Z0” → Zr /0 c0 ;“phi0” → œ0 ;“k0a” → k0a. The dashes become shorter for higher list entries of œ0 ; the curves are arranged from left to right (at low k0 a) in the order of these entries. The axial wave number there is km = 0.

1

Re{Zs/Z0}, {phi0}={5., 10., 15., 20., 40., 60.}

0.8 0.6 0.4 0.2

2

4

k0a

6

8

10

Radiation of Sound

F

301

Im{Zs/Z0}, {phi0}={5., 10., 15., 20., 40., 60.}

1 0.8 0.6 0.4 0.2

2

F.7

4

k0a

6

8

10

Plane Piston Radiators


A plane surface A, surrounded by a plane, hard baffle wall, oscillates with a constant velocity v. A general scheme of evaluation for the radiation impedance Zr can be designed for surfaces A with convex border lines. C hi

z R

P(x,y)

x0

xlo

y dA0 x

y0 xhi

C lo

The evaluation applies the field impedance ZF (x, y) on the radiating surface: j ZF = Z0 2

=

1 2

k02 A

e−j k0 R d(k02 A) R k0 Chi (x0 )

k0 xhi

d(k0 x0 ) k0 xlo

k0 Clo (x0 )

cos k0R sin k0R +j k0 R k0 R

(1)

d(k0 y0),

302

F

1 Zr = Z0 2 k02A

Radiation of Sound k0 xhi

k0 Chi (x)

d(k0x) k0 xlo

k0 Clo (x)

ZF (x, y) d(k0 y). Z0

(2)

Circular piston radiator with radius a: Zr S1 (2k0a) J1 (2k0a) +j , =1− Z0 k0 a k0 a

(3)

where J1 (z) is a Bessel function and S1 (z) a Struve function. Approximation for low k0 a (with x = 2k0 a; for about x < 4; range depends on number of terms): Zr x2 x4 x6 = − + − +...; 2 2 Z0 2 · 4 2 · 4 · 6 2 · 4 · 62 · 8 x3 x5 Zr 4 x = − 2 + 2 2 − +... . Z0 3 3 · 5 3 · 5 · 7

(4)

Approximation for high k0a (with x = 2k0a, for about x > 4 ): 2 Zr =1− Z0 x

2 · sin (x − /4) x

;

2 Zr 4 · sin (x + /4) . 1− = Z0 x x

Correspondence in the diagram below: “Zs/Z0” → Zr /0c0 ; “k0a” → k0a. Solid line: Re{Zr /0 c0 }, dashed line: Im{Zr /0 c0 }. Zs/Z0 ; circular piston 1.4 1.2 1 0.8 0.6 0.4 0.2 2

4

k0a

6

8

10

(5)

Radiation of Sound

F

303

Oscillating free circular disk with radius a; oscillation normal to disk: The sound field is described in oblate spheroidal co-ordinates (, ˜ , œ) [generated by rotation of the elliptic-hyperbolic cylinder co-ordinates (, ˜ ) around the short axis of the ellipses], in relation to the Cartesian co-ordinates: z = a · sinh · cos ˜

x cos = a · cosh · sin ˜ · œ. y sin

;

(6)

The co-ordinate value = 0 describes a circular disk with radius a normal to the z axis: ∞ Zr −8j k0 a he0n (−j k0 a , j sinh ) = Z0 9 d he0n (−j k0 a , j sinh )/d =0 n=1,3,... ·

d1 (−j k0 a| 0, n) S0n (−j k0a , cos ˜ ), 0n

(7)

where S0n (, ˜ ) is an azimuthal spheroidal function; he0n (, z) is an even radial spheroidal function of the third kind; the term d1 (−jk0 a|0, n) comes from the expansion of S0n (, ˜ ) in associated ∞ Legendre functions S0n (, ˜ ) = dm (| 0, 1) · T0m (˜ ) (8) m=1,3,...

+1

S20n (, ˜ ) d˜ .

and 0n from

0n =

Approximation for low k0a:

Zr 16 8 ≈ (k0a)4 + j k0 a. 2 Z0 27 3

−1

(9) (10)

Elliptic piston in a baffle wall: The ellipse has a long axis 2a and a short axis 2b; the ratio of the axes is = b/a. z

y ϑ ϕ

2b

x

2a

Some evaluations in the literature start from the Bouwkamp integral ( > Sect. F.2) with the following far field directivity function of the radiated sound:

J1 (k0 sin ˜ a2 cos2 œ + b2 sin2 œ) . D(˜ , œ) = 2 k0 sin ˜ a2 cos2 œ + b2 sin2 œ

(11)

304

F

Radiation of Sound

One solution for the real part of the radiation impedance Zr = Zr + j · Zr is: ∞ Zr (k0 a)2m · 2 F1 −m ; = k0 a · k0 b Z0 (m + 1) ! (m + 2) ! m=0

1 2

; 1 ; ‹2

;

‹ 2 = 1 − 2 ,

(12)

where 2 F1 (, ; ‚; z) is the hypergeometric function. The numerical errors become large for k0 a 1. A solution suited for numerical integration is: /2

2 Zr = 1 − k02ab Z0 /2

Zr

2 = k02 ab Z0

0

0

J1(2B) dœ B3

;

B = k0a cos2 œ + 2 sin2 œ, (13)

S1 (2B) dœ, B3

where J1 (z) is a Bessel function and S1 (z) a Struve function. The numerical integration can be avoided by an expansion of the integrands. This leads to the following iterative evaluation: nhi Zr 2 = (k0a) /2 + cn · In , Z0 n=2 (14) 2 a) −(k 0 · c c1 = (k0a)2 /2 ; cn = ; I0 = 1/ ; I1 = 1 ; In = 2 In /, ∗) n · (n + 1) n−1 4 k0 b Zr = Z0 2

nhi 4 16 2 I − (k0a) · I1 + cn · In , 3 0 45 n=2

c1 = −16 (k0a)2 /45 I0 = K(‹ 2)

;

;

cn = −4 (k0a)2 / ((2n + 1)(2n + 3)) ,

I1 = E(‹ 2 )

;

In =

(15)

2n − 2 2n − 3 2 (1 + 2) · In−1 − · In−2 2n − 1 2n − 1

with K(z), E(z) the complete elliptic integrals of the first and second kind. The upper summation limit should be nhi ≥ 2(k0a + 1). A further solution for the real component of Zr is: nhi Zr J1 (2k0a) − (1 − ) · J2 (2k0 a) − =1− cˆn · În · J1+n (2k0a), Z0 k0 a k0 a n=2

(16) (1 − 2 )k0 a · cˆ n−1 , n

2n − 3 1 2n − 3 1 · În−1 − · În−2 , + Iˆ0 = 1/ ; Iˆ1 = 1/ (1 + ) ; Iˆn = 2 1− 2n − 2 1 − 2 2n − 2 cˆ1 = (1 − 2 ) · k0a

;

cˆn =

where Jn (z) are Bessel functions. ∗)


Radiation of Sound

F

305

Correspondence in the diagrams below: “Zs/Z0” → Zr /Z0 ; “k0a” → k0a; dashes become shorter for higher positions in the parameter list {} = {b/a} = {0.25, 0.5, 1}. Re{Zs/Z0(b/a)}, elliptic piston , {b/a}={0.25, 0.5, 1.} 1.4 1.2 1 0.8 0.6 0.4 0.2 2

1

4

k0a

6

8

10

Im{Zs/Z0(b/a)}, elliptic piston , {b/a}={0.25, 0.5, 1.}

0.8 0.6 0.4 0.2

2

4

k0a

6

8

10

Rectangular piston in a baffle wall: The rectangle has a long side a and a short side b; the side length ratio is = b/a.

z

y ϑ ϕ

b

a

x

306

F

Radiation of Sound

Some evaluations in the literature start from the Bouwkamp integral ( > Sect. F.2) with the far field directivity function of the radiated sound [with si(z) = (sinz)/z]: D(˜ , œ) = si k0 a/2 · sin ˜ · cos œ · si k0b/2 · sin ˜ · sin œ . (17) A first form of the radiation impedance Zr = Zr + j · Zr is: sin k0 a cos k0 a − 1 Zr Ci(k0a) − + =1 + Z0 k0 a (k0a)2 2 1 sin k0 b cos k0b − 1 − Ci(k0b) − + I1 (k0a , ), + k0 b (k0 b)2 sin k0a cos k0 a − 2 Zr Si(k0a) + =− + Z0 (k0 a)2 k0 a 2 sin k0 b cos k0b − 2 1 Si(k0b) + − I2 (k0a , ) + − 2 (k0b) k0 b

(18)

with Ci(z), Si(z) the integral cosine and sine functions and the integrals: 1 1 I1 (k0 a , ) = Ci(k0a x2 + 1/2) + 2 Ci(k0b x2 + 2) · (1 − x) dx, 0

1 1 I2 (k0 a , ) = Si(k0a x2 + 1/2) + 2 Si(k0b x2 + 2) · (1 − x) dx.

(19)

0

A second form of the radiation impedance Zr = Zr + j · Zr is:

2 Zr 2 2 2 =1 − · 1 + cos k0 a 1 + + k0 a 1 + · sin k0 a 1 + Z0 (k0a)2 2 − cos (k0 a) − cos (k0b) + · Ia (k0a , ),

2 Zr 2 − k a 1 + 2 · cos k a 1 + 2 = · sin k a 1 + 0 0 0 Z0 (k0a)2 2 + k0a (1 + 1/) − sin (k0a) − sin (k0b) − · Ib (k0 a , ),

(20)

with the integrals

Ia (k0a , ) =

√ √ +1/ +1/ 2 1 − /x · cos (x k0 a ) dx + 1 − 1/(x)2 √ √

· cos (x k0 a ) dx,

1/

(21)

Radiation of Sound

F

307

√ √ +1/ +1/ Ib (k0a , ) = 1 − /x2 · sin (x k0 a ) dx + 1 − 1/(x)2 √ √

1/

· sin (x k0a ) dx. A modification of these formulas leads to a fast numerical evaluation: √ √ 2 Zr · 1 + cos (k0 a2 + b2) + k0 a2 + b2 · =1 − 2 Z0 k0 ab √ 2 · Îa , · sin (k0 a2 + b2) − cos (k0a) − cos (k0 b) + √ √ Zr 2 · k0 (a + b) + sin (k0 a2 + b2 ) − k0 a2 + b2 = 2 Z0 k0 ab √ 2 · Iˆb · cos (k0 a2 + b2 ) − sin (k0a) − sin (k0b) −

(22)

with the integrals √

1+(b/a)2

Iâ =

√

1+(a/b)2

1

− 1/x2

· cos (x k0a) dx +

1

√

1 − 1/x2 · cos (x k0b) dx,

1

1+(b/a)2

Iˆb =

√ 1 − 1/x2 · sin (x k0a) dx +

1

1+(a/b)2

(23) 1 − 1/x2 · sin (x k0 b) dx.

1

The component integrals are of the forms: ˜Ia (A, B) =

B B 1 − 1/x2 · cos (Ax ) dx ; ˜Ib (A, B) = 1 − 1/x2 · sin (Ax ) dx. 1

(24)

1

They can be evaluated iteratively: ˜Ia (A, B) = I−1 +

∞

∞

A2n A2n+1 · I2n−1 ; ˜Ib (A, B) = A · I0 + · I2n (25) (−1) (−1)n (2n) ! (2n + 1) ! n=1 n=1 n

with start values and recursion for the Im : Bm−1 2 m−1 (B − 1)3/2 + · Im−2 , m+2 m+2 √ √ B√ 2 1 = B2 − 1 − arccos (1/B) ; I0 = B − 1 − ln (B + B2 − 1). 2 2

Im = I−1

(26)

308

F

Radiation of Sound

An approximation for large k0a (> 5) and not too small b/a is: Zr 2 2 cos (k0a − /4) cos (k0b − /4) =1− + − [1 − cos k0a − cos k0b] 3/2 3/2 Z0 (k0a) (k0b) k02 ab (27) √ 2 (a2 + b2 )3/2 9 2 sin (k0 a − /4) sin (k0b − /4) + − + sin (k0 a2 + b2), 8 (k0a)5/2 (k0 b)5/2 (k0ab)3 2 2 (a + b) 2 sin (k0 a − /4) sin (k0 b − /4) Zr + − = + [sin k0a + sin k0 b] Z0 k0ab (k0a)3/2 (k0b)3/2 k02ab (28) √ 2 (a2 + b2 )3/2 9 2 cos (k0a − /4) cos (k0b − /4) + cos (k0 a2 + b2 ). + − 8 (k0 a)5/2 (k0 b)5/2 (k0ab)3 Correspondence in the diagrams below:“Zr/Z0” → Zr /Z0 ;“k0a” → k0a; dashes become shorter for higher positions in the parameter list {} = {b/a} = {0.25, 0.5, 1}. Re{Zr/Z0(b/a)}, rectangul.piston , {b/a}={0.25, 0.5, 1.} 1.4 1.2 1 0.8 0.6 0.4 0.2 2

4

k0a

6

8

10

Im{Zr/Z0(b/a)}, rectangul.piston , {b/a}={0.25, 0.5, 1.} 1 0.8 0.6 0.4 0.2

2

4

k0a

6

8

10

Radiation of Sound

F.8

F

309

Uniform End Correction of Plane Piston Radiators


The normalised end correction of a radiator is defined from its radiation reactance Zr by: Zr = , a k0a · Z0

(1)

where a is any side length. Thus /a equals the tangent of the curve of Zr /Z0 over k0 a at the origin k0a = 0. If one takes a = A3/4 · U1/2 ,

(2)

where A is the area and U is the periphery of the piston surface, then the curves of Zr /Z0 over k0 a coincide at the origin k0a = 0 for different shapes of the surface, assuming its border line is convex. So one can deduce end corrections for piston shapes with unknown solutions for Zr from end corrections of shapes with known solutions.

F.9

Narrow Strip-Shaped, Field-Excited Radiator


A plane radiator is called “field excited” if its vibration pattern agrees with that of an obliquely incident plane wave at the surface.

z Θ

y +a/2

x Φ a/2

The object here is an infinitely long strip of width a in a hard baffle wall, the strip by a plane wave with polar angle Ÿ of incidence and azimuthal angle ¥ with the strip axis. If either ¥ = 0 (then a is unlimited), or ¥ = 0, and a Š0 , the oscillation velocity of the strip surface can be assumed to be constant across the strip: v(x, y) = V0 · e−j kx x

;

kx = k0 · sin Ÿ · cos ¥ .

(1)

310

F

According to

|v| = const,

Radiation of Sound > Sect. F.1, because of

1 Zr = A

A

1 ZF dA = V0 a

+a/2

p(y, 0) dy

(2)

−a/2

with the field impedance ZF = p(x, y, 0)/v(x, y) and p(x, y, z) = p(y, z) · e−j kx x . The lateral sound pressure distribution is: k0 Z0 V0a p(y, z) = 2

+∞ −∞

√2 2 2 sin (ky a/2) e−j (ky y+z k0 −kx −ky ) dky , ky a/2 k02 − kx2 − ky2

(3)

and therewith the radiation impedance: Zr k0 a = Z0 2

+∞ −∞

sin (ky a/2) ky a/2

2

dky . k02 − kx2 − ky2

(4)

In a different form: k0 Zr = 2 Z0 k a

ka

(ka − |u|) · H(2) 0 (|u|) du

;

k2 = k02 − kx2 = k02(1 − sin2 Ÿ · cos2 ¥ )

(5)

0

with the Hankel function of the second kind H(2) 0 (z). After analytical evaluation of the integral: 2j H(2) Zr 1 (ka) + = k0a H(2) (ka) − 0 Z0 ka (ka)2 (2) (2) H1 (ka) · S0 (ka) − H0 (ka) · S1 (ka) + 2

(6)

or as real and imaginary parts J1 (ka) Zr + J1 (ka) · S0 (ka) − J0 (ka) · S1 (ka) , = k0a J0 (ka) − Z0 ka 2 2 Y1 (ka) Zr − = −k0 a Y0(ka) − + Y1 (ka) · S0 (ka) − Y0 (ka) · S1 (ka) , Z0 ka (ka)2 2

(7)

where Jn (z) is a Bessel function, Yn(z) a Neumann function and Sn (z) a Struve function.

Radiation of Sound

F

311

Approximation for small ka (with c = 0.57721, Euler’s constant):

Zr (ka)2 (ka)4 (ka)2 (ka)4 + 1− + = k0 a 1 − Z0 6 64 3 45 2 4 4 (ka) 1 (ka) (ka) − + 1 − (ka)2 + , − 2 16 192 9

(8)

Zr 2 k0 a (ka)2 (ka)4 (ka)2 (ka)4 ka = 1− + − 1− + ln +c Z0 9 225 3 45 2

(ka)4 1 3(ka)2 (ka)2 (ka)4 + + (ka)2 − + 1 − (ka)2 + · 1− 4 64 4 128 9

2 4

2 4 (ka) 1 5(ka) 10(ka) 1 (ka) ka +c − + − + − . · ln 2 2 16 192 4 64 2304

(9)

F.10 Wide Strip-Shaped, Field-Excited Radiator


A plane radiator is called “field excited” if its vibration pattern agrees with that of an obliquely incident plane wave at the surface. z

ϑ

Θ a/2

Φ

y

ϕ +a/2 x

The object here is an infinitely long strip of width a in a hard baffle wall, the strip is excited by a plane wave with polar angle ˜ of incidence and azimuthal angle œ with the normal to the strip axis.

F

312

Radiation of Sound

Notice the different co-ordinates and angles as compared to

> Sect. F.9:

cos ¥ = sin œ · sin ˜ , cos Ÿ = cos ˜ / 1 − sin2 œ · sin2 ˜ , cos ˜ = sin ¥ · cos Ÿ, √ sin œ = cos ¥ / 1 − sin2 ¥ · cos2 Ÿ.

(1)

Radiation impedance: Zr C = Z0 sin ¥ C = A + jB =

;

b = k0a · sin ¥

b 0

1−

x · cos (x · sin Ÿ) · H(2) 0 (x) dx, b

(2)

where H(2) 0 (x) is a Hankel function of the second kind. After power series expansion of the factor to the Hankel function in the integrand: A=

∞ n=0

−

(−1)n

1 2n + 2

1 sin2n Ÿ · b2n+1 2 · 1 F2 1/2 + n ; 1 , 3/2 + n ; −b /4 (2n) ! 2n + 1 2 1 + n ; 1 , 2 + n ; −b F /4 , 1 2

(3)

∞ 2n 2n+1 −1 4 1 n sin Ÿ · b 2 (−1) B= · ln 2 · 1 F2 1 + n ; 1 , 2 + n ; −b /4 n=0 (2n) ! b 2n + 2

1 2 − 1 F2 1/2 + n ; 1 , 3/2 + n ; −b /4 2n + 1 2 · 2 F3 1 + n , 1 + n ; 1 , 2 + n , 2 + n ; −b2 /4 + 2 (2n + 2) 2 2 − 1/2 + n , 1/2 + n ; 1 , 3/2 + n , 3/2 + n ; −b · F /4 2 3 (2n + 1)2 with hypergeometric functions 1F2 (a1 ; b1 , b2; z) and 2 F3 (a1 , a2 ; b1 , b2, b3 ; z). Correspondence in the diagram below:“Zr/Z0” → Zr /Z0 ;“theta” → ˜ ;“phi” → œ;“k0a” → k0 a; solid line: real part; dashed line: imaginary part.

Radiation of Sound

F

313

Zr/Z0 , wide strip, theta=45., phi=45. 1.4 1.2 1 0.8 0.6 0.4 0.2 2

4

6

k0a

8

10

F.11 Wide Rectangular, Field-Excited Radiator


A plane radiator is called “field excited” if its vibration pattern agrees with that of an obliquely incident plane wave at the surface. z ϑi

y +b/2 x

ϕi a/2

A +a/2

b/2

The object here is a rectangle A with side lengths a,b in a hard baffle wall, the rectangle is excited by a plane wave with polar angle ˜i of incidence and azimuthal angle œi with the axis parallel to side a. Velocity pattern on A: v(x, y) = V0 · e−j (kx x+ky y), kx = k0 · sin ˜i · cos œi = k0 · ‹x , ky = k0 · sin ˜i · sin œi = k0 · ‹y .

(1)

314

F

Radiation of Sound

The sound pressure field is: p(x, y, z) =

j k0Z0 2

v(x0 , y0) A

e−j k0 R dA0 R

;

R=

(x − x0 )2 + (y − y0 )2.

(2)

The definition of the radiation impedance Zr with the radiated power gives a first form: Zr =

j k0Z0 2A

dA

A

A

e−j k0 R −j (kx (x0 −x)+ky (y0 −y)) e dA0 ; R2 = (x0 − x)2 + (y0 − y)2 . (3) R

The fact that |v(x, y)| = const on A and that, therefore, the radiation impedance follows from the average field impedance with the Fourier transform of the velocity distribution leads to a form with fewer integrations: V(k1 , k2) =

v(x0, y0 ) e−j (k1 x0 +k2 y0 ) dx0 dy0

A

sin ((k1 + kx ) a/2) sin ((k2 + ky ) b/2) = V0ab (k1 + kx ) a/2 (k2 + ky ) b/2

(4)

using 1 = k1/k0; 2 = k2 /k0: Zr k0 a · k0b = Z0 42

+∞

−∞

sin ((‹x − 1)k0 a/2) (‹x − 1)k0 a/2

sin ((‹y − 2)k0 b/2) · (‹y − 2)k0 b/2

2

2

d1 d2 1 − 21 − 22

(5) .

The third form starts from the Bouwkamp integral ( > Sect. F.2): Zr k0 a · k0 b = Z0 4 2

/2+j ∞

2

|D(˜ , œ)|2 · sin ˜ d˜

dœ 0

(6)

0

with the far field directivity function

k0 a (sin ˜i cos œi − sin ˜ cos œ) sin 2 D(˜ , œ) = k0 a (sin ˜i cos œi − sin ˜ cos œ) 2

k0 b (sin ˜i sin œi − sin ˜ sin œ) sin 2 . · k0 b (sin ˜i sin œi − sin ˜ sin œ) 2

(7)

Radiation of Sound

F

315

The second form can be transformed into: 2j Zr = Z0 k0a · k0 b

k0 a x=0

√2 2 k0 b e−j x +y dx (k0 a − x) (k0b − y) cos (‹x x) cos (‹y y) dy. x2 + y 2

(8)

0

This becomes, for normal sound incidence with ˜i = 0; ‹x = ‹y = 0 Zr 2j = Z0 k0a · k0 b

k0 a x=0

√2 2 k0 b e−j x +y dx (k0 a − x) (k0b − y) dy. x2 + y 2

(9)

0

The double integral can be transformed by substitution of variables into: ⎡ ⎤ arctg(b/a) /2 Zr 2j ⎢ ⎥ = I(k0 a/ cos œ) dœ + I(k0b/ sin œ) dœ⎦ ⎣ Z0 k0a · k0 b 0

(10)

arctg(b/a)

with the intermediate integrals: R I(R) =

U + V · r + W · r2 cos (r) · cos (r) · e−j r dr,

(11)

0

U = k 0 a · k0 b

;

V = −(k0 a · sin œ + k0b · cos œ)

= ‹x · cos œ

;

= ‹y · sin œ.

;

W = sin œ · cos œ,

See the reference for an analytical procedure to solve the integrals contained in I(R). Correspondence in the following diagrams: “Zr/Z0” → Zr /Z0; “thetai” → ˜i ; “k0a” → k0 a; parameters: b/a = 3; œi = 0; the dashes become shorter with increasing position of the parameter value of ˜i in the list {˜i }. Re{Zr/Z0(thetai)}, {thetai}={0., 30., 45., 60., 85.} 1.4 1.2 1 0.8 0.6 0.4 0.2 2

4

k0a

6

8

10

316

1

F

Radiation of Sound

Im{Zr/Z0(thetai)}, {thetai}={0., 30., 45., 60., 85.}

0.8 0.6 0.4 0.2

2

F.12

4

k0a

6

8

10

End Corrections


See > Sect. F.1 for the definition of end corrections. End corrections represent the inertial near fields at expansions (orifices) of the cross section available for the sound wave. End corrections are mostly of interest for small k0 a, where a is a characteristic lateral dimension of the orifice.End corrections are influenced by the shape of the orifice and of the space which is available for the sound wave behind the orifice. Therefore in general the orifices on both sides of a“neck”must be distinguished (exterior and interior end correction). The relations of the end correction of an orifice with area A to the radiation impedance Zr = Zr + j · Zr and the oscillating mass Mr are =

mr Z Mr Z Z = = mr = r = r 0 A 0 –0 A –0 k0Z0

;

Zr = . a k0a · Z0

(1)

Radiation of Sound

F

317

Table 1 Oscillating mass Mr of simple oscillating bodies Object

Mr 1 Mr = 3 0 V 1 + (k0 a)2

Monopole sphere

−−−−−−! 3 0 V

Remarks A = 4a2 V = 4a3 =3

k0 a1

2aŒ

Mr = 3 0 V

Oscillating sphere

2 + (k0 a)2 4 + (k0 a)4

−−−−−−! 23 0 V k0 a1

1 n+1 1 −−−−−−−−−−! 3 0 V 2 2 (k k0 a n +1 0 a)

Sphere in nth mode

Mr −−−−−−−−−−−! 3 0 V

Monopole cylinder

Mr −−−−−−! −2 0 V ln(k0 a)

k0 a j2n−1j

k0 a 1

2aŒ

Oscillating cylinder

Cylinder in n-th mode

Mr −−−−−−! 2 0 V k0 a 1

Mr −−−−−−! 2 0 V k0 a 1

1 n

A = 2a V = a2

318

F

Radiation of Sound

Table 2 End corrections l=a of orifices ` a 0:785 = =4 < `=a 8=3 = 0:85

Object 2aø

` 8 2 = [1 − (k0 a)2 a 3 15 8 + (k0 a)4 ] 525

Circle in baffle wall

Tube orifice in free space

Remarks

a = radius

(0:65 to 0:69) a2 =Š0

a = radius

`=a = 2=[1 + (k0 a)2 ] ! 2

a = radius

2aø

Half monopole sphere in baffle wall 2aø

` = [1 + cos ˜] = [2 (1 + (k0 a)2 )] a

Orifice on sphere

2ϑ

a = radius a sin ˜ = orifice radius

a

Elliptical orifice in baffle wall

b

a

` 16 ` ` = 2 K(1 − 2 ) ; = a 3 b a 16 2 4 + 2 2 K(1 − ) ln 2 − 8 4 0 < 0:641; 11 + 52 2 7 + 92 0:641 1 p ` U=8 + (Š0 =2) ”0 (2k0 S=) =2 4 sin(x cos ) sin2 d ”0 (x) =

a = small b = large half axis = a=b < 1

K(1 − 2 )

Orifice in tube wall 2aø

0

Circular fence in tube

2aø

2bø

x2 =8 ; x 1 l=a −0:0445 728 − 0:728 326 x − 0:177 078 x2 + 0:0339 531 y+ + 0:00810 471 y 2 − 0:00100 762 xy = (a=b)2 ; x = lg ; y = lg(b=Š0 )

U = 2a = periphery S = a2 = area of orifice a = fence radius b = tube radius

Radiation of Sound

F

319

Table 2 continued Object l=a 8=3

Free circular disk

` a

Remarks a = radius

2aø

Rectangular orifice in baffle

2 1 − (1 + 2 )3=2 l = + a 3 2 p 2 1 ln + 1 + 2 2b +

1 p + ln 1 + 1 + 2

` 1 1 sin n˜ 2 ln(k0 a) − = 2a˜ n n˜ 2

2a

Slit on a cylinder

n1

2aø

2ϑ

Rectangular orifice in baffle wall b

a

Expansion of a flat duct b

a

Grid of slits L-a

a

1 1 1 + cot `=a = ln tg 2 4 2 4 1 ; 0 Š 2 n a/L 0 −1 n L

(5)

Radiation of Sound

F

321

2 1.5

Δ a ∞ sin 2 (nπa / L) = ⋅∑ a L n =1 (nπa / L)3

1

π a 1 π a 1 Δ 1 )] ) + cot( ≈ ln[ tan( 4 L 4 L 2 2 π a

0.5 0 -3

-2.5

-2

-1.5 x=lg(a/L)

-1

-0.5

0

Influence of higher modes in the neck of a slit grid plate: Width and distance of slits as above: the slits are in a plate of thickness d; radiation reactance of a back orifice:

Z Zrb = Zr0 · 1 + rb = Zrb · (1 − 10F(x,y) ), Zr0

Z F(x, y) = lg − sh = f (x) · (1 + g(y)), Zsh0 x = lg(a/L)

;

(6)

y = lg(d/a)

with (in −3 ≤ x < 0 and −1 ≤ y ≤ 1): f (x)

= −1.739 68 + 1.484 35 (x + 1.5) − 1.842 30 (x + 1.5)2 + 0.292 538 (x + 1.5)3 + 0.428 402 (x + 1.5)4,

g(y)

= H(−y) · [0.00 259 355 y − 0.0758 181 y 2

H(−y) =

+ 0.330 845 y 3 + 0.226 933 y 4 ], 1; y≤0 0;

y > 0.

(7)

F

322

Radiation of Sound

⎛ ΔZ ′rb ′ ⎞ lg − ⎜⎝ Z ′′ ⎠ r0

L/λ0=0.1

0

-2

-4

-6

1

0 -1

0

lg(a/L)

lg(d/a)

-2 -3 -1

Relative change of radiation reactance of a slit in a slit grid due to higher modes in the neck of the slit plate. Interior end correction of the slit orifice in a slit resonator array: No losses and only a plane wave in the slit (i.e. narrow slit). The resonators repeat in the y direction with a period length L = a + b. Lateral wave numbers in the volume: ! ‚0 = jk0 ; Re{‚n } ≥ 0

‚n = k0 or

n

Š0 L

2 − 1,

(8)

Im{‚n } ≥ 0.

Impedance of the back orifice: s2i Zsh a k0 a/L + j2 , = −j Z0 tan(k0t) L ‚i tanh(‚i t)

(9)

i>0

s0 = 1

;

si =

sin(i a/L) . i a/L

(10)

Radiation of Sound

F

323

L=b+a b/2 Q

y a x

V b/2

d

t

L/λ0=0.4 Δ b /a 2.5 2

1

1

0 -3

0.5 -2 lg (a / L )

0

lg (t/L)

-0.5

-1 0 -1

Influence of the shape parameter t/L on the interior end correction of the slit in a slit resonator

324

F

Radiation of Sound

The first term (outside the sum) is the spring reactance of the volume; thus the sum term is the mass reactance at the interior orifice. The back side end correction therefore is: s2i 2 1 b = . a L ‚i tanh(‚i t)

(11)

i>0

Interior end correction of the slit orifice in a slit resonator array with higher modes in the neck: Geometrical parameters as above. b0 /a interior end correction from above with only plane wave in the neck: b0 b ≈ (x) · [1 + f (y)] · [1 + g(z)], a a a d t x = lg ; y = lg ; z = lg , L a L f (y) = 0.001 448 29 · y + 0.002 555 10 · y 2 + 0.034 305 10 · y 3 + 0.015 682 99 · y 4,

(12)

g(z) = −0.000 932 290 · z − 0.007 672 04 · z2 − 0.019 259 72 · z3 − 0.018 048 39 · z4 . L/λ0=0.1 ; a/L=0.1 0 %

g(z)

-0.5 f(y) -1.0 -1.5 -2.0 -1

-0.75 -0.5 -0.25

0

0.25

0.5

0.75

1

z=lg (t/L) ; y=lg (d/a)

Influence (in per cent) of shape factors d/a and t/L on the interior end correction, with higher neck modes taken into account

Radiation of Sound

F

325

Interior orifice impedance of a slit in a slit array, with viscous and caloric losses in the neck taken into account: Let Zb0 be the back orifice impedance without losses. The back orifice impedance Zb = Zb + j · Zb can be approximated with: " " # # Zb0 Zb Zb0 Zb 10F (x) 10F (x) f[Hz]a[m] = √ = √ , 1+ √ 1+ √ ; ; x = lg 3 3 Z0 Z0 Z Z (a/L)3/2 a[m] · a/L a[m] · a/L 0 0 (13) F (x) = −4.641 06 + 0.435 993 x + 0.0142 851 x2 + 0.000 461 347 x3 , F (x) = −2.266 65 − 0.492 331 x − 0.000 719 182 x2 − 0.001 0208 x3 . Interior orifice impedance of a slit in a slit array in contact with a porous absorber layer (i.e. t = 0 in the sketch): Let the characteristic propagation constant and wave impedance of the porous material be a , Za . Air gap thickness t = 0. ¡ = flow resistivity of the porous material. (14) —n = k0 (sin Ÿ + n Š0 /L)2 + (a /k0)2 . Impedance Zb of the back slit orifice:

a a Za k0 sin (n a/L) 2 Zb =2 coth (—n s) . Z0 L k0Z0 n>0 —n n a/L Back orifice end correction:

s a Za sin (n a/L) 2 coth (—n s) b = −2j . a L k0 Z0 n>0 n a/L —n s y

x

Θ

a L

d

t

s

(15)

(16)

326

F

Radiation of Sound

Interior end correction of a slit in a slit array in contact with a porous absorber layer: b a Za =j a k0 Z0 · (0.0389998 + 0.454066 · x − 0.345328 · x2 − 0.125386 · x3 − 0.0143782 · y + 0.00418541 · y 2 + 0.0170766 · y 3 − 0.0142094 · z − 0.0715597 · z2 + 0.0915584 · z3 − 0.0115326 · x · y − 0.0195509 · x · z − 0.0595634 · y · z ), x = lg (a/L); y = lg (¡ s/Z0 ); z = lg (s/L).

(17)

Interior orifice impedance of a slit in a slit array with an air gap t between the slit plate and a porous absorber layer: Geometrical and material parameters as well as —n as above: ‚0 = j k0 cos Ÿ

;

‚n = k0 (sin ‡ + n Š0 /L)2 − 1.

(18)

Impedance Zb of back side orifice: Zb Z0

k0 sin (n a/L) 2 1 + rn e−2‚n t 1 + r0 e−2j k0 t , + 2j 1 − r0 e−2j k0 t ‚ n a/L 1 − rn e−2‚n t n>0 n

=

a L

=

k0Z0 a Za k0Z0 1+j a Za 1−j

rn

—n tanh (—n s) ‚n . —n tanh (—n s) ‚n

(19)

(20)

The first term in the brackets is the front side impedance of the porous layer transformed to the plane of the back side orifices of the slit plate. Therefore the second term (sum term) is the mass impedance Zbm of the oscillating mass of the back side orifice. The rn are the modal reflection factors at the front side of the porous layer. The end correction of the back slit orifice is: 1 sin (n a/L) 2 1 + rn e−2‚n t −j Zbm b = =2 . a k0 a Z0 ‚ L n a/L 1 − rn e−2‚n t n>0 n

(21)

Correspondence and parameters in the following diagrams: “F” → L/Š0 ; parameters: a/L = 0.25; d/a = 1; s/L = 1; R = ¡ · s/Z0 = 1; porous layer of glass fibres.

Radiation of Sound

F

327

Real (solid line) and imaginary (dashed line) part of interior end correction b /a for t/a = 0.01 (other parameters given above). The real part represents a mass reactance if it is positive; at negative values it represents the influence of the porous material on the spring reactance of the volume. The negative imaginary part represents a flow resistance

As above, but with a larger distance t/a = 1 between plane of orifices and absorber layer

F.13

Piston Radiating Into a Hard Tube

See also: Mawardi (1951)

A circular piston with diameter 2a oscillates with a velocity amplitude v0 in a hard end surface of a hard, circular tube with diameter 2R. S = a2 = piston area.

F

328

Radiation of Sound

r

v0

2aØ

z

2RØ

Sound pressure on the piston surface z = 0: – 0 J0 (k0n r) · J1 (k0na) · a p(r, z = 0) = v0 Z0 + j 2 S n≥1 2 k0n k0n − k02 · J20 (k0n a)

(1)

with k0 = –/c0 ; kmn = n-th root of Jm (kmnR) = 0. The second term vanishes in the special case a = R with J1 (k0na) = 0. The radiation impedance Zs is: Zs = Z0 + j – 0

n≥1

F.14

J2 (k0na) 1 . 2 2 k0n k0n − k02 · J20 (k0n a)

(2)

Oscillating Mass of a Fence in a Hard Tube

See also: Iwanov-Schitz/Rscherkin (1963)

A hard tube with diameter 2R is driven by a plane wave with velocity amplitude v0 from a piston in a distance to a thin fence with aperture diameter 2a; S = a2 . r

v0

2aØ 2RØ z

MI , MII are the oscillating masses of the fence orifice towards the piston and towards the tube, respectively: MI = 4S0 R

J2 (xm a) · coth (xm ) 1

m≥1

(xm R)3 · J20 (xm R)

;

MII = 4S0 R

J21 (xm a) (xm R)3 · J20 (xm R) m≥1

with xm the roots of J0 (xm R) = 0. In the limit MI −−−−→ MII . →∞

(1)

Radiation of Sound

F.15

F

329

A Ring-Shaped Piston in a Baffle Wall

See also: Antonov/Putyrev (1984)

A ring with interior radius r0 and exterior radius r1 oscillates in a baffle wall. The ring surface area is AR = (r12 − r02); the circle areas are A0 = r02 ; A1 = r12; the radius ratio = r0 /r1 with 0 ≤ < 1; the area ratio = AR /A0 = (r1 /r0)2 − 1.

z ϑ

A

r r0 r1

The mechanical radiation impedance Zs = Zs + j · Zs (force/velocity) is evaluated by: −j k0 r j k0 Z0 e dA1 dA 2 r AR ⎡ARr ⎤

1 r1 4 4 = 2Z0 ⎣ 1 − J0 (2k0r) − Is · r dr + j S0 (2k0r) − Ic · r dr⎦

Zs =

r0

(1)

r0

with J0 (z) the Bessel function, S0(z) the Struve function of zero order, and the integrals: Is Ic

arcsin (r0 /r)

= 0

sin (k0r · cos ˜ ) cos (k0 r · cos ˜ )

· sin

k0 r02 − r2 sin2 ˜ d˜ .

(2)

Approximation for low frequencies k0r0 1 and k0r1 1: 2 8 k0 r0 Zs 2 2 2 ≈ A0 (k0r0 ) + j (1 + )(1 − E()) + (1 − )K() Z0 2 3 2

(3)

with E(), K() the complete elliptic integrals of the first and second kinds, respectively. For low frequencies k0 r0 1 and k0r1 1 and a slender ring 0 < < 0.6: 16 3 Zs A0 2 k0 r0 (1 − 0.25 ) ln + . ≈ Z0 2 2

(4)

F

330

Radiation of Sound

Special case of a small circular piston radiator, i.e. r0 → 0 and k0r1 1: 8 Zs (k0r1 )2 +j k 0 r1 . ≈ A1 Z0 2 3

(5)

Far field of a ring-shaped piston radiator with elongation amplitude a: J1 (k0 r1 sin ˜ ) 1 r1 2 J1 (k0 r0 sin ˜ ) p(r, ˜ ) ≈ − a Z0 k0 r1 2 −2 · e−j k0 r . 2 r k0 r1 sin ˜ k0r sin ˜

F.16

(6)

Measures of Radiation Directivity

Let p(r, ˜ , œ) be the sound pressure generated by a radiator in the far field, k0r 1.

|p(r, ˜ , œ)| p(r, ˜ , œ) Directivity factor: D0 (˜ , œ) = or = (1) p(r, ˜0 , œ0 ) |p(r, ˜0 , œ0 )| where p(r, ˜0 , œ0 ) is the sound pressure in a reference direction (mostly the direction of some axis of symmetry of the radiator). Directivity coefficient:

D20 (˜ , œ) =

|p(r, ˜ , œ)|2 |p(r, ˜0 , œ0 )|2

(2)

Directivity value:

Dm (˜ , œ) =

|p(r, ˜ , œ)|2 |p(r, ˜ , œ)|2 ˜ ,œ

(3)

Directivity index:

DL0 (˜ , œ) = 10 · lg

|p(r, ˜ , œ)|2 |p(r, ˜0 , œ0)|2

(4)

Directivity:

DLm (˜ , œ) = 10 · lg

|p(r, ˜ , œ)|2 |p(r, ˜ , œ)|2 ˜ ,œ

(5)

With . . .˜ ,“ the average over the directions ˜ and “. Sharpness of directivity pattern is given as the angle between the normal to the radiator and the direction for which the intensity decreases to 1/2 of the maximum value.

F.17

Directivity of Radiator Arrays

See also: Skudrzyk, Ch. 26 (1971)

z γ r0 P r x S

ρ

ϕ ds

y

Radiation of Sound

F

331

The far field p of a plane radiator with area S and normal velocity distribution V(x, y) in an infinite baffle wall can be evaluated with the Huygens-Rayleigh integral:

V(x, y) · e−j k0 r ds r S j k0Z0 e−j k0 r0 V(x, y) · e+j k0 (x cos (r0 ,x)+y cos (r0 ,y)) ds, = 2 r0

p=

j k0Z0 2

(1)

S

where r0 is the radius from a reference point on the radiator to the field point P. If the velocity V has a constant phase on the radiator, the sound pressure attains its maximum P0 in the direction normal to the radiator: P0 =

j k0Z0 e−j k0 r0 ·Q 2 r0

;

Q=

V ds.

(2)

S

Describe the sound pressure in other directions with the directivity factor D: p = D · P0 with: 1 e+j k0 (x cos (r0 ,x)+y cos (r0 ,y)) dQ ; dQ = V · ds D= Q S (3) 1 1 +j k0 cos œ sin ‚ = e dQ −−−−−→ cos (k0 cos œ sin ‚) dQ symm. Q Q S

S

(see the graph for ‚, œ). The last relation holds for a radiator with a central axis of symmetry. In the case of an array with small elementary radiators having conphase volume flows Qn the integral is replaced by a sum: D=

1 Qn · e+j k0 (xn Q n

cos (rn ,xn )+yn cos (rn ,yn ))

;

Q=

Qn .

(4)

n

Two point sources with equal volume flow Qi at x = 0 and x = d: D = ej k0 d/2·cos(r,x) · cos (k0 d/2 · cos (r, x)) = ej k0 d/2·sin ‚ · cos k0 d/2 · sin ‚ . Maxima of |D| are at angles ‚ with d sin ‚ = 2Œ · Š0 /2 minima occur at odd multiples of Š0 /2.

;

Œ = 1, 2, 3, . . . ;

Point sources equally spaced along a line: The n point sources spaced at intervals d again are conphase and of equal strength. n−1

D=

sin (n) 1 j Œk0 d sin ‚ e = ej (n−1) n Œ=0 n · sin

;

= 12 k0 d sin ‚.

(5)

F

332

Radiation of Sound

Zeroes of the directivity are at angles ‚ with sin ‚ = ŒŠ0 /nd; the principal maximum (with unit value) is at ‚ = 0; the angles for the following maxima are at: (2Œ + 1) 2nd

sin ‚ =

;

Œ = 1, 2, 3, . . .

(6) DŒ =

with values at the maxima:

1 1 = . n sin n sin ((2Œ + 1)/(2n))

Densely packed linear array:

1

D = ej 2 k0 sin ‚

With = n d the length of the array:

sin

1 k0 sin ‚ 2

1 k0 sin ‚ 2

(7)

.

(8)

Densely packed circular array: The circle has the radius a; the elementary volume flow dQ = Q0 ds = Q0 · adœ is constant along the circle. 2

1 D= 2

ej k0 a

sin ‚ cos œ

dœ = J0 ()

;

= k0a sin ‚

(9)

0

Sources at constant intervals along a circle: Let n point sources with equal volume flow Q be distributed with equal intervals on a circle with radius a. r0 = radius from circle centre to field point P; ‚ = angle between circle axis and r0 ; œ = angle between the x axis in the plane of the circle and the projection of r0 on the circle plane. D = J0 (k0a sin ‚) + 2 jn Jn (k0a sin ‚) · cos (nœ) + 2 j2n J2n (k0a sin ‚) · cos (2nœ) + . . .

(10)

Circular piston in a baffle wall: The piston radius is a.Elementary volume flow dQ = Q0 ·r dr dœ; x = r·cos œ; y = r·sin œ. 1 D= 2 a 2 = 2 a

a 2

0 a

0

cos (k0r cos œ sin ‚) r dr dœ 0

J1 (k0a sin ‚) r J0 (k0 r sin ‚) dr = 2 k0a sin ‚

(11)

Radiation of Sound

F

333

Rectangular piston in a baffle wall: The side lengths are 2a, 2b; the elementary volume flow dQ = Q0 · dx dy. D = D1 · D2 1 D1 = 2a D2 =

1 2b

+a

ej k0 x

−a +b

cos (r,x)

sin (k0 a cos (r0, x)) k0a cos (r0 , x)

dx =

(12)

ej k0 y

−b

cos (r,y)

sin k0b cos (r0 , y) dy = k0 b cos (r0, y)

Rectangular plate, clamped at opposite edges, vibrating in its fundamental mode: Let the plate be in a one-dimensional vibration with (approximate) velocity distribution: V(y) = V0 · 1 − y 2 /b2 ,

(13)

where 2a is the length of the supported edges and 2b that of the other two edges.Average 2 8 velocities: V = V0 ; V2 = V20 , (14) 3 15 D=

3 2

sin − cos

;

= k0b sin ‚.

(15)

Rectangular plate, free at opposite edges,vibrating in its fundamental resonance: Let the plate be in a one-dimensional vibration with (approximate) velocity distribution: V(y) = V0 · 1 − 2y 2 /b2 .

(16)

√ The nodal lines (V = 0) are at y = ±b/ 2. The average velocities are: 1 V = V0 3 12 D= 2

;

V2 =

7 2 V, 15 0

3 sin sin − cos −

(17)

;

= k0 b sin ‚.

(18)

334

F

Radiation of Sound

Circular membrane and plate: Let the radius be a.The velocity distribution of the fundamental mode can be represented by a power series: $ $ 2 V() = V0 + V1 1 − 2 a2 + V2 1 − 2 a2 + . . . ,

(19)

1 1 1 Vn D = V 0 + V 1 + V2 + . . . + 2 3 n+1 (20) J1 () J2 () Jn+1 () −1 n+1 ; = k0a sin ‚. + 2 · 1 ! · V1 2 + . . . + 2 · n ! · Vn n+1 · 2V0 For a velocity distribution

V() = V0 · J0 (kB )

(21)

with the bending wave number kB on the radiator: D=

a 1 kB a 2 2 a J1 (kB a) kB − k02 sin ‚ · kB J0 (k0a sin ‚) J1 (kB a) − k0 sin ‚ J1 (k0a sin ‚) J0 (kB a) .

(22)

If the membrane or plate is supported at its edge, i.e. J0 (kBa) = 0: D=

kB2

kB2 J0 (k0 a sin ‚). − k02 sin ‚

(23)

Circular radiator with radial and azimuthal nodal lines: Develop the velocity distribution into a Fourier series: V(, œ0 ) =

Vm () · cos (mœ0 )

(24)

m≥0

with radial nodal lines for integer m > 0, and circular nodal lines at Vm () = 0. Write the far field pressure as:

p(r, ‚, œ) =

e−j k0 r Km (‚, œ). r m≥0

(25)

The directivity factor of a sum term then is: Dm (‚, œ) =

pm (r, ‚, œ) Km (‚, œ) 2 · Km(‚, œ) 2 · Km (‚, œ) = = = ; S = a2 , (26) p0 (r, 0, œ) K0 (0, œ) Q V S j m/2

a

Km (‚, œ) = cos (mœ) · e

Jm (k0 sin ‚) · Vm () · d. 0

(27)

Radiation of Sound

F

335

Introducing the integral transform (which is tabulated for many Vcm()): 1 a2

fm (Š) =

a Jm (Š) · Vm () · d,

(28)

Km (‚, œ) = a2 · jm · fm (k0 sin ‚) · cos (mœ).

(29)

0

one gets:

The directivity factor D(‚, ˜ ) is the sum of the Dm (‚, ˜ ). Array of finite size radiators: If all radiators have the same directivity factor Da , and the similar array with point sources has the directivity factor D0 , then the array with finite size radiators has the directivity factor D = Da · D0 .

F.18

(30)

Radiation of Finite Length Cylinder

See also: Skudrzyk, Ch. 21 (1971)

A cylinder with radius a and length 2b oscillates on its circumference with the velocity V and is at rest on its end caps. The centre of the cylinder is the origin of a cylindrical co-ordinate system (, z, œ) and of a spherical system (r, ‡, œ). Three angular sections are distinguished: (1) 0 ≤ ‡ ≤ ‡0 = arctan (a/b) (2) ‡0 < ‡ < − ‡0 (3) − ‡0 ≤ ‡ ≤ . The switch functions are defined for the section i: Hi = 1 for ‡ in i; Hi = 0 else. z 1 a θ0 θ 2b

r ρ y

ϕ

x 3

P 2

(1)

336

F

Radiation of Sound

Field formulation: p(r, ‡) = −j k0 Z0

an · Pn (cos ‡) · h(2) n (k0 r)

(2)

n=0,2,4...

with Pn (z) = Legendre polynomials; h(2) n (z)= spherical Hankel functions of second kind. The coefficients an are the solutions of the linear system of equations: an · (¥m , ¥n ) = V · (¥m , H2) ; m = 0, 2, 4, . . .

(3)

n=0,2,4...

with the integrals: (¥m , ¥n ) = b

2

(¥m , H2 ) = a2

‡0

¥m∗(1) (‡)

/2 d‡ d‡ + a ¥m∗(2) (‡) · ¥n(2) (‡) 2 , cos ‡ sin ‡

sin ‡ · ¥n(1) (‡) 3

0 /2

2

‡0

¥m∗(2) (‡)

‡0

(4)

d‡ sin2 ‡

containing the functions (primes indicate the derivative with respect to the argument):

¥n(1) (‡) = k0 cos ‡ · hn(2) (k0 b/cos ‡) · Pn (cos ‡) +

sin2 ‡ cos ‡ (2) · hn (k0b/cos ‡) · Pn (cos ‡), b

¥n(2) (‡) = k0 sin ‡ · hn(2) (k0a/sin ‡) · Pn (cos ‡) −

sin2 ‡ cos ‡ (2) · hn (k0a/sin ‡) · Pn (cos ‡), a

(5)

¥n(3) (‡) = −k0 cos ‡ · hn(2) (−k0 b/cos ‡) · Pn (cos ‡) +

sin2 ‡ cos ‡ (2) · hn (−k0b/cos ‡) · Pn (cos ‡). b

In the far field: p(r, ‡) = k0 Z0

e−j k0 r k0 r

jn · an · Pn (cos ‡).

(6)

n=0,2,4...

The corresponding result for an infinite cylinder which oscillates on a length 2b and is hard outside this band: p(r, ‡) = V

2k0bZ0 e−j k0 r sin (k0b cos ‡) 1 k0 r k0b cos ‡ sin ‡ · H0(2) (k0 a sin ‡)

with H(2) 0 (z) the Hankel function of second kind and order zero.

(7)

Radiation of Sound

F.19

F

337

Monopole and Multipole Radiators

See also: Morse/Ingard, Ch. 7 (1968)

Monopole: A point source is placed at the origin of a spherical co-ordinate system with a volume flow amplitude q (outward). j k0 Z0 e−j k0 r q . 4 r

p(r) j . 1− v = vr = Z0 k0 r

0 1 2 2 . w= |q| (k0 r) + (4r2 )2 2

Sound pressure:

p(r) =

Particle velocity: Energy density:

|p(r)|2 . 2Z0

Effective intensity:

I = Ir =

Radiated (effective) power:

¢ = 4r2 · Ir =

Radiant energy in a shell of unit thickness: Reactive energy outside the radius r:

(1) (2) (3) (4)

0 –2 2 |q| . 8 c0

0 k02 2 |q| . 4 0 |q|2 . E = 8 r E =

(5) (6) (7)

If the source has a finite radius a Š0 : Surface impedance (outward):

Zs =

Z0 p(a) k0a (k0 a + j) = = Z0 . vr (a) 1 − j/k0a 1 + (k0a)2

(8)

Let a monopole source with volume flow amplitude q be at r0 = (x0 , y0, z0 ): Sound pressure in r = (x, y, z): with

g(r|r0 ) =

−j k0 R

e 4R

;

p(r) = j k0 Z0 · q · g(r|r0)

R2 = |r − r0|2 = (x − x0 )2 + (y − y0 )2 + (z − z0 )2 .

(9) (10)

Dipole: Two monopoles with opposite sign of the volume flow q at a mutual distance d Š0 .

ϑ q

r +

d – –q

P

338

F

Radiation of Sound

Dipole strength:

D = q · d.

(11)

Sound pressure:

1 j 1 e−j k0 r 2 p(r) = j k0Z0 · q · g(r| d) − g(r| − d) = −k0 Z0 D 1− · cos ˜ . 2 2 4 r k0 r

(12)

Velocity components: vr = −k02 D

e−j k0 r 4 r

1−

j k0 r

−

2 (k0r)2

· cos ˜ ; v˜ = −j k0 D

e−j k0 r 4 r2

1−

j k0 r

· sin ˜ .(13)

2 1 1 + 3 cos2 ˜ k02D + cos2 ˜ + . 4 r 2(k0r)2 2(k0r)4

2 Z0 k02D Ir = cos2 ˜ . 2 4 r

Effective energy density:

w = 0

Effective intensity: Effective power:

¢=

0 –4 Z0 42 2 |D| = |D|2 . 2 3Š04 24 c30

(14) (15) (16)

0 k02 2 (17) |D| . 12 0 |D|2. (18) Reactive energy outside the radius r: E = 12 r3 A dipole corresponds to a small hard sphere with radius a Š0 oscillating back and forth in the direction of the dipole axis with a maximum surface velocity Ud in that direction.

D −j k0 a 1 2 1 + j k0 a − (k0a) . (19) Maximum velocity: Ud = e 2a3 2 j k0Z0 · D −j k0 a Driving force: Fd = 1 − j k0 a . (20) e 3 k0 a + j Fd 2a3 k0 Z0

. (21) = Mechanical driving impedance: Zd = 1 Ud 3 1 + j k0 a − (k0 a)2 2 Radiant energy in a shell of unit thickness:

E =

A dipole centred at the point r0 = (x0 , y0, z0 ) with dipole strength vector D = (Dx , Dy , Dz ), with R = r − r0 and R having the spherical angles ˜R , œR has the sound pressure field: p(r) = j k0Z0 · D · g(r|r0 )

;

g(r|r0 ) = (gx , gy , gz );

gx = sin ˜R cos œR · |g– |

;

gy = sin ˜R sin œR · |g– |

|g– | =

j k0 e−j k0 R 4 R

1−

j k0 R

(22) ;

gz = cos ˜R · |g–|, (23)

.

Radiation of Sound

F

339

Lateral quadrupole: For d Š0 , with Dxy = q · d2 .

(24)

Sound pressure field: p = −j k03 Z0 · Dxy

x y e−j k0 r 4 r3

1−

3 3j − k0r (k0r)2

.

(25)

y –q

P

q

–

+

+

– –q

r

d q

x

d

Linear quadrupole: The two central monopoles collapse to a volume flow −2q. For d Š0 , with Dxx = q · d2 .

(26)

Sound pressure field: p = −j k03 Z0 · Dxx y r q +

–q –q –– d

F.20

q +

1 j e−j k0 r x 2 3x2 − r2 + − . 4 r r r2 k0 r (k0 r)2

(27)

P

x

d

Plane Radiator in a Baffle Wall

See also: Heckl (1977)

A plane radiator with either dimensions L × B in Cartesian co-ordinates (x1 , x2 , x3 ) or radius a in polar co-ordinates (R, ‡, œ) is contained in a hard baffle wall. A point on the radiator is at ( 1, 2 ). The radiator area is, respectively, S = L · B = · a2 .

340

F

Radiation of Sound

θ ξ ξ

ϕ θ

Geometrical relations: r2 = (x1 − 1 )2 + (x2 − 2)2 + x32 ,

(1)

R2 = x12 + x22 + x32 , x1 = R · sin ‡ · cos œ

;

x2 = R · sin ‡ · sin œ

;

x3 = R · cos ‡.

(2)

Quantities: v( 1, 2) vˆ (k1 , k2) p(x1 , x2 , x3 ) ¢ kb = 2/Šb k1 , k2 rq , ” pL , ¢L v0 Zs mw † g(x)

given velocity distribution of the radiator Fourier transform of v( 1, 2) sound pressure in a field point effective sound power radiated towards one side wave number of radiator bending wave bending wave number components in directions x1 , x2 polar co-ordinates r, œ of a point on the source in polar co-ordinates sound pressure and effective power radiated by a line source velocity amplitude of the radiator radiation impedance radiation efficiency oscillating medium mass bending wave loss factor envelope of the radiator velocity distribution

Sound pressure in a far field point, i.e. k0L2 /R 1 or R · Š0 > L2 : j k0Z0 e−j k0 r Cartesian: p(x1 , x2 , x3) = d 1 d 2 , v( 1, 2 ) 2 r S

(3)

Radiation of Sound

polar:

p(R, ‡, œ) =

j k0 Z0 e−j k0 R 2 R

F

341

v( 1, 2 ) · ej k0 sin ‡ ( 1 cos œ+ 2 sin œ) d 1 d 2 . (4)

S

Using the wave number spectrum vˆ (k1 , k2) of the radiator pattern: k0Z0 p(x1 , x2, x3 ) = 42

+∞ −∞

vˆ (k1 , k2) k02

− k12

−

k22

· ej (k1 x1 +k2 x2 ) · e−j x3

√

k02 −k12 −k22

dk1 dk2 .

(5)

Long source (in x2 direction): +∞

k0 Z0 pL (x1 , x3 ) = 2

−∞

√2 2 vˆ (k1) · ej k1 x1 · e−j x3 k0 −k1 dk1 . k02 − k12

(6)

Radiator with radial symmetry (index r): pr (R, ‡) =

j k0 Z0 · vˆ r (k0 sin ‡) · e−j k0 R . 2R

(7)

Wave number spectrum of radiator velocity pattern: rectangular (Fourier transform): +∞ vˆ (k1, k2) =

−j (k1 1 +k2 2 )

v( 1 , 2) · e −∞

1 v( 1 , 2) = 2 4

+∞

d 1 d 2 = v( 1, 2) · e−j (k1 1 +k2 2 ) d 1 d 2 , S

(8)

vˆ (k1 , k2) · e+j (k1 1 +k2 2 ) dk1 dk2

−∞

with radial symmetry (Hankel transform): ∞ vˆ (k1, k2) → vˆ r (kr ) = 2

v(rq ) · J0 (kr rq ) · rq drq

;

0

kr = k12 + k22

(9)

1 = rq · cos ” . 2 = rq · sin ”

;

Effective sound power ¢ radiated towards one side: ¢

¢

=

=

=

1 2Z0

/22

k02Z0 82

0

% % % p(R, ‡, œ) %2 · R2 sin ‡ dœ d‡ ,

(10)

0

/22 0

% % % vˆ (−k0 sin ‡ cos œ , −k0 sin ‡ sin œ) %2 sin ‡ dœ d‡

0

⎧ +∞ ⎫ ⎨ ⎬ % % k0Z0 % vˆ (k1 , k2) %2 dk1 dk2 Re . ⎩ 82 k02 − k12 − k22 ⎭ −∞

(11)

F

342

Radiation of Sound

Special case of line source: k0Z0 ¢L = 4

+/2 +k0 % % % % k0Z0 % vˆ (k1 ) %2 dk1 % vˆ (k0 cos • ) %2 d•. = 2 2 4 k0 − k 1

(12)

−/2

−k0

Special case of source with radial symmetry: k 2 Z0 ¢r = 0 4

/2 % % % vˆ r (k0 sin ‡) %2 · sin ‡ d‡.

(13)

0

Radiation impedance Zs (¢ is complex power; . . . indicates average): S Definition: ¢ = · Zs · v 2 1 , 2 , 2 +∞

k0 Z0 Zs = 4 v 2 1 , 2

−∞

% % % vˆ (k1, k2 ) %2 dk1 dk2 . k02 − k12 − k22

(14)

(15)

Radiation efficiency : Definition: =

Re{Zs } , Z0

(16)

% % % vˆ (k1 , k2 ) %2 dk1 dk2 k02 − k12 − k22 2

= k0 ·

+∞

k12 +k22 k0

+∞

% % % vˆ (k1, k2 ) %2 dk1 dk2 .

(19)

−∞

Useful substitutions for evaluation: Set

k1 → k0 cosh (z cos œ)

makes

dk1 dk2 k12 + k22 − k02

For line sources, set makes

dk1 k12 − k02

;

k2 → k0 cosh (z sin œ)

= k0 · cosh z · dz dœ.

k1 → k0 cosh (z) = dz.

;

k2 → 0

(20) ;

S→B (21)

Radiation of Sound

Velocity Pattern

Range

F

343

Transform

Fourier transforms of some 1-dimensional velocity patterns: z ≡ k1 L/2 v( 1 ) = v0

v( 1 ) = v0 1 − 2| 1/L|

| 1| < L/2

vˆ (k1)/v0 = L · sin z/z

| 1| < L/2

vˆ (k1)/v0 = L/2 · (sin (z/2)/(z/2))2

2 v( 1 ) = v0 1 − (2 1/L)2

| 1| < L/2

v( 1 ) = v0 2 1 /L

| 1| < L/2

2 $ v( 1 ) = v0 3 2 1 /L − 1 2

| 1| < L/2

v( 1 ) = v0 e−| 1 |

| 1| < ∞

vˆ (k1) = L 24 z−5 − 8 z−3 · sin z v0 − 24 z−4 · cos z vˆ (k1)/v0 = −j L sin z /z2 − cos z /z vˆ (k1) = L 3 cos z /z2 v0 + 1/z − 3/z3 sin z vˆ (k1) 2 = 2 v0 + 21

Hankel transforms of some velocity patterns with radial symmetry: v(rq ) = v0

rq < a

v(rq ) = v0 J0 (kb rq )

rq < a

v(rq ) = v0 e− rq

rq < ∞

v(rq ) = v0 e−p

2 2 rq

vˆ r (kr ) J1(kr a) = 2 v0 a 2 kr a vˆ r (kr ) = v0 a 2 kr a J0 (kb a) J1(kr a) − kb a J0(kr a) J1 (kb a) 2 (kr a)2 − (kb a)2 vˆ r (kr ) = 2 2 2 v0 a ( + kr2 )3/2 vˆ r (kr ) 2 −kr2 /4 p2 = e 2 v0 a 2 p2

rq < ∞

v(rq ) = v0 /rq

rq < ∞

vˆ r (kr ) 2 = v0 a 2 kr

v(rq ) = v0 /rq‰

rq < ∞

vˆ r (kr ) (1 + ‰/2) −(‰+2) k = 2‰+2 2 v0 a (−‰/2) r

Radiator with nearly periodic velocity pattern vM (x1 ): vM(x1 ) = v0 · g(x1 ) · v0 vˆ M (k1) = 2 =

cos (kb x1 ) , sin (kb x1 )

gˆ (‹) ƒ(k1 − kb − ‹) + ƒ(k1 + kb − ‹) d‹

v0 gˆ (k1 − kb ) ± gˆ (k1 + kb ) · 2

(22)

1 , j

with ƒ(x) the Dirac delta function and gˆ (k) the Fourier transform of the envelope g(x1 ).

F

344

F.21

Radiation of Sound

Ratio of Radiation and Excitation Efficiencies of Plates

See also: Heckl (1964)

Consider two “experiments”: • Plate excited at a point with a force F radiates a sound power ¢; • Plate excited by a diffuse sound field with pressure p vibrates with a velocity v. Define the radiation efficiency a by

¢eff = a · F2eff ;

(1)

define the excitation efficiency b by

2 veff

(2)

Then:

a Z0 k02 = . b 4

=b·

p2eff .

(3)

Consider two “experiments”: • Plate excited by a line source with a force FL radiates a sound power ¢L ; • Plate excited by a diffuse sound field with pressure p vibrates with a velocity v. Define the radiation efficiency by

¢L,eff = · F2L,eff ;

(4)

define the excitation efficiency by

2 veff = · p2eff .

(5)

Then:

Z0 k0 = . 4

(6)

F.22

Radiation of Plates with Special Excitations

See also: Ver, Ch. 11 (1971)

Let S be the area of a plate in a baffle wall, v 2 S the average of the squared vibration velocity, ¢ the sound power radiated into one half-space, with ¢ = Z0 S · · v 2S /2 the radiation efficiency, fcr the critical frequenc, P the Poisson ratio, cD the dilatation wave velocity, h the plate thickness, and m = p h the surface mass density. Infinite plate with a free bending wave kB at f > fcr : =1

$ $ 1 − (kB/k0)2 = 1 1 − (fcr /f ).

(1)

Infinite plate (without losses) excited by a point force with amplitude F for f fcr (with m = p h surface mass density):

¢≈

0 2 c0

F 2 m

2 1−

0 tan−1 k0 m

⎧

0 F 2 ⎪ ⎪ ⎪ ⎨ 2 c 2 m ; k0m 0 ≈ ⎪ 0 (k F)2 ⎪ ⎪ ⎩ 0 ; 24 Z0

k0 m 1 0 k0 m

1. 0

(2)

Radiation of Sound

F

345

Infinite plate (without losses) with a point velocity source with amplitude v for k0 m /o 1: ¢≈

2 0 c30 2 v . 3 fcr2

(3)

Radius a of the equivalent ideal piston radiator (with same v 2S and unit efficiency): a=

$ 8 3 · Šcr = 0.286 · Šcr

;

Šcr = c0 /fcr .

(4)

Far field for a point force acting on an infinite plate with force amplitude F (R = radius from excitation point to field point, ˜ its polar angle): For a thin plate without losses (f < 0.7 · fcr ): p(R, ˜ ) =

j k0 e−j k0 R ·F· · 2 R

cos ˜ . k0 m 1+j cos ˜ · 1 − (f /fcr)2 sin4 ˜ 0

(5)

For a thick plate without losses (f > 0.7 · fcr ): p(R, ˜ ) =

e−j k0 R j k0 ·F· 2 R

1 + œ(˜ ) cos ˜ · , k0m 1 − P ( cD sin ˜ )2 1 + œ(˜ ) + j 1+ 1− · œ(˜ ) 0 24 c20 2(k0h)2 2 sin ˜ − (c0 /cD )2 . œ(˜ ) = 2 (1 − P )

(6)

For plates with a loss factor † substitute cD → cD · (1 + j · †/2); fcr → fcr · (1 + j · †/2).

(7)

References Antonov, S.N.; Putyrev,V.A.: Sov. J. Phys.Acoust. 30, 429–432 (1984)

Iwanov-Schitz, K.M., Rschevkin, S.N.: Acustica 13 403–406 (1963)

Bouwkamp, C.J.: Diffraction Theory. Philipps Research Report 1, 251–277 (1945/46)

Mawardi, O.K.: J. Acoust. Soc. Am. 23, 571–576 (1951)

Heckl, M.: Acustica 37, 155–166 (1977)

Mechel, F.P.: Schallabsorber, Vol. I, Ch. 9: Radiation impedances Hirzel, Stuttgart (1989)

Heckl, M.: Frequenz 18, 299–304 (1964)

346

F

Radiation of Sound

Mechel, F.P.: Schallabsorber, Vol. II, Ch. 22 Collection of end corrections Hirzel, Stuttgart (1995)

Skudrzyk, E.: The Foundations of Acoustics, Chs. 21 and 26 Springer, New York (1971)

Mechel, F.P.: Schallabsorber, Vol. I, Ch. 10: Radiation impedance of field pattern excited radiators Hirzel, Stuttgart (1989)

Ver, I.: In: Noise and Vibration Control, Ch. 11 McGraw-Hill, New York (1971)

Morse, P.M., Ingard, K.U.: Theoretical Acoustics, Ch. 7 McGraw-Hill, New York (1968)

G Porous Absorbers If no extra reference is given in the sections of this chapter, see Mechel (1995). The aims of the sections in this chapter are twofold: (1) derive the characteristic propagation constant a and the characteristic wave impedance Za of a plane wave in the porous material as functions of structure data and (2) derive the flow resistivity ¡ of the material as function of structure data because the flow resistivity is the most useful material parameter for the evaluation of a ,Za .Different models of a porous material will be displayed; special attention will be given to fibrous materials. See also the sections about scattering in random media in the chapter “Scattering of Sound” for propagation constant and wave impedance in fibrous and granular media.

G.1

Structure Parameters of Porous Materials

The definition of structure parameters depends on the model theory in which they are applied (see the relevant sections for specific definitions). Especially, see the chapter about Biot’s theory for special parameters of that theory. This section describes the structure parameters which come from the theory of the“quasi-homogeneous material” (see next section) because they are most often used to describe qualitatively porous absorber materials; the theory of the quasi-homogeneous material is the most simple theory. Volume porosity V , massivity ‹: The volume porosity V is the ratio of air volume contained in the porous material to the total volume; it is given by V = 1 − a /m with a the bulk density of the porous absorber material and m the density of the (dense) matrix material.For glass or mineral fibre materials a value m ≈ 2250[kg/m3] may be used. For some considerations it is advantageous to apply the massivity ‹ = 1 − V . The following table gives ranges of V for some materials. Structure factor ” : The structure factor is the most ambiguous quantity in porous material theories. It is defined in the theory of the quasi-homogeneous material as ” = V /S with S = surface porosity of a cut through the material. Its value depends on the type of the pore shapes. The pore volume Vp is given by the integral of the pore surface Sp over a distance x normal to the considered pore surface: S x Vp = Sp (x) dx ; ” = . (1) S x

348

G

Porous Absorbers

Table 1 Ranges of porosity v of some materials Material

v from v to

Mineral fibre materials

0.92

0.99

Foams

0.95

0.995

Felts

0.83

0.95

Wood-fibre board

0.65

0.80

Wood-wool board

0.50

0.65

Porous render

0.60

0.65

Pumice concrete

0.25

0.50

Pumice fill

0.65

0.85

Gravel and stone chip fill

0.25

0.45

Ceramic filtres

0.33

0.42

Brick

0.25

0.30

Sinter metal

0.10

0.25

Fire-clay

0.15

0.35

Sand stone

0.02

0.06

Marble

ca. 0.005

A model of an open-cellular foam consisting of cubic cells with connecting pores has the values: V ≈ 1 − 3

t d

;

”=

(1 + a/2) (d − 3t) . a2

Two end corrections = a/2 have been added to the neck length t. a d

t

(2)

Porous Absorbers

G

349

The following table gives the volume and surface porosities V , S and the structure factor ” of some regular model structures; a is the width of the pores, d is their distance. Table 2 Porosities v , s and structure factor ” for regular models Structure

V

S

”

Flat pores, longitudinal or inclined

a/d

a/d

1

Square pores, longitudinal or inclined

(a/d)2

(a/d)2

1

Round pores, longitudinal or inclined

(a/d)2 /4

(a/d)2 /4

1

Square fibres, longitudinal or inclined

2a=d − (a=d)2

2a=d − (a=d)2

1

Round fibres, longitudinal or inclined

1 − (a=d)2 =4

1 − (a=d)2 =4

1

Square fibres, transversal

2a=d − (a=d)2

a/d

2 − a=d

Round fibres, transversal

1 − (a=d)2 /4

1 − a=d

1 + a=d + (1 − =4)(a=d)2

Array of cubes

3a=d − 3(a=d)2 + (a=d)3

2a=d − (a=d)2

3=2 − 3a=(4d) + (a=d)2 =8

Flow resistivity ¡; absorber variable E: The flow resistivity ¡ of a porous material is its flow resistance per unit thickness for stationary flow with low velocity V (about V = 0.05 [cm/s]). For a material test sample of thickness dx: ¡=−

1 dP dx V

(3)

with dP the static pressure difference across the sample in flow direction. Theories mostly determine the interior velocity Vi in a pore for a given pressure difference. This “internal” flow resistivity ¡i is related with the flow resistivity ¡ of the sample by ¡ = ¡i /. A suitable non-dimensional quantity R is the ratio of the flow resistance ¡·d of a layer of thickness d with the free field wave impedance Z0 : R = ¡ · d/Z0 . According to R=

¡d , –0 · Š0 /2

(4)

350

G

Porous Absorbers

this is the ratio of the flow resistance to the mass reactance of a layer of air with thickness Š0 /2. An important non-dimensional quantity for the evaluation of the characteristic data a , Za is the absorber variable (f = frequency): E=

0 f . ¡

(5)

One distinguishes with fibrous materials consisting of parallel fibres the flow resistivity¡|| if the flow is parallel to the fibres and ¡⊥ if the flow is transversal to the fibres. Empirical data by Sullivan for parallel fibre materials with mono-valued fibre radii a follow the relations: ¡|| = 3.94 ·

† ‹ 1.413 1 + 27 ‹ 3 a2 1 − ‹

(6)

and

⎧ † ‹ 1.531 ⎪ ⎪ 10.56 ⎪ ⎨ a2 (1 − ‹)3 ¡⊥ = ⎪ † ‹ 1.296 ⎪ ⎪ ⎩ 6.8 2 a (1 − ‹)3

;

a ≈ 6 − 10 [‹m] , (7)

;

a ≈ 20 − 30 [‹m].

a = fibre radius; † = dynamic viscosity; ‹ = 1 − = massivity Semi-empirical data (analytical relation fitted to experimental values) for fibre materials with mono-valued fibre radii a and random fibre orientation give:

√ † ‹2 ‹ 4/3 + 2 ¡ = 4 · 2 0.55 . (8) a (1 − ‹) (1 − ‹)3 Empirical data for fibrous materials with random fibre radius distribution and random fibre orientation can be approximated by: † 3.2 ‹ 1.42 ; glass fiber material , (9) ¡= 2 · a 4.4 ‹1.59 ; mineral fiber material.

G.2 Theory of the Quasi-homogeneous Material An equivalent network is designed for a homogeneous material, taking into account a finite volume porosity V , a structure factor ” of randomly oriented pores and a relaxation time constant ‘ for heat exchange between air and matrix material.A possible vibration of the matrix, induced by the friction between air and matrix, can also be included.

Porous Absorbers

ρ0 χ σV

vx

σ V (κ − 1) 2 ρ 0c 0

σV 2 ρ 0c 0

p

G

351

Ξ

ρa

τ ρ0 c20 σ V (κ − 1)

Equivalent network for unit length of a porous material The first transversal branch represents the compressibility of the air in the material; the second transversal branch represents the relaxation due to heat exchange with the matrix; the first longitudinal element represents the inertia of the air in the pores (modified by the structure factor ”); the second longitudinal element represents the friction. If matrix vibration is included, the bulk density a of the material is parallel to ¡. This can be taken into account by an effective resistivity: ¡ → ¡eff =

j–a · ¡ . j–a + ¡

(1)

Characteristic values: a v ¡ ‰−1 · 1−j , =j ” 1+ k0 1 + j–‘ –0 ” 1 Za = Z0 v

”

v ¡ 1−j –0 ”

/

‰−1 1+ . 1 + j–‘

k0 Z0 0 –

= = = =

free field wave number; free field wave impedance; density of air; angular frequency;

‰ ¡ ” V

= = = =

adiabatic exponent of air; flow resistivity; structure factor; volume porosity;

‘

= heat relaxation time constant

(2)

(3)

352

G

Porous Absorbers

Introducing E = 0 f /¡ and E0 = 0 f0/¡ with f0 = relaxation frequency,the characteristic values are: Za a v 1 1 + jE/E0 v ‰ + jE/E0 ; . (4) =j · ”−j = · ”−j k0 1 + jE/E0 2 E Z0 v ‰ + jE/E0 2 E E0 → 0 belongs to an isothermal sound wave in the material; E0 → ∞ belongs to an adiabatic sound wave; E0 = 0.1 is a typical value for mineral fibre materials. 10 Γa′ / k 0 Γa′′/ k 0 − 1 Z a′ / Z 0 − 1 − Za′′ / Z0 1

σ = 0.95 ; χ = 1.3 ; E o = 0.1

0.1 0.001

0.01

0.1

E

1

Thick lines: theory of quasi-homogeneous material; thin lines: measured values

G.3

Rayleigh Model with Round Capillaries

The Rayleigh model has for a long time been the only existing model for porous materials. This model consists of parallel circular capillaries with radius a and mutual distance d in a bloc of the matrix material. The sound propagation in the capillary is determined with viscous and thermal losses at the capillary wall taken into account. See > Ch. J, “Duct Acoustics”, for sound propagation in capillaries. The arrangement and mutual distance of the capillaries is supposed to be such that a prescribed porosity is obtained (e.g. = a2 /d2 for a square arrangement), even if the value of the porosity cannot be realised physically (in a square arrangement the realisable porosity is ≤ /4 = 0.785), because only a single capillary is considered. The theory gives the propagation constant a of the density wave (which is considered to be the propagation constant of sound in the porous material) and the “interior” axial wave impedance Zi of the density wave in a capillary. Its relation to the desired wave impedance Za of the porous material is Za = Zi /.

Porous Absorbers

G

The normalised characteristic values of a cell are: a Zi eff Ceff eff Ceff =j · ; = k0 0 C0 Z0 0 C0

353

(1)

with the ratios of the effective air density and air compressibility (index 0 indicates free field values): Ceff 1 eff ; = = 1 + (‰ − 1) · J1,0 (k0 a) (2) 0 1 − J1,0 (kŒ a) C0 and the function J1,0 (z) = 2

1 z2 z 2 z2 z2 J1(z) =2· ... ; n = 1, 2, 3 . . . z · J0 (z) 2− 4− 6− 8 − . . . 2 · n − ...

(3)

with Bessel functions Jn (z) and the continued-fraction expansion of their ratio in the last term (the expansion must go until |z|2/(2n) 1). The asymptotic approximation 3 + tan z − 2 4 8z (4) J1,0 (z) ≈ 1 z tan z − 1+ 8z 4 may be applied for large arguments |z|. The squares of the used wave numbers are: √ 2 = kŒ2 · ‰ Pr. kŒ2 = −j –/Œ ; k0 ‰ Œ †

= adiabatic exponent; = kinematic viscosity; = dynamic viscosity;

Pr ‹ C0 ¡

= = = =

(5)

Prandtl number; cell massivity; 1/(0 c20 ); flow resistivity

The argument kŒ a should be replaced with an argument which can be applied also for material with random pore radius distribution.This is the“absorber variable”E = 0 f /¡. For its application, the flow resistivity ¡i in a pore must be determined. The flow velocity profile in a circular capillary is, with the static pressure gradient dP/dx: V(r) = −

1 dP 2 (a − r2 ). 4† dx

(6)

The flow resistivity ¡i is defined with the volume flow Q through the capillary crosssection area S: ¡i =

−dP/dx · S 8† = 2 Q a

;

Q=−

a4 dP . 8† dx

(7)

G

354

Porous Absorbers

Thus the desired relations between the arguments are: (kŒ a)2 = −16j E

;

(k0a)2 = −16j ‰ Pr E.

(8)

If in a material test the (exterior) flow resistivity ¡ is determined, its relation to the interior flow resistivity is ¡ = ¡i / with the porosity of the material. The continued-fraction approximation (up to the fifth fraction) of the characteristic values is with the variable E:

2 75 + 30j [8 + Pr(5 + 3‰)] E − 2 [90 + Pr(480 + 288‰) + . . . i = −‰ k0 2E {75j − 40(1 + 6‰ Pr) E − . . . + Pr 2 ‰(80 + 10‰)] E2 − 43 j Pr[45 + 27‰ + ‰ Pr(84 + 8‰)] E3 + . . . −22 j ‰ Pr(64 + 45‰ Pr) E2 + . . .

(9)

+124 ‰ Pr2 (8 + ‰) E4 , +483 (‰ Pr)2 E3 }

Zi Z0

2 =

225 + 720j (1 + ‰ Pr) E − 182[15 + . . . 1 ‰ 2E {225j − 30[4 + Pr(15 + 9‰)] E − 62 j Pr[40 + 24‰ + . . . +‰ Pr(128 + 15‰ Pr)]E2 − 8643j‰ Pr(1 + ‰ Pr) E3 + 3244 (‰ Pr)2 E4 . +‰ Pr(40 + 5‰)] E2 + 163 ‰ Pr2 (8 + ‰) E3 }

(10)

σ=0.98

10

Re{Γa/k0} Im{Γa/k0} – 1

1

Re{Za/Z0} – 1 – Im{Za/Z0} 0.1 0.001

0.01

0.1

E

1

Comparison of the characteristic values from the Rayleigh model (curves) with measurements at a technical mineral fibre absorber (points)

G.4

Model with Flat Capillaries

The good agreement between characteristic values from measurements with fibrous absorber materials and the theory of the Rayleigh model is rather surprising because

Porous Absorbers

G

355

the Rayleigh model has a somehow “inverted” geometry as compared to a fibre material. Therefore the simpler model with flat capillaries (instead of round capillaries) may have a chance, also. The model consists of parallel flat capillaries, 2h wide and in a mutual distance d, in a bloc of the matrix material. The sound propagation in the capillary is determined with viscous and thermal losses at the capillary walls taken into account. See > Sects. J.1 and J.2 in Ch. J,“Duct Acoustics”, for sound propagation in capillaries. The arrangement and mutual distance of the capillaries is supposed to be such that a prescribed porosity is obtained ( = 2h/d). The theory gives the propagation constant a of the density wave (which is considered to be the propagation constant of sound in the porous material), and the “interior” axial wave impedance Zi of the density wave in a capillary. Its relation to the wanted wave impedance Za of the porous material is Za = Zi /. The normalised characteristic values of a capillary are: eff Ceff a =j · k0 0 C0

Zi = Z0

;

eff 0

Ceff C0

(1)

with the ratios of the effective air density and air compressibility (index 0 indicates free field values): eff = 0

1 tan(kŒ h) 1− kŒ h

;

Ceff tan(k0 h) = 1 + (‰ − 1) . C0 k0h

(2)

The squares of the used wave numbers are: kŒ2 = −j –/Œ

;

2 k0 = kŒ2 · ‰ Pr.

(3)

‰ = adiabatic exponent; Œ † Pr C0

= = = =

kinematic viscosity; dynamic viscosity; Prandtl number; 1/(0c20 );

¡ = flow resistivity The argument kŒ h should be replaced with an argument which can be applied also for material with random pore width distribution. This is the“absorber variable”E = 0 f /¡. For its application, the flow resistivity ¡i in a pore must be determined. The flow velocity profile in a flat capillary is, with the static pressure gradient dP/dx and y the transversal co-ordinate: V(y) = −

1 dP 2 (h − y 2). 2† dx

(4)

356

G

Porous Absorbers

The flow resistivity ¡i is defined with the volume flow Q through the capillary crosssection: −dP/dx · 2h 3† ¡i = = 2 Q h

h ;

Q=2

V(y)dy = − 0

2 dP 3 h . 3† dx

(5)

Thus the wanted relations between the arguments are: |kŒ h|2 = 6E

;

(kŒ h)2 = −6j E

;

(k0h)2 = −6j ‰ Pr E.

(6)

If in a material test the (exterior) flow resistivity ¡ is determined, its relation to the interior flow resistivity is ¡ = ¡i / with the porosity of the material. σ=0.95

10


1

Re{Za/Z0} – 1 – Im{Za/Z0} 0.1 0.001

0.01

0.1

E

1

Comparison of the characteristic values from the model with flat capillaries (curves) with measurements at a technical mineral fibre absorber (points)

G.5

Longitudinal Flow Resistivity in Parallel Fibres

The geometry of the fibre bundle models agrees better with the geometry of real fibrous materials. First a regular arrangement of the fibres is used in the theory, then the diameters and distances of the fibres are randomised. The direction of flow and sound relative to the fibres must be treated separately, when it is parallel to the fibres or transversal.

Porous Absorbers

2R 2a

G

357

Vi

Δz

Regular arrangement: Fibres of equal radius a are arranged in a regular array with cells of radius R around each fibre. The flow velocity Vi is parallel to the fibres. The cell radius R is adjusted to give the porosity : = 1 − (a/R)2 = 1 − N · a2

with N = number of fibres in unit area.

(1)

Velocity profile of viscous flow in a cell (with C = a free factor; z axis in flow direction): 2

r 1 2 R ln − r − a2 . (2) vz (r) = C 2 a 4 Flow resistivity, with definition ¡|| = (−∂p/∂z) vz : (3) 4† ‹ (4) ¡|| = (−∂p/∂z) vz = 2 1 a 2‹ − ln ‹ − ‹ 2 − 1, 5 2 with † = dynamic viscosity of air; ‹ = 1 − = massivity. Random arrangement: A material sample contains I cells,i = 1, 2, . . ., I with random variation of the fibre radius ai and/or the cell cross-section area Si = Ri2 , which gives a massivity ‹i = (ai /Ri )2 of each cell. The resulting flow resistivity is: 4† ¡|| = I . (5) 2 Si 2 Ri · · (2 ‹i − 0, 5 ‹i − ln ‹i − 1, 5) Si i=1 i

One can collect the fibre radii ai in groups around radius group values am with a relative frequency qm and correspondingly the cell radii Ri in groups around group radii Rn with relative frequency pn . The flow resistivity is: ¡|| =

2 4† < am > .

4

2 1 am am am 4 − 1, 5 ‹ qm pn · Rn 2 − − 2 ln Rn 2 Rn Rn m,n

Technical fibrous absorbers mostly have Poisson distributions of the am and Rn .

(6)

358

G

G.6

Longitudinal Sound in Parallel Fibres

Porous Absorbers

Suppose a fibre model as in

>

Sect. G.4 with sound propagation parallel to the fibres.

The sound field in each cell is evaluated with viscous and thermal losses (at the isothermal fibre surface) taken into account. The propagation constant of the density wave is a ; the axial wave impedance of the density wave is Zi (interior wave impedance); its relation to the characteristic impedance Za (exterior wave impedance) of a material sample is Za = Zi /, with = porosity of the material. The normalised characteristic values of a cell are: eff Ceff eff Ceff a Zi =j · ; = k0 0 C0 Z0 0 C0

(1)

with the ratios of the effective air density and air compressibility (index 0 indicates free field values): 1 eff = 0 1 − H1,0(kŒ a ; ‹)

;

Ceff = 1 + (‰ − 1) · H1,0(k,0a ; ‹) C0

(2)

and the function H1,0(x, ‹) =

√ √ −2 ‹ J1 (x/ ‹) · Y1 (x) − J1 (x) · Y1(x/ ‹) √ √ x (1 − ‹) J1 (x/ ‹) · Y0 (x) − J0 (x) · Y1(x/ ‹)

(3)

with Bessel functions Jn (z) and Neumann √ functions Yn(z). The squares of the used wave 2 numbers are kŒ2 = −j –/Œ ; k0 = kŒ2 · ‰ Pr. (4) ‰ = adiabatic exponent; Œ = kinematic viscosity; Pr = Prandtl number; ‹ = cell massivity; C0 = 1/(0c20 ) The replacement of the argument kŒ a with the absorber variable E = 0 f /¡ is performed with the result of the previous > Sect. G.5: 1 2‹ − ln ‹ − ‹ 2 − 1.5 1 1 2‹ − ln ‹ − 1.5 2 2 |kŒ a| ≈ |kŒ a|2 , E= 8 ‹ 8 ‹ |kŒ a|2 = 8 E

‹

1 2‹ − ln ‹ − ‹ 2 − 1.5 2

≈ 8 E

‹ . 2‹ − ln ‹ − 1.5

(5) (6)

The present model also permits one to determine the relaxation (angular) frequency –0 of the heat transfer between sound wave and matrix (–0 was needed in > Sect. G.2, but it was not possible there to determine its magnitude). The relaxation is marked by a

Porous Absorbers

G

359

deep minimum of Im{Ceff /C0} in the plane of |kŒ a|2 and massivity ‹. Its position gives the relations: 2 3 –0 2 a = 10(1.69843 + 2.58454 log ‹ + 0.541408 (log ‹) + 0.0726279 (log ‹) ) , |kŒ a|2 = Œ (7) 2 3 Œ Œ 2 (1.69843 + 2.58454 log ‹ + 0.541408 (log ‹) + 0.0726279 (log ‹) ) –0 = 2 |kŒ a| = 2 · 10 . a a σ=0.98

10


1

Re{Za/Z0} – 1 – Im{Za/Z0} 0.1 0.001

0.01

0.1

E

1

Comparison of the characteristic values from the model of a regular arrangement of longitudinal fibres (curves) with measured data from a technical mineral fibre absorber The model of parallel fibres with longitudinal sound propagation can be easily randomised with respect to the fibre radii ai and cell radii Ri or cell areas Ai = Ri2 . Their random values are counted in groups of interval widths a, A around average group values am , An. If their relative frequencies qm , pn have the forms of Poisson distributions with distribution parameters Š, qm = e−Š

2 am

Šm m!

n n!

pn = e−

;

2

= (Š + 2Š + 1/4) a

2

;

;

m, n = 0, 1, 2, . . . ,

(8)

An = ( + 1/2) A

with the total and the group massivities ‹=

2 > < am < An >

;

‹mn = ‹

Š2

+ 1/2 (m + 1/2)2 , + 2Š + 1/4 n + 1/2

(9)

the effective group densities and compressibilities are: 1 mn = 0 1 − H1,0 (kŒ am , ‹mn )

;

Cmn = 1 + (‰ − 1) H1,0(k0 am , ‹mn ). C0

(10)

G

360

Porous Absorbers

The characteristic values are evaluated from the statistically relevant density s and compressibility Cs by: a s Cs s Cs Zi =j · ; = (11) k0 0 C0 Z0 0 C0 with 1

<

s = 0

m,n

qm pn < m,n

<

mn

0

qm pn

;

Cs = C0

m,n

qm pn

< m,n

Cmn C0

,

(12)

qm pn

where the upper symbol < at the summation sign indicates that the summation is performed only if ‹mn < 1; otherwise it is skipped. Typical values of the distribution parameters are Š = 2.5 (deviations thereof have little influence on the result) and = 4.5. μ=0.02 ; λ=2.5 ; Λ=4.5

10

Re{Γ/k0} 1

Re{Zi/Z0} − 1 Im{Γ/k0}-1 0.1

-Im{Zi/Z0}

0.01 0.001

0.01

0.1

|kν a|2

1

Characteristic values from model with parallel fibres and longitudinal sound propagation if the fibre radii and the cell radii are Poisson distributed

G.7 Transversal Flow Resistivity in Parallel Fibres The model is as depicted in > Sect. G.5, but the flow velocity Vi is normal to the fibres. Analytical studies (see Mechel, 1995) have shown that a cell model, in which the cell

Porous Absorbers

G

361

“walls” are immaterial symmetry surfaces for the disturbance flow field (u, v), can be used also for this transversal flow as an approximation (in fact there is a windward-tolee unbalance, but it is small for massivity values as they are found for technical fibre absorbers). = U · (1 + u, v). The cell and the incoming flow U are as in the sketch. The flow field is V The disturbance flow v and its radial derivative are supposed to be zero on the cell surface. Besides the Cartesian co-ordinates (x, y), also cylindrical co-ordinates (r, œ) are used (œ = 0 on the x axis). 2R

Ø

y

x

U Ai

Ø

2a

General solutions with the present conditions of symmetry of the Navier–Stokes equation are: u(kr, œ) = vx /U C0 cos œ − m [Cm (kr)m−1 · cos(m − 1)œ − Dm (kr)−m−1 kr m≥1 1 n · cos(m + 1)œ] − ekr cos œ · An · [(cos nœ + cos(n + 1)œ) 2 kr n≥0

=−

(1)

· In (kr) − cos œ cos nœ In−1 (kr)] + Bn · [(cos nœ +

n cos(n + 1)œ) Kn (kr) − cos œ cos nœ Kn−1 (kr)], kr

v(kr, œ) = vy /U C0 sin œ m [Cm (kr)m−1 · sin(m − 1)œ + Dm (kr)−m−1 − kr m≥1 1 n · sin(m + 1)œ] − ekr cos œ · An · [ sin(n + 1)œ 2 kr n≥0

=−

· In (kr) − sin œ cos nœ In−1 (kr)] + Bn · [

n sin(n + 1)œ) Kn (kr) + sin œ cos nœ Kn−1(kr)] kr

(2)

362

G

Porous Absorbers

with the flow parameter k = U/(2Œ) (Œ = kinematic viscosity), the modified Bessel functions Im (z), Km (z) of the first and second kind, and yet undetermined coefficients An , Bn , Cn , Dn . These are determined from the boundary conditions u(ka, œ) kr=ka = −1 ; v(ka, œ) kr=ka = 0 ; ur (kr, œ)

kr=kR

=0

which, with a summation up to m, n = 3, gives a linear system of equations of the following form:

;

∂uœ (kr, œ) = 0, ∂kr kr=kR ⎡ c1,1 c1,2 . . . c1,14 ⎢ ⎢ c2,1 c2,2 . . . c2,14 ⎢ .. .. ⎢ .. ⎣ . . . c15,1 c15,2 . . . c15,14

(3)

c1,15 c2,15 .. . c15,15

⎤ ⎡ ⎥ ⎢ ⎥ ⎢ ⎥·⎢ ⎥ ⎢ ⎦ ⎣

A0 A1 .. . D3

⎤

⎡

⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎦ ⎣

−1 0 .. . 0

⎤ ⎥ ⎥ ⎥(4) ⎥ ⎦

with the list of unknown coefficients {A0, A1 , A2 , A3 , B0 , B1 , B2 , B3 , C0 , C1 , C2 , C3 , D1 , D2 , D3 , }

(5)

and the matrix coefficient rows ci = {ci1 , ci2 , ci3 , . . ., ci15 }:

(6)

c1 = { − (1/2 + ka2 /8) · I0 (ka) + (ka/4 + ka3 /32) · I1 (ka), (1/4 + 3ka2 /32) · I0 (ka) − (5ka/16 + ka3 /32) · I1 (ka), (ka/8 + ka3 /48) · I1 (ka) − ka2 /12 · I2 (ka), ka2 /32 · I2 (ka) − ka3 /96 · I3 (ka), −(1/2 + ka 2 /8) · K0 (ka) − (ka/4 + ka3 /32) · K1(ka), −(1/4 + 3ka2 /32) · K0(ka) − (5ka/16 + ka3/32) · K1 (ka), −(ka/8 + ka3 /48) · K1 (ka) − ka2 /12 · K2 (ka), −ka2 /32 · K2 (ka) − ka3 /96 · K3(ka), 0, −1, 0, 0, 0, 0, 0};

c2 = { − (ka/2 + ka3 /16) · I0 (ka) + (1/2 + 3ka2 /16) · I1 (ka), (3ka/8 + 5ka3 /96) · I0 (ka) − (3/4 + 11ka2/48) · I1 (ka), (1/4 + ka2 /8) · I1 (ka) − (3ka/8 + ka3 24) · I2 (ka), (ka/8 + 5ka3 /192) · I2 (ka) − 3ka2 /32 · I3 (ka), −(ka/2 + ka3 /16) · K0 (ka) − (1/2 + 3ka2 /16) · K1(ka), −(3ka/8 + 5ka3 /96) · K0(ka) + (−3/4 − 11ka2 /48) · K1 (ka), (−1/4 − ka2 /8) · K1 (ka) − (3ka/8 + ka3 /24) · K2 (ka), −(ka/8 + 5ka3/192) · K2(ka) − 3ka2 /32 · K3(ka), −1/ka, 0, −2ka, 0, 0, 0, 0};

Porous Absorbers

c3 = { − ka2/8 · I0 (ka) + (ka/4 + ka3/24) · I1 (ka), (1/4 + ka2 /8) · I0 (ka) − (1/(2ka) + 3ka/8 − ka3/24) · I1 (ka), (ka/4 + 7ka3/192) · I1 (ka) − (1 + 3ka2 /16) · I2 (ka), (1/4 + 3ka2 /32) · I2 (ka) − (7ka/16 + ka3 /32) · I3 (ka), −ka2 /8 · K0(ka) + (−ka/4 − ka3 /24) · K1 (ka), −(−1/4 ka2 /8) · K0 (ka) − (1/(2ka) + 3ka/8 + ka3 /24) · K1(ka), −(ka/4 + 7ka3/192) · K1 (ka) − (1 + 3ka2/16) · K2 (ka), −(1/4 + 3ka2/32) · K2 (ka) − (7ka/16 + ka3 /32) · K3(ka), 0, 0, 0, −3ka2, ka−2 , 0, 0}; c4 = { − ka3/48 · I0 (ka) + ka2 /16 · I1 (ka), (ka/8 + 5ka3 /192) · I0 (ka) − (1/4 + 3ka2 /32) · I1 (ka), (1/4 + 3ka2 /32) · I1 (ka) − (ka−1 + ka/2 + ka3 /32) · I2 (ka), (ka/4 + ka3 /32) · I2 (ka) − (5/4 + 7ka2 /32) · I3 (ka), −ka3 /48 · K0(ka) − ka2 /16 · K1(ka), −(ka/8 + 5ka3/192) · K0 (ka) − (1/4 + 3ka2 /32) · K1 (ka), −(1/4 + 3ka2/32) · K1 (ka) − (ka−1 + ka/2 + ka3 /32) · K2 (ka), −(ka/4 + ka3 /32) · K2(ka) − (5/4 + 7ka2/32) · K3 (ka), 0, 0, 0, 0, 0, 2/ka3, 0}; c5 = {ka3 /96 · I1 (ka), ka2 /32 · I0 (ka) − (ka/16 + ka3 /96) · I1 (ka), (ka/8 + ka3 /48) · I1 (ka) − (1/2 + ka2 /8) · I2 (ka), (1/4 + 3ka2 /32) · I2 (ka) − (3/(2ka) + 5ka/8 + ka3 /32) · I3 (ka), −ka3 /96 · K1(ka), −ka2 /32 · K0(ka) − (ka/16 + ka3/96) · K1 (ka), −(ka/8 + ka3 /48) · K1(ka) − (1/2 + ka2/8) · K2(ka), −(1/4 + 3ka2/32) · K2 (ka) − (3/(2ka) + 5ka/8 + ka3 /32) · K3 (ka), 0, 0, 0, 0, 0, 0, 3/ka4};

G

363

364

G

Porous Absorbers

c6 = {(1/2 + ka2 /16) · I1 (ka), (ka/8 + ka3 /96) · I0 (ka) − (1/4 + ka2 /48) · I1 (ka), −I1 (ka)/4 − ka/8 · I2 (ka), −(ka/8 + ka3 /192) · I2 (ka) − ka2 /32 · I3 (ka), −(1/2 + ka2 /16) · K1 (ka), −(ka/8 + ka3 /96) · K0 (ka) − (1/4 + ka2/48) · K1 (ka), K1 (ka)/4 − ka/8 · K2 (ka), (ka/8 + ka3 /192) · K2(ka) − ka2 /32 · K3(ka), −1/ka, 0, 2ka, 0, 0, 0, 0}; c7 = {(ka/4 + ka3 /48) · I1 (ka), (1/4 + ka2 /16) · I0 (ka) − (1/(2ka) + ka/8) · I1 (ka), ka3 /192 · I1 (ka) − (1/2 + ka2/16) · I2 (ka), −(1/4 + ka2 /32) · I2 (ka) − 3ka/16 · I3 (ka), −(ka/4 + ka3 /48) · K1 (ka), −(1/4 + ka2 /16) · K0 (ka) − (1/(2ka) + ka/8) · K1 (ka), −ka3 /192 · K1 (ka) − (1/2 + ka2 /16) · K2 (ka), (1/4 + ka2 /32) · K2(ka) − 3ka/16 · K3(ka), 0, 0, 0, 3ka2, 1/ka2, 0, 0}; c8 = {ka2 /16 · I1 (ka), (ka/8 + ka3 /64) · I0 (ka) − (1/4 + ka2 /32) · I1 (ka), (1/4 + ka2 /32) · I1 (ka) − (1/ka + ka/4) · I2 (ka), −(3/4 + 3ka2 /32) · I3 (ka), −ka2 /16 · K1(ka), −(ka/8 + ka3 /64) · K0 (ka) − (1/4 + ka2/32) · K1 (ka), −(1/4 + ka2 /32) · K1 (ka) − (1/ka + ka/4) · K2(ka), −(3/4 + 3ka2 /32) · K3(ka), 0, 0, 0, 0, 0, 2/ka3, 0};

Porous Absorbers

G

c9 = {(1/2 + kR2 /8) · I1 (kR) − (kR/4 + kR3/32) · I0 (kR), (kR/4 + kR3/32) · I0 (kR) − (1/2 + kR2/8) · I1 (kR), kR2 /16 · I1 (kR) − (kR/4 + kR3 /48) · I2 (kR), kR3 /96 · I2 (kR) − kR2 /16 · I3 (kR), −(1/2 + kR2 /8) · K1 (kR) − (kR/4 + kR3/32) · K0 (kR), −(kR/4 + kR3 /32) · K0(kR) − (1/2 + kR2/8) · K1 (kR), −kR2 /16 · K1 (kR) − (kR/4 + kR3/48) · K2 (kR), −kR3 /96 · K2 (kR) − kR2 /16 · K3(kR), −1/kR, 0, 0, 0, 0, 0, 0}; c10 = {(kR/2 + kR3 /16) · I1 (kR) − (1/2 + 3kR2 /16) · I0 (kR), (1/2 + 3kR2/16) · I0 (kR) − (1/(2kR) + 9kR/16 + 5kR3 /96) · I1 (kR), (kR/4 + kR3/24) · I1 (kR) − (3/4 + 5kR2/24) · I2 (kR), kR2 /16 · I2 (kR) − (5kR/16 + 5kR3/192) · I3 (kR), −(kR/2 + kR3 /16) · K1(kR) − (1/2 + 3kR2/16) · K0(kR), −(1/2 + 3kR2 /16) · K0 (kR) − (1/(2kR) + 9kR/16 + 5kR3/96) · K1 (kR), −(kR/4 + kR3 /24) · K1(kR) − (3/4 + 5kR2/24) · K2(kR), −kR2 /16 · K2 (kR) − (5kR/16 + 5kR3 /192) · K3 (kR), 0, −1, 0, 0, 1/kR2, 0, 0}; c11 = {kR2 /8 · I1 (kR) − (kR/4 + kR3/24) · I0 (kR), (kR/4 + kR3/24) · I0 (kR) − (1/2 + kR2/6) · I1 (kR), (1/2 + kR2/8) · I1 (kR) − (1/kR + kR/2 + 7kR3/192) · I2 (kR), (kR/4 + kR3/32) · I2 (kR) − (1 + 3kR2/16) · I3 (kR), −kR2 /8 · K1 (kR) − (kR/4 + kR3 /24) · K0(kR), −(kR/4 + kR3 /24) · K0(kR) − (1/2 + kR2/6) · K1 (kR), −(1/2 + kR2 /8) · K1 (kR) − (1/kR + kR/2 + 7kR3/192) · K2(kR), −(kR/4 + kR3 /32) · K2(kR) − (1 + 3kR2 /16) · K3(kR), 0, 0, −2kR, 0, 0, 2/kR3, 0};

365

366

G

Porous Absorbers

c12 = {kR3/48 · I1 (kR) − kR2/16 · I0 (kR), kR2 /16 · I0 (kR) − (3kR/16 + 5kR3/192) · I1 (kR), (kR/4 + kR3 /32) · I1 (kR) − (3/4 + 5kR2/32) · I2 (kR), (1/2 + kR2 /8) · I2 (kR) − (3/(2kR) + 5kR/8 + kR3 /32) · I3 (kR), −kR3 /48 · K1 (kR) − kR2 /16 · K0(kR), −kR2 /16 · K0 (kR) − (3kR/16 + 5kR3 /192) · K1 (kR), −(kR/4 + kR3 /32) · K1(kR) − (3/4 + 5kR2/32) · K2(kR), −(1/2 + kR2 /8) · K2 (kR) − (3/(2kR) + 5kR/8 + kR3 /32) · K3(kR), 0, 0, 0, −3kR2, 0, 0, 3/kR4}; c13 = {kR/8 · I0 (kR) + (1/2 + kR2 /16) · I1 (kR), (−1/(2kR) + kR/16 + kR3/96) · I0 (kR) + (1/kR2 + kR2 /48) · I1 (kR), −(3/4 + kR2 /24) · I1 (kR) + 3/(2kR) · I2 (kR), −(5kR/16 + kR3/192) · I2 (kR) + 5/8 · I3 (kR), kR/8 · K0 (kR) − (1/2 + kR2 /16) · K1 (kR), (1/(2kR) − kR/16 − kR3 /96) · K0(kR) + (1/kR2 + kR2 /48) · K1(kR), (3/4 + kR2 /24) · K1(kR) + 3/(2kR) · K2 (kR), (5kR/16 + kR3 /192) · K2(kR) + 5/8 · K3(kR), 0, 0, 0, 0, −2/kR3, 0, 0}; c14 = {(1/4 + kR2 /16) · I0 (kR) + (kR/4 + kR3 /48) · I1 (kR), (kR2 · I0 (kR))/24 + (kR · I1 (kR))/24, (−1/kR − kR/4 + kR3 /192) · I1 (kR) + (1/4 + 3/kR2 + kR2 /192) · I2 (kR), (−1 − kR2 /8) · I2 (kR) + (3/kR + kR/8) · I3 (kR), (1/4 + kR2 /16) · K0(kR) + (−kR/4 − kR3/48) · K1 (kR), −(kR2 · K0 (kR))/24 + (kR · K1(kR))/24, (1/kR + kR/4 − kR3/192) · K1 (kR) + (1/4 + 3/kR2 + kR2 /192) · K2(kR), (1 + kR2/8) · K2(kR) + (3/kR + kR/8) · K3 (kR), 0, 0, 2, 0, 0, −6/kR4, 0};

Porous Absorbers

G

367

c15 = {(kR · I0 (kR))/8 + (kR2 · I1 (kR))/16, (kR/16 + kR3/64) · I0 (kR) + (kR2 · I1 (kR))/32, (−1/4 − kR2 /32) · I1 (kR) + I2 (kR)/(2kR), (−3/(2kR) − (3kR)/8) · I2 (kR) + (3/4 + 6/kR2) · I3 (kR), (kR · K0(kR))/8 − (kR2 · K1(kR))/16, (−kR/16 − kR3 /64) · K0 (kR) + (kR2 · K1 (kR))/32, (1/4 + kR2/32) · K1 (kR) + K2 (kR)/(2kR), (3/(2kR) + (3kR)/8) · K2 (kR) + (3/4 + 6/kR2) · K3(kR), 0, 0, 0, 6kR, 0, 0, −12/kR5}. Identities: ka11 = (ka)11 ;

kRn = (kR)n .

to the The diagram below shows the field of the magnitude |V/U| of the total velocity V incoming velocity U for ka = 0.001; R/a = 10. |V(r/a, ϕ)/U| ; ka= 0.001 ; R/a= 10

U

y/a x/a

Field of total velocity magnitude around a fibre in a regular fibre array with transversal incoming flow U On each fibre acts a pressure force Fxp and a viscous force Fxv in the x direction: 2 2 œ ∂V Fxp = − a p dœ ; Fxv = − a† · · cos ˜ dœ. ∂r 0

0

r=a

(7)

G

368

Porous Absorbers

Their values are:

Fxp C0 = · + 2 ka C2 , 1 2 ka U · 2a 2

Fxv

1 2 U · 2a 2

= − ·

(8)

! 1 ka2 ka4 I0 (ka) ka + + 16 192 6144

1 ka6 ka ka3 5 ka5 1 ka2 ka4 + + + I1 (ka) · A0 + − + + + I0 (ka) 4 32 768 36864 4ka 32 256 36864

1 3 ka2 ka2 5 ka4 ka4 ka6 + + (ka) · A + − + I + + − I1 (ka) 1 1 2 ka2 128 9216 8 48 3072 92160

5 ka 13 ka3 11 ka5 ka3 ka5 3 I2 (ka) · A2 + − I2 (ka) − + + + + 4 ka 1536 23040 32 1536 61440

1 11 ka4 ka2 ka4 5 I3 (ka) · A3 + ka K0 (ka) − + + + 16 30720 16 192 6144

1 ka2 ka4 1 ka6 ka ka3 5 ka5 − + + + K1 (ka) · B0 + − − − K0 (ka) 4 32 768 36864 4ka 32 256 36864

3 ka2 ka2 5 ka4 ka4 ka6 1 + (ka) · B + + K + + − K1 (ka) + 1 1 2 ka2 128 9216 8 48 3072 92160

5 ka 13 ka3 11 ka5 ka3 ka5 3 K2 (ka) · B2 + K2 (ka) − + + + + 4 ka 1536 23040 32 1536 61440 "

1 11 ka4 5 K3 (ka) · B3 − 3 · D1 . − + 16 30720 ka

+

(9)

For small ka 1 a power series development of the modified Bessel functions in Fxv gives the approximation: ! Fxv = − · [0.1875 ka + 0.0520833 ka2 ] · A0 1 2 U · 2a 2 +[2.63111 · 10−8 ka + 0.0130208 ka3 ] · A1 +[−0.093750 ka − 0.0260416 ka3] · A2 +[1.31556 · 10−7 ka − 0.013021 ka3] · A3 +[−0.25/ka−0.076978 ka+0.0116033 ka3 −(0.1875 ka +0.0520833 ka3) ln ka] · B0 3

(10) 3

+[0.5/ka − 0.125/ka + 0.0312499 ka + 0.00198205 ka +(2.63111 · 10−8 ka + 0.0130208 ka3 ) ln ka] · B1 +[1.5/ka3 + 4.500 · 10−8 /ka + 0.0502076 ka − 0.00766781 ka3 +(0.09375008 ka + 0.0260416 ka3 ) ln ka] · B2 +[2.5/ka3 + 7.500 · 10−8 /ka − 0.02500025 ka − 0.00235060 ka3 " 1 −7 3 +(1.315556 · 10 ka − 0.0130210 ka ) ln ka ] · B3 − 3 · D1 . ka

Porous Absorbers

G

369

The test sample of a model material with regular fibre arrangement is supposed to have the dimensions Dx in the flow direction and Dy normal to the flow and to the fibres. The flow resistivity ¡⊥ is given by the sum of the viscous forces on the fibres in the sample: 2† · ¡⊥ =

I

Fxv,i (kai , kRi ) ai 0 U2 i=1 Dx Dy (1 − ‹) kai

(11)

with † the dynamic viscosity, ‹ = 1 − the massivity of the material, and Dx Dy = I · Ri2 . The terms under the sum in the numerator are nearly independent of kai , so they are a function c(Ri /ai ) of the remaining parameter Ri /ai . This function can be evaluated by (a regression through the values from the cell model):

R R = ef (x) ; x = ln , c a a 0.0250214 0.322560 1.78839 (12) f (x) = 0.865823 + − + x3 x2 x − 0.530524 x + 0.0604543 x2 − 0.00312698 x3 . ka=0.0001

100 c(R/a)

10

1

0.1 1

10

100

R/a

1000

Function c(R/a) of the sum terms in the numerator of ¡⊥. Points: from the model theory; curve: regression. The curves coincide if ka is varied by powers of ten If the fibre and cell radii are randomised with relative frequencies qm of the fibre radius ai in the counting group with mean value am , and correspondingly pn the relative frequency of Ri in the counting group around Rn , the flow resistivity ¡⊥ is:

2‹ < a2 > Rn = · . (13) qm pn · c ¡⊥ † (1 − ‹) m,n am

G

370

Porous Absorbers

If the relative frequencies are Poisson distributions: Šm n ; pn = e− · m! n! then the needed radius ratios are: Š2 + 2Š + 1/4 n + 1/2 1 Rn . = √ am m + 1/2 ‹ 2 + 2 + 1/4 qm = e−Š ·

;

m, n = 0, 1, 2, . . . ,

(14)

(15)

1 Ξ⊥〈a2〉/η

0.1

λ=4 ; Λ=12 λ=4 ; Λ=2

λ=4 ; Λ=4

0.01

experimental

0.001 0.001

0.01

0.1

μ

1

Flow resistivity ¡⊥ of a bundle of parallel fibres for transversal flow. Computed curves with different parameters of the Poisson distributions, and shaded range of experimental data

G.8 Transversal Sound in Parallel Fibres The model of the porous material is illustrated in > Sect. G.5,but the sound propagation is transversal to the fibres, which are supposed to be at rest. An elementary cell and the used co-ordinates are shown here. The direction of the incident wave is vi . The sound field in a cell is evaluated with density and viscous and thermal waves taken into account.

Porous Absorbers

G

371

The cell radius R is determined so that,with the given fibre radius a,the desired massivity ‹ is obtained. vi z

r ϕ x

2aø 2Rø

In a first model (closed-cell model) the cell surfaces are adiabatic surfaces of symmetry for the field scattered at a fibre (the scattered fields do not penetrate the cell surface, but the surface is transparent for the incident wave); in a second model the scattered field freely propagates through the fibre bundle (open-cell model); this model is ultimately used to evaluate multiple scattering between fibres (multiple scattering model). The advantage of the multiple scattering model is that it permits random fibre distances. The characteristic propagation constant i and wave impedance Zi of the density wave in the fibre bundle are determined from the effective density eff and effective compressibility Ceff by: Zi i Ceff eff Ceff eff =j · ; = , (1) k0 0 C0 Z0 0 C0 which in turn are evaluated by the integrals:

# ¢ ¥i + ¥ + ¥ · cos œds ¢ eff , = 2 R 0 # ∂¦z ∂¦z cos œ + r sin œ dr [¥i + ¥ + ¥ ] · cos œds − dœ ∂œ ∂r 0

(2)

a

2 R Ceff = C0

k2

A0

vir (R, œ)dœ 0

. ¢ (¥e + ¥ ) + ¥ dA ¢

(3)

372

G

Porous Absorbers

Here ¥i is the potential function of the incident density wave; ¥ , ¥ are the potentials of the scattered density and thermal waves; ¦z is the z component of the vector potential of the scattered viscous wave; k , k , kŒ are the free field wave numbers of the density, thermal and # viscous waves, respectively; for the factors ¢ , ¢ see > Sects. J.1–J.3. The integrals . . . ds are taken over the surface of the cell; the integral over A0 is over the cell area. The field components of the scattered field are formulated as Fourier series over œ; the index = , stands for the density and thermal wave types: Closed-cell model: Fourier series formulations of the component fields: ¥ (r, œ) = =

n=0

n=0

[A n H(2) n (k r) + B n Jn (k r)] · cos(nœ) ¥ rn(r) · cos(nœ)

;

= , ,

(4)

¦ = {0, 0, ¦z } , ¦z (r, œ) = =

n=0

n=0

[AŒn H(2) n (kŒ r) + BŒn Jn (kŒ r)] · sin(nœ)

(5)

¦Œrn (r) · sin(nœ).

The boundary conditions at the cell and fibre surfaces are (for each Fourier series term):

(R) + k R ¥rn (R) − n ¦zrn(R) = 0 k R ¥rn

−n [¥rn (R) + ¥rn (R)] + n [k R ¥rn (R) + kR ¥rn (R)] − (kŒ R)2 ¦zrn (R) = 0

(6)

Ÿ

(R) + k R ¥rn (R) = 0, k R ¥rn Ÿ

n [¥rn (a) + ¥rn (a)] − kŒ a ¦zrn (a) = −n ¥ern (a)

k R ¥rn (a) + k R ¥rn (a) − n ¦zrn(a) = −k a ¥ern (a)

(7)

Ÿ ¥rn (a) = −¥ern (a), ¥rn (a) + Ÿ (for the factors Ÿ , Ÿ see > Sects. J.1–J.3; the prime indicates the derivative with respect to the argument of the radial functions in the field terms). The inhomogeneous linear systems of equations can be solved for the amplitudes of the Fourier terms (see Mechel (1995), Ch. 12) for more details). The integrals for eff and Ceff need only the terms n = 0 for the effective compressibility and n = 1 for the effective density (other

Porous Absorbers

G

373

integrals vanish); so the final equations are (Ai is the arbitrary amplitude of the incident density wave) as follows: a a (2) J1 (k a)] + A1 [H(2) H (k a)] + . . . 1 (k R) − R R 1 a a (2) (−2jAi + B1 )[J1 (k R) − J1 (k a)] + A1 [H(2) H1 (k a)] + . . . 1 (k R) − R R % a (2) a ¢ $ (2) ... + A1[H1 (k R) − H1 (k a)] + B1 [J1 (k R) − J1 (ka)] ¢ R R a a H(2) (k a)] + B1 [J1 (kR) − J1 (k a)] − . . . . . . + A1 [H(2) 1 (k R) − R 1. . . R , a a (2) −AŒ1 [H(2) (k R) − (k a)] − B [J (k R) − (k a)] H J Œ Œ Œ1 1 Œ 1 Œ 1 R 1 R (−2jAi + B1 )[J1 (k R) −

eff = 0

(8)

k R J1 (k R) Ai

Ceff = C0

k R

kR

(Ai + B0 )[yJ1(y)]k a + A0 [yH(2) 1 (y)]k a + . . . ... 2 %. ¢ k $ (2) k R k R ... + A0[yH1 (y)]k a + B0 [yJ1(y)]k a ¢ k2

(9)

with [f (y)]ba = f (b) − f (a). The free field wave numbers k , k, kŒ can be reduced to k0 = –/c0 (c0 = adiabatic sound velocity) and kŒ2 = −j –/Œ (Œ = kinematic viscosity; see section “Sound in capillaries”), and kŒ in turn can be related to the flow resistivity ¡ of a transversal fibre bundle with the absorber variable E = 0 f /¡ by the relation

4‹ R |kŒ a| = ·c ·E 1−‹ a 2

(10)

with the massivity ‹ and the function c(R/a) from

> Sect. G.7.

The real part of the characteristic impedance from the transversal fibre closed-cell model typically shows a resonance because of the assumedly identical interactions of the fibres with the cell surfaces in all cells (the closed cell model cannot handle the time lag of scattering during the propagation through the bundle). Open-cell model: The potentials of the incident density wave ¥i and of the scattered waves ¥ ; = , Œ (scattered density and temperature wave) as well as the scattered viscosity wave ¦ are formulated as: ¥i (r, œ) = Ai · e−jk x = Ai =

n≥0

n≥0

ƒn (−j)n · Jn (k r) · cos(nœ)

¥irn (r) · cos(nœ)

;

ƒn =

1; 2;

n=0 , n>0

(11)

G

374

Porous Absorbers a=2 μm ; μ=0.02

10

Re{Γ/k0} 1 Im{Γ/k0}-1

-Im{Za/Z0} 0.1

Re{Za/Z0} - 1

0.01 0.001

0.01

0.1

E

1

Characteristic values of a transversal fibre bundle evaluated with the closed-cell model of regular arrangement ¥ (r, œ) = =

n≥0

n≥0

A n H(2) n (k r) · cos(nœ) ¥ rn (r) · cos(nœ)

;

= , ,

¦ = {0, 0, ¦z }, AŒn H(2) ¦z (r, œ) = n (kŒ r) · sin(nœ) n≥0 = ¦Œrn (r) · sin(nœ).

(12)

(13)

n≥0

The scattered field amplitudes follow from the system of linear equations (n = 0, 1) which represent the boundary conditions: (2) (2) A n n H(2) n (k a) − AŒn [kŒ a Hn−1 (kŒ a) − n Hn (kŒ a)] =,

= −n ƒn (−j)n Jn (k a) Ai =,

(2)

(2) A n [k a Hn−1 (k a) − n H(2) n (k a)] − AŒn n Hn (kŒ a)

= − ƒn (−j)n [k a Jn−1 (k a) − n Jn (k a)] Ai An H(2) n (k a) +

Ÿ n An H(2) n (k a) = − ƒn (−j) Jn (k a) Ai , Ÿ

(14)

Porous Absorbers

G

375

and with them the effective density eff and effective compressibility Ceff are evaluated from: a a (2) −2jAi [J1 (k R) − J1 (k a)] + A1 [H(2) H1 (k a)] + . . . 1 (k R) − eff R R = a a 0 −2jAi [J1 (k R) − J1 (k a)] + A1 [H(2) H(2) (k a)] + . . . 1 (k R) − R R 1 (15) a (2) ¢ H1 (ka)] A1 [H(2) ... + 1 (k R) − ¢ R , a (2) a (2) (2) H1 (kŒ a)] . . . + A1[H1 (k R) − H1 (k a)] − AŒ1 [H(2) 1 (kŒ R) − R R Ceff = C0

(2) Ai k R J1 (k R) + A0 k R H(2) 1 (k R) + A0 kR H1 (kR) 2 ¢ k k R k R k R Ai [yJ1 (y)]k a + A0 [yH(2) (y)] + A0 [yH(2) 1 1 (y)]k a k a ¢ k2

(16)

with [f (y)]ba = f (b) − f (a). The characteristic values and Zi of the propagation constant and wave impedance of the sound wave propagating through the open-cell model fibre bundle approach realistic values only at relatively high values of |kŒ a|2 = –a2 /Œ; at low values the presence of the fibres is under-weighted. This is the consequence of the assumedly missing scatter interaction between neighbour fibres at low frequencies, where the viscous boundary layer at the fibres fills the whole interspace between the fibres in reality. Nevertheless, the open-cell model is needed for the evaluation of the scattered field in the multiple scattering model. a=2 μm ; μ=0.02 1 Im{Γ/k0}-1 Re{Γ/k0} 0.1 -Im{Zi/Z0} Re{Zi/Z0} – 1 0.01 0.001

0.01

0.1

|kν a|2

1

Characteristic values in a transversal fibre bundle evaluated with the open-cell model, which contains only single scattering at a fibre

376

G

Porous Absorbers

Multiple scattering model: This model considers the scattering at a reference fibre (in position Pi ) at which not only the incident wave pi is scattered but also the scattered fields coming from neighbour fibres (in positions Pj ). ϕj i

Pj

ϕj

pi

x

rj

χ ri j

P ϕi j

ri

α

ϕi

pi

x

Pi

A minimum distance d between the reference fibre and the neighbour fibres is assumed. The fibres outside the range d may be arranged randomly with a porosity or massivity ‹ or fibre number density N of the transversal fibre bundle: = 1 − ‹ = 1 − N · a2 .

(17)

The“radiating”scattered fields, i.e. the scattered fields which assumedly propagate freely through the bundle, are indicated with a prime (with r, œ the co-ordinates centred in the scattering fibre): ∞ A n · H(2) = , ¥ (r, œ) = n (k r) · cos nœ ; n=0

¦z (r, œ) =

∞ n=0

(18) AŒn ·

H(2) n (kŒ r)

· sin nœ

;

AŒ0 = 0.

The total scattered field in a point of immission P is indicated with a double prime; it is obtained by integration over the contributions of all neighbour fibres to the reference fibre. If the point of immission P is in position Pi of the reference fibre, the total scattered field is: 2 ∞

¥ (ri , œi ) = dœij N(rij )· ¥ (rj , œj ) rij drij a

0

=N

¦z

(ri , œi ) =

n≥0

A n · 0

2

∞ dœij

n≥0

(19) H(2) n (k rj ) rij cos nœj drij dœij ,

d

N(rij )· ¦z (rj, œj ) rij drij

a

0

=N

2 ∞

2 ∞

AŒn · 0

d

(20) H(2) n (kŒ rj ) rij · sin nœj drij dœij .

Porous Absorbers

G

377

(the first expressions also permit a variable fibre density function N(rij )). After application of the addition theorem for Hankel functions, to transform all co-ordinates to those of the reference fibre, one gets for the total scattered field around the reference fibre: ¥ (ri , œi ) = A n · [H(2) n (k ri ) · cos(nœi ) n≥0

+ (−1)n N

2

∞ m=−∞

∞ dœij

0

rij · H(2) n+m (k rij )

d

· Jm (k ri ) · cos ((n + m)œij − m œi )drij ],

¦z (ri , œi ) =

n≥0

AŒn ·

[H(2) n (kŒ ri )

+ (−1)n N

· sin(nœi )

2

∞ m=−∞

(21)

∞ dœij

0

(2)

rij · Hn+m (kŒ rij ) · Jm (kŒ ri ) d

· sin ((n + m)œij − m œi )drij ] with the geometrical relations œj i = œij + & rj = ri2 + rij2 − 2 ri rij cos(œij − œi ) œj = œj i + arctg

(22)

ri sin(œij − œi ) . rij − ri cos(œij − œi )

In the above expressions for ¥ , ¦z all integrals over œij disappear except those with m + n = 0; then the integral returns the value 2. Thus the total scattered field at the reference fibre becomes: ¥ (ri , œi ) = A n · cos nœi · [H(2) n (k ri ) n≥0

− ¦z (ri , œi ) =

2N k d H(2) 1 (k d) · Jn (k ri )], k 2

n≥0

−

(23)

AŒn · sin nœi · [H(2) n (kŒ ri )

2N kŒ d H(2) 1 (kŒ d) · Jn (kŒ ri )]. kŒ2

(24)

With these expressions plus the incident wave the boundary conditions at the reference fibre are satisfied. Using the abbreviations: (2)

Sn (k y) = H(2) n (k y) − 2 = , , Œ A n A n = , Ai

;

H1 (k R) · Jn (k y), k R

y = a, R,

(25)

378

G

Porous Absorbers

the needed scattered field amplitudes (normalised with the amplitude Ai of the incident density wave) are: Ÿ k a J0 (k a) S1 (ka) Ÿ A 0 = − , Ÿ −k a S1 (k a) S0 (k a) + k a S0 (k a) S1 (ka) Ÿ Ÿ −k a J1 (k a) S0 (k a) + k a J0 (k a) S1 (k a) , A 0 = − Ÿ Ÿ −k a S1 (k a) S0 (ka) − ka S0 (k a) S1 (ka) Ÿ k a J (k a) S (k a) − k a Œ 1 0 Œ kŒ a J0 (k a) S0 (kŒ a) + k a J0 (k a) S1 (kŒ a) , A 1 = −2j k a kŒ a S0 (ka) S0 (kŒ a) − k a S0 (ka) S1 (kŒ a) − kŒ a S1 (k a) S0 (kŒ a) −k a J1 (k a) S0 (k a) +

A Œ1 = −2j

(26)

k a J1 (k a) S0 (k a) − k a J0 (k a) S1 (k a) . k a kŒ a S0 (ka) S0 (kŒ a) − k a S0 (ka) S1 (kŒ a) − kŒ a S1 (k a) S0 (kŒ a)

With these the effective density eff and compressibility Ceff are evaluated from: ⎡ ⎤−1 $ % a

S · S (k R) − (k a) A 1 Œ 1 Œ Œ1 eff ⎢ ⎥ R $ $ % %⎦ , = ⎣1 − a a

0 −2j J1 (k R) − J1 (k a) + A1 · S1 (k R) − S1 (k a) R R k a J1 (k a) + A 0 · k a S1 (k a) + . . . Ceff = 1− C0 k R J1 (k R) + A 0 · S1 (k R) + A 0 · k R S1 (k R) ' ( 2 ¢ k

. . . + A0 · (k a S1 (ka) − k R S1 (kR)) + k R S1 (kR) −1 ¢ k2 . ...

(27)

(28)

Because mostly k a 1, the following approximations are possible: J1 (k a) ≈1 k a · J0 (k a)

;

J1 (k R) 1 ≈√ J1 (k a) ‹

S1 (k R) 2−‹ ≈ (k a)2 1 3/2 S1 (k a) 4 ‹ (1 − ‹)

;

S1 (k a) 1−‹ ≈− k a · S0 (ka) 2‹

Ÿ · k2 ≈ −(‰ − 1) Ÿ · k2

;

¢ · k2 ¢ · k2

(29)

1.

If one introduces the abbreviation: 2‹ 1−‹ √ 2 ‹ =− 1−‹

K1,0(k a, ‹) = −

S1 (k a) 1 S1 (k R) · 1− √ k a S0 (k a) ‹ S1 (k a) √ ‹ S1 (k a) − S1 (k R) · , k a · S0 (k a) ·

(30)

Porous Absorbers

then the effective material data finally become with good precision: Ceff eff 1 ; ≈ ≈ 1 + (‰ − 1) · K1,0 (k a, ‹) 0 1 − K1,0(kŒ a, ‹) C0

G

379

(31)

(with ‰ = adiabatic exponent), and from these the characteristic values as usual are: eff Ceff eff Zi Ceff i =j · ; = . (32) k0 0 C0 Z0 0 C0 The following diagram shows the components of the characteristic data over |kŒ a|2 = –a2 /Œ evaluated with these expressions. Numerical tests show that the ratio d/R of the minimum distance d of neighbour fibres to the reference fibre and the radius R of a cell around that fibre is important; the (reasonable) value d/R = 1 is recommended. a=2 [μm] , μ=0.02 , d/R=1 10

Re{Γ/ k0} Im{Γ/ k0} – 1

1

0.1

Re{Zi/Z0} – 1 – Im{Zi/Z0}

0.01 0.001

0.01

0.1

| kν a|2

1

Components of the characteristic values of a transversal fibre bundle, evaluated with multiple scattering The connection of the independent variable |kŒ a|2 with the flow resistivity variable E = 0 f /¡ is required. This is given by ( > Sect. G.7): 4‹ |kŒ a|2 = · c(x) · E, 1−‹

R R = ef (x) ; x = ln = −0.5 ln(‹), c a a (33) 0.0250214 0.322560 1.78839 − 0.530524 x f (x) = 0.865823 + − + x3 x2 x + 0.0604543x2 − 0.00312698x3.

380

G

Porous Absorbers

The multiple scattering model of a transversal fibre bundle only contains the function K1,0(k a, ‹) in the characteristic values. This function can be approximately represented by: Re{K1,0 (x, y)} = 0, 51 − 0, 49 · tanh (2, 9 · dr (x, y)), Im{K1,0 (x, y)} =

h(x, y) 3/2

cosh

(2, 5 · di (x, y))

, (34)

dr (x, y) = −0.5903666 − 0.8220386 · x + 0.5694317 · y, di (x, y) = −0.494366 − 0.802776 · x + 0.59628 · y, h(x, y) = −0.4959784 − 0.1499322/x + 0.003230626 · x. With these relations and

(kŒ a)2 = −j |kŒ a|2

;

(ka)2 ≈ ‰ Pr ·(kŒ a)2

(35)

(Pr = Prandtl number), the characteristic values of a transversal fibre bundle (with multiple scattering) can easily be evaluated. See the comparison below between exact evaluation and approximation.

a=2 [μm] , μ=0.02 , d/R=1 10 Re{Γ/ k0} Im{Γ/ k0} – 1

1

0.1

Re{Zi/Z0} – 1 – Im{Zi/Z0}

0.01 0.001

0.01

0.1

| kν a|2

1

Comparison of the components of the characteristic values in a transversal fibre bundle from exact evaluation (thick lines) and approximation (thin lines)

Porous Absorbers

G.9

G

381

Effective Wave Multiple Scattering in Transversal Fibre Bundle

As in > Sect. G.8 (with which this section has a number of similarities), the porous material consists of parallel fibres (at rest) with fibre radius a and fibre number density N. A sound wave propagates inside the fibre bundle normal to the fibres. Whereas it was assumed in > Sect. G.8 that the propagating wave is a density wave, here it is assumed to be a less specified“effective”wave with potential ¥E having propagation constant = jke. The propagating wave (exciting wave) may be some hybrid wave of a density wave, a thermal wave or a viscous wave. The propagation constant and wave impedance Ze of this wave shall be determined. See > Sect. G.8 for the assumed co-ordinates. As before, the index = , indicates a density or thermal wave with scalar potential functions ¥ , and ¦z is the z component of the vector potential of a viscous wave. It is further assumed as before that the nearest neighbours of a reference fibre in the position (ri , œi ) are at a distance d from it. Exciting wave formulation: ¥E (r, œ) = AE · e− x = AE · e−jke r

= AE ·

n≥0

cos œ

= AE ·

n≥0

(−j)n ƒn Jn (ke r) cos(nœ)

¥Ern (r) · cos(nœ)

;

ƒn =

(1)

1; n=0 2 ; n > 0.

Scattered field formulation (marked with a prime): ¥ (r, œ) = ¦z (r, œ) =

n≥0

n≥0

A n H(2) n (k r) cos(nœ) = AŒn H(2) n (kŒ r) sin(nœ) =

n≥0

n≥0

¥ n (r) · cos(nœ)

¦Œn (r) · sin(nœ)

; ;

= , , AŒ0 = 0.

(2)

Total scattered field at a reference fibre in (ri , œi ) (marked with a double prime): ¥

(ri , œi )

=N

¦z

(ri , œi ) = N

n≥0

n≥0

2 A n

∞ dœij

0

d

2 AŒn

∞ dœij

0

e−jke rij cos œij · H(2) n (k rj ) cos(nœj ) · rij drij , (3) e−jke rij cos œij · H(2) n (kŒ rj ) sin(nœj ) · rij drij ,

d

and after application of the addition theorem for Hankel functions and integration: ¥ (ri , œi ) =

n≥0

cos(nœi ) · {A n H(2) n (k ri )

2N n (−j) J (k r ) · jm B mn · A m } n i k 2 − ke2 m≥0 ¥ rn (ri ) · cos(nœi ), = −

n≥0

(4)

382

G

¦z (ri , œi ) =

n≥0

Porous Absorbers

sin(nœi ) · {AŒn H(2) n (kŒ ri )

2N (−j)n Jn (kŒ ri ) · jm BŒmn · AŒm } 2 2 k Œ − ke m≥0 = ¦zrn (ri ) · sin(nœi ) +

(5)

n≥0

with the following abbreviations for both = , : (2) B mn = ke d · [H(2) m+n (k d) Jm+n−1 (ke d) + Hm−n (k d) Jm−n−1 (ke d)] (2) − k d · [H(2) m+n−1 (k d) Jm+n (ke d) + Hm−n−1 (k d) Jm−n (ke d)],

BŒmn = ke d ·

[H(2) m+n (kŒ d) Jm+n−1 (ke d)

−

(6)

H(2) m−n (kŒ d) Jm−n−1 (ke d)]

(2) − kŒ d · [H(2) m+n−1 (kŒ d) Jm+n (ke d) − Hm−n−1 (kŒ d) Jm−n (ke d)].

These scattered waves plus the exciting wave have to satisfy the boundary conditions at the reference fibre (which is supposed to be isothermal): ⎧ 2 N ⎪ ⎪ n · [An H(2) (−j)n Jn (k a) · jm Bmn · Am ] ⎪ n (k a) − 2 2 ⎪ k − k ⎪ e ⎪ m≥0 ⎪ ⎪ ⎪ 2 N ⎪ n (2) ⎪ ⎨ + n · [An Hn (k a) − 2 (−j) Jn (ka) · jm Bmn · Am ] 2 k − k e vœ (a, œi ) = 0: (7) m≥0 ⎪ ⎪ 2 N ⎪ ⎪ (−j)n J n (kŒ a) · jm BŒmn · AŒm] − kŒ a · [AŒn H (2) ⎪ n (kŒ a) + 2 ⎪ ⎪ kŒ − ke2 ⎪ m≥0 ⎪ ⎪ ⎪ ⎩ = −n ƒn (−j)n Jn (ke a) · AE , ⎧ 2 N ⎪ ⎪ k a · [An H (2) (−j)n J n (k a) · jm Bmn · Am ] ⎪ n (k a) − 2 2 ⎪ k − k ⎪ e ⎪ m≥0 ⎪ ⎪ ⎪ 2 N ⎪ ⎪ ⎨ +k a·[An H (2) (−j)n J n (k a)· jm Bmn ·Am ] n (k a)− 2 k − ke2 vr (a, œi ) = 0: m≥0 ⎪ ⎪ 2 N ⎪ n (2) ⎪ H (k a) + (−j) J (k a) · jm BŒmn · AŒm ] −n · [A ⎪ Œn Œ n Œ n ⎪ ⎪ kŒ2 − ke2 ⎪ m≥0 ⎪ ⎪ ⎪ ⎩ n

= −ƒn (−j) kea Jn (ke a) · AE ,

(8)

⎧ 2 N ⎪ ⎪ [An H(2) (k a) − 2 (−j)n Jn (ke a) · jm Bmn · Am ] ⎪ n 2 ⎪ k − k ⎪ e ⎪ m≥0 ⎨ T(a, œi ) 2 N Ÿ n =0 : ·[An H(2) (k a) − 2 (−j) J (k a)· jm Bmn ·Am ] + n n ⎪ T0 ⎪ Ÿ k − ke2 ⎪ m≥0 ⎪ ⎪ ⎪ ⎩ = −ƒn (−j)n Jn (ke a) · AE .

(9)

Porous Absorbers

With a solution for the amplitudes A n , AŒn evaluate R ¢ ¥r1 (r) r · ¥Er1 (r) + ¥r1 (r) + ¢ eff a = , 0 [r · (¥Er1 (r) + ¥r1 (r) + ¥r1 (r) − ¦zr1 (r) )]Ra Ceff C0

=

[¥ˆ Er0 (r) + ¥ˆ r0 (r) + ¥ˆ r0 (r)]Ra 2 ¢ k2 eff k ˆ Er0 (r) + ¥ˆ r0 (r) + ¥ ¥ˆ r0 (r) 0 ke2 ¢ k2

with [f (y)]ba = f (b) − f (a) and ¥ˆ (r) =

r

G

383

(10)

R a

k 2 r · ¥ (r) dr.

a=2 [μm] , μ=0.02 , d/R=1 , mmax=4 10

Re{Γ/ k0} Im{Γ/ k0} – 1 1

0.1

Re{Zi/Z0} – 1 – Im{Zi/Z0} 0.01 0.001

0.01

0.1

| kν a|2

1

Comparison of the components of characteristic values in a transversal fibre bundle, evaluated iteratively for an effective propagating wave (thick lines, from this section), and with a propagating density wave (thin lines, from > Sect. G.8) The evaluation must proceed iteratively because the equations from the boundary conditions contain (besides the unknown amplitudes) the unknown wave number ke . Begin the iteration with an approximation = jke from > Sect. G.8; solve for a first approximation of A n ,AŒn ; insert into the expressions for eff , Ceff ; evaluate the next approximation for the propagating wave from: Ze eff Ceff eff Ceff =j · ; = , (11) k0 0 C0 Z0 0 C0

G

384

Porous Absorbers

and resume the cycle of iteration. Apply in the evaluations, with the massivity ‹: a √ = ‹ R

2N 2‹ = . 2 2 (k a) − (ke a)2 − ke

;

(12)

k 2

The diagram above compares the components of the characteristic values from the present iterative evaluation (with an “effective” propagating wave) with values from > Sect. G.8 (with the density wave as the propagating wave). Up to mmax = 4 orders of scattering were used. Notice that the theoretical curves of this section reproduce the humps of the experimental points in the figure from > Sect. G.4. The present method, although much more complicated numerically than the method of > Sect. G.8, is well suited to evaluate particle velocity profiles around a fibre in a transversal fibre bundle. |kνa|2=1.024 , μ=0.02 , mmax=8 vE

|v (r/a, ϕ)/vE|

1.5

1

0.5

R/a 5

0 -R/a -5

0 0 x/a

y/a

-5 5

R/a -R/a

Profile of the magnitude of the total particle velocity around a fibre in a fibre bundle with transversal sound propagation

Porous Absorbers

G

385

| kνa|2=1.024 , μ=0.02 , mmax=4

vE

0.4

|vy (r/a, ϕ)/vE|

0.2 0 R/a

-0.2

5

-0.4 -R/a

0 -5

y/a

0 x/a

-5 5

R/a -R/a

Profile of the real component of the transversal particle velocity around a fibre in a fibre bundle with transversal sound propagation

G.10 Biot’s Theory of Porous Absorbers

See also: Tolstoy (1992), a survey in Mechel (1995)

Whereas in other sections of this chapter the matrix of the porous material is assumed to be rigid, it may be elastic in Biot’s theory. The consequence of this elasticity is the onset of additional wave types by the coupling between the matrix and the enclosed fluid in the pores. Biot’s theory is not a“terminated”theory; he took some fundamental relations between flow and sound from the theory of circular capillaries, which was available at his time, instead of results from possibly better suited models which are now available. The price for the wider range of application of Biot’s theory is a number of specially defined material parameters, for which Biot has given prescriptions on how to measure them. Fundamental assumptions: • The matrix is homogeneous and isotropic in scales which are larger than the scale of the pores. • The pores are interconnected. • The size of the pores is small compared to considered volume elements dV and small compared to the wavelength.

386

G

Porous Absorbers

Fundamental equations: Let u s be the movement (elongation) of a solid, averaged over a volume element dV; let u f be the average movement of the fluid in dV. Equations of motion: ss

ff

∂ 2 u s ∂ 2 u f + = P · grad div u s + Q · grad div uf sf ∂t2 ∂t2

∂ u f ∂ us − N · rot rot u s + bF(–) · − , ∂t ∂t ∂ 2 u f ∂ 2 u s + sf = R · grad div u f 2 ∂t ∂t2

+ Q · grad div u s − bF(–) · Strain-stress equations: ‘sij

= [(P − 2N) · div u s + Q · div u f ] ƒij + N ·

∂ u f ∂ us − . ∂t ∂t

∂usi ∂usj + ∂xj ∂xi

(1)

(2)

,

(3)

‘fij = − p ƒij = [R · div u f + Q · div u s ] ƒij , where usi are the components of u s in the direction of the co-ordinate xi ; ‘s is the tension on the matrix absorber in a unit area; ‘f is the tension on the fluid in a unit area; i, j = 1, 2, 3 denote co-ordinates; ‘s33 is the x3 component of the tension on the matrix acting on a surface normal to x3 ; and ‘s13 denotes a shear tension in the x1 direction on a surface normal to x3 . ƒij is the Kronecker symbol. mn are effective mass densities if m = n; they are coupling coefficients between a solid and a fluid if m = n. A, P, Q, R, N are elastic constants, introduced by Biot. A corresponds to the first Lamé constant of a material matrix, N to its second Lamé constant. In most cases P = A + 2N holds. Q evidently is a coupling coefficient between matrix and fluid; it determines also the coefficient R by the relation R = −Q · e/—, where e = div u s and — = div u f are the strains of the solid and the fluid, respectively. There are three coupling coefficients between solid and fluid in the equations: sf , bF(–), and Q. With = volume porosity of the porous material, the effective densities are: ss = (1 − ) s − sf

;

ff = 0 − sf ,

(4)

where s is the density of the (compact) solid and 0 = f is the density of the fluid. The coupling density sf describes the extra inertia of a relative motion between solid and fluid: sf = − (” − 1) 0 .

(5)

The term ” represents the tortuosity of the pores in the material (it corresponds to the structure factor in older theories; see below for its determination); it is a pure form factor.

Porous Absorbers

G

387

The term bF(–) is a coupling factor of the viscous forces; it is associated with the relative velocity ∂ u f /∂t−∂ us /∂t of fluid and solid.It is mainly determined by the flow resistivity ¡ of the material; it is frequency dependent, whereby the transition from an approximately parabolic flow profile at low frequencies into an approximately rectangular profile at high frequencies can be described. Biot has taken the quantity bF(–) from the observation that the effective densities eff in flat and in circular capillaries have about the same frequency dependence, up to a “stretching” of the frequency axis with a factor c [see the sections on sound in capillaries in > Ch. J, “Duct Acoustics”; see there also for the definition of J1,0 (z)].√Thus he took from circular capillaries (a = pore radius in the capillary model, kŒ = −j (–/Œ), Œ = kinematic viscosity): 2¡ 4

c kŒ a · J1,0 (c kŒ a) . (6) 2 1− J1,0 (c kŒ a) c kŒ a The equivalent radius a of the capillary for a given porous material can be taken from: 8† ” ¡= 2 († = dynamic viscosity), (7) a where ¡ the measured flow resistivity, the porosity and ” the tortuosity. The matching factor c in the argument of J1,0 (z) changes between c = 1 for cylindrical pores and c = (4/3)1/2 for flat pores; values for triangular and square pores have been evaluated. b F(–) = −

A different method (from Johnson et al.) to determine bF(–) is: 1/2

4j ”2†0 – 8†” 1/2 2 ; =c . (8) b F(–) = ¡ 1 + ¡ 2 2 2 ¡ The coupling factor Q is called the potential coupling factor. It takes into account the fact that the pressure in the pores may change, even if div u f = 0 in the pores, by a dilatation of the matrix. If only shear stresses act on the absorber, no dilatation takes place; the strain-stress equations reduce to:

∂usi ∂usj + (9) ‘sij = N ; ‘fij = 0, ∂xj ∂xi thus N is the shear modulus of the matrix. Suppose a sample of the porous material is coated with a thin, limp foil, allowing a connection between the inside and outside space, and exerts a static pressure Ps on the coated sample. Whereas the matrix may be deformed, the pressure in the pores remains constant.

388

G

Porous Absorbers

The strain-stress equations simplify to:

4 −Ps = P − N · div u s + Q · div u f ; 0 = R · div u f + Q · div u s . 3

(10)

After elimination of u f : Q2 4 , Kb = P − N − 3 R

(11)

where Kb = −Ps /div u s is the bulk compression modulus of the matrix.

(12)

Suppose further that a material sample is subjected (in a tank) to a hydrostatic pressure pf . The forces acting on the matrix and the pores are (1 − ) · pf and · pf , respectively. The strain-stress equations become:

4 −(1 − ) pf = P − N · div u s + Q · div u f ; − pf = R · div u f + Q · div us . (13) 3 With Ks the compression modulus of the compact matrix material and Kf the compression modulus of the fluid in the pores, one gets: P=

(1 − )(1 − − Kb /Ks ) Ks + (Ks /Kf ) Kb 4 + N, 1 − − Kb /Ks + Ks /Kf 3

Q=

(1 − − Kb /Ks) Ks , 1 − − Kb /Ks + Ks /Kf

R=

2 Ks . 1 − − Kb /Ks + Ks /Kf

(14)

In many materials Ks Kb and Ks Kf ; then the equations simplify to: P = Kb +

4 (1 − )2 N+ Kf 3

;

Q = (1 − ) Kf

;

R = Kf .

(15)

The compression modulus Kf of the fluid in the pores changes from isothermal compression at low frequencies (if the heat capacity and heat conduction of the matrix material are much larger than those of the fluid) to adiabatic compression at high frequencies. This transition is taken from the model of circular capillary pores (with the same fitting parameter c from above for other pore shapes): Kf = 0 c20 / [1 + (‰ − 1) J1,0(c k0a) ],

(16)

where a is the equivalent pore radius, ‰ is the adiabatic exponent, k0 = ‰Pr · kŒ , Pr is a Prandtl number, kŒ is the viscous wave number, and J1,0 (z) (see > Ch. J, “Duct Acoustics”). Wave equations: The solution of the fundamental equations in principle consists of a triple of two longi f in the tudinal compressional waves and a transversal shear wave. The strain fields u s , u solid and the fluid are described by scalar and vector potentials: u s = grad ¥ + rot H

;

u f = grad ¦ + rot G.

(17)

G

Porous Absorbers

Insertion into the fundamental equations gives: ∂ 2¥ ∂2¦ ss 2 + sf = P · div grad ¥ + Q · div grad ¦ + bF(–) · ∂t ∂t2 ff

∂2¦ ∂ 2¥ + = R · div grad ¦ + Q · div grad ¥ − bF(–) · sf ∂t2 ∂t2

and ∂ 2H ∂ 2G + bF(–) · ss 2 + sf 2 = N · grad div H ∂t ∂t ( ' ∂H ∂G ∂2G ∂ 2H − . = −bF(–) · ff 2 + sf ∂t ∂t2 ∂t ∂t

'

∂• ∂¥ − ∂t ∂t ∂¥ ∂¦ − ∂t ∂t

389

,

( ∂H ∂G − , ∂t ∂t

(18)

(19)

With a time factor ej –t and the abbreviations: bF(–) bF(–) bF(–) bF(–) ; ˜ ff = ff − j ; ˜ sf = sf + j ; ”˜ = ” − j ˜ ss = ss − j , (20) – – – –0 one gets: −–2 (˜ss ¥ + ˜ sf ¦ ) = P · ¥ + Q · ¦ ,

(21)

2

−– (˜ff ¦ + ˜ sf ¥ ) = R · ¦ + Q · ¥ . Elimination of ¦ yields: ¦ =

(PR − Q2 ) · ¥ + –2 (˜ss R − ˜ sf Q) · ¥ , –2 (˜ff Q − ˜ sf R) 2

2

2

(22) 4

(PR − Q ) · ¥ + – (˜ss R + ˜ ff P − 2˜sf Q) · ¥ + – (˜ss ˜ ff −

˜ sf2 )

· ¥ = 0.

The last equation is formally interpreted as a product of two wave equations ( − k12 ) · ( − k22 ) ¥ = 0 with a solution ¥ = ¥1 + ¥2 of which the sum terms obey: (23) ( − k12 ) ¥1 = 0 2 = k1,2

;

( − k22 ) ¥2 = 0,

√ –2 [(˜ss R + ˜ ff P − 2˜sf Q) ± D ], 2 2 (PR − Q )

(24)

D = (˜ss R + ˜ ff P − 2˜sf Q)2 − 4 (PR − Q2 )(˜ss ˜ ff − ˜ sf2 ). There are two compressional waves ¥1,2 in the solid with characteristic wave numbers k1,2. The scalar potential ¦ for the sound wave in the fluid can be written as: ¦

= ‹ 1 ¥1 + ‹ 2 ¥2 ,

2 ˜ ss R − ˜ sf Q − (PR − Q2) k1,2 /–2 . ‹1,2 = ˜ ff Q − ˜ sf R

(25)

390

G

Porous Absorbers

H can be derived similarly. One finds: The vector potentials G,

˜ –2 ˜ ss ˜ ff − ˜ sf2 = − sf · H = ‹3 · H ; H − = − k32 H = 0, G ·H ˜ ff ˜ ff N k32

–2 ˜ ss ˜ ff − ˜ sf2 = N ˜ ff

˜ sf ‹3 = − . ˜ ff

;

(26)

Special case: weak coupling, densities of solid and fluid very different In many porous absorber materials the porosity is ≈ 1, the tortuosity is 1 ≤ ” ≤ 2, the density of the solid material is ˜ ss ˜ ff , and also ˜ ss ˜ sf . Then Biot’s parameters simplify to: 4 P ≈ Kb + N; 3

Q ≈ 0;

R ≈ Kf ,

(27)

the wave numbers of the compressional waves approximately become:

˜ sf2 ˜ ss ˜ ss ˜ ff − ˜ sf2 ˜ ff 2 2 2 2 ; k2 ≈ – , k1 ≈ – − − P R˜ss R R˜ss and the amplitude ratio of the shear wave to the compressional waves is:

P ˜ ff ˜ sf ˜ss P ˜ff ˜ sf ‹1 ≈ − − + − ; ‹2 ≈ − . R ˜ sf ˜ ss ˜ sf R ˜ sf ˜ ss

(28)

(29)

The condition for this weak coupling is satisfied at frequencies above f = 2¡/2 (1 − ) s . The effective densities are, under the mentioned conditions: ˜ ss ≈ (1 − ) s − j

bF(–) –

;

˜ ff ≈ 0 − j

bF(–) –

;

˜ sf ≈ j

bF(–) . –

One gets for not too high flow resistivities ¡:

˜ sf2 ˜ sf2 ˜ ss ˜ ff + ; k22 ≈ –2 − . k12 ≈ –2 P R ˜ ss R − ˜ ff P ˜ ss R − ˜ ff P

(30)

(31)

The amplitude ratios are then: ‹1 ≈

˜ ss − P k12/–2 P ˜ sf , ≈ −˜sf ˜ ss R − ˜ ff P

˜ ss − P k22/–2 (˜ss R − ˜ ff P)2 + PR ˜sf ≈ . ‹2 ≈ −˜sf (˜ss R − ˜ ff P) R ˜ sf

(32)

The wave number of the shear wave approximates the value for that wave in the evacuated matrix: k32 ≈

–2 (1 − ) s N

;

‹3 ≈ 0.

(33)

Porous Absorbers

G

391

Input impedance of a porous layer with rigid backing (as an example) A plane wave is incident under a polar angle Ÿ on a plane layer of a porous material with thickness d. The graph below shows schematically the three excited waves in the layer.

The three waves are formulated as three displacement potential functions; œ3 is the not of the shear wave: vanishing component of the vector potential H In forward direction: œn = An · e−jkn (x1 wn1 +x3 wn3 ) ;

n = 1, 2, 3.

(34)

Waves reflected at the hard wall: •n = Bn · e−jkn (x1 wn1 −x3 wn3 ) ;

n = 1, 2, 3.

(35)

From Snell’s law, where kn are the Biot wave numbers: ) k0 sin Ÿ = wn1 kn ; wn3 = 1 − wn1 .

(36)

The layer is in close contact with the hard wall at x3 = d; thus us1 (x1 , d) = us3 (x1 , d) = uf 3 (x1 , d) = 0.

(37)

Special case of an open front side: i.e. no cover sheet on the absorber layer Let p be the sound pressure and v3 the particle velocity in the x3 direction in front of the absorber; therefore (with the common factor in x1 dropped and an unit amplitude of the incident wave supposed): v3 =

cos Ÿ (1 − R); Z0

p = 1 + R,

(38)

where R is the reflection factor of the layer. Boundary conditions on the front side are: v3 = j– [ (1 − ) us3 + uf 3 ]

;

−(1 − ) p = ‘s33

;

−p = ‘f33 .

(39)

G

392

Porous Absorbers

The field formulations, when inserted into the boundary conditions at the back and front sides, give the following system of six equations: us1 = − j k1 w11 (A1e−jk1 w13 d + B1 e+jk1 w13 d ) − j k2 w21 (A2e−jk2 w23 d + B2 e+jk2 w23 d ) + j k3 w33 (A3e−jk3 w33 d − B3 e+jk3 w33 d ) = 0, us3 = − j k1 w13 (A1e−jk1 w13 d − B1 e+jk1 w13 d ) − j k2 w23 (A2e−jk2 w23 d − B2 e+jk2 w23 d ) − j k3 w31 (A3e−jk3 w33 d + B3 e+jk3 w33 d ) = 0, uf 3 = ‹1 [−j k1w13 (A1 e−jk1 w13 d − B1 e+jk1 w13 d )]

‘s33

+ ‹2 [−j k2 w23 (A2 e−jk2 w23 d − B2 e+jk2 w23 d )] + ‹3 [−j k3 w31 (A3 e−jk3 w33 d + B3 e+jk3 w33 d )] = 0, ∂us3 = (P − 2N) div u s + Q div u f + 2N ∂x3 = (P − 2N + ‹1 Q)(−k12 (A1 + B1 ))

(40)

+ (P − 2N + ‹2Q)(−k22 (A2 + B2 )) 2 2 + 2N [−k12 w13 (A1 + B1 ) − k22 w23 (A2 + B2 ) − k32 w31 w33 (A3 − B3 ) ] = − (1 − )p, ‘f33 = Q div u s + R div u f 2 = (Q + ‹1R)(−k12 w13 (A1 + B1 )) 2 2 + (Q + ‹2R)(−k2 w23 (A2 + B2 )) ‘s13

= −p,

∂us1 ∂us3 =N + ∂x3 ∂x1 = N[−k12 w11 w13 (A1 − B1 ) − k22 w21 w23(A2 − B2 ) − k32(A3 + B3 )] = 0.

Together with the relation for the reflection factor these equations comprise a system of seven equations for A1 , B1 , A2 , B2 , A3 , B3 and R. After a numerical solution, the input impedance Z of the absorber layer can be evaluated from: 1 1+R Z = . Z0 cos Ÿ 1 − R

(41)

Special case of an adhesive foil or membrane on the front side Let be the surface mass density of the foil, S its bending stiffness, T its tension in the case of a membrane. Let P1 be immediately in front of the absorber, P2 a point on the surface of the absorber, and P3 a point immediately behind the front side of the absorber. The equations of motion of the foil are: −–2 u3(P2 ) = ‘s33 (P3 ) + ‘f33 (P3 ) − ‘f33 (P1 ) + T 2

−–

u1(P2 ) = ‘s13 (P3 )

∂ 2 u1(P2 ) +S . ∂x32

∂ 2 u3 (P2 ) , ∂x32

(42)

Porous Absorbers

G

393

With ‘f33(P1 ) = −p these equations can be transformed into: −–2 2 u1(P2 ) = ‘s13 (P3 )

(43)

−–2 1 u3(P2 ) = ‘s33 (P3 ) + ‘f33 (P3 ) − ‘f33 (P1 ), with 1 = − T k02 sin2 Ÿ/–2

2 = − S k02 sin2 Ÿ/–2 .

;

(44)

The boundary conditions at the front side now become: us3 (P3 ) = uf 3(P3 ) = u3(P2 ) = u3(P1 )

;

us1 (P3 ) = u1 (P2 ).

(45)

Together with the boundary conditions at the back side one has seven equations; ((ui )n is the component in the direction xn of ui ): (j–1 − Z) · j–u3 (P2 ) − ‘s33 (P3 ) − ‘f33 (P3 ) = 0, −–2 2 · (u1)1 (P3 ) − ‘s13 (P3 ) = 0, (u1 )3 (P2 ) = (u2)3 (P2 ), (46) (u1 )1 = 0; (u2 )3 = 0; (u1 )3 = 0 at x3 = d, −‘f33 (P1 ) . Z= j– u3 (P1 ) The numerical solution of the system of equations, after insertion of the field formulations, gives the amplitudes of the wave components and the input impedance Z. 10

10 Θ=0°

Θ=60°

Re{Z / Z0 }

Re{Z / Z 0 }

0

0

Im {Z / Z 0 } Im {Z / Z 0 }

-10

0

1

2

3

-10 4 5 0 f [kHz]

1

2

3

4 5 f [kHz]

Measured (points) and evaluated (curves) components of the input impedance Z of an open layer of PU foam, d = 2 [cm] thick 1 |R| 0.8 0.6 0.4 open surface covered surface

0.2 0 0

1

2

3

4

f [kHz]

5

Measured (points) and evaluated (curves) magnitude of the reflection factor R of a d = 2 [cm] thick PU foam, once with open surface, once with a cover foil

394

G

Porous Absorbers

G.11 Empirical Relations for Characteristic Values of Fibre Absorbers

See also: Mechel (1995), Mechel/Grundmann (1982)

For a great number of glass fibre and mineral fibre absorber material from different producers and with a wide range of bulk densities the characteristic values, i.e. the propagation constant a and wave impedance Za ,as well as the flow resistivity ¡ were carefully measured in Mechel/Grundmann (1982). The materials could be subdivided (from an acoustical point of view) into three product groups: glass fibre products, mineral fibre products (rockwool) and basalt wool products. This section shows experimental values for the components of the normalised charac

teristic values an = a /k0 = an + j · an and Zan = Zan /Z0 = Z an + j · Z

an over the “absorber variable” E = 0 f /¡ (0 = density of air; f = frequency; ¡ = flow resistivity).

It is advantageous (according to a proposal by Delany and Bazley) to plot an , an − 1,

Zan − 1, −Zan as functions of E, because the experimental data then group around simple curves, and to derive empirical relations for these quantities by regressions through the data. The following table gives for the three product groups average values of (1) the fibre diameter d,(2) the distribution parameter of a Poisson distribution of the diameters,to which the empirical distribution can be best matched, (3) the shot content (per weight, for shot with diameters > 100[‹m] ), and (4) the content of an organic binder (per weight). Table 1 Average technical data of fibre absorber materials Product group

Average fibre diameter d‹m

Distribution Shot content parameter % (d = 1‹m) ( > 100‹m)

Binder content %

Glass fibre

5.2

5.3

Sect. G.2) The characteristic values are evaluated from: eff Ceff Za 1 eff Ceff =j · ; Zan = = , an = k0 0 C0 Z0 0 C0 · g(E) eff =” − j 0 2E

;

Ceff ‰ + 1 · j E/E0 = , C0 1 + j E/E0

g(E) = ‚0 + ‚1 E + ‚2 E2 with fitted parameters for three product groups; = 1 − ‹ = porosity:

(1)

400

G

Porous Absorbers

Table 1 Coefficients for the relations (1) Parameters

Glass fibre

Basalt wool

Rockwool

”

1.3

1.3

1.3

‹

0.02

0.02

0.02

‰

1.40 + 0.15 j 1.40 + 0.10 j

1.40 + 0.15 j

1

1

1.1

1.1

E0

0.125

0.10 (0.07 for Zan )

0.125

‚0

1.09872

0.976206

1.12140

‚1

0.333239

2.18474

1.49953

‚2

1.62642

−1:26275

0.468552

The next diagram shows, as an example, the normalised propagation constant in glass fibre materials; the points are floating averages over experimental data, the thick solid lines come from the fitted theory, while the thin dashed curves belong to the original theory. Glass fiber

10

Γ ′an 1 Γ ″an-1

0.1 0.001

0.01

0.1

E

1

Components of the normalised propagation constant in glass fibre materials; points: floating average of experimental data,thick lines:fitted theory,thin lines:original theory

Porous Absorbers

G

401

Fitted model of flat capillaries ( > Sect. G.4) The characteristic values are evaluated from: Za 1 eff Ceff eff Ceff = an = j · ; = Zan = / , k0 0 C0 Z0 0 C0 ) tan −6j ‰ Pr a2 E 1 eff Ceff ) = = 1 + (‰ − 1) ) ; , 0 C0 tan −6j a1 E −6j ‰ Pr a2 E 1− ) −6j a1 E

(2)

where Pr is a Prandtl number and = 1 − ‹ is the porosity and fitted parameters for three product groups: Table 2 Coefficients for relations (2) Material

Char. Value

‹

‰

a1

a2

Glass fibre

an

–

1.60 + 0.1 j

1

1

Zan

0.05 1.40 + 0.1 j

1

1.5

an

–

Zan

0.05 1.40 + 0.1 j

0.9 1.7

an

–

1

Zan

0.05 1.40 + 0.1 j

Basalt wool Rockwool

1.40 + 0.15 j 1 1.70 + 0.1 j

1 0.6

0.9 1.7

The diagrams show the characteristic values for rockwool fibre materials (points: from floating averages over experimental data; curves: from fitted capillary model). Rockwool 10

Γ ′an 1 Γ ″an-1

0.1 0.001

0.01

0.1

E

1

402

G

Porous Absorbers

Rockwool

10

− Z ″an 1 Z ′an-1 0.1 0.001

0.01

0.1

E

1

References Mechel, F.P., Grundmann, R.: Akustische Kennwerte von Faserabsorbern,Vol. I, Bericht BS 85/83 (1983); Materialdaten, Vol. II, Bericht BS 75/82 (1982), Berichte des Fraunhofer-Instituts f¨ ur Bauphysik, Stuttgart Mechel,F.P.: Schallabsorber,Vol.II,Hirzel,Stuttgart (1995)

Tolstoy, I. (ed.): Acoustics, Elasticity, and Thermodynamics of Porous Media; Twenty-one Papers by M.A.Biot,Acoustical Society of America,American Institute of Physics, Melville, NY (1992)

H Compound Absorbers Sound absorbers, except simple porous layers, are compound absorbers, i.e. they consist of elements in special arrangements, such as air volumes, foils (either limp or elastic, tight or porous), membranes, plates (either stiff or elastic, tight or porous) mostly with perforations (“necks”) in the shape of e.g. slits or circular holes, porous absorber layers, etc. The aim mostly is to evaluate the input admittance G of such absorbers, or impedance Z = 1/G, because it is this quantity with which absorbers enter into acoustical computations. Many compound absorbers, in turn, are arrays of elementary absorbers, such as arrays of Helmholtz resonators, and they have an inhomogeneous surface, e.g. the neck areas of Helmholtz resonators and the hard plate between the necks. One must distinguish what the input admittance stands for, either the neck area or the whole array. If the lateral dimensions of an array element are small compared to the wavelength Š0 (typically < Š0 /4), the performance of an absorber in most applications can be equivalently described by an average admittance (or“homogenised”admittance),which is the average of the local admittances in an array. Otherwise the array must be treated as a periodic structure with the admittance profile along the surface explicitly taken into account. A further distinction concerns the radiation impedance Zr = Zr +j·Zr of the absorber, or more distinctly the radiation resistance Zr , whether or not it is included in the absorber impedance Z. This distinction comes from the general equivalent network of a source and an absorber. The network consists of a pressure source, with Pi the sound pressure of the incident wave and the internal source impedance Zr , and the absorber with input impedance Ze . For a plane wave with polar angle Ÿ of incidence the radiation resistance is Zr = Z0 / cos Ÿ. It does not contain information about the absorber. The radiation reactance, however, contains the oscillating mass of necks and therefore influences the tuning of resonators. Thus it makes sense to attribute Zr to the source and Zr to the absorber. It should be noted that this attribution is a matter of convention, and therefore one must examine absorber formulas for the convention being used. It will be indicated, at the end of some sections in this chapter about absorber elements, how the absorber element is introduced into the equivalent chain network of a multilayer absorber. Most technical sound absorbers can be described with such an equivalent network ( > Sect. C.5). Some sections below will mainly give chains of equations which lead to the finally desired input admittance G or input impedance Z by iterated insertion.

H

404

Compound Absorbers

v

2Pi

Ze

Z″r

Z′r

v

Zr

2Pi – Z ′ r·v

2Pi

Ze

Possible attributions of the radiation impedance Zr = Zr + jZr to the absorber input impedance Ze .

H.1

Absorber of Flat Capillaries


See

> Sect. J.1 for sound

in flat capillaries.

A plane sound wave is incident on a layer of thickness d (with hard backing) consisting of thin plates with mutual distance 2h. The arrangement has a surface porosity . The plates first are assumed to be normal to the back wall. Viscous and thermal losses are considered in the capillaries between the plates.

k0

ϑ

2h

ϕ

y x d

z With reflection factor r on the front side and transmission factor t from outside to inside of the capillaries:

Compound Absorbers

H

405

Sound wave in front of the absorber: p = pe + pr = A · e−j(kx x+ky y) (e−jkz z + r e+jkz z ) kx = k0 sin ˜ cos œ

;

ky = k0 sin ˜ sin œ

(1) ;

kz = k0 cos ˜ .

Sound wave inside the capillaries: pa = A · t · e−ax x · cosh(az (z − d)) · e−jky y ( − a2)pa = 0

ax2 + az2 = a2

;

ax = jkx = jk0 sin˜ cosœ vaz =

−1 ∂pa a Za ∂z

;

vax = 0

;

az = k0 ;

a k0

2

(2) − sin2 ˜ cos2œ

vay = 0,

where a is the propagation constant in a flat capillary, Za = Zi /, and Zi is the wave impedance in a flat capillary ( > Sect. J.1). Input impedance: Za a /k0 Z = coth(k0d · az /k0) Z0 Z0 az /k0

⎧ 1 Za /Z0 ⎪ ⎪ ⎪ ⎪ ⎨ k0d a /k0

; ˜ = 0 or œ = ±/2 Za /Z0 · a /k0 1 = −−−−−→ Za /Z0 · a /k0 1 k0 d1 k0 d (a /k0 )2 − sin2 ˜ cos2 œ ⎪ ⎪ ; œ=0 ⎪ ⎪ ⎩ k0d (a /k0)2 − sin2 ˜

−−−−−→ k0 d1

Za /Z0 · a /k0

(a /k0)2 − sin2 ˜ cos2 œ

=

⎧ Za /Z0 ⎪ ⎪ ⎨

;

Za /Z0 · a /k0 ⎪ ⎪ ⎩ (a /k0)2 − sin2 ˜

Reflection factor r and absorption coefficient as usual: Z −1 Z0 r= Z cos˜ +1 Z0 cos˜

;

(˜ , œ) = 1 − |r|2

˜ = 0 or œ = ±/2 ;

œ=0

.

(3)

H

406

Compound Absorbers σ=0

.95

; d/λ 0=0.1

; R= Ξ

d/Z 0=0.2

α(ϑ,ϕ) 0.25

0.25 0 0.25

0

0 0.25 0.25

Absorption coefficient (˜ , œ) as function of direction of sound incidence. The lamellae distance 2h is given by the normalised flow resistance R of the arrangement Next, the lamellae are assumed to be inclined with ˜0 . k0 ϑ

2h

ϕ

y x d

z

ϑ0

The effective depth changes to deff = d/ cos ˜0 ; the outside wave impedance changes to Z0 = Z0 / cos ˜0 .

Compound Absorbers

H

407

Thus Z(˜0 = 0, d) 1 Z(˜0 = 0, deff ) → Z0 cos ˜0 Z0

(4)

and Z(0, deff ) Z/Z0 − 1 Z0 → r= Z(0, deff ) Z/Z0 + 1 Z0

cos ˜ −1 cos ˜0 . cos ˜ +1 cos ˜0

(5)

The inclination of the lamellae (indicated in the next diagram) has no immediate influence on the sound absorption (˜ , œ) as a function of angles of incidence. σ=0.

95 ; d/λ 0=0.0

5 ; R=0 .2

α(ϑ,ϕ)

; ϑ 0=60°

0.3

0.3 0 0

0.3 0

0.3 0.3

Absorption coefficient (˜ , œ) as function of direction of sound incidence,with inclined lamellae. The lamellae distance 2h is given by the normalised flow resistance R of the arrangement

H.2

Plate with Narrow Slits


“Narrow” slits means only plane waves are considered in the neck channels (in contrast to“wide”slits in the next section,where higher modes are assumed in the necks),but they are still wide enough so that viscous and thermal losses in the necks may be neglected. Consider an array of parallel slits with width a and mutual distance L in a (rigid) plate with thickness d. Excitation is by a plane wave with normal incidence and amplitude Ae .

408

H

Compound Absorbers

There are three sound zones I, II, III. Field formulation in zone II: pII (x, y) = Be−jk0 x + Cejk0 x 1 −jk0 x − Cejk0 x Be Z0 vIIy (x, y) = 0

(1)

vIIx (x, y) =

Field in zone I in front of the plate: pI (x, y) = Ae e−jk0 x + A0ejk0 x + 2

n>0

An e‚n x cos(†n y)

1 ‚n ‚n x −jk0 x jk0 x − A0 e + 2j An e cos(†n y) vIx (x, y) = Ae e Z0 k0 n>0 vIy (x, y) = −

(2)

†n ‚n x 2j An e sin(†ny) Z0 n>0 k0

Field in zone III behind the plate: pIII (x, y) = D0 e−jk0 x + 2

n>0

Dn e−‚n x cos(†ny)

1 ‚n −‚n x −jk0 x vIIIx (x, y) = D0 e − 2j Dn e cos(†n y) Z0 k0 n>0 vIIIy (x, y) = −

2j †n −‚n x Dn e sin(†ny) Z0 n>0 k0

(3)

Compound Absorbers

H

409

Wave numbers and propagation constants: †0 = 0 ; ‚0 = jk0 ;

2n Š0 = k0 · n ; n = 1, 2, . . . L L Š0 2 2 2 n ‚n = †n − k0 = k0 − 1 ; n = 1, 2, . . . L

†n =

(4)

From particle velocity boundary conditions: A0 = Ae − An = −j Dn = j

a · (B − C) L

a k0 sin · (B − C) L ‚n

(5)

a k0 sin e‚n d · (B e−jk0 d − C e+jk0 d ) L ‚n

with sin =

sin(na/L) ; na/L

;

n = 0, 1, 2, . . .

si0 = 1.

From matching average sound pressures in the slit orifices: B (1 + S) e+jk0 d = Ae 2S cos(k0d) + j(1 + S2 ) sin(k0d)

(6)

−(1 − S) e−jk0 d C = Ae 2S cos(k0d) + j(1 + S2 ) sin(k0d) cos(k0d) + jS sin(k0d) B−C =2 Ae 2S cos(k0 d) + j(1 + S2 ) sin(k0 d) 2 B e−jk0 d − C e+jk0 d = Ae 2S cos(k0d) + j(1 + S2 ) sin(k0d)

k0 a 2 s . 1 + 2j with the abbreviation S = L ‚i i

(7)

(8)

i>0

Front side orifice impedance Zsf : B+C S + j tan(k0d) < pII (0, y) > Zsf = = ; = Z0 Z0 < vIIx (0, y) > B − C 1 + j S · tan(k0 d)

(9)

the last expression has the typical form of a (normalised) impedance (here S ) which is transformed by a transmission line of length d. Back side orifice impedance Zsb : Be−jk0 d + Ce+jk0 d < pII (d, y) > Zsb = −jk d = = S, Z0 Z0 < vIIx (d, y) > Be 0 − Ce+jk0 d

(10)

H

410

Compound Absorbers

⎤

⎡

⎢ ⎥ sin(n a/L) 2 ⎥ Zsb 1 a ⎢ ⎥. ⎢1 + 2j = ⎥ Z0 L ⎢ n a/L ⎦ ⎣ Š0 2 n>0 −1 n L

(11)

The first term in the brackets represents the radiation resistance. If one subtracts in the front side orifice the sound pressure of the equivalent source 2Pi = 2Ae , then: < pII (0, y) − 2Ae > B + C − 2Ae = = −S, Z0 < vIIx (0, y) > B−C

(12)

i.e. the orifice impedances on both sides are symmetrical. End correction of the orifice: ∞ ∞ Zsb 1 S2n a sin2 (na/L) 1 = S" ≈ = · = . a k0a Z0 k0 a n=1 n L n=1 (na/L)3

If the summation is approximated by an integration a 1 a 1 1 ≈ ln tan + cot a 2 4 L 2 4 L

(13)

(14)

and from a numerical regression x

= lg(a/L)

= −0.395 450 · x + 0.346 161 · x2 + 0.141 928 · x3 + 0.0200 128 · x4 . a

(15)

2 1.5

Δ = 1 π a

1 0.5 0

∑ n≥1

si2n n

Δ 1 1 π a π a a = π ln 2 [tan( 4 L ) + cot( 4 L ) ]

-3

-2.5

-2

-1.5 -1 x=lg(a/L)

-0.5

0

End correction of a slit in an array; points: summation; solid line: regression; dashed line: integration

Compound Absorbers

H.3

H

411

Plate with Wide Slits


See

> Sect. H.2 for the arrangement, co-ordinates and field zones.

In contrast to > Sect. H.2, higher modes are assumed in the neck channels. The field formulations in the three zones are as follows. Zone I: pI (x, y) = Ae e−jk0 x +

n≥0

ƒn An e‚n x cos(†ny) ,

Z0 vIx (x, y) = Ae e−jk0 x + j Z0 vIy (x, y) = −2j Zone II: pII (x, y) =

n≥0

n>0

An

n≥0

ƒ n An

‚n ‚n x e cos(†n y) , k0

(1)

†n ‚n x e sin(†n y) . k0

[Bn e−j‰n x + Cn ej‰n x ] cos(—n y) ,

‰n [Bn e−j‰n x − Cn ej‰n x ] cos(—n y) , n≥0 k0 —n [Bn e−j‰n x + Cn ej‰n x ] sin(—n y) . Z0 vIIy (x, y) = −j k n≥0 0

Z0 vIIx (x, y) =

Zone III: pIII (x, y) =

n≥0

(2)

ƒn Dn e−‚n x cos(†n y) ,

‚n −‚n x e cos(†n y) , k0 n≥0 †n −‚n x Dn e sin(†n y) Z0 vIIIy (x, y) = −2j k0 n>0 1; n=0 1; m=n with ƒn = ; ƒmn = 2; n>0 0; m= n Z0 vIIIx (x, y) = −j

ƒn Dn

(3)

(4)

and lateral wave numbers †0 = 0

;

†n =

2n Š0 = k0 · n L L

;

—0 = 0

;

—n =

2n Š0 = k0 · n a a

as well as axial propagation constants Š0 2 2 2 n − 1 ; Re{‚n } ≥ 0 or Im{‚n } ≥ 0 , ‚0 = jk0 ; ‚n = †n − k0 = k0 L Š0 2 ; Im{‰n } ≤ 0 or Re{‰n } ≥ 0 . ‰0 = k0 ; ‰n = k02 − —2n = k0 1 − n a

(5)

(6)

H

412

Compound Absorbers

Mode coupling coefficients in the orifice planes are as follows: 1 sm,n = a 1 = 2

+a/2

cos(†m y) cos(—n y) dy

;

m = 0, 1, 2, . . .

−a/2

sin((†m − —n ) a/2) sin((†m + —n ) a/2) + (†m − —n ) a/2 (†m + —n ) a/2

m a/L sin (m a/L) (m a/L)2 − (n)2 a a n m · sin m a (−1) L L n = 0, ; m = = a 2 L − n2 m L and the special cases are: = (−1)n

sm,n =

a 1 sin(m a/L) for m = n = 0 ; s0,0 = 1 ; sm,0 = ; s0,n>0 = 0. 2 L m a/L

(7)

(8)

Boundary conditions for the particle velocities at the zone limits give (m = 0, 1, 2, . . .):

k0 a ‰n sm,n · (Bn − Cn ) , Am = −j −ƒ0,mAe + ‚m L n≥0 k0 (9) a k0 ‰n −j‰n d +j‰n d −‚m d Dm e =j sm,n · (Bn e − Cn e ). L ‚m n≥0 k0 The boundary conditions for the sound pressure yield (m = 0, 1, 2, . . .): 1 (Bm + Cm ) = ƒ0,mAe + ƒn sn,m · An , ƒm n≥0 1 (Bm e−j‰m d + Cm e+j‰m d ) = ƒn e−‚n d sn,m · Dn . ƒm n≥0

(10)

Instead of solving these systems for Am , Bm , Cm , Dm the auxiliary quantities Xn , Yn are introduced: Xn± : = Bn ± Cn

;

Yn± : = Bn e−j‰n d ± Cn e+j‰n d

with intrinsic relations: 1 + e−2j‰n d e−j‰n d − 2Y · , Xn+ = Xn− · n− 1 − e−2j‰n d 1 − e−2j‰n d e−j‰n d 1 + e−2j‰n d − Y · . Yn+ = 2Xn− · n− 1 − e−2j‰n d 1 − e−2j‰n d The Bn , Cn follow from: 1 1 Bn = (Xn+ + Xn− ) = (Yn+ + Yn− )e+j‰n d , 2 2 1 1 Cn = (Xn+ − Xn− ) = (Yn+ − Yn− )e−j‰n d . 2 2

(11)

(12)

(13)

Compound Absorbers

H

413

A coupled system of equations for Xn− , Yn− is obtained with the form n≥0

amn · Xn− + cm · Ym− = bm · Ae

cm · Xm− +

n≥0

amn · Yn− = 0

; m = 0, 1, 2, . . . , ; m = 0, 1, 2, . . .

(14)

and the coefficients am,n = j cm

a ‰n ƒm,n 1 + e−2j‰m d Sm,n + , L k0 ƒm 1 − e−2j‰m d

2 e−j‰m d =− ƒm 1 − e−2j‰m d

;

(15)

bm = ƒ0,m + s0,m = 2ƒ0,m

with the abbreviations Sm,n =

i≥0

ƒi

k0 k0 si,m · si,n = −j ƒ0mƒ0n + 2 · si,m · si,n ; ‚i i>0 ‚i

(16)

m, n = 0, 1, 2, . . . The normalised backside orifice impedance is: Zsb Y0+ < pII (d, y) > = = Z0 Z0 < vIIx (d, y) > Y0− ⎤ ⎡ k0 2 ‰n Yn− k0 a =j ⎣ ƒi s + · ƒi si0 sin ⎦ . L ‚i i0 n>0 k0 Y0− ‚i i≥0

(17)

i≥0

The first term in the brackets is just the orifice impedance of a neck with only plane waves in it; thus the second term is a correction term for the influence of higher modes in the neck. The normalised front side orifice impedance is: Zsf < pII (0, y) > X0+ = = Z0 Z0 < vIIx (0, y) > X0− ⎤ ⎡ 2Ae a k0 2 ‰n Xn− k0 = −j ⎣ ƒi s + · ƒi si0 sin ⎦ . B0 − C0 L ‚i i0 n>0 k0 X0− ‚i i≥0

(18)

i≥0

After subtraction of the sound pressure of the equivalent source (the first term in the last expression) the orifice impedances on both sides remain symmetrical.

H

414

Compound Absorbers

The slit impedances above were defined with the average sound pressure and axial particle velocity. The slit radiation impedances, which are defined with the radiated power ( > Sect. F.1) are: ∗ 1 ‰n∗ pII (d, y) · Z0 vIIx (d, y) dy Yn+ Y∗n− Zrb n≥0 ƒn k0 a = , = Z0 1 ‰n 2 ∗ 2 Z20 vIIx (d, y) · vIIx (d, y) dy k |Yn− | 0 n≥0 ƒn a (19) ∗ ∗ 1 ‰ pII (0, y) · Z0 vIIx (0, y) dy n ∗ Xn+ Xn− ƒ k Zrf n 0 n≥0 a = = . Z0 1 ‰n 2 ∗ Z20 vIIx (0, y) · vIIx (0, y) dy |Xn− |2 ƒ k n≥0

a

Δ Im{Z

lg ( −

Im{Z

sb

0

L/λ0=0.1 ; nhi=6 ; ihi=24

}

}

n

)

sb0

0

-2

-4 -6

1

0 -1 lg(a/L)

0

lg(d/a)

-2 -3 -1

Relative change of the imaginary part of the neck orifice impedance Zsb due to higher modes in the neck, as compared to the impedance Zsb0 with only plane waves in the neck

Compound Absorbers

H

415

The relative change of the orifice reactance Zsb (and therefore also for the end correction) due to higher modes as compared with the orifice reactance Zsb0 with only plane waves in the neck ( > Sect. H.2) is: Z Zsb = Zsb0 · 1 + sb = Zsb0 · (1 − 10F(x,y)) , Zsb0 F(x, y) = lg(− x = lg(a/L)

(20)

Zsb ) = f (x) · (1 + g(y)) ; Zsb0

;

y = lg(d/a)

with functions f(x) and g(y) in the ranges −3 ≤ x < 0 and 1 ≤ y ≤ 1: f (x) = −1.739 68 + 1.484 35 (x + 1.5) − 1.842 30 (x + 1.5)2 +0.292 538 (x + 1.5)3 + 0.428 402 (x + 1.5)4 ,

g(y) = H(−y) · 0.00 259 355 y − 0.0758 181 y 2 +0.330 845 y 3 + 0.226 933 y 4

;

H(−y) =

(21) 1; 0;

y≤0 . y>0

The influence of higher modes is small if a/L < 0.25, and only if d/a 1 does the plate thickness becomes sensible.

H.4

Dissipationless Slit Resonator


Parallel slits in a neck plate and air volumes V behind them form an array of slit resonators. Excitation is by a plane wave with normal incidence and amplitude Ae . First, higher modes will be assumed in the necks, then the special case of only plane waves in the neck will be treated. The field formulations with higher modes remain as in in zone III: ƒn Dn cosh(‚n (x − d − t)) cos(†ny) , pIII (x, y) = n≥0

> Sect. H.3,except in the volumes

‚n sinh(‚n (x − d − t)) cos(†n y) , k0 n≥0 †n Dn cosh(‚n (x − d − t)) sin(†n y) Z0 vIIIy (x, y) = −2j k0 n>0

Z0 vIIIx (x, y) = j

ƒnDn

(with wave numbers and propagation constants from

> Sect. H.3).

(1)

416

H

Compound Absorbers

The boundary conditions give for the auxiliary quantities Xn± , Yn± : Xn± : = Bn ± Cn

;

Yn± : = Bn e−j‰n d ± Cn e+j‰n d ,

Bn =

1 1 (Xn+ + Xn− ) = (Yn+ + Yn− )e+j‰n d , 2 2

Cn =

1 1 (Xn+ − Xn− ) = (Yn+ − Yn− )e−j‰n d , 2 2

Xn+ = Xn− ·

(2)

1 + e−2j‰n d e−j‰n d − 2Y · , n− 1 − e−2j‰n d 1 − e−2j‰n d

Yn+ = 2Xn− ·

e−j‰n d 1 + e−2j‰n d − Y · . n− 1 − e−2j‰n d 1 − e−2j‰n d

a coupled system of linear equations: amn · Xn− + cm · Ym− = bm · Ae ; m = 0, 1, . . . , n≥0

cm · Xm− +

n≥0

dmn · Yn− = 0

;

m = 0, 1, . . .

(3)

with coefficients am,n = j

a ‰n ƒm,n 1 + e−2j‰m d Sm,n + , L k0 ƒm 1 − e−2j‰m d

dm,n = j

a ‰n ƒm,n 1 + e−2j‰m d Tm,n + , L k0 ƒm 1 − e−2j‰m d

cm = −

2 e−j‰m d ƒm 1 − e−2j‰m d

;

bm = ƒ0,m + s0,m = 2ƒ0,m ,

(4)

Compound Absorbers

wherein the Sm,n are defined as in Tm,n =

i≥0

ƒi

k0 sii,m · sii,n ‚i tanh(‚n t)

;

> Sect. H.3, as well

417

as the sm,n , and

m, n = 0, 1, 2, . . . .

The amplitudes Am , Dm are given by:

k0 a ‰n −ƒ0,m Ae + sm,n · Xn− Am = −j ‚m L n≥0 k0

(5)

(6)

‰n a k0 Dm = j sm,n · Yn− . L ‚m · sinh(‚m t) n≥0 k0 The back side orifice impedance Zsb is:

‰n Yn− a Zsb a/L =j T0,n = −j T0,0 + Z0 L k Y tan(k 0− 0 t) n>0 0 ⎡ ⎤ ‰n Yn− k0 si,0 si,n k0 s2i,0 a ⎦. ƒi + + j ⎣2 L ‚i tanh(‚i t) n>0 k0 Y0− ‚i tanh(‚i t) i>0

H

(7)

i≥0

The last term in the first line is the spring reactance of the resonator volume when it is driven by a piston of width a. Therefore the first term in the second line is the mass reactance of the back side orifice. With only plane waves in the neck (i.e. narrow necks and/or low frequencies) substitute B0 = B;

C0 = C;

Bn>0 = Cn>0 = 0

(8)

to get for the back side orifice impedance Zsb : s2i,0 a a/L a k0 Zsb = j T0,0 = −j +j2 . Z0 L tan(k0 t) L ‚i tanh(‚i t)

(9)

i>0

The front side orifice impedance Zsf has the form of the impedance of a free plate (see > Sect. H.2; see there for S): Zsf B+C 2 Ae Zsb /Z0 + j tan(k0 d) < pII (0, y) > = = −S= . = Z0 Z0 < vIIx (0, y) > B − C B − C 1 + j (Zsb /Z0) tan(k0 d)

(10)

The back side end correction b /a can be defined from the back side neck reactance Zsb by: Zsb b 1 a/L = . (11) +j a k0 a Z0 tan(k0t) The influence of the shape parameter t/L is of interest. The back side end correction sensibly differs from the end correction of a free plate (i.e. from the front side end correction) only for rather small values of t/L; then it is larger than the front side end correction.

H

418

Compound Absorbers

L/λ0=0.4 Δ b /a 2.5 2

1 1

0 -3

0.5 -2 lg (a / L )

0

lg (t/L)

-0.5

-1 0

-1

Influence of the shape parameter t/L on the end correction b /a of the back side neck orifice (towards the resonator volume) For slit resonators with higher modes in the neck the back orifice end correction is (in the range −1 ≤ y, z ≤ 0): b ≈ (x) · [1 + f (y ; z0 , x0 )] · [1 + g(z ; y0 , x0 )] ; a a d t a ; y = lg ; z = lg , x = lg L a L f (y ; z0 , x0) ≈ g(z ; y0 , x0) ≈ 0 x0 = −1

;

for

y ≥ y0

;

z ≥ z0 ;

y0 = z0 = 0, 3 ≈ lg(2),

f (y ; z0 , x0) = 0.001 448 29 · y + 0.002 555 10 · y 2 + 0.034 305 10 · y 3 + 0.015 682 99 · y 4 , g(z ; y0 , x0 ) = −0.000 932 290 · z − 0.007 672 04 · z2 − 0.019 259 72 · z3 − 0.018 048 39 · z4 , where /a is the end correction for a free slit in an array.

(12)

Compound Absorbers

H.5

H

419

Resonance Frequencies and Radiation Loss of Slit Resonators


The slit resonator is a special form of a Helmholtz resonator. Let V be the volume of the resonator, Q the cross-section area of the neck, then the resilience F, the oscillating mass M and the angular resonance frequency –0 of the resonator usually are given as (0 = density of air; c0 = sound velocity): F =

V Q20 c20

;

M = 0 Q (d + + b ) ,

1

–0 = √ = c0 F·M

Q Q ≈ c0 . V · (d + + b ) V · (d + 2)

(1)

This formula is known to return seriously false results for some parameter combinations. If the resonance condition is defined by zero reactance of the front side orifice impedance Zsf , then it is (for slit resonators with only plane waves in the neck): S + T + j (1 + ST) tan(k0d) ! Zsv + S = Im =0, Im Z0 1 + jT tan(k0 d)

k0 a sin(na/L) S= 1 + 2j ; s0 = 1 , ; sn = s2 (2) L ‚n n na/L n>0

a s2n a k0 =j T=j L n≥0 ‚n tanh(‚n t) L

k0 a/L s2n − +2 tan(k0t) ‚ tanh(‚n t) n>0 n

.

Because of the periodicity of tan(k0d) one must further demand that the zero value be crossed with positive slopes (in order to avoid anti-resonances), i.e. transition from spring to mass-type reactance with increasing frequency. The resonance condition then is: b a b a/L k0 a + − + 1−(k0 a)2 · + · tan(k0d) = 0. (3) k0 a a a tan(k0t) a a L a tan(k0t) For low frequencies with tan(k0 d) ≈ k0d, tan(k0 t) ≈ k0 t: a 2 t d a d a 4 d 2 a 2 − = 0, +2 + − 2 Š0 a a a L a a Š0 a a L

(4)

and neglecting further the term with (a/Š0 )4 , the lowest resonance is approximately: L ≈ Š0

t 2 L

1 d +2 a a

a 2 d + L a a

.

(5)

H

420

Compound Absorbers

A better approximation is obtained with a continued fraction expansion of tan z: L ≈ Š0

t 2 L

1 1 t d +2 + a a 3 L

,

(6)

or with volume V, volume cross-section area QV , neck cross-section area Q: Q –0 ≈ c0 ! . 1 V · (d + + b + QV/QV2 ) 3

(7)

This form of the resonance formula may be compared with the traditional formula (1). A resonance formula for the lowest resonance with a higher precision is: √ L/t v − v 2 − 4uw L ≈ Š0 2 2u /a 1 t/L (a/L)2 t/L 2 (a/L) 1+ ; v = 1+ + +3 ; w= . u= t/L a t/L 3 d/a a d/a t/L d/a

(8)

2

A slit resonator in an array has a radiation loss corresponding to its back radiation (reflection). Its radiation loss factor † is given by: †=

R R = R · –0 F = " , –0 M M F

(9)

or with approximations for the circuit elements at resonance: a2 Lt l 1 t d R = 0 c0 ; F = 2 2 ; M = 0 a2 +2 + , L a 0 c0 a a 3 L R =! †= √ M/F

H.6

(10)

t/L d/a + 2l/a +

1 t/L 3

.

Slit Array with Viscous and Thermal Losses

See also: Mechel Vol. II, Ch. 19 (1995)

The object is an array of slits in a free plate with thickness d; the width of the slits is a, and their mutual distance L. The sketch shows the combination with a resonator volume V; it further shows zones of the sound field, field quantities for which boundary conditions exist, and mode amplitudes as well as mode wave numbers of the field formulations. A plane sound wave with amplitude Ae is incident normally on the plate.

Compound Absorbers

H

421

A simplification is applied: the thermal wave component is neglected in the reflected field in zone I and in the transmitted field in zone III (however, viscous waves are considered in these zones). The full triple of density wave ( = ), viscous wave ( = Œ), and thermal wave ( = ) is applied in the necks. These waves satisfy the wave equations (see sections about sound in capillaries in > Ch. J,“Duct Acoustics”): ( + k2 ) ¥ = 0 kŒ2 = −j

– Œ

;

,

= , ;

2 = −j ‰ k2 ≈ k0

( + kŒ2) ¦ = 0 ,

– = ‰ Pr · kŒ2

;

k2 ≈ k02 =

– . c0

(1)

The particle velocity v , the sound pressure p and the oscillating (absolute) temperature are: v = −grad(¥ + ¥) + rot¦ , p = ¢ · ¥ + ¢ · ¥ , P0

(2)

T = Ÿ · ¥ + Ÿ · ¥ T0 with scalar potential functions ¥ ; = , ; and a vector potential ¦ . See the sections about capillaries for the coefficients ¢, Ÿ .

422

H

0 c0 ‰ Œ † Pr P0 , T0

= = = = = = = =

Compound Absorbers

density; adiabatic sound velocity; adiabatic exponent; kinematic viscosity; dynamic viscosity; temperature conductivity; Prandtl number; athmospheric pressure and temperature

Particle velocity components for boundary conditions are: vx = −

∂(¥ + ¥ ) ∂¦z + ∂x ∂y

;

vy = −

∂(¥ + ¥ ) ∂¦z − . ∂y ∂x

(3)

Numerical coefficients: 1 ; n=0 1 ; m=n ƒn = ; ƒm,n = . 2; n>0 0 ; m = n

(4) ¥e = Ae · e−jk x .

Incident plane wave (formulated as a potential function): Field potential formulations: ¥I (x, y) = Ae · e−jk x + in Zone I :

in Zone II:

¦I (x, y) =

n≥0

¥II (x, y) =

n≥0

ƒn An · e+‚n x · cos(†n y) ,

ƒn AŒn · e+‚Œn x · sin(†n y)

=,

;

(6)

AŒ0 = 0 .

[B · e− x + C · e+ x ] · cos(— y) ,

(7)

¦II (x, y) = [BŒ · e− x + CŒ · e+ x ] · sin(—Œ y) . ¥III (x, y) =

in Zone III:

¦III (x, y) =

n≥0

n≥0

(5)

ƒn Dn · e−‚n x · cos(†n y) , ƒn DŒn · e−‚Œn x · sin(†n y)

;

(8)

DŒ0 = 0 .

with wave numbers and propagation constants: †n = n

2 2 ; n = 0, 1, 2 . . . ; ‚n = †n2 − k2 ; = , L

;

‚0 = jk ,

(9)

—2 = 2 + k2 ; = , , Œ and the known solution (see sections about capillaries) of the characteristic equation Ÿ Ÿ tan(—Œ h) 2 + — h · tan(— h) − −1 — h · tan(— h) = 0. (10) ( h) Ÿ —Œ h Ÿ Relations between amplitudes: Ÿ cos(— h) B B = − Ÿ cos(— h)

;

Ÿ cos(— h) 1− B ; BŒ = − —Œ cos(—Œ h) Ÿ

(11)

Compound Absorbers

Ÿ cos(— h) C = − C Ÿ cos(— h)

Ÿ cos(— h) C . CŒ = + 1− —Œ cos(—Œ h) Ÿ

;

H

423

(12)

So only B , C must be determined in the set of B , C . Mode-coupling coefficients: 1 sn = a Sn

cos(†ny) dy = −h

1 = a =

Rn

+h

+a/2

−a/2

1 cos(— y) · cos(†n y) dy = 2

;

s0 = 1

sin(— − †n )h sin(— + †n )h + (— − †n )h (— + †n )h

†n h · sin(†n h) · cos(— h) − — h · cos(†n h) · sin(— h) (†n2 − —2 )h2

1 = a =

sin(†n h) sin(n a/L) = †nh n a/L

+a/2

−a/2

1 sin(— y) · sin(†n y) dy = 2

sin(— − †n )h sin(— + †n )h − (— − †n)h (— + †n )h

(13)

— h · sin(†n h) · cos(— h) − †n h · cos(†n h) · sin(— h) (†n2 − —2 )h2

with special cases: sin(— h) sin(†n a) 1 ; n = 0 ; Sn = 1+ ; †n = — ; Sn = 1 ; n = — = 0 — h 2 †n a sin(†n a) 1 n =0 1− ; †n = — . = 0; ; Rn = — = 0 2 †na

Sn = Rn

The auxiliary quantities

X± = B ± C

;

1 + e−2 d e− d − 2Y · , − 1 − e−2 d 1 − e−2 d

X+

=

X− ·

Y+

=

2X− ·

B

=

C

=

with intrinsic relations

from which follow

Y± = B e− d ± C e+ d

e− d 1 + e−2 d − Y · − 1 − e−2 d 1 − e−2 d

1 (X+ + X− ) = 2 1 (X+ − X− ) = 2

1 (Y+ + Y− ) e+ d , 2 1 (Y+ − Y− ) e− d 2

(14)

(15)

(16)

(17)

H

424

Compound Absorbers

are solutions of the following coupled system of equations:

# $ 1 + e−2 d X− · ƒn sn Vn − U ƒn sn Wn 1−U 1 − e−2 d n≥0 n≥0 # $ e− d 1−U ƒn sn Vn = 2U · Ae , − Y− · 2 1 − e−2 d n≥0 # $ # $

e− d 1 + e−2 d 1−U 1−U ƒn sn Vn − Y− · ƒn sn Vn X− · 2 1 − e−2 d 1 − e−2 d n≥0 n≥0 ƒn sn Wn = 0. −U

(18)

n≥0

The other mode amplitudes follow from: An = ƒ0,n · Ae + Vn · X+ + Wn · Y−

(19)

Dn e−‚n d = Vn · Y+ − Wn · Y− with the coefficients ¢ Ÿ cos(— h) −1 , U = S0 − S0 ¢ Ÿ cos(— h) Ÿ cos(— h) †n a Vn = — Rn − — Rn 2 L †n − ‚n ‚Œn Ÿ cos(—h) 2 Ÿ cos(— h) , − RŒn 1 − —Œ Ÿ cos(—Œ h) Ÿ cos(— h) Ÿ cos(— h) a ‚Œn Wn = − S . − S S 1 − n n Œn L †n2 − ‚n ‚Œn Ÿ cos(— h) Ÿ cos(—Œ h)

(20)

Introducing the abbreviations Ÿ cos(— h) Ÿ cos(— h) − SŒn 1 − , Ÿ cos(— h) Ÿ cos(—Œ h) Ÿ cos(— h) Ÿ cos(— h) 2 dn = — Rn − — Rn − RŒn 1 − ; Ÿ cos(— h) —Œ Ÿ cos(—Œ h)

en = Sn − Sn

(21) d0 = 0,

the coefficients can be written as Vn =

†n a dn 2 L †n − ‚n ‚Œn

;

Wn =

‚Œn a en . 2 L †n − ‚n ‚Œn

The back side slit impedance Zsb then is: U ƒn sn Wn 2 2 2 k 1 − k /k0 1 Zsb n≥0 = −j 2 Z0 k0 1 − ‰ k2 /k0 U e0 1 − U ƒn sn Vn n≥0

(22)

(23)

Compound Absorbers

and the front side slit impedance Zsf : k2

Zsf = +j Z0 k0

2 1 − k2/k0 2 1 − ‰ k2 /k0

tanh( d) · 1 − U

ƒn sn Vn − U

H

425

ƒn sn Wn n≥0 n≥0 1

. (24) U e0 ƒn sn Vn − tanh( d) · U ƒn sn Wn 1−U n≥0

n≥0

Using the approximations, which are possible for |— h| 1 ; | h|2 |k h|2 ; = , Œ: S0 = S0

sin(— h) ≈1, — h

cos(— h) tan(— h) 1−j 1−j ≈ S0 ≈√ , −−−−−−→ S0 √ cos(— h) (— h) |— h|1 2 |kh| 2 |kh|

1 a a U≈ ; W0 ≈ − e0 = − e0 ; e0 ≈ S0 S0 L ‚0 L jk0

1−j

(25)

1−j 1− √ ≈1 − √ (26) 2 |k h| 2 |kh|

one gets:

√ 2 sn Wn a jk0 2 |kŒ h| Zsb n>0 ≈ − √ . Z0 L sn Vn 2 |kŒ h| − (1 − j) 1 − 2

(27)

n>0

The first term is the normalised radiation resistance; in the second term the first two fractions have about unit value for not too narrow slits. The end correction can be evaluated from: Im{Zsb } = . a Z0 k0a

(28)

Ratios of the components of the slit impedance Zsb with losses to these of the slit impedance Zsb0 without losses

426

H

Compound Absorbers

Loss factor of the oscillating mass of a slit when viscous and thermal losses in the neck are taken into account The components of the slit impedance Zsb = Zsb + j · Zsb can approximately be evaluated from those of the slit impedance Zsb0 = Zsb0 + j · Zsb0 without losses by: $ # Zsb Zsb0 f [Hz]a[m] 10F (x) ; x = lg = · 1+ √ , √ 3 Z0 Z0 (a/L)3/2 a[m] · a/L $ # Zsb0 Zsb 10F (x) (29) ; = · 1+ √ √ Z0 Z0 a[m] · 3 a/L F (x) = −4.641 06 + 0.435 993 x + 0.0142 851 x2 + 0.000 461 347 x3 , F (x) = −2.266 65 − 0.492 331 x − 0.000 719 182 x2 − 0.001 0208 x3 .

H.7

Slit Resonator with Viscous and Thermal Losses


See the scheme drawing in

> Sect. H.6.

The field formulations remain as in > Sect. H.6, except the field in the ¥III (x, y) = ƒn Dn · cosh(‚n (x − d − t)) · cos(†n y) Zone III:

n≥0

¦III (x, y) =

n≥0

ƒn DŒn · sinh(‚Œn (x − d − t)) · sin(†n y) ; DŒ0 = 0.

(1)

Compound Absorbers

The system of equations for the auxiliary quantities X− , Y− (see

X− ·

#

1 + e−2 d 1 − e−2 d

1−U

$ ƒn sn Vn − U

n≥0

>

H

427

H.6) is:

ƒn sn Wn

n≥0

# $ e− d ƒn sn Vn = 2U · Ae , − Y− · 2 1−U 1 − e−2 d n≥0

# $ # $ e− d 1 + e−2 d ƒn sn Vn − Y− · ƒn sn Vn X− · 2 1−U 1−U 1 − e−2 d 1 − e−2 d n≥0 n≥0 ƒn sn Wn = 0 −U

(2)

n≥0

with the new coefficients Vn =

a †n dn 2 L †n − ‚n ‚Œn · tanh(‚n t)/ tanh(‚Œn t)

Wn =

a ‚Œn / tanh(‚Œn t) en L †n2 − ‚n ‚Œn · tanh(‚n t)/ tanh(‚Œn t)

and all other terms as in

V0 = 0 ,

;

(3)

> H.6.

The back and front orifice impedances Zsb , Zsf become:

ƒn sn Wn Zsb 1 n≥0 = −j 2 Ue 1−U Z0 k0 1 − ‰ k2 /k0 ƒn sn Vn 0 k2

U

2 1 − k2 /k0

n≥0

k2

Zsf = +j Z0 k0

2 1 − k2 /k0 2 1 − ‰ k2 /k0

tanh( d) · 1 − U

1

U e0

1−U

n≥0

n≥0

ƒn sn Vn − U

ƒn sn Vn

− tanh( d) · U

n≥0

n≥0

(4)

ƒn sn Wn . ƒn sn Wn

Compared with the results of > Sect. H.6 only the substitutions Vn → Vn , Wn → Wn take place, which correspond to the substitutions ‚n → ‚n · tanh (‚n t) , ‚Œn → ‚Œn / tanh (‚Œn t).∗) Let ZsM be the mass reactance part of the back orifice impedance Zsb with losses, and let Zsb0 be the mass reactance part of the free slit plate without losses ( > Sect. H.2), then the relative change of the reactance can be evaluated with: x = lg(f [Hz]) ∗)

;

y = lg(a[m])


;

z = lg(a/L)

;

u = lg(t/L)

(5)

428

H

Compound Absorbers

from ZsM lg − 1 = Zsb0 − 2.240408 − 0.1580984 · x + 0.00688292 · x2 + 0.0225970 · x3 − 0.7868117 · y + 0.3117230 · y 2 + 0.0739239 · y 3 + 0.7621584 · z + 0.4961154 · z2 + 0.1579759 · z3 − 1.113747 · u + 1.609799 · u 2 − 2.026946 · u 3 + 0.2694603 · x · y + 0.1078516 · x · y 2 + 0.0741470 · x2 · y + 0.1401039 · x · z + 0.00720527 · x · z2 − 0.0424421 · x2 · z + 0.0937094 · u · x − 0.0519085 · u · x2 + 0.7279337 · u 2 · x − 0.1959382 · y · z − 0.00180315 · y 2 · z − 0.0587445 · y · z2 − 1.014977 · u · y − 0.1716795 · u · y 2 + 1.373450 · u 2 · y + 0.1977607 · u · z − 0.1665151 · u 2 · z + 0.0690112 · u · z2.

(6)

The loss factor † can be evaluated with: x = lg(a[m])

;

y = lg(a/L)

;

z = lg(t/L)

;

u = lg(d/a)

(7)

in the range 0.0025 ≤ a ≤ 0.02[m];

0.025 ≤ a/L ≤ 0.4;

0.25 ≤ t/L ≤ 2.0;

0.25 ≤ d/a ≤ 4.0

from lg(†) = − 3.42990 − 0.567811 · x − 0.405786 · y + 0.395143 · z + 0.0811464 · z2 − 0.0337095 · u + 0.0871987 · u 2 + 0.0168052 · u · x + 0.0184409 · u 2 · x − 0.225751 · u · y + 0.0404207 · u2 · y − 0.143725 · u · z − 0.0130437 · u 2 · z − 0.132369 · u · z 2 − 0.114934 · x · y − 0.0195440 · x · z − 0.000512528 · x · z2 + 0.0682123 · y · z + 0.0335370 · y · z2 + 0.0534482 · u · x · y + 0.0314014 · u2 · x · y − 0.00696663 · u · x · z + 0.0156990 · u 2 · x · z − 0.0805588 · u · y · z − 0.00456461 · u 2 · y · z − 0.0409866 · x · y · z + 0.0150972 · u · x · y · z + 0.0183208 · u 2 · x · y · z + 0.597378 · u 2 · z2 − 0.0122410 · u · x · z2 + 0.00565404 · u 2 · x · z2 − 0.0586390 · u · y · z2 + 0.454270 · u 2 · y · z2 − 0.00757341 · x · y · z2 − 0.000668200 · u · x · y · z2 + 0.00446479 · u 2 · x · y · z2.

(8)

Compound Absorbers

H

429

A regression for the lowest resonance frequency f 0 [Hz] is with the same variables and range: lg(f0 [Hz]) = 1.624303 − 0.321020 · u − 0.128558 · u 2 − 1.046357 · x + 0.0110806 · u · x + 0.0100787 · u 2 · x + 1.041716 · y − 0.0927421 · u · y + 0.00399800 · u 2 · y − 0.0841638 · x · y + 0.0258718 · u · x · y + 0.0178172 · u 2 · x · y − 0.623277 · z + 0.136128 · u · z − 0.0220866 · u 2 · z − 0.0107052 · x · z − 0.00112857 · u · x · z − 0.000679359 · u2 · x · z − 0.057692 · y · z + 0.0667389 · u · y · z − 0.0262524 · u 2 · y · z − 0.0204744 · x · y · z − 0.00198288 · u · x · y · z + 0.00124123 · u 2 · x · y · z − 0.0806259 · z2 + 0.083414 · u · z2 + 0.00318264 · u 2 · z2 + 0.00264497 · x · z2 − 0.00150123 · u · x · z2 − 0.00265546 · u 2 · x · z2 − 0.0218893 · y · z2 + 0.0362045 · u · y · z2 − 0.00795119 · u 2 · y · z2 + 0.00570269 · x · y · z2 − 0.00580403 · u · x · y · z2 − 0.00346127 · u 2 · x · y · z2.

H.8

(9)

Free Plate with an Array of Circular Holes, with Losses


The object is a (rigid) plate with thickness d, containing circular holes with diameter 2a in a hexagonal arrangement. A plane wave with normal incidence has the amplitude Ae . The sketch shows the arrangement with a symmetry cell behind each hole.In this section the length of the cell is t = ∞. The radius b of the cell is fixed so that the cells cover all the backside area of the plate (so also square arrays can be treated with this model).

2bø

2aø

t

d

430

H

Compound Absorbers

Zones I and III are supposed to be dissipationless; the necks (zone II) have viscous and thermal losses, or the neck walls may be absorbent. The sketch (with a resonator volume for the next section) shows field quantities which have to satisfy boundary conditions at the indicated surfaces, mode amplitudes in the zones, and wave numbers. The fundamentals of this section correspond widely to those of > Sect. H.7 for slit arrays with losses, except the losses in zones I and III are neglected here (their effect is unimportant).

Sound field formulations: pI (x, r) = Ae e−jk0 x + in Zone I:

n≥0

An e‚n x J0 (†n r) n≥0

An

‚n ‚n x e J0(†n r) k0

(1)

[Bn e−n x + Cn en x ] J0 (—n r)

Z0 vIIx (x, r) = −j pIII (x, r) =

in Zone III:

n≥0

Z0 vIx (x, r) = Ae e−jk0 x + j pII (x, r) =

in Zone II:

n≥0

n [Bn e−n x − Cn en x ] J0 (—n r) k 0 n≥0

(2)

Dn e−‚n x J0 (†nr)

Z0 vIIIx (x, r) = −j

n≥0

Dn

‚n −‚n x e J0 (†n y) k0

(3)

The sound field in the neck (zone II) is formulated as a mode sum. Some cases can be treated as follows:

Compound Absorbers

H

431

(1) Use the formulation as it is if the neck walls are absorbent with an admittance G and higher neck modes shall be considered. Solve the characteristic equation —n a

J1 (—n a) = j k 0 a Z0 G J0 (—n a)

(4)

for a sufficiently large set of wave numbers —n and determine the axial propagation constants n from n2 = —2n − k02 . (2) The neck wall is absorbent, but the neck is narrow, so that only the fundamental neck mode must be retained: determine —0 , 0 as above and set Bn>0 = 0 and Cn>0 = 0. (3) The neck wall is hard, and higher neck modes shall be considered: proceed as in (1), but with the characteristic equation for G = 0. (4) The neck wall is hard, and only a plane wave is assumed in the neck: proceed as in (2), but with —0 , 0 for G = 0. (5) The neck wall is hard, and only the fundamental capillary mode shall be considered (the neck is very narrow; viscous and thermal losses in it shall be considered): take for 0 the propagation constant in circular capillaries (see sections about capillaries in > Ch. J,“Duct Acoustics”) and evaluate —0 from —20 = 02 + k02 . (5) (6) A somewhat exotic model assumes a very narrow neck, but with higher capillary modes. Either solve the characteristic equation of circular capillaries for a set of higher-mode propagation constants (which is not easy) or solve —n a

J1 (—n a) = j k 0 a Z0 G J0 (—n a)

(6)

with an equivalent Z0 G: Ÿ Ÿ J1 (k a) J1 (kŒ a) (k a)2 1 − − k a j Ÿ kŒ a J0 (kŒ a) Ÿ J0 (k a) Z0 G = Ÿ J1(kŒ a) k0 a 1−2 1− Ÿ kŒ a J0 (kŒ a)

(7)

for a set of —n and then n from n2 = —2n − k02 .

(8)

Relations between wave numbers and propagation constants are:

‚n2 = †n2 − k02

;

n2 = —2n − k02 .

zn = †n b are solutions (n = 0, 1, 2, . . .) of J1 (z) = 0 with z0 = 0.

(9)

432

H

Compound Absorbers

Mode-coupling coefficients: Tn,m

1 = 2 a

a J0 (—n r) J0 (†m r) · r dr

a a a − zm J0 (—n a) J1 zm —n a J1 (—n a) J0 zm b b b = a 2 2 (—n a)2 − zm b J0 (—n a) J0 (†ma) J1 (†m a) = j k0a Z0 G − †m a (—n a)2 − (†m )2 J0 (†m a) a 1 J1 (—m a) J0 (—m a) = · j k 0 a Z0 G Tm,0 = 2 J0 (—m r) · r dr = a —m a (—m a)2 0 J0 (—m a) J1(†i a) 2 J0 (—n a) a Z G−† a Tm,i Tn,i = J20 (†i a) j k 0 0 i (—m a)2 −(†i a)2 (—n a)2 −(†i a)2 J0(†i a) 0

(10)

and 1 Rm = 2 a

a 0

1 2 (k0a)2 2 J0 (—n r) J0 (—m r) · r dr = J0 (—m a) 1 − (Z0 G) . 2 (—m a)2

(11)

The boundary conditions lead to a coupled system of equations for the auxiliary quantities; Xn± : = Bn ± Cn

;

Yn± : = Bn e−n d ± Cn en d

(12)

with intrinsic relations Xn+ = Xn− ·

1 + e−2n d e−n d − 2Y · , n− 1 − e−2n d 1 − e−2n d

(13)

e−n d 1 + e−2n d − Y · . Yn+ = 2Xn− · n− 1 − e−2n d 1 − e−2n d The system of equations has the form (m = 0, 1, . . .): amn · Xn− + cm · Ym− = bm · Ae n≥0

cm · Xm− +

n≥0

(14)

amn · Yn− = 0

with coefficients: a 2 k 1 + e−2m d n 0 am,n = 2 Tm,i Tn,i + ƒm,n 2 b k0 1 − e−2m d ‚ J (z ) i≥0 i 0 i cm = −2 Rm

e−m d 1 − e−2m d

;

bm = Tm,0

k0 1+j ‚0

.

(15)

433

… … …

0 cm

0

Xn–

bm =

•

0

...

… cm

Yn–

0

…

…

0

amn

…

M

amn

…

M m=0

H

n=0…....…N n=0………N

m=0

… … …

…………… ……………

Compound Absorbers

The mode amplitudes follow from solutions with: ⎡ ⎤ a 2 k0 i ⎣ƒ0,n · jAe − 2 Ti,n · Xi− ⎦ , An = b k0 ‚n J20 (zn ) i≥0

1 1 Bn = (Xn+ + Xn− ) = (Yn+ + Yn− )e+n d , 2 2 1 1 Cn = (Xn+ − Xn− ) = (Yn+ − Yn− )e−n d , 2 2 a 2 k e‚n d 0 i Ti,n · Yi− . Dn = 2 b ‚n J20 (zn ) i≥0 k0 The front side and back side orifice impedances Zsf , Zsb are obtained from: Tn,0 · Xn+ Zsf < pII (0, r) >a n≥0 = =j , n Z0 Z0 < vIIx (0, r) >a Tn,0 · Xn− n≥0 k0 Tn,0 · Yn+ Zsb < pII (d, r) >a n≥0 = =j . n Z0 Z0 < vIIx (d, r) >a Tn,0 · Yn− n≥0 k0

(16)

(17)

If only the fundamental mode n = 0 is retained in the neck, the system of equations becomes: a0,0 · X0− + c0 · Y0− = b0 · Ae or

X0− =

b0 a0,0 2 a0,0 − c20

a0,0 = 2 with

;

;

c0 · X0− + a0,0 · Y0− = 0

Y0− =

−b0 c0 2 a0,0 − c20

a 2 k 1 + e−20 d 0 0 T T + , 0,i 0,i b k0 1 − e−20 d ‚i J20 (zi ) i≥0

b0 = T0,0

k0 1+j ‚0

;

e−0 d c0 = −2 R0 . 1 − e−20 d

(18) (19)

(20)

H

434

Compound Absorbers

The end correction /a of an orifice follows from:

Im {Zsb /Z0} = . a k0 a

(21)

Analytical results for necks with viscous and thermal losses in a free plate can be represented by the regression: = − 0.0454 728 − 0.728 326 x − 0.177 078 x2 + 0.0339 531 y a (22)

+ 0.00810 471 y 2 − 0.00100 762 xy, = lg () = lg (a2 /b2)

x

;

y = lg (b/Š0).

a=0.001 [m] ; d/a=1 ; nhi=12

Δ / a 0.7 0.6

0.4

lg (b/λ0)

0.2 0.1 -2. -1.69 -1.38 -1.07 -0.764 -0.455

-0.762 -1.08 -1.40 -1.72 -2.04 -2.36 -2.68 -3. -0.30 2 lg (a/b)

End correction /a of an orifice of an array of circular necks in a free plate, with losses in the neck; points: analytical solution; curves: regression

Compound Absorbers

H.9

H

435

Array of Helmholtz Resonators with Circular Necks


The arrangement is as shown in > Sect. H.8, but now with a finite length t of the cells behind the necks. The fields in zones I and III are formulated as in > Sect. H.8. The field formulation in zone III now is: pIII (x, r) = Dn cosh(‚n (x − d − t)) J0 (†n r) , n≥0

Z0 vIIIx (x, r) = j

n≥0

Dn

‚n sinh(‚n (x − d − t)) J0(†n y) k0

(1)

with wave numbers as in > Sect. H.8. The auxiliary quantities Xn− , Yn− of that section are now solutions of the following coupled system of equations (m = 0, 1, 2, . . .): n≥0

amn · Xn− + cm · Ym− = bm · Ae ,

cm · Xm− +

n≥0

(2)

dmn · Yn− = 0

with the coefficients am,n , bm , cm from dm,n = 2

> Sect. H.8, and

a 2 k coth(‚ t) 1 + e−2m d n 0 i Tm,i Tn,i + ƒm,n . 2 b k0 1 − e−2m d ‚i J0 (zi )

(3)

i≥0

The new mode amplitudes Dn are evaluated with solutions by: Dn = 2

a 2 b

i k0 Ti,n · Yi− . ‚n sinh(‚n d) · J20 (zn ) k0

(4)

i=0

The front side and backside orifice impedances Zsf , Zsb are: Tn,0 · Xn+ Zsf < pII (0, r) >a n≥0 = =j , n Z0 Z0 < vIIx (0, r) >a Tn,0 · Xn− n≥0 k0 Tn,0 · Yn+ < pII (d, r) >a Zsb n≥0 = =j . n Z0 Z0 < vIIx (d, r) >a Tn,0 · Yn− n≥0 k0

(5)

If losses can be neglected in the neck, i.e. —n a = zn with zn the solutions of J1 (zn ) = 0; n = 0, 1, 2, . . .; z0 = 0, then: †n b = zn

;

†0 = 0

;

‚n2 = †n2 − k02

;

‚0 = jk0 ,

—n a = zn

;

—0 = 0

;

n2 = —2n − k02

;

0 = jk0

(6)

H

436

and Rm

1 = 2 a

Tn,m =

1 a2

a 0

Compound Absorbers

1 J20 (—m r) · r dr = J20 (—m a) , 2

a J0 (—n r) J0 (†m r) · r dr 0

a a a − zm J0 (—n a) J1 zm —n a J1 (—n a) J0 zm b b b = a 2 2 (—n a)2 − zm b †m a J0 (—n a) J1 (†m a) =− , (—n a)2 − (†m )2 Tn,0

1 = 2 a

1 T0,m = 2 a

a J0 (—n r) · r dr = 0

a J0 (†m r) · r dr = 0

J1 (—n a) =0 —n a J1 (†ma) †m a

;

n>0,

;

m>0,

(7)

1 2 further: bm = ƒ0,m. The orifice impedances then become: T0,0 =

< pII (d, r) >a Y0+ Zsb = = Z0 Z0 < vIIx (d, r) >a Y0−

;

Zsf < pII (0, r) >a X0+ = = . Z0 Z0 < vIIx (0, r) >a X0−

(8)

The equivalent network of a Helmholtz resonator can be conceived as in the diagram. ZR is the radiation resistance of the front orifice; Z1 , Z2 represent the neck; ZF is the spring reactance of the resonator volume. Then Mf , Mb represent the oscillating masses of the front and back orifices. sin (k0 d) ; ZF = −j cot (k0t) Z1 = j Z0 sin (k0 d) ; Z2 = −j Z0 1 − cos (k0d)

(9)

with = (a/b)2 the surface porosity of the neck plate. Im {ZM /Z0} –M/Z0 = = . (10) The end corrections are given by: a k0 a k0 a The front side end correction f /a is that of the front side orifice of a free plate ( > Sect. H.8). With the back side orifice impedance Zsb written as: ZMb Zsb = − j cot(k0t). (11) Z0 Z0

Compound Absorbers

I

II

III

Mf

Z1

Mb

ZR

Z2 Z2

H

437

ZF

Zsf

Zsb

the interior end correction can be represented for t/b ≥ 0.5 by: x = lg = 2 lg(a/b)

y = lg(b/Š0 )

;

b = − 0.0481 939 − 0.731 823 x − 0.179 629 x2 + 0.0342 687 y a + 0.00818 059 y 2 − 0.00101 281 xy. The resonance (angular) frequency –0 follows from the resonance condition: f b f b k0 a + − cot(k0 t) + 1 − (k0 a)2 a a a a f cot(k0t) · tan(k0 d) = 0 + k0 a a

(12)

(13)

with an approximation for k0d 1 and k0 t 1 (with Sa the neck cross-section area and V the resonator volume): (k0 a)2 ≈

d a

"

–0

H.10

≈ c0

d t

d f f b d + + + a t a a a

(14)

Sa . V (d + f + b ) + Sa d f

Slit Resonator Array with Porous Layer in the Volume, Fields


The arrangement consists of a stiff plate at a distance t + s of a hard wall, which contains an array of parallel slits with width a and a mutual distance L. Behind the plate is a porous layer with thickness s (backed by the wall) and a distance t to the plate. A plane wave with amplitude Ae is obliquely incident under a polar angle Ÿ. Special cases, such as Ÿ = 0; t = 0; s = ∞, will be considered. A great interest lies in the influence of the porous layer on the back side end correction.

H

438

Compound Absorbers

The arrangement is treated as a periodic structure with period length L. The field in the necks is composed of mode sums in a hard duct (viscous and thermal losses generally can be neglected compared with the losses introduced by the porous layer). As a special case, only plane waves in the necks will be assumed also. y I

IV

II III

x

Θ a L

d t

Below is

ƒm =

s

1; m=0 . 2; m>0

(1)

Oblique incidence; t > 0; higher neck modes Field formulation in zone I: pI (x, y) = Ae e−j (kx x+ky y) +

+∞ n=−∞

An · e‚n x · e−j n y

+∞ kx ‚n Z0 vIx (x, y) = Ae e−j (kx x+ky y) + j An · e‚n x · e−j n y k0 k0 n=−∞

with and

kx = k0 cos Ÿ ;

ky = k0 sin Ÿ

2 Š0 = k0 sin Ÿ + n ; L L ‚n2 = 2n − k02 ; ‚0 = j kx = j k0 cos Ÿ; ‚n = k0 (sin ‡ + n Š0 /L)2 − 1. 0 = ky = k0 sin Ÿ;

(2)

n = 0 + n

(3)

The necks in zone II are numbered with Œ = 0, ±1, ±2, . . ., beginning with the neck which contains the x axis. The local co-ordinate in the Œ-th neck is yŒ = y − Œ · L with |yŒ | ≤ a/2: ∞ yŒ 1 − pII (x, yŒ ) = e−j0 ŒL (Bm e−j ‰m x + Cm e+j ‰m x ) cos m a 2 m=0 (4) ∞ ‰m yŒ 1 −j ‰m x +j ‰m x −j0 ŒL − (Bm e − Cm e ) cos m Z0 vIIx (x, yŒ ) = e a 2 m=0 k0

Compound Absorbers

with

‰m =

⎧ 2 ⎨ k0 − (m/a)2 ; ⎩

−j (m/a)2 − k02;

m ≤ mg

H

439

; mg = INT(k0 a/) = INT(2a/Š0 ). (5)

m > mg

The index limit mg defines the transition from propagating modes to cut-off modes. In the air gap of zone III (with wave numbers as in zone I): pIII (x, y) = (Dn e−‚n y + En e+‚n x ) · e−j n y , n≥0

Z0 vIIIx (x, y) = −j

‚n (Dn e−‚n y − En e+‚n x ) · e−j n y . n≥0 k0

(6)

In the absorber layer, zone IV: pIV (x, y) = Fn cosh(—n (x − d − t − s)) · e−j n y , n≥0

Z0 vIVx (x, y) = −

k0 Z0 —n Fn sinh(—n (x − d − t − s)) · e−j n y a Za n≥0 k0

(7)

with the characteristic propagation constant a and wave impedance Za of the porous material and —n = 2n + a2 = k0 (sin Ÿ + n Š0 /L)2 + (a /k0)2 . (8) Auxiliary amplitudes are introduced: Xm± =: Bm ± Cm

;

Ym± =: Bm e−j‰m d ± Cm e+j‰m d

(9)

with intrinsic relations Xm+ = Xm−

1 + e−2j‰m d e−j‰m d − 2Y , m− 1 − e−2j‰m d 1 − e−2j‰m d

Ym+ = 2 Xm−

e−j‰m d 1 + e−2j‰m d − Y m− 1 − e−2j‰m d 1 − e−2j‰m d

(10)

and giving the other amplitudes by: 1 1 (Xm+ + Xm− ) = (Ym+ + Ym− ) e+j‰m d , 2 2 1 1 Cm = (Xm+ − Xm− ) = (Ym+ − Ym− ) e−j‰m d , 2 2

∞ ‰m k0 a ƒ0,n cos Ÿ − Xm− sm,n , An = j ‚n 2L m=0 k0

Bm =

∞ e+‚n d ‰m a k0 Dn = j Ym− sm,n , −2‚ t n 2L ‚n 1 − rn e k0 m=0

En = Dn · rn e−2‚n (d+t) , Fn =

1 ( Dn e−‚n (d+t) + En e+‚n (d+t) ). cosh (—n s)

(11)

H

440

Compound Absorbers

Here the modal reflection factors at the zone boundary III–IV are: k0Z0 a Za rn = k0Z0 1+j a Za

—n tanh (—n s) ‚n . —n tanh (—n s) ‚n

1−j

(12)

The Xn− , Yn− are solutions of the coupled system of linear equations ∞

am,n Xn− + cm Ym− = bm

∞

;

n=0

dm,n Yn− + cm Xm− = 0

(13)

n=0

with the coefficients: am,n

∞ a ‰n k0 ƒm,n 1 + e−2j ‰m d =j sm,i sn,i + 2 (−1)m , 2L k0 ‚i ƒm 1 − e−2j ‰m d i=−∞

∞ −2j ‰m d a ‰n k0 1 + ri e−2‚i t m ƒm,n 1 + e s s + 2 (−1) , m,i n,i 2L k0 ‚i 1 − ri e−2‚i t ƒm 1 − e−2j ‰m d

dm,n = j

(14)

i=−∞

cm = −

m

4 (−1) ƒm

j m/2

sm,n = e

e−j ‰m d 1 − e−2j ‰m d

= = and

bm = 2 sm,0 ,

sin ((m + n a)/2) sin ((m − n a)/2) + (−1)m (m − n a)/2 (m + n a)/2

n a/2 =2 · (n a/2)2 − (m/2)2

;

sin (n a/2) ;

m = even

−j cos (n a/2); m = odd

2

;

n a = m ,

m=0

ejm/2

;

n a = m ,

m = 0

2

; −n a = m ,

m=0

(−1)m ejm/2

; −n a = m ,

m = 0

s−m,n = sm,n .

(15)

Oblique incidence; t > 0; higher neck modes; infinite layer thickness t → ∞ Change the field formulation in zone IV to: Fn e−—n x e−j n y , pIV (x, y) = n≥0

Z0 vIVx (x, y) =

k0Z0 —n Fn e−—n x e−j n y , a Za n≥0 k0

substitute in rn :

tanh(—n s) → 1,

(16)

Compound Absorbers

and evaluate the amplitudes from: The other expressions remain.

H

Fn = e—n (d+t) ( Dn e−‚n (d+t) + En e+‚n (d+t) ).

441

(17)

Normal incidence; t > 0; higher neck modes It is not advisable to treat this case as a special case Ÿ = 0 of the above results because the anti-symmetrical modes for oblique incidence (odd m) will vanish and the matrix will get a banded structure. Field in zone I: pI (x, y) = Ae e−j k0 x +

n≥0

Z0 vIx (x, y) = Ae e with

ƒn =

1 2

‚n An e‚n x cos (†n y) k0 n≥0 % 2 n=0 ; ‚n = †n2 − k02 ; ; †n = n n>0 L

−j k0 x

ƒn An e‚n x cos (†ny) ,

; ;

+j

(18)

ƒn

‚0 = j k0 .

(19)

Field in zone II: pII (x, y) = ( Bm e−j‰m x + Cm e+j‰m x ) cos (2m y/a) , m≥0

‰m ( Bm e−j‰m x − Cm e+j‰m x ) cos (2m y/a) m≥0 k0 2 k0 − (2m/a)2 ; m ≤ mg ‰m = ; ‰0 = k0, −j (2m/a)2 − k02 ; m > mg

Z0 vIIx (x, y) =

with

(20)

(21)

and the limit index for cut-off: mg = INT(a/Š0 ). Field in zone III: ƒn (Dn e−‚n x + En e+‚n x ) cos (†n y) , pIII (x, y) = n≥0

Z0 vIIIx (x, y) = −j

n≥0

ƒn

‚n (Dn e−‚n x − En e+‚n x ) cos (†n y). k0

(22)

Field in zone IV: ƒn Fn cosh (—n (x − d − t − s)) cos (†n y) , pIV (x, y) = n≥0

Z0 vIVx (x, y) = − with

—n =

k0 Z0 —n ƒn Fn sinh (—n (x − d − t − s)) cos (†n y) a Za n≥0 k0

†n2 + a2 ;

—0 = a .

(23)

(24)

H

442

Compound Absorbers

Mode-coupling coefficients: 1 Sm,n : = a

+a/2

cos (2m y/a) cos (†n y) dy −a/2

sin (m − n a/L) sin (m + n a/L) + m − n a/L m + n a/L

=

1 2

=

−(−1)m n a/L sin (n a/L) ; m2 − (n a/L)2

=

sin (n a/L) ; n a/L

= 0;

m > 0;

= 1;

m=n=0

m = 0;

;

m, n > 0 ;

m, n > 0 m = n a/L (25)

n≥0

n=0

1 ; m = n a/L = 0. 2 The auxiliary amplitudes Xn− ,Yn− from above again are solutions of two coupled systems of equations as above, but with the following coefficients: =

am,n = j

a ‰n k0 ƒm,n 1 + e−2j‰m d ƒi Sm,i Sn,i + , L k0 ‚i ƒm 1 − e−2j‰m d i≥0

1 + ri e−2‚i t ƒm,n 1 + e−2j‰m d a ‰n k0 dm,n = j ƒi Sm,i Sn,i + , L k0 ‚i 1 − ri e−2‚i t ƒm 1 − e−2j‰m d

(26)

i≥0

cm = −

e−j‰m d 2 ; ƒm 1 − e−2j‰m d

bm = 2 ƒ0,m.

The mode amplitudes follow from solutions of this system as

k0 a ‰n ƒ0,m − Sn,m Xn− , Am = j ‚m L n≥0 k0 Bm =

1 1 (Xm+ + Xm− ) = (Ym+ + Ym− ) e+j‰m d , 2 2

Cm =

1 1 (Xm+ − Xm− ) = (Ym+ − Ym− ) e−j‰m d , 2 2

Dm = j

+‚m d

a e L 1 − rm e−2‚m t

k0 ‰n Sn,m Yn− , ‚m n≥0 k0

Em = rm e−2‚m (d+t) Dm , Fm =

1 ( Dm e−‚m (d+t) + Em e+‚m (d+t) ). cosh (—m s)

(27)

Compound Absorbers

H

443

Normal incidence; t > 0; only plane waves in the neck The system of equations to be solved simplifies to: a0,0 · X0− + c0 · Y0− = b0

;

c0 · X0− + d0,0 · Y0− = 0

(28)

or X0− =

b0d0,0 a0,0d0,0 − c20

;

Y0− =

−b0 c0 a0,0 d0,0 − c20

;

X0− d0,0 =− Y0− c0

(29)

with coefficients a0,0

a k0 sin(i a/L) 2 =j ƒi − j cot(k0d) , L ‚i i a/L i≥0

d0,0

a k0 sin(i a/L) 2 1 + ri e−2‚i t =j ƒi − j cot(k0 d) , L ‚i i a/L 1 − ri e−2‚i t

(30)

i≥0

c0 =

j sin(k0d)

;

b0 = 2.

The modal reflection factors are: k0Z0 a Za rn = k0Z0 1+j a Za 1−j

—n tanh(—n s) ‚n —n tanh(—n s) ‚n

(31)

and the mode amplitudes B0 =

1 1 (X0+ + X0− ) = (Y0+ + Y0− ) e+j k0 d , 2 2

1 1 C0 = (X0+ − X0− ) = (Y0+ − Y0− ) e−j k0 d , 2 2 k0 a sin (n a/L) X0− , An = j ƒ0,n − ‚n L n a/L Dn = j

k0 sin (n a/L) a e+‚n d Y0− , −2‚ t n L 1 − rn e ‚n n a/L −2‚n (d+t)

En = rn e Fn =

Dn ,

1 ( Dn e−‚n (d+t) + En e+‚n (d+t) ). cosh (—n s)

(32a)

(32b)

H

444

Compound Absorbers

Normal incidence; t = 0; higher neck modes If the neck plate is in contact with the porous layer, zone III is obsolete; the amplitudes Dn , En are not needed. The axial function in zone IV changes to cosh (—n (x − d − s)); sinh (—n (x − d − s)). The system of equations for Xn− , Yn− has the following coefficients: am,n = j

a ‰n k0 ƒm,n 1 + e−2j‰m d ƒi Sm,i Sn,i + , L k0 ‚i ƒm 1 − e−2j‰m d i≥0

dm,n = j

a ‰n a Za k0 ƒm,n 1 + e−2j‰m d ƒi Sm,i Sn,i coth (—i s) + , L k0 k0 Z0 —i ƒm 1 − e−2j‰m d

(33)

i≥0

cm = −

e−j‰m d 2 ƒm 1 − e−2j‰m d

;

bm = 2 ƒ0,m.

The amplitudes Fm follow from: Fn =

‰m a a Za k0 Sm,n · Ym− , L k0 Z0 —n · sinh (—n s) m≥0 k0

(34)

the other amplitudes as above. Normal incidence; t = 0; only plane waves in the neck The coefficients of the two equations for X0− , Y0− become: a0,0 d0,0

a k0 sin (i a/L) 2 1 + e−2j k0 d e−j k0 d = j ƒi + ; b = 2; c = −2 , 0 0 L ‚i i a/L 1 − e−2j k0 d 1 − e−2j k0 d i≥0 (35) a a Za k0 sin (i a/L) 2 1 + e−2j k0 d = ƒi coth (—i s) + . L k0Z0 —i i a/L 1 − e−2j k0 d i≥0

H.11

Slit Resonator Array with Porous Layer in the Volume, Impedances


The object here is the same as in

>

Sect. H.10.

The intention in this section is to evaluate the average impedance Z (without radiation resistance ZR ) of an array of slit Helmholtz resonators with a porous absorber layer in the resonator volume by a chain of equations, which represent a simple equivalent network. Some elements of the network are described with the help of the field evaluations in the previous > Sect. H.10.

Compound Absorbers

I

II

Mf

ZN1

ZR

III

Mb

ZN2

Z

g1

Za1

Zg2

Za2

sb

445

IV

Z

Z

sf

H

W a

W at

The symmetrical ¢-fourpoles with ZN1 , ZN2 ; Zg1 , Zg2 ; Za1 , Za2 represent, respectively, the neck, the air gap between the neck plate and the absorber layer, and the absorber layer. Wa is the input impedance of the absorber layer; Wat is that impedance transformed to the back side surface of the neck plate; Zsf , Zsb are the orifice impedances of the front side orifice and back side orifice. Mf , Mb are the oscillating masses of the two orifices. The equations are as follows: L Zsf Z j k0 a + , = Z0 a a Z0 j tan (k0 d) + Zsb /Z0 Zsf = , Z0 1 + j Zsb /Z0 · tan (k0d) ZMsb a Wat Zsb = + ; Z0 Z0 L Z0

ZMsb b , = j k0 a Z0 a

Wat j tan (k0 t) + Wa /Z0 ; = Z0 1 + j Wa /Z0 · tan (k0t)

(1)

Wa Za /Z0 = Z0 tanh (a s)

where /a is the front side end correction; the third line is a defining equation for the back side end correction b /a using the back side orifice impedance Zsb evaluated from the sound field as given in > Sect. H.10 for different conditions. Some of these conditions will be considered below. There an = a /k0; Zan = Za /Z0 are the normalised characteristic values of the porous material.

H

446

Compound Absorbers

Normal sound incidence, absorber layer in contact with neck plate, t = 0, plane wave in the neck The back side orifice impedance Zsb is (with quantities from & ' pII (d, y) a Y Zsb ' = 0+ = & Z0 Z0 vIIx (d, y) a Y0− = −2 =

a L

=

a L

> Sect. H.10):

e−jk0 d 1 + e−2jk0 d d0,0 1 + e−2jk0 d − = d − 0,0 1 − e−2jk0 d c0 1 − e−2jk0 d 1 − e−2jk0 d a Za k0 sin (n a/L) 2 ƒn coth (—n s) k0Z0 n≥0 —n n a/L

a Za k0 sin (n a/L) 2 Za /Z0 +2 coth (—n s) . tanh (a s) k0Z0 n>0 —n n a/L

(2)

For the absorber layer in contact with the neck plate, as assumed here, Wa = Wat ; therefore, evidently: a a Za k0 sin (n a/L) 2 ZMb =2 coth (—n s) (3) Z0 L k0 Z0 n>0 —n n a/L and

s a Za sin (n a/L) 2 coth (—n s) b −j ZMb = −2j = a k0 a Z0 L k0 Z0 n>0 n a/L —n s eff s sin (n a/L) 2 coth (—n s) =2 a 0 a n>0 n a/L —n s

with

a Za j eff = , k0 Z0 a 0

(4)

where a is the porosity of the porous material and eff its (acoustical) effective density. A set of variables for b /a is: F=

L Š0

;

R=

¡s Z0

;

E=

0 f s F = ; ¡ L R

a ; L

d ; a

t ; L

s . L

(5)

A regression through analytically evaluated end correction values for the absorber layer in contact with the neck plate is: a Za b =j · ( 0.0389998 + 0.454066 · x − 0.345328 · x2 − 0.125386 · x3 a k0Z0 − 0.0143782 · y + 0.00418541 · y 2 + 0.0170766 · y 3 − 0.0142094 · z − 0.0715597 · z2 + 0.0915584 · z3 − 0.0115326 · x · y − 0.0195509 · x · z − 0.0595634 · y · z ) x = lg (a/L)

;

y = lg (R) = lg (¡ s/Z0 )

;

z = lg (s/L).

(6)

Compound Absorbers

H

447

F=0.0316; R=1.256; nhi=30

–Im

Δb/a Γan Zan

1.5 1.2 0.8 1

0.4

0.5

0 -2

0 z = lg (s/L) -1.5 -1 x = lg (a/L)

-0.5 -0.5

-1 -0.2

Back side orifice end correction b /a if the absorber layer is in contact with the neck plate; points: analytic evaluation; curves: regression

Normal sound incidence, t > 0, plane wave in the neck The back side orifice impedance Zsb is (with quantities from

> Sect. H.10):

' & pII (d, y) a Zsb Y0+ ' = = & Z0 Z0 vIIx (d, y) a Y0− d0,0 1 + e−2jk0 d e−jk0 d 1 + e−2jk0 d − = d0,0 − −2jk d −2jk d 1 − e 0 c0 1−e 0 1 − e−2jk0 d a k0 sin (n a/L) 2 1 + rn e−2‚n t =j ƒn L n≥0 ‚n n a/L 1 − rn e−2‚n t

k0 sin (n a/L) 2 1 + rn e−2‚n t a 1 + r0 e−2jk0 t = + 2j L 1 − r0 e−2jk0 t ‚ n a/L 1 − rn e−2‚n t n>0 n = −2

(7)

and ZMb a k0 sin (n a/L) 2 1 + rn e−2‚n t = 2j . Z0 L n>0 ‚n n a/L 1 − rn e−2‚n t

(8)

The variation of ZMb in the parameter space makes it unsuited for the definition of an end correction with a representation by regression in this case.

H

448

ZMb a/l · Wat 0.56

Compound Absorbers

F=0.1; R=10; s/L=10; nhi=10

ZMb a/l · Wat

F=0.1; R=100; s/L=10; nhi=10

0.96 0.8

0.4

0.4

0.2

0 -1.

-2. -1.5 -0.8

-1. lg (t/L) -0.5

-0.6

-0.4 -0.2 lg (a/L)

0.

0 -1.

-2. -1.5 -1. lg (t/L) -0.5

-0.8

0.

-0.6 -0.4 lg (a/L) -0.2

0.

0.

Normal sound incidence, infinite absorber layer in contact with neck plate, t = 0; s = ∞, plane wave in the neck The impedance ZMb of the back orifice oscillating mass is (with quantities from > Sect. H.10): sin (n a/L) 2 ZMb 1 a = 2 an Zan (9) 2 Z0 L n a/L (n Š0 /L)2 + an n>0 From the comparison with the corresponding impedance ZMb0 of a free neck plate, and assuming n · Š0 /L |an |2; n · Š0 /L 1 for all n ≥ 1, one gets: ZMb ZMb0 . ≈ −j an Zan = k0a an Zan Z0 Z0 a With a higher order approximation for Š0 /L 1: a sin ( a/L) 2 ZMb ZMb0 ≈ −j an Zan +2 an Zan Z0 Z0 L a/L $ # 1 1 × − 2 (Š0 /L)2 + an (Š0 /L)2 − 1 $

# 1 sin ( a/L) 2 1 −1 , + ≈ k0a an Zan a a/L 1 − (L/Š0 · an )2

(10)

(11)

where /a is the front side orifice end correction. Normal sound incidence, infinite absorber layer, s = ∞, air gap between neck plate and porous layer, t > 0 plane wave in the neck The back orifice oscillating mass impedance is: ZMb a k0 sin (n a/L) 2 1 + rn e−2‚n t = 2j , Z0 L n>0 ‚n n a/L 1 − rn e−2‚n t

(12)

Compound Absorbers

and the modal reflection factors in this case are:

rn = 1 −

with the wave number ratio

—n = ‚n

consequently:

1 + rn e−2‚n t = 1 − rn e−2‚n t

—n j an Zan ‚n

( 1+

H

—n j an Zan ‚n

449

2 (n Š0 /L)2 + an , 2 (n Š0 /L) − 1

j —n tanh (‚n t) an Zan ‚n . j —n tanh (‚n t) + an Zan ‚n

(13) (14)

1+

Under the condition and ensuing approximations —n 1 ≈ 1 ; ‚n t ≈ k0t · n Š0 /L = 2n t/L, (n Š0 /L)2 2 ; |an | ‚n

(15)

(16)

j j tanh (2n t/L) 1+ 1 + rn e an Zan an Zan one gets ≈ = 1. (17) −−−−−−→ j j t/L > 1 1 − rn e−2‚n t tanh (2n t/L) + 1+ an Zan an Zan ZMb0 ZMb ≈ . Thus for t/L > 1/: Z0 Z0 −2‚n t

1+

An approximation of higher order is: Zmsh → − −−−−−−−−−−−−− Z0 s→∞ Š0 /L 1 ⎡ ⎢ ⎢ ⎢ 1 j k0 a ⎢ ⎢ a + ⎢ ⎣ ⎡ ⎢ 1 ≈ j k0 a ⎢ ⎣ a +

H.12

⎞ ⎤ 2 (Š0 /L)2 + an ⎠ (1 − tanh (2 t/L)) ⎥ ⎥ (Š0 /L)2 − 1 ⎥ sin ( a/L) ⎥ ⎥ a/L ⎥ (Š0 /L)2 + an2 j ⎦ tanh (2 t/L) + an Zan (Š0 /L)2 − 1 ⎤ j 2 1 − 1 + (an L/Š0 )2 (1 − tanh (2 t/L)) ⎥ sin ( a/L) an Zan ⎥. ⎦ j a/L 2 tanh (2 t/L) + 1 + (an L/Š0 ) an Zan ⎛

j ⎝1 − 2 an Zan

(18)

Slit Resonator Array with Porous Layer on Back Orifice


The object here is similar to that in > Sects. H.10 and H.11,but a (possibly thin) absorber layer covers the back side orifices and allows an air space deeper in the resonator volume. This is a common method for introducing an additional loss to Helmholtz resonators. A (stiff) plate of thickness d contains an array of parallel slits, width a, mutual distance L, each of which is backed with a resonator volume V of depth t. A porous material layer

H

450

Compound Absorbers

of thickness s < t is in contact with the back side of the plate. The characteristic values of the layer material are a , Za , and in normalised form an = a /k0; Zan = Za /Z0 . If the layer becomes a cloth, wire mesh, felt, etc., the limit transition s → 0 is made with its flow resistance R = ¡s/Z0 kept constant. A normally incident plane wave with amplitude Ae is assumed. y I

L

II III

IV

a x

d

s

t

The field formulations in zones I and II are taken from

> Sect. H.10.

Field formulation in zone III: ƒn (Dn e−—n x + En e+—n x ) cos (†n y) , pIII (x, y) = n≥0

Z0 vIIIx (x, y) =

(1)

k0Z0 —n ƒn (Dn e−—n x − En e+—n x ) cos (†n y) a Za n≥0 k0

Field formulation in zone IV: ƒn Fn cosh (‚n (x − d − t)) · cos (†n y) , pIV (x, y) = n≥0

Z0 vIVx (x, y) = j

n≥0

ƒn

(2)

‚n Fn sinh (‚n (x − d − t)) · cos (†n y) k0

with wave numbers and propagation constants †n = 2n/L ; ‚n = †n2 − k02 ; ‚0 = j k0 , ⎧ 2 k0 − (2m a/L)2 ; m ≤ mg ⎪ ⎪ ⎨ ‰m = ; ⎪ −j (2m a/L)2 − k02 ; m > mg ⎪ ⎩ —n = †n2 + a2 ; —0 = a .

mg = INT(a/Š0 ) ;

‰0 = k0 ,

(3)

Compound Absorbers

H

451

The limit order mg separates cut-on and cut-off (i.e. radiating and non-radiating) spatial harmonics. Modal reflection factors at the back surface of the porous layer are: j an Zan rn = j an Zan

—n coth (‚n (t − s)) − 1 ‚n . —n coth (‚n (t − s)) + 1 ‚n

(4)

The auxiliary amplitudes Xn± , Yn± are defined as in solutions of the coupled system of equations: ∞

am,n Xn− + cm Ym− = bm

n=0

;

∞

>

Sect. H.10. Xn− and Yn− are

dm,n Yn− + cm Xm− = 0

(5)

n=0

with the coefficients am,n = j

a ‰n k0 ƒm,n 1 + e−2j‰m d ƒi Sm,i Sn,i + L k0 ‚i ƒm 1 − e−2j‰m d i≥0

dm,n =

1 + ri e−2—i s ƒm,n 1 + e−2j‰m d a a Za ‰n k0 ƒi Sm,i Sn,i + L k0Z0 k0 —i 1 − ri e−2—i s ƒm 1 − e−2j‰m d

(6)

i≥0

−2 e−j‰m d ; bm = 2 ƒ0,m ƒm 1 − e−2j‰m d 1; m=0 1; m=n ; ƒm,n = where ƒm = 2; m>0 0; m= n cm =

and Sm,n is found in

> Sect. H.10.

The field term amplitudes follow from the solutions Xn− , Yn− as:

k0 a ‰m An = −j Sm,n · Xm− − ƒ0,n Ae , ‚n L m≥0 k0 1 (Xm+ + Xm− ) = 2 1 Cm = (Xm+ − Xm− ) = 2 Bm =

Dn =

1 (Ym+ + Ym− ) e+j‰m d , 2 1 (Ym+ − Ym− ) e−j‰m d , 2

(8)

(9)

‰m a a Za k0 e—n d Sm,n · Ym− , L k0Z0 —n 1 − e−—n s [1 + rn e−2—n s − e−—n s ] m≥0 k0

En = Dn e−2—n (d+s) [1 − e—n s + rn e−—n s ] , Fn =

(7)

1 [Dn e−—n (d+s) + En e+—n (d+s) ]. cosh (‚n (t − s))

(10)

452

H

Compound Absorbers

The back orifice impedance Zsb becomes:

sin (i a/L) 2 k0 1 + ri e−2—i s a Zsb 1 + r0 e−2a s = Zan + 2an , Z0 L 1 − r0 e−2a s i a/L —i 1 − ri e−2—i s

(11)

i>0

and the impedance ZMb of the oscillating mass at the back orifice becomes: sin (i a/L) 2 k0 1 + ri e−2—i s a ZMb = 2 an Zan . Z0 L i a/L —i 1 − ri e−2—i s

(12)

i>0

In the limit s → 0 for thin orifice covers:

sin (i a/L) 2 k0 a Zsb R − j cot (k0t) + 2j = coth (‚i t) . Z0 L i a/L ‚i

(13)

i>0

This is the value for an empty resonator volume except the term R for the flow resistance of the thin cover. If R has no reactive component, the tuning of the resonator is not changed either by the additional resistance or by the end correction of the interior orifice. If the porous foil can freely oscillate, the substitution R → Reff should be made, with Reff =

1 j–mf · ¡s j k0 s · f /0 · R = , Z0 j–mf + ¡s j k0 s · f /0 + R

(14)

where mf is the surface mass density of the foil, and f the density of the foil material.

H.13

Slit Resonator Array with Porous Layer on Front Orifice


> Sections H.10–H.12 deal with additional damping of Helmholtz resonators with porous layers. This section considers an arrangement of the porous layer which gives the possibility of combining the broad-band absorption of a porous layer with the peak absorption of a resonator.

In contrast to the previous sections, the porous layer now is placed on the front side of the resonator array. A plane wave with amplitude Ae is normally incident. The characteristic values of the layer material are a , Za , and in normalised form an = a /k0; Zan = Za /Z0. If the layer is a cloth, wire mesh, felt, etc., the limit transition s → 0 is made with its flow resistance R = ¡s/Z0 kept constant.

Compound Absorbers

H

453

y I

III II

IV

a

L x

s d

t

The field formulations in the zones are as follows: Zone I: pI (x, y) = Ae e−jk0 x +

n≥0

ƒn An e‚n x cos (†n y) ,

Z0 vIx (x, y) = Ae e−jk0 x + j

n≥0

ƒn

‚n An e‚n x cos (†ny) . k0

(1)

Zone III: pIII (x, y) =

n≥0

ƒn (Dn e−—n x + En e+—n x ) cos (†n y) , k0Z0 —n ƒn (Dn e−—n x − En e+—n x ) cos (†n y) . a Za n≥0 k0

Z0 vIIIx (x, y) =

(2)

Zone II: pII (x, y) =

m≥0

Z0 vIIx (x, y) =

(Bm e−j‰m x + Cm e+j‰m x ) cos (2m y/a) ,

‰m (Bm e−j‰m x − Cm e+j‰m x ) cos (2m y/a) . m≥0 k0

(3)

Zone IV : pIV (x, y) =

n≥0

ƒn Fn cosh (‚n (x − d − t)) cos (†n y) ,

Z0 vIVx (x, y) = j

n≥0

ƒn

‚n Fn sinh (‚n (x − d − t)) cos (†ny) . k0

(4)

H

454

Compound Absorbers

The wave numbers and propagation constants are: †n = 2n/L; ‚n = †n2 − k02; ‚0 = j k0 , ⎧ 2 k0 − (2m a/L)2 ; m ≤ mg ⎪ ⎪ ⎨ ‰m = ; ⎪ −j (2m a/L)2 − k02 ; m > mg ⎪ ⎩ —n = †n2 + a2 ; —0 = a .

mg = INT(a/Š0 );

‰0 = k 0 ,

(5)

The boundary conditions give the set of equations for the mode amplitudes: ‚n k0Z0 —n An e−‚n s = (Dn e+—n s − En e−—n s ) , k0 a Za k0 a ‰m k0 Z0 —n (Dn − En ) = Sm,n (Bm − Cm ) , a Za k0 L m≥0 k0 a ‰m ‚n Fn sinh (‚n t) = j Sm,n (Bm e−j‰m d − Cm e+j‰m d ) , k0 L m≥0 k0

ƒ0,nAe e+jk0 s + j

(6)

ƒ0,nAe e+jk0 s + An e−‚n s = Dn e+—n s + En e−—n s , 1 (Bm + Cm ) = ƒn Sm,n (Dn + En ) , ƒm n≥0 1 (Bm e−j‰m d + Cm e+j‰m d ) = ƒn Sm,n Fn cosh (‚n t) ƒm n≥0 with the mode coupling coefficients: 1 Sm,n : = a

+a/2

cos (2m y/a) cos (†n y) dy −a/2

=

1 2

=

−(−1)m n a/L sin (n a/L); m2 − (n a/L)2

sin (m − n a/L) sin (m + n a/L) + m − n a/L m + n a/L

sin (n a/L) ; = n a/L = 0;

m > 0;

= 1;

m=n=0

=

1 ; 2

m = 0;

;

m, n > 0 ;

m, n > 0 m = n a/L (7)

n≥0

n=0

m = n a/L = 0.

Henceforth only plane waves are supposed to exist in the necks, i.e. Bm>0 = Cm>0 = 0.

Compound Absorbers

H

455

The equations of the boundary conditions simplify to: . ‚n k0Z0 —n Dn e+—n s − En e−—n s , An e−‚n s = k0 a Za k0

ƒ0,nAe e+jk0 s + j

(8)

a k0 Z0 —n (Dn − En ) = S0,n (B0 − C0 ) , a Za k0 L ‚n a Fn sinh (‚n t) = j S0,n (B0 e−jk0 d − C0 e+jk0 d ) , k0 L ƒ0,nAe e+jk0 s + An e−‚n s = Dn e+—n s + En e−—n s , B0 + C0 =

n>0

ƒn S0,n (Dn + En ) ,

B0 e−jk0 d + C0 e+jk0 d = with

(9)

n≥0

ƒn S0,n Fn cosh (‚n t)

S0,n = sin(na/L)/(na/L).

(10)

The auxiliary amplitudes X0± , Y0± are introduced: X0± =: B0 ± C0 ;

Y0± =: B0 e−jk0 d ± C0 e+jk0 d

(11)

with intrinsic relations X0+ = X0−

1 + e−2jk0 d e−jk0 d − 2Y , 0− 1 − e−2jk0 d 1 − e−2jk0 d

Y0+ = 2 X0−

(12)

e−jk0 d 1 + e−2jk0 d − Y0− −2jk d 1−e 0 1 − e−2jk0 d

and 1 (X0+ + X0− ) = 2 1 C0 = (X0+ − X0− ) = 2

B0 =

1 (Y0+ + Y0− ) e+jk0 d , 2 1 (Y0+ − Y0− ) e−jk0 d . 2

(13)

X0− and Y0− are solutions of the two equations:

a ƒn S20,n 1 Y0− − cot (k0d) + X0− = 0 coth (‚n t) + L n≥0 ‚n /k0 sin (k0d) and Y0−

(14)

ƒn S20,n a j + X0− − j cot (k0d) + an Zan coth (—n s) sin (k0 d) L — /k0 n≥0 n

+

j ‚n /k0 ‚n —n 1 2 sinh (—n s) −j coth (—n s) an Zan k0 k0

=

2 Zan ejk0 s . sinh (a s) + Zan cosh (a s)

(15)

456

H

Compound Absorbers

With the solutions, the amplitudes other than B0 and C0 may be evaluated from the boundary conditions. The average input impedance Z of the arrangement will be given by: & ' pIII (−s, y) L Z 2 ejk0s cosh (a s) − a/L · X0− ' = Zan = & Z0 Z0 vIIIx (−s, y) L 2 ejk0 s sinh (a s) + a/L Zan · X0− (16) a/L Zan (1 + Zan coth (a s) ) = Zan coth (a s) − jk s · X0− . 2 e 0 sinh (a s) + a/L Zan · X0− The first term in the last line is the input impedance (normalised) of the porous layer when it has a hard back; therefore the second term is a correction due to the backing by the resonator array.

H.14

Array of Slit Resonators with Subdivided Neck Plate


The intention with arrangements as described in this (and the following) section may be twofold: first introduce new, differently tuned resonances, then introduce losses. The object is an array of Helmholtz resonators with slit-shaped necks. The neck plate is subdivided, allowing a narrow gap between the parts. The graphs show two possibilities of realisation. The plate parts are supposed to be stiff in this section. The average surface impedance Z or the average surface admittance G = 1/Z of the arrangement shall be evaluated with the method of equivalent networks. The equivalent network is shown in the third graph in a p/q-analogy (q = volume flow; the circuit elements are admittances).

y

d1 vx

a d2

t L

s

Compound Absorbers

f b Z1 G2 Gf 2 Zsb Gy

= = = = = = =

H

457

front side end correction; end correction of neck orifice towards volume; entrance impedance of the first neck; admittance of the second neck and air gap in parallel; entrance admittance of second neck; output impedance of back orifice of second neck; entrance admittance of air gap

ωρ

ωρ

ωρ

The chain of equations is (G is evaluated without radiation resistance): a/L Z1 1 + j tan (k0d1 ) · Z0 G2 ; = ; f Z1 Z0 j tan (k0d1 ) + Z0 G2 + j k0 a a Z0 s 1 + j tan (k0 d2 ) · Zsb /Z0 Z0 G2 = Z0 Gf 2 + 2 Z0 Gy ; Z0 Gf 2 = ; a j tan (k0 d2 ) + Zsb /Z0 Z0 G =

−j a/L b Zsb = + j k0 a + Rres Z0 tan (k0t) a

;

Gy =

(1)

1 tanh (y ). Zy

Here f and b may be taken from a previous section on end corrections in a resonator array; Rres denotes a possibly added (normalised) resistance representing additional losses in the back side orifice and/or in the resonator volume. y , Zy are the characteristic propagation constant and wave impedance, respectively, in a flat capillary of width s (see sections on capillaries in > Ch. J,“Duct Acoustics”). If a poro-elastic foil (see later sections on foils) with effective surface mass density meff tightly covers the entrance orifice of the second neck, and if the air gap length is different on both sides of the neck, → 1 , 2 , then evaluate Gf 2 and G2 from: – meff j tan (k0d2 ) + Zsb /Z0 1 =j + , Z0 Gf 2 Z0 1 + j tan (k0d2 ) · Zsb /Z0 s Z0 G2 = Z0 Gf 2 + (Z0 Gy1 + Z0 Gy2 ). a

H.15

(2)

Array of Slit Resonators with Subdivided Neck Plate and Floating Foil in the Gap


The object of this section is similar to that in the previous > Sect. H.14, but the analysis is rather different. A freely floating poro-elastic foil (see later sections on foils) with

458

H

Compound Absorbers

effective surface mass density meff is placed in the air gap between the parts of the neck plate. From a technical point of view this is one way of protecting mechanical sensible foils; from an analytical point of view, sound transmission through the foil over all of its length must be considered. The analysis assumes (possibly different) air gap thicknesses s1 , s2 in front of and behind the foil. The necks have the shown shapes only for analytical reasons: they indicate that no shift of the co-ordinates in front of and behind the neck plate is expressively considered. Schematic arrangement:

y

y s

a

d1

pi1

meff, Rf s1

pi2

d2

L

L

Schematic network:

ω

ωρ

f b Zsf 1 Zsf 2 Zsb G1 G2 Gy1, Gy2

= = = = = = = =

front orifice end correction; end correction of orifice towards volume; entrance impedance of first neck; entrance impedance of second neck; exit impedance of second neck; admittance at exit of first neck; load admittance of foil; input admittances of air gaps

s2 t

Compound Absorbers

H

459

Detail in the air gaps:

The chain of equations for the average surface admittance G = 1/Z of the arrangement is, with the assumption that sound transmission through the foil in the gaps can be neglected, as follows: 1 = (jk0 a · f /a + Zsf 1 /Z0 ) · L/a , Z0 G 1 + j tan (k0 d1 ) · Z0 G1 Zsf 1 = , Z0 j tan (k0 d1 ) + Z0 G1 Z0 G1 =

Z0 s1 + 2 · Z0 Gy1 , Zs1 a

Zs1 j– meff 1 = + , Z0 Z0 Z0 G2 Z0 G2 =

(1)

Z0 s2 + 2 · Z0 Gy2 , Zsf 2 a

j tan (k0 d2 ) + Zsh /Z0 Zsf 2 = , Z0 1 + j tan (k0 d2 ) · Zsh /Z0 −j a/L Zsb + j k0a · b /a + Rres , = Z0 tan (k0t) Gy1 =

1 tanh (y1 ) Zy1

y1 , y2 Zy1 , Zy2 f /a b /a meff Rres

= = = = = =

;

Gy2 =

1 tanh (y2 ). Zy2

capillary propagation constants in the gaps; capillary wave impedance in the gaps; end correction of front side orifice; end correction of orifice towards resonator; effective surface mass density of foil; normalised resistance for possible additional loss in the volume

(2)

460

H

Compound Absorbers

The two gaps on each side in fact are coupled wave guides; the differential equations for which are: p1 − p2 = j–m · vx , −

∂vy1 vx − = j–Ceff ,1 · p1 ∂y s1

;

−

∂vy2 vx + = j–Ceff ,2 · p2 , ∂y s2

−

∂p1 = j–eff · vy1 ∂y

;

−

∂p2 = j–eff · vy2 , ∂y

(3)

where the effective air densities eff ,i and air compressibilities Ceff ,i follow from the capillary propagation constants y,i and wave impedances Zy,i (i = 1, 2) by: y,i eff ,i Ceff ,i =j · k0 0 C0

;

Zy,i = Z0

eff ,i 0

(

Ceff ,i . C0

(4)

One gets two coupled, inhomogeneous wave equations for the fields in the air gaps: ∂ 2 p1 + –2 eff ,1 Ceff ,1 − ∂y 2 ∂ 2 p2 + –2 eff ,2 Ceff ,2 − ∂y 2

eff ,1 eff ,1 · p2 , · p1 = − ms1 ms1 eff ,2 eff ,2 · p2 = − · p1 ms2 ms2

(5)

with solutions satisfying the symmetry conditions pi = Ai · cosh(a y) + Bi · cosh(b y)

;

i = 1, 2 .

(6)

The characteristic equation of the system has the solutions:

a2 = b2 =

2 y1

eff ,2 2 eff ,1 − y2 m s2 m s1 : = 2. eff ,2 eff ,1 − m s2 m s1

(7)

Therefore the gap fields are: pi (y) = Ai · cosh ( y)

;

vyi (y) = −Ai

· sinh ( y) j–eff ,i

;

i = 1, 2,

(8)

and the gap input admittances are: Gyi = −

vyi () = tanh ( ). pi () j–eff ,i

(9)

Compound Absorbers

H

461

I m {Z0G}

0.75 f

0.5

0.25 0 -0.25 -0.5 -0.75

0 0.2 0.4 0.6 0.8 1 1.2 1.4 Re {Z0G}

Normalised surface admittance Z0 G of a slit resonator array with subdivided neck plate and a floating poro-elastic foil in the gap, for increasing frequency f. Parameters: Rf = 0.35; fcr · d = 12[Hz · m]; f = 2750[kg/m2]; df = 0.0001[m]; L = 0.05[m]; a = 0.02[m]; s = 0.002[m]; d = 0.02[m]; t = 0.1[m]; d1 /d = 0.5; s1 /s = 0.45; ¡ = 125[Pa · s/m2 ]

1.5 Re {Z0G}

1 0.5 0 -0.5

I m {Z0G}

-1 -1.5

10

1k

100

f [Hz]

4k

Frequency response curves of the real and imaginary parts of Z0 G for the arrangement from above The load admittance G1 of the back orifice of the first neck (which is needed in the second line of the chain of equations) follows as: ⎡ Z0 G 1 = j

Z0 –meff

⎢ ⎢ ⎢ ⎣

2

1

–meff s2 – 1− +j (Z0 Gsv2 + 2 Gy2 ) –2 Z0 a ⎤ # $ 2 ⎥ –meff s1 – ⎥ − 1− +j · 2 Gy1 ⎥ . ⎦ –1 Z0 a

(10)

462

H

Compound Absorbers

If different gap lengths 1 , 2 are used on opposite sides of the neck, substitute 2

si si Z0 Gyi () → (Z0 Gyi (1 ) + Z0 Gyi (2 ) ); a a

i = 1, 2.

(11)

In the parameters of the above example are s = s1 + s2 ; d = d1 + d2 ; Rf = normalised flow resistance of the poro-elastic foil; df = foil thickness; f = foil material density (the foil is of aluminium); fcr · d = product of foil thickness and critical frequency.

H.16

Array of Slit Resonators Covered with a Foil


An array of slit resonators is covered with a poro-elastic foil of effective surface mass density meff (for ease of writing also meff → m). The width of the air gaps between the foil and the neck plate may be different on both sides of a neck and also the lengths of the gaps: s = (s1 + s2 )/2; L = 1 + 2 + a.

α

α

α

α

α

α

The resonator neck may be excited in three ways: (1) via the left-hand-side foil ( = 1); (2) via the right-hand-side foil ( = 2); (3) via the foil in the orifice range ( = 3). A separate index = 1, 2, 3 indicates the foil ranges (left, right, centre). The three ways of excitation can be considered as three sources. According to Helmholtz’s source superposition theorem, the general state is a superposition of three states in which only one source is active and the other two are short-circuited. The following graph shows a schematic p/q-network of the arrangement with the three sources.

Compound Absorbers

ω

ωρ

H

463

ωρ

ω

ω ω

Let eff ,, Ceff , be the effective air density and air compressibility in a flat capillary with hard and rigid walls having lateral dimension s ; they can be evaluated from the capillary propagation constant y and wave impedance Zy (see sections on sound in capillaries in > Ch. J,“Duct Acoustics”) by: 2 = − –2 eff , Ceff , y

;

j– eff , = y Zy .

(1)

The supports of the foil in the above sketch must not be solid supports; their acoustical role can be played by positions of sound field symmetry. The differential equations in the gaps are: −

∂vy vx + = j–Ceff , · pi , ∂y s

∂pi = j–eff , · vy − ∂y

;

(2)

= 1, 2, 3

;

= 1, 2 ; i = 1, 2.

They lead to the wave equations ∂ 2 pi vx + –2 eff , Ceff , · pi + j–eff , ∂y 2 s =

∂ 2 pi vx 2 − y · pi + j–eff , = 0. ∂y 2 s

(3)

Or, with the Kronecker symbol ƒ,: ƒ, · p − pi = j–meff · vx

;

1 (ƒ, · p − pi ) , j–meff

vx =

eff , ∂ 2 pi eff , 2 · pi = −ƒ, − + · p; = 1, 2, 3; = 1, 2 ; i = 1, 2. y 2 ∂y meff s meff s

(4)

Suitable solutions are: pi (y) = A cosh ( y) + ƒ, · B · p vy (y) = −

;

= 1, 2, 3

;

A sinh ( y) = − A sinh ( y) j–eff , y Zy

= 1, 2 , (5)

H

464

Compound Absorbers

with 2 2 = y +

B =

2 y

eff , y Zy 2 = y + s meff j–s meff

eff , /smeff = + eff , /smeff

1 . –meff y s 1+j Z0 Zy /Z0

(6)

The boundary conditions of continuity in the orifice range lead to: B1 A11 = − p 2 B2 A21 = + p 2

2s1 U2 + WV2 U1 V 2 + U 2 V 1 2s2 U2 − WV2 U1 V 2 + U 2 V 1

; ;

B1 A12 =+ p 2 B2 A22 =− p 2

2s1 U1 − WV1 , U 1 V2 + U 2 V1 2s2 U1 + WV1 , U 1 V2 + U 2 V1

(7)

cosh (2 2 ) A32 cosh (1 1 ) A31 = ; = p U1 cosh (2 2 ) + U2 cosh (1 1 ) p U1 cosh (2 2 ) + U2 cosh (1 1 ) with s = (s1 + s2 )/2 W=1 −

– –0

2

;

–20 =

1 0 c20 , = meff s C0 meff s

(8a)

+ j–meff Gsv

s 1 –meff sinh ( ) + W cosh ( ) ; = 1, 2 , a y Zy 2 sa sinh ( ) + s cosh ( ) ; = 1, 2. V = j –0 2 y Zy

U = j

The average surface admittance of the arrangement is defined as: ⎡ 1 1 1 ⎣ G= (vx11 + vx21 + vx31 ) dy + a · (vx13 + vx23 + vx33 ) L p 0 ⎤ 2 + (vx12 + vx22 + vx32) dy ⎦ .

(8b)

(9)

0

One gets with the above solutions: Z0 L2 L1 Z0 G = j B1 + B2 −1+ –meff L L a sinh (1 1 ) A11 A21 A31 + + · cosh (1 1 ) + + p p p 1 1 2L a sinh (2 2 ) A12 A22 A32 + + · cosh (2 2 ) . + + p p p 2 2 2L

(10)

Compound Absorbers

H

In the special case of symmetrical gaps, i.e. s1 = s2 = s; 1 = 2 = : sinh ( ) a A11 A21 A31 Z0 Z0 G = j −1 + B + + + · 2 + cosh ( ) , –m p p p L A11 B 2sU + WV A21 B 2sU − WV A31 1 =− ; =+ ; = p 4 UV p 4 UV p 2U

465

(11)

with 2 = y2 + W=1 −

y Zy eff = y2 + s meff j–s meff – –0

;

B=

eff /smeff = y2 + eff /smeff

1 , (12) –meff y s 1+j Z0 Zy /Z0

2 + j–meff Gsv ,

1 s sinh ( ) + W cosh ( ) , –meff a y Zy 2 sa –0 sinh ( ) + s · cosh ( ) V =j 2 y Zy U =j

(13)

A11 A21 A31 1 − BW + + = , p p p 2U

and

so that finally in this special case: ⎡ Z0 G = j

Z0 –meff

⎤ tanh ( ) a +2 ⎢ ⎥ ⎢−1 + B + (1 − BW) ⎥ L ⎣ y tanh ( ) ⎦ 1 + j–smeff W+2 a Zy ⎡

=j

Z0 –meff

⎤ tanh ( ) a +2 ⎢ ⎥ L ⎣−1 + B + (1 − BW) ⎦ . tanh ( ) 2 2 W + 2 1 − – /–eff a

(14)

with –2eff = 1/(meff s · Ceff ) the square of the foil resonance (angular) frequency of the cover foil with the effective air compressibility in the gap. See also: Mechel, Vol. II, Ch. 26 (1995) for a discussion of the result and of a possible parameter non-linearity in the gaps.

H.17

Poro-elastic Foils

See also: Mechel, Vol. II, Ch. 26 (1995); Mechel (2000)

Poro-elastic foils may be tight or porous, limp or elastic. Their common features are

466

H

Compound Absorbers

(1) lateral homogeneity in scales which are comparable with the free bending wave length, (2) incompressibility (at least approximate). Thus the description below of poro-elastic foils uniformly covers materials like limp metal or resin foils, thin porous layers, clothes, gauzes, felts, wire mesh, perforated metal sheets, and elastic plates (with or without perforations). Poro-elastic foils must not be plane; they also may have the form of curved shells. It is assumed that the foil is placed in a co-ordinate surface of a co-ordinate system in which the wave equation is separable. We apply orthogonal co-ordinates {x1 , x2 , x3 } for a general survey and assume that the foil occupies a co-ordinate surface {x1, x2 } and that the co-ordinate x3 is normal to the foil, which is at the position x3 = . It seems that Cremer has introduced the term Trennimpedanz (partition impedance) for the quantity ZT defined by: ZT =

p pfront (x1 , x2 , ) − pback (x1 , x2 , ) = , v v(x1, x2 )

(1)

where pfront , pback are the sound pressures in front of and behind the foil, respectively, and v is the velocity of the foil which is counted positive in the direction front → back. The boundary conditions to be applied at a foil are: vfront = vback = v

;

pfront − pback = ZT · v.

(2)

Begin with a tight and limp foil: ZT = j – mf = j – f df , where mf is the surface mass density of the foil, df the foil thickness, and f the foil material density. First generalisation: Porous limp foil: The flow resistance £ = ¡ · df (¡ = flow resistivity of the foil material) acts in parallel with the mass reactance of the foil: Zt =

j – mf · £ = j – mf j – mf + £

1 = j – mp . j – mf 1+ £

So this first generalisation is performed by the substitution:

mf → meff ,p =

(3)

mf . j – mf 1+ £

(4)

Second generalisation: Tight elastic foil: The oscillation of the foil (which indeed is a thin plate) obeys the bending wave equation

j– x1 ,x2 x1 ,x2 − kB4 v = · p, B

(5)

Compound Absorbers

H

467

where x,y is the Laplace operator in the indicated co-ordinates, kB is the wave number of the free bending wave on the plate, B is the bending stiffness, and p = pfront − pback is the driving sound pressure difference. With k0 f 4 2 mf ; = , (6) kB = – B kB fcr where – = 2f and f cr = (critical) coincidence frequency, one immediately gets:

2 p f 1 x1 ,x2 x1 ,x2 v , ZT = = j – mf · 1 − · vp fcr v k04

2 2 ZT f f mf 1 x1 ,x2 x1 ,x2 v mf = j k0 = j k0 · 1− · · 1− sin4 ” Z0 0 fcr k04 v 0 fcr

(7)

(8)

with an effective angle ” of sound incidence on the plate (see below). Third generalisation: Elastic foil with bending wave losses: Energy dissipation in the foil can be taken into account by a loss factor † introducing a complex modulus B → B · (1 + j†). This leads to:

. ZT = Zm F · 1 − F2 sin4 ” − j † F2 sin4 ” Z0

2 2 –cr mf f f − j† ; ; Zm = j = Zm F · 1 − fc fc Z0

f F= , fcr

(9)

where Zm is the normalised inertial impedance of the plate at the critical frequency f cr and f c is the coincidence frequency at the incidence angle ”, with f cr = fc · sin2 ”. So elasticity and bending losses of the foil can be taken into account by using an effective surface mass density:

2 2 f f − j† . (10) mf → meff ,e = mf · 1 − fc fc Fourth generalisation: Combine porosity and elasticity effects: Substitute

mf → meff ,p,e =

meff ,e . j – meff ,e 1+ £

(11)

Cylindrical shell: The co-ordinate system is {x1, x2 , x3 } → {˜ , z, r}. The value = a is the radius of the shell. The sound fields separate into factors (vp = velocity of the shell) p(r, ˜ , z) = R(r) · T(˜ ) · U(z)

;

vp (a, ˜ , z) = A · T(˜ ) · U(z).

(12)

468

H

Compound Absorbers

⎧ ±j k z ⎨ e z The form of U(z) may be any of, or a linear combination of: U(z) = cos (kz z) . (13) ⎩ sin (kz z) The shape of R(r) may be one of the cylinder functions / 0 (2) Zn (kr r) = Jn (kr r) , Yn (kr r) , H(1) n (kr r) , Hn (kr r) . Then T(˜ ) for fields which are periodic in ˜ is:

T(˜ ) =

(14) cos (n˜ ) . (15) sin (n˜ )

The Laplace operators in cylindrical co-ordinates are: =

∂2 1 ∂2 1 ∂ ∂2 + + + ∂r2 r ∂r r2 ∂˜ 2 ∂z2

;

˜ ,z =

1 ∂2 ∂2 + . a2 ∂˜ 2 ∂z2

(16)

The bending wave equation is satisfied with the above field factors, when the following secular equation holds: k02 = kz2 + kr2

;

1 = (kz /k0)2 + (kr /k0)2 = sin2 Ÿ + cos2 Ÿ.

(17)

The angle Ÿ is between the wave vector and the radius. The two-dimensional Laplace operator gives, together with the Bessel differential equation for the Z(i) n (kr r): 2 n (18) ˜ ,z p(a, ˜ , z) = − 2 + kz2 · p(a, ˜ , z). a 2 2 2 2 ˜ ,z ˜ ,z vp n n 2 2 4 Therefore: = + k z = k0 + sin Ÿ . (19) vp a2 (k0 a)2 Comparing this with the form for ZT /Z0 leads to the effective angle ” of incidence: 2 1/2 n 2 sin ” = + sin Ÿ . (20) (k0 a)2 Spherical shell: Spherical co-ordinate system {r, ˜ , œ} and the correspondence {x1 , x2, x3 } → {˜ , œ, r}. The factors of the field are: p(r, ˜ , œ) = R(r) · T(˜ ) · P(œ)

;

vp (a, ˜ , z) = A · T(˜ ) · P(œ)

(21)

with R(r) being spherical Bessel functions and T(˜ ) (associated) Legendre functions of the first and second kind: / 0 (2) R(r) = zm (k0r) = jm (k0 r) , ym (k0 r) , h(1) (22) m (k0 r) , hm (k0 r) ; n n Pm (cos ˜ ) Pm cos(n œ) T(˜ ) = ; P(œ) = . (23) n n sin(n œ) Qm (cos ˜ ) Qm The Laplace operators are: =

1 ∂2 1 2 ∂ 1 ∂2 ∂ ∂2 + + + + , ∂r2 r ∂r r2 ∂˜ 2 r2tg ˜ ∂˜ r2 sin2 ˜ ∂œ2

(24a)

Compound Absorbers

∂2 1 1 ∂2 1 ∂ + + , a2 ∂˜ 2 a2 tg ˜ ∂˜ a2 sin2 ˜ ∂œ2 2 ˜ ,z ˜ ,z vp 4 m(m + 1) = k0 and therefore: vp (k0 a)2 ˜ ,œ =

with the effective angle ” of incidence given by: m(m + 1) 1/2 sin ” = . (k0a)2

H

469

(24b) (25)

(26)

Because of Pnm (cos ˜ ) ≡ 0; n > m, the angle of incidence is ” = 0 for the “breathing sphere” m = n = 0, which is plausible. See Mechel, Acta Acustica, 86 (2000) for elliptic-cylindrical and hyperbolic-cylindrical shells. Partition impedance of membranes: Membranes get their bending stiffness from the tension in their plane (only plane membranes in the (x, y) plane; the method could also be applied to blown-up balloons). The inhomogeneous wave equation of a membrane is: -

. . j– 2 + km pfront − pback vm = T

;

km = – M/T

(27)

with the surface mass density M and the tension T of the membrane. The partition impedance follows immediately: jT 2 pfront − pback T vm 2 2 2 ZT = kmx + kmy (28) = + km = − km vm j – vm – for a pattern of the membrane velocity:

H.18

vm (x, y) = A · e±j kmx x · e±j kmy y .

(29)

Foil Resonator


A foil resonator consists of a foil having a (effective) surface mass density mf at a distance t to a hard wall; the interspace may be filled with air or (partially) with a porous material. The surface impedance Z for normal sound incidence is: ⎧ ; air −j cot (k0 t) ⎪ ⎨ Z mf + = j k0 t Zan Z0 0 t ⎪ ⎩ ; completely porous material tanh (an k0 t)

(1)

with an = a /k0 , Zan = Za /Z0 the normalised characteristic values of the porous material. The (angular) resonance frequency with air in the volume is, under condition k0 t 1: √ (2) –0 = c0 0 /(mf t) ; f0 ≈ 600/ mf t (f0 in Hz ; m in kg/m2 ; t in cm ).

H

470

Compound Absorbers

If k0t 1 does not hold, the resonance equation is (for the nth resonance): –n t –n t 0 t · tan = . c0 c0 mf

(3)

An approximation for the lowest resonance solution is: –0 t 2 105 + 45 (0t/mf ) − 11025 + 5250 (0t/mf ) + 1605 (0t/mf )2 . = c0 20 + 2 (0 t/mf )

(4)

For oblique sound incidence with the polar angle of incidence Ÿ, substitute Z0 → Z0 / cos Ÿ ; k0 → k0 · cos Ÿ. As long as k0 t · cosŸ 1, the resonance changes to –0 → –0 /cosŸ. For oblique incidence on a foil resonator with porous material in the volume: mf Z an Zan % = j k0 t . +% Z0 0 t 2 + sin2 Ÿ · tanh (k t 2 + sin2 Ÿ) an 0 an

(5)

2 10 0

10

g/ [k m

ω2

5

ω1

m [kg/m2]

10 0

m2 ]= 0 50

The lowest three resonances –0 , –1 , –2 (with air in the volume) can be read from the following nomograph. Enter the nomograph on its vertical axis with the value m of the foil surface mass density; proceed horizontally to the line for the distance t; proceed vertically to one of the (thick) lines for –0 , –1 , –2 and from there horizontally to the line for the value of m; then vertically to the horizontal axis, where you can read the resonance frequency f 0 (or f1,f 2 ).(The example in the nomograph is for m = 0.5[kg/m2]; t = 2[cm]; giving f 0 = 700[Hz].)

ω

0. 0. 2 0. 1 05 0. 0. 02 01

0

0. 1 5

2

10

1

0 ]= 5 0 2 0 1

cm

t[

0.1

5 2 1 5 0,

0.01

1

10

100

1k

10 k f0 [Hz]

Compound Absorbers

H.19

H

471

Ring Resonator

Ring resonators are used,for example,in mufflers and in low-frequency silencer sections, e.g. in gas turbine run-up and test cells. The aim of this section is to evaluate the orifice input admittance G. The input orifice may be covered with a poro-elastic foil ( > Sect. H.17) with a partition impedance Zs (normalised). The normalised admittance G then is: G=

Gv , 1 + Zs Gv

(1)

where Gv is the (normalised) input admittance of the ring-shaped neck. A porous absorber layer in zone IV (rs ≤ r ≤ ra ) is supposed to be made locally reacting by partitions.

Corresponding to the low-frequency application,only the fundamental cylindrical mode is supposed to propagate in the field zones. Field formulations in the zones (with Bessel, Neumann, and Hankel functions): (1) pI = a0 · H(2) 0 (k0 r) + b0 · H0 (k0 r)

Zone I:

= (a0 + b0 ) · J0 (k0 r) + j · (a0 − b0 ) · Y0 (k0r) ,

(2)

Z0 vrI = −j · (a0 + b0 ) · J1 (k0r) + (a0 − b0 ) · Y1 (k0r) . pII = c0 · J0 (k0r) + d0 · Y0 (k0r) , Zone II:

Z0 vrII = −j · [c0 · J1 (k0r) + d0 · Y1 (k0r)] .

(3)

pIII = e0 · J0 (k0 r) + f0 · Y0(k0 r) , Zone III:

Z0 vrIII = −j · [e0 · J1 (k0 r) + f0 · Y1(k0 r)] .

(4)

pIV = g0 · J0 (−j an r) + h0 · Y0 (−j an r) , Zone IV :

Z0 vrIV =

j

g0 · J1 (−j an r) + h0 · Y1(−j an r) . Zan

with an , Zan the normalised characteristic values of the porous material.

(5)

H

472

Compound Absorbers

The radial input admittance (normalised) Ga of the absorber layer (zone IV) is: Ga =

j J1 (j an k0rs ) · Y1(j an k0 ra ) − J1(j an k0 ra ) · Y1 (j an k0 rs ) . an J0 (j an k0rs ) · Y1(j an k0 ra ) − J1(j an k0 ra ) · Y0 (j an k0 rs )

(6)

Boundary conditions: The interior neck orifice is additionally loaded with the mass impedance Zm = j · k0a · i /a of the interior orifice end correction. r = rs :

!

−j [c0 · J1 (k0 rs ) + d0 · Y1 (k0 rs )] = Ga · [c0 · J0 (k0 rs ) + d0 · Y0 (k0rs )]

(7)

r = rd ; I − II: velocity: !

− j [c0 · J1 (k0 rd ) + d0 · Y1 (k0rd )] = a

−j (a0 + b0 ) · J1 (k0rd ) + (a0 − b0 ) · Y1 (k0rd ) L a −j 1 − [e0 · J1 (k0rd ) + f0 · Y1 (k0rd )] L

(8)

pressure: (a0 + b0 ) · J0 (k0rd ) + j (a0 − b0 ) · Y0 (k0rd )

! −Zm · −j (a0 + b0 ) · J1 (k0 rd ) + (a0 − b0 ) · Y1(k0 rd ) = c0 · J0 (k0rd ) + d0 · Y0(k0 rd ) r = rd ; II − III: pressure:

(9) !

e0 · J0 (k0 rd ) + f0 · Y0 (k0 rd ) = c0 · J0 (k0 rd ) + d0 · Y0 (k0rd ) r = ri : !

e0 · J1 (k0 ri ) + f0 · Y1(k0 ri ) = 0.

(10)

Setting the arbitrary amplitude a0 = 1, the neck input admittance is: Gv = −j

(1 + b0 ) · J1 (k0 ri ) + j (1 − b0) · Y1 (k0ri ) . (1 + b0 ) · J0 (k0ri ) + j (1 − b0 ) · Y0(k0 ri )

(11)

Compound Absorbers

H

473

Example of Re{Z0 G}, Im{Z0G} of a low-tuned ring resonator for a turbine test cell; with input parameters for computation as listed. (*Duct*) ri[m] = 3. ra[m] = 4.9 (*Cell*) L[m] = 1.0 a[m] = 0.25 d[m] = 0.2 t[m] = 0.2 s[m] = 1.5 (*Absorber*) s[m] = 1.5 ¡[Pas/m2 ] = 500

Ring resonator as above, but the necks additionally covered with a tight aluminium foil, df = 1 mm thick

474

H

Compound Absorbers

So one must solve the boundary condition equations for b0 =: −N/D with N= 1 . −Ji(k0 ri ) Ga J0 (k0 rs ) + j J1(k0 rs ) J1(k0 ri ) Y0 (k0 rd ) − J0(k0 rd ) Y1 (k0 ri ) (1) · J1 (k0 rs ) Y0 (k0 rd ) − J0 (k0 rd ) Y1 (k0 rs ) −j H(1) 0 (k0 rd ) + Zm H1 (k0 rd ) 2 3 + Ga J0 (k0 rd ) Y0 (k0 rs ) − J0(k0 rs ) Y0 (k0 rd ) + j J0(k0 rd ) Y1 (k0 rs ) − J1 (k0 rs ) Y0 (k0 rd ) (12) 2 · −j a/L · J0 (k0 rd ) J1(k0 ri ) H(1) 1 (k0 rd ) J1 (k0 ri ) Y0 (k0 rd ) − J0 (k0 rd ) Y1 (k0 ri ) . − J1 (k0 ri ) J1 (k0 ri ) J1(k0 rs ) Y0 (k0 rd ) + (a/L − 1) J0(k0 rd ) Y1 (k0 rd ) 34 (1) − J0 (k0 rd ) Y1 (k0 ri ) J1 (k0 rs ) + (a/L − 1) J1 (k0 rd ) · −j H(1) 0 (k0 rd )+Zm H1 (k0 rd ) D= 1 J1 (k0 ri ) Ga J0 (k0 rs ) + j J1(k0 rs ) J1 (k0 ri ) Y0 (k0 rd ) − J0 (k0 rd ) Y1 (k0 ri ) (2) · J1 (k0 rs ) Y0 (k0 rd ) − J0 (k0 rd ) Y1 (k0 rs ) j H(2) 0 (k0 rd ) − Zm H1 (k0 rd ) 3 2 + Ga J0 (k0 rd ) Y0 (k0 rs ) − J0(k0 rs ) Y0 (k0 rd ) − j J1(k0 rs ) Y0 (k0 rd ) − J0 (k0 rd ) Y1 (k0 rs ) (13) 2 (2) · −j a/L · J0 (k0 rd ) J1(k0 ri ) H1 (k0 rd ) J1(k0 ri ) Y0 (k0 rd ) − J0(k0 rd ) Y1 (k0 ri ) + J1 (k0 ri ) J1(k0 ri ) J1(k0 rs ) Y0 (k0 rd ) + (a/L − 1) J0(k0 rd ) Y1 (k0 rd ) 34 (2) . − J0 (k0 rd ) Y1 (k0 ri ) J1 (k0 rs ) + (a/L − 1) J1 (k0 rd ) · j H(2) 0 (k0 rd ) − Zm H1 (k0 rd )

H.20 Wide-Angle Absorber, Scattered Far Field See also: Mechel, Vol. III, Ch. 5 (1998); Mechel, Acustica 81 (1995); Schroeder/Gerlach (1977)

The focus of this section originally was conceived as wide-angle diffusers, Schroeder & Gerlach (1977), but the unavoidable losses make the discussion here applicable to effective absorbers. In principle,“diffuser” and “absorber” are contradictions in se. This section is more concerned with the “diffuser”, in that it describes mainly the scattered far field (see next > Sect. H.21 for other field ranges). The focus of our discussion here is on 1-D or 2-D arrays of Š/4-resonators. The depth tk of the resonators varies in one of two possible pseudo-random manners (see below). Mostly, the arrangement is composed of groups of resonators, and the pseudo-random variation of tk is within the group; then the object has a periodic structure. Three indices will be used: k for the number of a resonator in the group, m for the group and n for the resonator within the arrangement.

Compound Absorbers

pe L

ϑe

z

475

P

ϑ r w

2h

H

x

tk T k= 0 1 m=

1 N-1 0 2 N-1 1 -M/2

xn

1 N-1 0 2 N-1 1 -1

……… -1 n=–M/2·N –M/2·(N–1) ……… -1

0 0

1 N-1 0 2 N-1 1

1 N-1 2 N-1

0

M/2–1

1 1

QRD PRD

QRD ……… M/2·N–1 ……… M/2·(N–1) –1 PRD

1-D array The “classical” arrangements are as follows (N = prime number): QRD: quadratic residue diffuser c0 mod (k 2 , N) N –r

1-D:

tk =

2-D:

tk, =

;

c0 mod (k2 + 2 , N) N –r

k = 0, 1, . . . , N − 1 ;

k, = 0, 1, . . . , N − 1

(1) (2)

with –r = 2fr the “working (angular) frequency” and mod(a, b) = “modulo function” the remainder of a/b. PRD: primitive root diffuser 1-D:

tk =

c0 mod (k , N) N –r

;

k = 1, . . . , N − 1

(3)

with the “primitive root” of N. (mod(k , N) produces the numbers 1, 2, . . ., N − 1 in irregular sequence, if k = 1, 2, . . ., N − 1). The needed Helmholtz numbers for 1-D diffusers are: ⎧ ⎪ mod (k 2, N) ; k = 0, 1, . . . , N − 1; QRD f ⎨ N k0 tk = · ⎪ fr ⎩ mod (k , N) ; k = 1, . . . , N − 1; PRD. N

(4)

Cell centre co-ordinates are: ⎧ ⎨ n = −MN/2, . . . , +MN/2 − 1 1 1 (5) L = mT + k + L; m = −M/2, . . . , +M/2 − 1 QRD: xn = xk,m = n + ⎩ 2 2 k = 0, . . . , N − 1 ⎧ n = −M(N − 1)/2, . . . , ⎪ ⎪ ⎨ 1 1 + M(N − 1)/2 − 1 PRD: xn = xk,m = n + L = mT + k − L; m = −M/2, . . . , +M/2 − 1 ⎪ 2 2 ⎪ ⎩ k = 1, . . . , N − 1.

476

H

Compound Absorbers

The input admittance (normalised) G(xk ) of a chamber is: G(xk ) =

tanh(an k0tk ) , Zan

(6)

where an , Zan are the normalised propagation constant and wave impedance in the cell; the fundamental mode values in capillaries are used if viscous and thermal losses are taken into account (see sections on capillaries in > Ch. J,“Duct Acoustics”). Scattered far field for 2-D diffuser: The scattered far field ps for a 2-D diffuser can always be factorised: ps (r, ˜ , œ) = Pi · ¥s (˜e , œe |˜ , œ) ·

e−jk0 r . k0 r

The angular distribution ¥s is evaluated from: −j k02 cos ˜e G(x, y) · e−j (‹x x+‹y y) dx dy ¥s (˜e , œe |˜ , œ) = 2 (cos ˜e + G)

(7)

(8)

A

pe

P

z ϑ

ϑe

r

ϕ

ϕe

with

y

x

‹x = k0 (sin ˜e cos œe − sin ˜ cos œ) ,

(9)

‹y = k0 (sin ˜e sin œe − sin ˜ sin œ). If the surface A = (MT)2 of the arrangement has the homogeneous, averaged admittance G, then: ¥s =

−j k02 cos ˜e · < G > sin (‹x MT/2) sin (‹y MT/2) (MT)2 . 2 (cos ˜e + < G >) ‹x MT/2 ‹y MT/2

(10)

For the 2-D QRD with cell centre co-ordinates:

⎧ ⎨ m = −MN/2, . . . , +MN/2 1 1 = m+ L = mg T + k + L; mg = −M/2, . . . , +M/2 − 1 ⎩ 2 2 k = 0, . . . , N − 1 , ⎧ ⎨ n = −MN/2, . . . , +MN/2 1 1 ng = −M/2, . . . , +M/2 − 1 L = ng T + + L; = n+ ⎩ 2 2 = 0, . . . , N − 1.

xm = xk,mg

yn = x,ng

(11)

Compound Absorbers

H

477

(mg , ng = group indices ; k, = cell incices), the scattered far field distribution is: ¥s =

sin ‹x h sin ‹y h −j (‹x +‹y ) L/2 −j (k0 h)2 cos ˜e · · ·e (cos ˜e + < G >) ‹x h ‹y h −j (‹x mg +‹y ng ) T · e · G(k, ) e−j (‹x k+‹y ) L . mg , ng

(12)

k,

The difference of the QRD, as compared with a surface A of equal size and same average admittance G, is mainly produced by the factor of the last sum. In the following examples of |¥s | the cell orifices may contain a normalised flow resistance R (e.g. a wire mesh; see parameter list). The QRD with N = 11 is exceptional in that it shows a strong backscattering. ϑe=45°

|Φs| 3 0 2 -2 1

-4

0 -0.5

y

-6

0 x 0.5 -8

Scattered far field directivity of a QRD with N = 11. Input parameters: f = 285[Hz]; fr = 285[Hz]; ˜e = 45◦ ; œe = 90◦ ; N = 11; M = 4; h = 0.03[m]; w = 0.003[m]; R = 0; ˜ = 6◦ ; œ = 12◦ −

H

478

Compound Absorbers

ϑe=45°

|Φs| 0.3

0.2

0.1

0.2 0.1

0 -0.15 -0.1 -0.05 x 0 0.05 -0.2

0

y

-0.1

Angular distribution |¥s | of scattered far field with M = 4. Input parameters: f = 285[Hz]; fr = 285[Hz]; ˜e = 45◦ ; œe = 90◦ ; N = 7; M = 4; h = 0.03[m]; w = 0.003[m]; R = 0; ˜ = 6◦ ; œ = 12◦ −

ϑe=45°

|Φs| 1

0.75

0.5 0.75

0.25 0 -0.2 -0.1 0 x 0.1

0.5 0.25 0

y

-0.25

Angular distribution |¥s | of scattered far field with M = 8. Input parameters: f = 285[Hz]; fr = 285[Hz]; ˜e = 45◦ ; œe = 90◦ ; N = 7; M = 8; h = 0.03[m]; w = 0.003[m]; R = 0; ˜ = 6◦ ; œ = 12◦ −

Compound Absorbers

H

479

Scattered far field for 1-D diffuser: (œe = 0) Scattered far field: "

j e−jk0 r · ¥s (˜e |˜ ) · √ . 2 k0 r Directivity function (A = width of diffuser): ps (r, ˜ , œ) = −Pi

+k 0 A/2

¥s (˜e |˜ ) =

G(x) −k0 A/2

p(x, 0) jk0 x sin ˜ e d(k0 x) Pi

with sound pressure at the surface and surface admittance G(x): cos ˜e p(x, 0) = 2 e−jk0 x sin ˜e − Pi cos ˜e + G +k5 0 A/2 −jk0 (x −x) sin ˜e e−jk0 x sin ˜e G(x ) H(2) d(k0x ). 0 (k0 |x − x |) e

(13)

(14)

(15)

−k0 A/2

So ¥s has the form: ¥s (˜e |˜ ) = 2

+k5 0 A/2 −k0 A/2

G(x) ejk0 x (sin ˜ −sin ˜e ) d(k0 x) +k0 A/2

cos ˜e · − cos ˜e + < G >

G(x) G(y) ejk0 (x sin ˜ −y sin ˜e )

−k0 A/2

(16)

·H(2) 0 (k0 |x − y|) d(k0 x) d(k0 y) cos ˜e · I2 . cos ˜e + G The first integral I1 is for a QRD (for a PRD: summation k = 1, . . ., N − 1, and change sign in the exponent of the last factor in the third line below): sin (k0h (sin ˜ − sin ˜e )) I1 = 4 k0 h G(xn ) ejk0 xn (sin ˜ −sin ˜e ) k0h (sin ˜ − sin ˜e ) n : = I1 −

= 4 k0 h

sin (k0h (sin ˜ − sin ˜e )) j k0 L/2· (sin ˜ −sin ˜e ) e k0h (sin ˜ − sin ˜e )

M/2−1

·

ej mk0 T (sin ˜ −sin ˜e )

m=−M/2

N−1

(17)

G(xk ) ej k k0 L (sin ˜ −sin ˜e ) .

k=0

The second integral I2 is G(xn )G(xn ) · ej k0 (xn sin ˜ −xn sin ˜e ) · In,n , I2 = n,n

k0h In,n : =

(18) ej (x sin ˜ −y sin ˜e ) · H(2) 0 (|n − n | k0 L + x − y) dx dy.

−k0 h

See Mechel, Vol. III, Ch. 5 (1998) for the integration of In,n .

480

H

Compound Absorbers

H.21 Wide-Angle Absorber, Near Field and Absorption

See also: Mechel, Vol. III, Ch. 5 (1998); Mechel, Acustica 81 (1995)

The focus of this section is the same as in the previous > Sect. H.20. This section is mainly concerned with the field analysis near the absorber (diffuser) and its absorption. The parts of this section are: • 1-D absorber: • exterior field without losses; in the cells fundamental capillary mode; • exterior field without losses, in the cells higher capillary modes; • exterior field with losses; • 2-D absorber. A plane wave pe is incident with a polar angle ˜e . 1-D absorber: The absorber is composed of cell groups with width T = N· L (for QRD) or T = (N−1) · L (for PRD). The cell raster is L = 2h + w; 2h = cell width; w = thickness of walls between cells. The absorber is treated as a periodic structure with period length T. Fundamental capillary mode in the cells: Field in front of the absorber: pe (x, z) = Pe · ej (−xkx +z kz ) , p(x, z) = pe (x, z) + ps (x, z) kx = k0 sin ˜e with n = 0 + n

;

ps (x, z) =

+∞ n=−∞

An · e−‚n z · e−jn x

(1)

; kz = k0 cos ˜e ;

2 T

;

(2) ‚n = (sin ˜e + n · Š0 /T)2 − 1; ‚0 = j kz = j k0 cos ˜e . k0

Index range of radiating space harmonics (the other harmonics are surface waves): −

T T (1 + sin ˜e ) ≤ ns ≤ (1 − sin ˜e ). Š0 Š0

At the surface: ⎡ p(x, 0) = ⎣Pe + A0 +

(3)

2 ⎤ x −jn T ⎦ e−j kx x , An · e

(4)

n=0

⎡ −Z0 vz (x, 0) = ⎣(Pe − A0 ) cos ˜e + j

n=0

2 ⎤ x ‚n −jn T ⎦ e−j kx x . An · e k0

(5)

Boundary condition: (Pe − A0 ) cos ˜e + j

n=0

⎡ 2 ⎤ 2 x x − jn −jn ‚n T = G(x) · ⎣Pe + A0 + T ⎦ (6) An · e An · e k0 n=0

Compound Absorbers

H

481

or, with the Fourier analysis of G(x):

G(x) =

+∞

2 −jn x T gn · e

;

n=−∞

1 gn = T

T

2 +jn x T dx. G(x) · e

(7)

0

System of equations for the An (ƒm,n = Kronecker symbol): +nhi n=−nhi

‚n An · g−m−n − j ƒm,−n = Pe (ƒm,0 cos ˜e − g−m ) k0

With the admittance profile

G(xk ) =

in the kth cell with

k0tk =

;

m = −nhi , . . . , +nhi . (8)

tanh (an k0tk ) Zan

f · fr N

mod(k 2 , N) mod(k , N)

(9) ; ;

QRD , PRD

(10)

where an , Zan are the normalised capillary propagation constant and wave impedance. The Fourier components are: N−1

gn =

2h sin (2n h/T) G(xk ) e−jn(2k+1)/N T 2n h/T

;

QRD ,

k=0

(11)

N−1

2h sin (2n h/T) G(xk ) e−jn(2k−1)/(N−1) gn = T 2n h/T

;

PRD.

k=1

The absorption coefficient (˜e ) is: 2 A0 1 Ans 2 1 − (sin ˜e + ns Š0 /T)2, (˜e ) = 1 − − Pe cos ˜e n =0 Pe

(12)

s

where the summation index ns spans the range of radiating space harmonics, but not ns = 0. The second term |A0/Pe |2 represents the geometrical reflection |rg |2; therefore the third term represents the non-geometrical reflection |rs |2 by scattering. The following diagram shows (˜e ) over f; first,without losses in the exterior space taken into account (thick, full line), second, the exterior losses considered by the substitution k0 → k ,where k is the free field wave number of viscous and heat-conducting air (thin, dashed line). This substitution surely under-estimates by far the real losses; however, the strong modification of (˜e ) by the substitution indicates the sensitivity to losses.

H

482

1 α(ϑe) 0.8

Compound Absorbers

1-dim. PRD ; N=11 ; nhi=22

0.6 0.4 0.2 0 50 200

400

600

800

1000 1200 f [Hz]

Absorption coefficient (˜e ) of a 1-D PRD; solid: no losses in exterior space; dashed: with free field wave number k of lossy air used in the exterior space. Input parameters: fr = 285[Hz]; ˜e = 45◦ ; h = 0.03[m]; w = 0.003[m]; R = 0; nhi = 22; f = 20[Hz] This result suggests that with a small (normalised) additional flow resistance R in the orifices (e.g. by a wire mesh) a good absorber could be realised.

1

1-dim. mofif. PRD

α

0.8 0.6 0.4 0.2 0 50 200

|rs|2 400

600

800 1000 1200 f [Hz]

A 1-D PRD as above, but with an additional flow resistance R=0.4 in the cell orifices. Input parameters: fr = 285[Hz]; ˜e = 45◦ ; h = 0.03[m]; w = 0.003[m]; R = 0.4; N = 15; nhi = 30 The dependence of this high absorption on the polar angle ˜e of sound incidence is depicted in the next 3-D graph of (f , ˜e ) over frequency f and angle of incidence ˜e .

Compound Absorbers

H

483

1-dim. modif. PRD

α (f, ϑe) 1

0.8 0.6 0.4 0 0.2

20

0

40 1000

60

750 f [Hz]

500

ϑe

80

250 50

0 9

Absorption coefficient (˜e ) of the 1-D PRD from above, plotted over f and ˜e Input parameters: fr = 285[Hz]; ˜e = 0 − 90◦ ; h = 0.03[m]; w = 0.003[m]; R = 0.4; N = 15; nhi = 22 Losses and higher modes in the cells of a 1-D absorber: Assume density waves of lossy air in the exterior space, i.e. with a free field wave number k ≈ k0, and assume capillary wave modes in the cells. The field formulation in the k-th cell is: pk (x, z) = e−j0 xk

∞

Bk,n · cosh (n (z − tk )) · qn (x − xk )

(13)

n=0

with 0 = k · sin ˜e and the mode profiles ⎧ ⎨ cos (—n (x − xk )) ; n = ne qn (x − xk ) = ⎩ sin (—n (x − xk )) ; n = no

(14)

for the symmetrical even modes (ne = 0, 2, 4. . .) and the anti-symmetrical odd modes (no = 1, 3, 5. . .). The characteristic equation for the —n = —n of symmetrical modes is: 2

2

[(— h) − (k h) ]

Ÿ −1 Ÿ

tan

−

(— h)2 − (k h)2 + (kŒ h)2 + — h · tan — h √ ...

Ÿ % √ (— h)2 − (k h)2 + (k h)2 · tan . . . = 0, (15) Ÿ

484

H

where

√ . . . denotes the nearest root and

Compound Absorbers

(k h)2 1 − ‰ (k h)2 /(k0h)2 Ÿ = Ÿ (kh)2 1 − ‰ (k h)2 /(k0h)2

(16)

with the free field wave numbers – –h 2 (k0 h)2 = ; (kŒ h)2 = −j h2 ; (k0 h)2 = ‰ Pr ·(kŒ h)2 , c0 Œ ⎫ 1 4 ‰ (k h)2 ⎪ ⎪ ∓ + + [. . .]2 − 2 · {. . .} ⎬ (k0 h)2 3 (kŒ h)2 (k0 h)2 = , ⎪ 4 2‰ 1 ⎪ ⎭ + (k h)2 (k0h)2 ‰(k0 h)2 3 (kŒ h)2

(17)

where [. . . ] and {. . . } under the root repeat the corresponding expressions from outside the root. The characteristic equation for the —n = —n of anti-symmetrical modes is obtained by the substitutions cos → sin; sin → − cos; tan → − cot. Start values for the numerical solution of the characteristic equation are, for even modes, —0 h = — h; —n>0 h = n/2 and, for odd modes, —n>0 h = n/2. The field in the exterior space is formulated as: p(x, z) = pe (x, z) + ps (x, z)

;

pe (x, z) = Pe · ej (−xkx +z kz ) , +∞ An · e−‚n z · e−jn x , ps (x, z) =

(18)

n=−∞

where ps is periodic in x with a period length T, and with kx = k0 sin ˜e n = 0 + n

2 ; T

;

kz = k0 cos ˜e ; ‚n = (sin ˜e + n · Š0 /T)2 − 1; k0

‚0 = j kz = j k0 cos ˜e .

(19)

The boundary conditions at the surface give a linear system of equations for the amplitudes An of the reflected space harmonics: +nhi ‚m T ƒm,n j ƒm,0 cos ˜e + (1 − ƒm,0 ) An h k0 n=−nhi ihi N−1 Sm,i · Sn,i j (m−n) 2 xk /T i i e (−1) tanh (i tk ) −◦ (20) k Qi k i=0

ihi N−1 S T · S i m,i 0,i ej m 2 xk /T (−1)i tanh (i tk ) · Pe , = j ƒm,0 cos ˜e + ◦ h k Qi k i=0 where ƒm,n is the Kronecker symbol and the circle o at ◦ indicates that cells with depth tk = 0 (which exist for a QRD) are excluded from the summation. The amplitudes Bk,n follow, with a set of solutions An , from: (Pe + A0) R0,m + An e−jn 2 xk /T Rn,m Bk,m =

n=0

Qm cosh (m tk )

.

(21)

Compound Absorbers

H

These equations use mode norms and coupling coefficients: ⎧ sin 2—n h ⎪ +h ⎪ ⎨ 1 + 2— h ; n = ne = 0, 2, . . . 1 n Qn : = qn2 (y) dy = ⎪ h sin 2— ⎪ nh ⎩ 1− ; n = no = 1, 3, . . . −h 2—n h 1 Sm,n : = h

+h

jm y

e

cos (—n y) dy; sin (—n y)

·

−h

n = ne ; n = no

(22)

m = 0, ±1, ±2, . . .

1 [ (m + —n ) sin ((m − —n )h) + (m − —n ) sin ((m + —n )h) ]; n = ne 2 h (m − —2n ) 1 = [ (m + —n ) sin ((m − —n )h) − (m − —n ) sin ((m + —n )h) ]; n = no h (2m − —2n ) =

1 Rm,n : = h

=

+h

e−jm y ·

−h

⎧ ⎨ Sm,ne ⎩

−Sm,no

;

cos (—n y) dy sin (—n y)

;

n = ne n = no

485

;

m = 0, ±1, ±2, . . .

(23)

(24)

n = ne ;

n = no .

Limit values for m → ±—n are: Sm,ne → Qne ;

Sm,no → ±j Qno .

(25)

See Mechel, Vol. III, Ch. 5 (1998) for field evaluation with 2-D absorbers.

H.22 Tight Panel Absorber, Rigorous Solution

See also: Mechel (2001); Mechel (1997)

A tight, long, elastic panel is simply supported at its borders at x = ±c. Its thickness is d, the plate material density p , the elastic parameter for bending fcr d, with the critical frequency f cr , and the bending loss factor is †. The panel covers a back volume of depth t. The characteristic propagation constant and wave impedance in the back volume are a , Za (thus the back volume may be filled with air, i.e. a → j · k0; Za → Z0 , if t is not too small, or a , Za from a flat capillary for small t, or a , Za from porous materials if the back volume is filled with such material). The front side of the arrangement is flush with a hard baffle wall. A plane wave pe is incident (normal to the z axis) with a polar angle Ÿ.

486

H

Compound Absorbers

Field formulation in front of the absorber: p(x, y) = pe (x, y) + pr (x, y) + ps (x, y)

(1)

with pr the reflected wave after reflection at a hard plane y = 0 and ps the scattered wave. pe (x, y) = Pe · e−j kx x · e−j ky y kx = k0 sin Ÿ

;

;

pr (x, y) = Pe · e−j kx x · e+j ky y ;

(2)

ky = k0 cos Ÿ.

Field pa in the back volume, with the wave and impulse equations: -

. − a2 pa = 0

;

va =

−1 grad pa a Za

(3)

as the sum of volume modes: . ak · pak (x) · cos ‰k (y − t) , pa (x, y) =

(4)

k≥0

‰k c = j

%

pak ( ) =

(a c)2 + ‚k2 ,

⎧ 9 . ⎨ cos k 2 = cos (‚k )

;

k = 0, 2, 4, . . .

9 . sin k 2 = sin (‚k )

;

k = 1, 3, 5, . . .

⎩

vay ( , y = 0) =

;

‚k = k/2 ,

−1 ak ‰k · pak ( ) · sin (‰k t). a Za

(5)

(6)

k≥0

Plate vibration velocity V(x), or V( ) with = x/c: V( ) = vn (x) =

n≥1

Vn · vn ( ) ,

⎧ 9 . ⎨ cos n 2 = cos (‚n )

;

9 . sin n 2 = sin (‚n )

;

⎩

n = 1, 3, 5, . . . n = 2, 4, 6, . . .

(7) ;

‚n = n/2.

The feature of ps (x, 0) = ps ( ) to have a finite normal particle velocity vsy (x, 0) in −c ≤ x ≤ +c and zero normal velocity outside suggests the use of elliptic-hyperbolic cylinder co-ordinates (, ˜ ) for the formulation of that component field. These coordinates follow from the Cartesian co-ordinates (x, y) by the transformation x = c · cosh · cos ˜

;

y = c · sinh · sin ˜ .

(8)

x = ±c are the positions of the common foci of the ellipses and hyperbolic branches.

Compound Absorbers

H

487

At = 0: = x/c = cos ˜ v −−−→ −vy →0

v˜ −−−→ vx →0

1 grad p −−−→ →0 c sin ˜

∂p ∂p n

+ n

˜ . ∂ ∂˜

(9)

The wave equation in these co-ordinates is written as follows: . ∂ 2p ∂ 2 p + + (k0c)2 cosh2 − cos2 ˜ · p(, ˜ ) = 0. 2 2 ∂ ∂˜

(10)

It separates for p(, ˜ ) = U(˜ ) · W() into the two Mathieu differential equations: . d2 U(z) + b − 4q cos2 z · U(z) = 0 dz2 . d2 W(z) − b − 4qcosh2 z · W(z) = 0 2 dz

(11)

with q = (k0c)2 /4 and b a separation constant. Solutions are the Mathieu functions (see Mechel (1997) for these functions). The sum pe + pr can be expanded in Mathieu functions cem (˜ ), Jcm (): pe (, ˜ ) + pr (, ˜ ) = 4Pe

∞

(−j)m cem () · Jcm () · cem (˜ ).

(12)

m=0

A formulation of ps which has the mentioned features for each term is: ps (, ˜ ) = 4 = /2 − Ÿ

∞ m=0

;

Dm (−j)m cem () · Hc(2) m () · cem (˜ )

(13)

q = (k0 c)2 /4

The cem (˜ ) are “azimuthal Mathieu functions” which are even in ˜ at ˜ = 0, and the Hc(2) m () = Jcm () − j · Ycm () are associated “radial Mathieu functions”, or “MathieuHankel functions” of the second kind, which represent outward propagating waves and

H

488

Compound Absorbers

satisfy Sommerfeld’s far field condition. They are composed by the “Mathieu-Bessel” function Jcm () and the “Mathieu-Neumann” function Ycm () like the cylindrical “Hankel functions” of the 2nd kind. The Mathieu functions depend on the parameter q. Note for later use that cem (˜ ), Jcm (), Ycm () are real functions and Jcm (0) = 0, Hc(2) m (0) = −j · Ycm (0) (where the primes indicate derivatives with respect to ). It will be important for later evaluations that the cem (˜ ) are generated as a Fourier series∗) cem (˜ ) =

+∞

. A2s+p · cos (2s + p)˜

;

m = 2r + p

s=0

;

r = 0, 1, 2, . . . , p = 0, 1

(14)

so the real Fourier coefficients A2s+p are delivered by the computing program which generates the Mathieu function. The plate vibration modes vn ( ), the back volume modes pak ( ) and the Mathieu functions cem (˜ ) are normal functions in the range of the plate with norms: 51 2 51 2 2; k=0 , Npn = vn ( ) d = 1 ; Nak = pak ( ) d = 1; k>0 −1 −1 (15) 50 2 5 2 Nsm = cem (˜ ) d˜ = cem (˜ ) d˜ = 2 − 0 The remaining boundary conditions to be satisfied are: vsy ( ) = V( ) , vay ( ) = V( ) , pe ( ) + pr ( ) + ps ( ) − pa ( ) =

n≥1

(16) Vn ZTn · vn ( ),

where, in the last condition, ZTn are modal partition impedances of the panel:

# $ fcr d ZTn f ‚n 4 ‚n 4 2 2 = 2Zm F †F +j 1−F ; Zm = p . ; F= Z0 k0 c k0 c fcr Z0

(17)

The last condition assumes that the left-hand side is expanded in plate modes vn ( ) and that the condition holds term-wise. Multiplication of the first condition with sin ˜ · cem (˜ ) and integration with respect to ˜ over (−, 0) gives: Dm =

1 −k0 c Z0 Vn · Qm,n . m 2(−j) cem () · Ycm (0) n≥1

(18)

Multiplication of the second condition with pak ( ) and integration over −1 ≤ ≤ +1 gives: ak = ∗)

−a Za Vn · Sk,n ‰k · Nak · sin (‰k t) n≥1


;

k ≥ 0.

(19)

Compound Absorbers

H

489

Multiplication of the last condition with vn ( ) and integration over −1 ≤ ≤ +1 gives: ZTn · Z0 Vn Z0

=4 (−j)m cem () · Qm,n · Pe Jcm (0) + Dm Hc(2) ak · Sk,n · cos (‰k t) , (20) m (0) −

Npn

m≥0

k≥0

and after insertion of Dm , ak the linear system of equations for Z0 Vn (Œ = 1, 2, 3, . . .):

Z0 Vn

n≥1

⎧ ⎨

⎤⎫ (2) k0cZ0 2j a Za Hcm (0) Sk,Œ · Sk,n /Nak ⎬ ⎣ ⎦ · ƒn,Œ NpŒ − · Qm,Œ Qm,n + (2) ⎩ ZTŒ m≥0 Hcm (0) k0 Z0 ‰k c · tan (‰k t) ⎭ ⎡

(21)

k≥0

= 4Pe

Z0 (−j)m cem () · Qm,Œ · Jcm (0) ZTŒ m≥0

with the Kronecker symbol ƒn,Œ . After its solution, the amplitudes Dm , ak follow from above. These equations use the following mode-coupling coefficients: Sk,n

:=

5+1 −1

pak ( ) · vn ( ) d ,

⎧ 9 . cos k 2 = cos (‚k ) ⎪ ⎪ ⎨ 9 . pak ( ) = sin k 2 = sin (‚k ) ⎪ ⎪ ⎩ ⎧ 9 . cos n 2 = cos (‚n ) ⎪ ⎪ ⎨ 9 . vn ( ) = sin n 2 = sin (‚n ) ⎪ ⎪ ⎩

;

k = 0, 2, 4, . . .

;

k = 1, 3, 5, . . . ;

n = 1, 3, 5, . . .

;

n = 2, 4, 6, . . .

; ‚k = k/2 ,

(22)

; ‚n = n/2

with the values

Sk,n =

⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎨

;

ke & ne

;

k o & no

2 (−1)(ko −ne −1)/2 (−1)(ko +ne −1)/2 ⎪ + ⎪ ⎪ ⎪ k o − ne ko + ne ⎪ ⎪ ⎪ ⎪ (ke −no −1)/2 ⎪ (−1)(ke +no −1)/2 2 (−1) ⎪ ⎪ ⎩ − ke − no k e + no

;

k o & ne

;

k e & no

(23)

H

490

Compound Absorbers

and the coupling coefficients Qm,n

:=

5+1 −1

cem (arccos ) · vn ( ) d =

⎧ 9 . cos n 2 = cos (‚n ) ⎪ ⎪ ⎨ 9 . = sin n 2 = sin (‚n ) ⎪ ⎪ ⎩

vn ( )

cem (˜ ) =

+∞ s=0

-

A2s+p · cos (2s + p)˜

.

5 0

sin ˜ · cem (˜ ) · vn (cos ˜ ) d˜ ,

;

n = 1, 3, 5, . . .

;

n = 2, 4, 6, . . .

;

m = 2r + p

;

;

‚n = n/2 ,

(24)

r = 0, 1, 2, . . . p = 0, 1

with zero values if both m and n are even or odd, and for different parities of m, n:

i s 2 2 i i! Q2r,n = A2s J1/2(‚n ) + (1 − ƒ0,s) (−1) ‚n s≥0 (2i)! ‚ n i=1 (25) i−1 . : 4s2 − 4k 2 , · Ji+1/2 (‚n ) Q2r+1,n

k=0

s 2 i! A2s+1 J3/2 (‚n ) + (1 − ƒ0,s) (−1)i ‚n s≥0 (2i)! i=1 i i . 2 :2 2 (2s + 1) − (2k − 1) · · Ji+3/2 (‚n ) ‚n k=1

=

with Bessel functions of half-integer orders Ji+1/2 (z). The sound absorption coefficient (Ÿ) = ¢a /¢e is evaluated with the effective incident power ¢e (per unit panel length): ¢e =

c · cos Ÿ |Pe |2 , Z0

(26)

and the absorbed effective sound power: ⎧ +1 ⎫ ⎬ c ⎨ ∗ ¢a = Re (pe + pr + ps ) · vsy d ⎭ 2 ⎩ −1

/. ∗0 −4c 2 cem () · Ycm (0) · Re Pe · Jcm (0) + Dm · Hc(2) = m (0) · Dm . k0 c Z0 m≥0

(27)

Special case: back volume is locally reacting i.e. its input impedance (at y = 0) is Zb = Za · coth (a t).

(28)

The boundary conditions then become: pe ( ) + pr ( ) + ps ( ) = Vn (ZTn + Zb ) · vn ( ) , vsy ( ) =

n≥1

n≥1

Vn · vn ( ).

(29)

Compound Absorbers

H

491

The system of equations for the plate mode amplitudes Z0 Vn will be (Œ = 1, 2, 3, . . .):

2j k0c Hc(2) Z0 m (0) Z0 Vn · ƒn,Œ − Qm,Œ · Qm,n · ZTŒ + Zb m≥0 Hc(2) m (0) n≥1 = 4Pe

Z0 (−j)m cem () · Jcm (0) · Qm,Œ . ZTŒ + Zb m≥0

(30)

The amplitudes Dm of the scattered field are evaluated as above, as is the absorption coefficient. An alternative form for the sound absorption coefficient in this special case is: ; +1 c ∗ ¢a = Re Vn (ZTn + Zb ) · vn ( ) · Vn · vn ( ) d 2 n≥1 n≥1 −1 ZTn + Zb c (31) Npn Re · |Z0 Vn | 2 , = 2Z0 n≥1 Z0 ZTn + Zb 1 (Ÿ) = Re · |Z0 Vn /Pe | 2 . 2 cos Ÿ n≥1 Z0 The numerical examples for a plywood panel absorber use as constant parameters d = 6 mm, p = 700 kg/m3, fcr d = 20 Hz · m, † = 0.02; the back volume with depth t = 10 cm is filled with glass fibre material having a flow resistivity ¡ = 2500 Pa · s/m2 . The angle of sound incidence is Ÿ = 45◦ . The used upper limits for the modes are nhi = 10; khi = mhi = 8. The plots of (Ÿ) over the frequency f also contain (as dashed curves, for orientation) the absorption coefficient for an infinite panel.

αΘ

Sound absorption coefficient (Ÿ) for Ÿ = 45◦ of a plywood panel absorber with c = 0.2 m; modal analysis: solid line; infinite panel: dashed line.Input parameters:c = 0.2[m]; d = 0.006[m]; t = 0.1[m]; p = 700[kg/m3]; † = 0.02; Zr/ Z0 = 0; ¡ = 2500[Pa · s/m2 ]; nhi = 10; mhi = 8; khi = 8

492

H

Compound Absorbers

Magnitude of the sound pressure field for the above absorber, at f = 50 Hz; sound incidence is from the side of negative x/c values

αΘ

Sound absorption coefficient (Ÿ) for Ÿ = 45◦ of a plywood panel absorber with c = 0.5 m and locally reacting back volume; modal analysis: solid line; infinite panel: dashed line. Input parameters: c = 0.5[m]; d = 0.006[m]; t = 0.1[m]; p = 700[kg/m3]; † = 0.02; Zr/ Z0 = 0; ¡ = 2500[Pa · s/m2 ]; nhi = 10; mhi = 8; khi = 8

Compound Absorbers

H

493

H.23 Tight Panel Absorber, Approximations

See also: Mechel (2001)

The focus and symbols of this section are as in the previous > Sect. H.22. This section describes approximations which avoid the evaluation of Mathieu functions. The principal step in such approximations is the subdivision of the boundary value problem in two subtasks. The first subtask finds the plate mode amplitudes with the assumption that ps ( ) can be neglected compared to pe ( )+pr ( ). This sum is supposed to be the driving force on the front side for the plate motion. The assumption is plausible if the surface impedance of the plate is not too small (i.e.outside resonances).The second step then evaluates the absorbed power with the plate mode amplitudes Vn found in the first step. If the back volume is supposed to be bulk reacting (i.e. possible sound propagation parallel to the plate) the boundary conditions of the first subtask are: pe ( ) + pr ( ) − pa ( ) = 2Pe · e−j kx c − pa ( ) = Vn ZTn · vn ( ) , vay ( ) =

n≥1

n≥1

Vn · vn ( ),

(1)

where in the first equation the left-hand side is supposed to be expanded in plate modes vn ( ) so that modal plate partition impedances ZTn ( > Sect. H.22) can be applied. It should be noticed that this equation describes the excitation of the plate by a distributed force without radiation load on the side of excitation. Multiplication of the first equation with vŒ ( )(Œ = 1, 2, 3, . . .) and integration over −1 ≤ ≤ +1 yields the equations: VŒ ZTŒ NpŒ = 2Pe · RŒ − ak · Sk,Œ · cos (‰k t) (2) k≥0

with the mode coupling coefficients Sk,n from the previous section, and the new coefficients: ⎧ 4n (−1)n cos (kx c) ⎪ ⎪ +1 ; n = odd ⎪ ⎨ (n)2 − 4 (kx c)2 . (3) Rn : = e−j kx c · vn ( ) d = ⎪ ⎪ − 4j n (−1)n/2 sin (kx c) ⎪ −1 ⎩ ; n = even (n)2 − 4 (kx c)2 Insertion of the back volume mode amplitudes ak which follow from the second boundary conditions as in the previous > Sect. 22: −a Za Vn · Sk,n ; k ≥ 0 (4) ak = ‰k · Nak · sin (‰k t) n≥1 gives the following system of equations to be solved for Vn if the back volume is bulk reacting: ⎤ ⎡ Sk,n · Sk,Œ · cot (‰k t) Z a a ⎦ = 2Pe · RŒ Vn · ⎣ƒn,Œ · NpŒ − (5) Z ‰ · N ZTŒ TŒ k ak n≥1 k≥0

(ƒn,Œ is the Kronecker symbol).

H

494

Compound Absorbers

If the back volume is locally reacting, the only remaining boundary condition becomes: pe ( ) + pr ( ) = Vn (ZTn + Zb ) · vn ( ). (6) n≥1

Multiplication and integration as before gives the explicit expressions for Vn : Vn =

2Pe · Rn . (ZTn + Zb ) Npn

(7)

The second subtask determines the absorbed sound power, assuming that the plate velocity V( ), expanded in Vn · vn( ), is a given oscillation (i.e. again without consideration of a possible back reaction of radiation on the oscillation). In a first variant of this step one applies the product (pe ( ) + pr ( )) · V∗ ( ) for the evaluation of the power which (pe ( ) + pr ( )) feeds into the plate. Thus one makes the same error twice, because (pe ( ) + pr ( )) is not the true exciting pressure: ¢a1

c = 2

+1 Re

-

;

. ∗ pe ( ) + pr ( ) · Vn · vn ( )

d = Pe · c

n≥1

−1

0 / Re V∗n · Rn .

(8)

n≥1

In a second variant one takes into account the sound pressure radiated by the plate with the given velocity profile V( ) in a baffle wall. We write ps for the radiated sound (V( ) is counted positive in the direction oriented into the plate; thus the plate in fact energy of ps ). The absorbed power is given by the integral over - is a sink .for the pe ( ) + pr ( ) · V∗ ( ) + ps ( ) · V∗ ( ). The first term gives the power contribution ¢a1; the second term is the absorbed effective power ¢as due to ps . This can be obtained by (see Heckl (1977)): c ¢as = 2

+1 −1

k0 Z0 = 4

/

0

k0 Z0 Re ps ( ) · V ( ) d = 4 /2

∗

k0 −k0

|ˆv(k1 )| 2 dk1 k02 − k12

(9)

|ˆv(k0 cos •)| 2 d•,

−/2

where vˆ (k1 ) is the wave number spectrum of V(x), which follows by a Fourier transform (L = 2c is the plate width): +∞ +1 −j k1 x −j k1 x V(x) · e dx = V(x) · e dx = c V( ) · e−j k1 c· d , vˆ (k1) = −∞

L

(10)

−1

and the back transformation; 1 V(x) = 2

+∞ vˆ (k1) · e+j k1 x dk1 . −∞

(11)

Compound Absorbers

H

495

In the present application V( ) is the sum of terms Vn ·vn ( ). The wave number spectrum vˆ n (k1 ) of vn ( ) is: ⎧ 4n (−1)n cos (k1c) ⎪ ⎪ +1 ; n = odd ⎪ ⎨ c (n)2 − 4 (k1c)2 −j k1 c , (12) · vn ( ) d = vˆ n (k1 ) = c e ⎪ − 4j n (−1)n/2 sin (k1 c) ⎪ ⎪ −1 ⎩ c ; n = even (n)2 − 4 (k1c)2 and the contribution to the absorbed power becomes: k0 Z0 ¢as = 4

2 /2 Vn · vˆ n (k0 cos •) d•.

(13)

n≥1

−/2

The integral must be evaluated numerically. A third variant of the second subtask completes the absorbed intensity to (pe ( )+pr ( )+ ps ( )) · V∗ ( ) but does not worry about possible scattering at the border lines between plate and baffle wall. The scattered field ps is expanded in plate modes ps (x, y) = dn · vn (x) · fn (y). (14) n≥1

A plausible form for f n (y) representing outgoing waves is f n (y) = exp(j—n y); the terms satisfy the wave equation and Sommerfeld’s condition if (—n c)2 = (k0 c)2 − ‚n2 ; Im{—n } ≤ 0. From vsy ( ) = V( ) one gets: dn = −

k0 Z0 Vn . —n

(15)

The absorbed power is: c ¢a = 2 c ¢as = 2

+1 Re −1 +1

Re

-

; . ∗ pe ( ) + pr ( ) + ps ( ) · Vn · vn ( ) d = ¢a1 + ¢as , n≥1

dn · vn ( ) ·

V∗n

· vn ( )

; d

(16)

n≥1 n≥1 −1 k0 −c Re · Npn · |Z0 Vn | 2 . = 2Z0 n≥1 —n

The approximation results are acceptable, except in or near plate resonances. A very simple approximation, serving more for orientation than as approximation, assumes the plate to be infinitely wide (L → ∞). Then the absorption coefficient (Ÿ) follows from the reflection factor R as (Ÿ) = 1 − |R|2 with R=

(ZT + Zb ) · cos Ÿ − Z0 , (ZT + Zb ) · cos Ÿ + Z0

(17)

496

H

Compound Absorbers

where ZT is the plate partition impedance of an infinite panel:

. ZT = 2Zm F †F2 sin4 Ÿ + j 1 − F2 sin4 Ÿ Z0

;

F=

f fcr

;

Zm =

fcr d p , Z0

(18)

and Zb is the input impedance of the back volume, which is given, for a locally reacting volume, by: Zb = Za · coth (a t);

(19)

and for a bulk reacting volume by: an Zan Zb = · coth (k0t an cos ‡1) Z0 an cos ‡1 % 2 + sin 2 Ÿ. an cos ‡1 = an

H.24

;

an = a /k0

;

Zan = Za /Z0

(20)

Porous Panel Absorber, Rigorous Solution


The object here is like the absorber in > Sect. H.22, except now the panel is supposed to be perforated with a porosity . Field formulations and symbols will be taken from there. A tight, long, elastic panel is simply supported at its borders at x = ±c. Its thickness is d, the plate material density p , the elastic parameter for bending fcr d, with the critical frequency f cr , the bending loss factor is †. The panel covers a back volume of depth t. The characteristic propagation constant and wave impedance in the back volume are a , Za (thus the back volume may be filled with air, i.e. a → j · k0 ; Za → Z0 if t is not too small, or a , Za from a flat capillary for small t, or a , Za from porous materials if the back volume is filled with such material). The front side of the arrangement is flush with a hard baffle wall.A plane wave pe is incident (normal to the z axis) with a polar angle Ÿ.

To make the perforation tractable in the analysis,it is supposed that a“micro-structured” perforation is applied. This means that the diameter of the perforations and their distances are small compared with both the sound wave length and the panel width. We further suppose a homogeneous distribution of the perforation over the panel (possibly except narrow border areas).

Compound Absorbers

H

497

One first has to fix how the acoustic qualities of the perforation and of the perforated panel have to be defined. The perforation changes the mechanical parameters, effective plate material density p and bending modulus B of the plate: √ 1− p → p (1 − ) ; B → B · (1 − ) ; fcr d → fcr d √ . (1) 1− The indicated change in B is for nearly square holes and perforation raster; more sophisticated relations can be derived for other geometries. The symbol ZT will be used for the partition impedance of an equivalent tight plate, evaluated with these parameters and defined by p= ZT · vp , where vp is the velocity of this tight panel. The pores are characterised with an impedance Zr = Zr + j · Zr determined by p = Zr · vr , where p is the pressure difference driving the average velocity vr through the perforated plate at rest. Preferably one determines Zr experimentally (because the technical roughness of hole walls and the effect of rounding of the hole corners are difficult to describe analytically; an exception could be straight, very fine holes for which the real part Zr also can be determined precisely from the theory of capillaries), and the imaginary part Zr by evaluation from: k0 a d e i Zr + + , (2) = Z0 a a a where a is any representative hole dimension (mostly its radius) and e , i are the exterior and interior end corrections, respectively, where the important distinction should be made for the interior end correction whether the interior orifice ends in air or on a porous material. Zr can be realised either by the friction force and end corrections of the pores themselves and/or by the impedance of thin porous sheets (e.g. a fine wire mesh) covering the perforation orifice. In the latter case it must be distinguished whether the sheet is forcelocking with the plate or not. The assumption of the micro-structure of the perforation implies that the sound pressure distributions along the panel surfaces do not have significant ripples by the perforation pattern. The formulations of the component fields therefore remain as in > Sect. H.22. For the determination of the unknown amplitudes ak , Dm ,Vn one needs three boundary conditions as in > Sect. H.22, but now modified for the parallel volume flow through the panel and the pores. The sketch indicates, in a representative area element S, the distribution of the velocities on the plate and the holes.

H

498

Compound Absorbers

(1–σ)·V v

ZT/(1–σ) Zr σ·vh = vr Δp

The average velocity is: v = (1 − ) · V + · vh = (1 − ) · V + vr = (1 − ) · V + p/Zr .

(3)

The second sketch shows the equivalent network for the perforated panel. The effective impedance is: Zeff =

Zr · ZT /(1 − ) , Zr · ZT + (1 − )

(4)

and the first boundary condition becomes: p = Zeff · v =

Zr · ZT /(1 − )

(1 − ) · V + p/Zr , Zr + ZT /(1 − )

(5)

or with a transformation, if = 1: p

Zr · ZT Zr = · V. Zr + ZT /(1 − ) Zr + ZT /(1 − )

This corresponds to the boundary condition with tight plates in the effective plate partition impedance: ZTeff =

Zr · ZT . Zr + ZT /(1 − )

In these relations

p = pe + pr + ps − pa .

The other boundary conditions of matching velocities become: ! vsy . v = (1 − ) · V + p/Zr = vay

(6) > Sect. H.22

if we use

(7) (8)

(9)

Up to now it was tacitly assumed that the friction force on the plate could be neglected compared to the driving force from p. This assumption is plausible if the holes of the perforation are sufficiently wide (say a few millimetres) and the porosity is not too high. But for large porosity values with narrow pores or with a force-locking, vibrating wire

Compound Absorbers

H

499

mesh, the friction exerts an additional force Fr on the panel if the relative velocity vh − V = 0. The driving force on the panel in a section S becomes: F = Fp + Fr = ZT · V Sp + Zr · (vh − V) Sh , or, after division with the plate area Sp : 3 2 vr − ·V p = ZT + Zr 1− V p/Zr − ·V = ZT + Zr 1 − p/ZT ZT = ZT + Zr − · V. 1 − Zr

(10)

(11)

The expression in the last brackets replaces ZT in the above equations if friction force coupling must be taken into account. After these preparations one has the following boundary conditions: ZTeff n · Vn · vn ( ) , p =

(12)

n≥1

vsy = (1 − ) · V + p/Zr ,

(13)

vay = (1 − ) · V + p/Zr .

(14)

Combination of the first equation with the second or third gives: ZTeff n Vn · vn ( ) (1 − ) + , vsy = Zr n≥1 ZTeff n . vay = Vn · vn ( ) (1 − ) + Zr n≥1

(15)

(16)

Multiplication of eq. (15) with sin ˜ · cem (˜ ) and integration over 0 ≤ ˜ ≤ , and multiplication of the eq. (16) with pak ( ) and integration over −1 ≤ ≤ +1 using from > Sect. H.22: vsy ( = 0, ˜ ) =

vay ( , y = 0) =

−4 Dm (−j)m cem () · Ycm (0) · cem (˜ ), k0 c Z0 sin ˜ m≥0 −1 ak ‰k · pak ( ) · sin (‰k t) a Za

(17)

(18)

k≥0

gives, respectively: −2 ZTeff n m (−j) cem () · Ycm (0) · Dm = , Z0 Vn · Qm,n (1 − ) + k0 c Zr n≥1

(19)

500

H

Compound Absorbers

ZTeff n −Nak . ‰k · sin (‰k t) · ak = Z0 Vn · Sk,n (1 − ) + a Za /Z0 Zr n≥1

(20)

With p = pe + pr + ps − pa inserted into the boundary (12) condition and multiplication with vŒ ( )(Œ ≥ 1), integration over −1 ≤ ≤ +1 leads, with Dm , ak inserted from (19), (20) above, to the key system of equations for Z0Vn (an overbar over impedances indicates normalisation with Z0 ): Z0 Vn n≥1 2k0c (1 − ) 1 Hc(2) (0) · Qm,Œ · Qm,n · ƒn,Œ NpŒ + + (−j)m cem () m Ycm (0) ZTeff Œ Zr m≥0 (21) (1 − ) 1 cot (‰k t) − a Za + · Sk,Œ · Sk,n ZTeff Œ Zr k≥0 ‰k · Nak 4Pe = (−j)m cem () · Jcm (0) · Qm,Œ . ZTeff Œ m≥0 If one compares this system with the corresponding system of equations in for tight panels, one sees the transition due to the perforation: 1 (1 − ) 1 → + . ZTŒ ZTeff Œ Zr The amplitudes Dm , ak follow from (19), (20) with the solutions Z0 Vn as: k0 c ZTeff n Z0 Vn · Qm,n (1 − ) + , Dm = − 2 (−j)m cem () · Ycm (0) n≥1 Zr ZTeff n a Za /Z0 ak = − . Z0 Vn · Sk,n (1 − ) + Nak ‰k · sin (‰k t) n≥1 Zr

> Sect. H.22

(22)

(23)

The numerical examples are for a plywood panel absorber with d = 6 mm, p = 700 kg/m3, f cr d = 20 Hz · m, † = 0.02 and a porosity = 0.2; the panel width is L = 2c = 0.4 m; the back volume with t = 10 cm is filled with glass fibres with ¡ = 2500 Pa · s/m2 . Sound incidence is under Ÿ = 45◦ . The impedance of the pores is set to a fixed value Zr = 10 · Z0. The first diagram is evaluated without friction force coupling, the second diagram with friction force coupling. Both diagrams show as dashed curves the absorption coefficients for an infinite porous panel with otherwise identical parameters.

Compound Absorbers

H

501

alfa, bulk, perfor. 4

3

2

1

0 20

50

100

200 f[Hz]

500

1000

2000

Sound absorption coefficient (Ÿ) for a porous plywood panel absorber without friction coupling. Solid line: finite-size panel; dashed line: infinite panel. Input parameters: Ÿ = 45◦ ; c = 0.2 [m]; = 0.2; d = 0.006 [m]; t = 0.1 [m]; p = 700 [kg/m2]; fcrd = 20 [Hz · m]; † = 0.02; ¡ = 2500 [Pa · s/m2 ]; Zr /Z0 = 10

alfa, bulk, perfor. 1 0.8 0.6 0.4 0.2 0 20

50

100

200 f[Hz]

500

1000

2000

Sound absorption coefficient (Ÿ) for a porous plywood panel absorber as above, but with friction coupling. Solid line: finite-size panel; dashed line: infinite panel The difference in with or without friction coupling is rather high in the displayed examples because of the rather high value of Zr = 10 · Z0 ; it decreases with decreasing flow resistance of Zr .

502

H

Compound Absorbers

References Heckl, M.: Abstrahlung von ebenen Schallquellen. Acustica 37, 155–166 (1977) Mechel, F.P.: he wide-angle diffuser – a wide-angle absorber? Acustica 81, 379–401 (1995) Mechel, F.P.: Schallabsorber, Vol. II, Ch. 10: Sound in capillaries. Hirzel, Stuttgart (1995) Mechel, F.P.: Schallabsorber, Vol. II, Ch. 18: Plates with slits and resonators with slit plates, without losses. Hirzel, Stuttgart (1995)

Mechel, F.P.: Schallabsorber, Vol. II, Ch. 24: Helmholtz resonator with additional losses. Hirzel, Stuttgart (1995) Mechel, F.P.: Schallabsorber, Vol. II, Ch. 26: Foil absorbers Hirzel, Stuttgart (1995) Mechel, F.P.: Mathieu Functions; Formulas, Generation, Use. Hirzel, Stuttgart (1997) Mechel, F.P.: Schallabsorber, Vol. III, Ch. 5: Wideangle diffusors. Hirzel, Stuttgart (1998)

Mechel, F.P.: Schallabsorber, Vol. II, Ch. 19: Slit plates and slit plate resonators with viscous and caloric losses. Hirzel, Stuttgart (1995)

Mechel, F.P.: About the partition impedance of plates, shells, membranes. Acta Acustica 86, 1054– 1058 (2000)

Mechel, F.P.: Schallabsorber, Vol. II, Ch. 21: Perforated plates (circular holes), and resonators with perforated plate. Hirzel, Stuttgart (1995)

Mechel, F.P.: Panel absorber. J. Sound Vibr. 248, 43– 70 (2001)

Mechel, F.P.: Schallabsorber, Vol. II, Ch. 23: Slit plates and perforated plates on absorber layers. Hirzel, Stuttgart (1995)

Schroeder, M.R., Gerlach, R.E.: Diffuse sound reflection surfaces. 9th ICA, Madrid, paper D8 (1977)

I Sound Transmission This chapter deals with sound transmission through objects like porous absorber material layers (“noise barriers”), slits and holes in walls, wide passages which nevertheless are sound insulating (“noise sluices”), plates and multiple leaf walls, suspended ceilings, office fences, etc.

I.1 “Noise Barriers”


Flanking ducts,e.g.cable ducts or plenum ducts of suspended ceilings,may be the critical path for sound transmission between neighbouring rooms. It may be difficult to install a well-fitting partition wall in the duct, but it is easy to fill the duct to some length d with a “plug” of porous absorber material. Let a , Za be the characteristic propagation constant and wave impedance of the material.

Sound fields for normal incidence: In zone I:

p0 = e−j k0 x + r0 e+j k0 x ,

(1)

Z0 v0 = e−j k0 x − r0 e+j k0 x . In zone II:

In zone III:

pi = A · cosh (a x) + B · sinh (a x), −1 Z0 v i = [A · sinh (a x) + B · cosh (a x) ]. Za /Z0 pt = t e−j k0 x , Z0 vt = t e−j k0 x .

(2)

(3)

Za (1 − r0 ), (4) Z0 2 .(5) and for the sound transmission factor: te−jko d = Za Z0 sinh (a d) 2cosh (a d) + + Z0 Za The boundary conditions give:

A = 1 + r0

;

B=−

I

504

Sound Transmission

The transmission coefficient ‘ = |t e−j k0 d |2 , and from that the transmission loss R = −10 · lg‘ is: 2 R = −20 · lg 2 cosh (a d) + (Za /Z0 + Z0 /Za ) sinh (a d)

[dB].

(6)

It is advisable to take for the evaluation of a , Za an effective absorber variable Eeff which takes the vibration of the material matrix (with bulk density RG) into account: Eeff = 0 f /¡eff = E −

j 0 . 2 RG

(7)

The diagram compares sound transmission loss values R for layers of basalt wool with different thickness d, from measurements (points) and from the present evaluation. 0 RG ¡ E f

= = = = =

density of air; bulk density of porous material; flow resistivity of material; 0 f /¡; frequency

[dB] 70 d [m] = R 60 0.88 0.72 50 0.56 0.40 40 0.24 0.08 30

Basalt wool d [m] = 0.88 0.72 0.56

0.40

0.24

20

0.08

10 0 50

100

200

500

1k

2k

5k f [Hz]

10k

Oblique sound incidence: (under polar angle Ÿ) Internal angle Ÿi : (with an = a /k0; Zan = Za /Z0 ) j sin Ÿ, an 2 + sin2 Ÿ. cos Ÿi = an

sin Ÿi = an

(8)

Sound Transmission

I

505

Sound fields: pe = Pe · e−jk0 (x cos Ÿ + y sin Ÿ) , pr = r0 Pe · e−jk0 (−x cos Ÿ + y sin Ÿ) ,

(9)

pi = Pi1 · e−a (x cos Ÿi + y sin Ÿi ) + Pi2 · e−a (−x cos Ÿi + y sin Ÿi ) , pt = Pt · e−jk0 (x cos Ÿ + y sin Ÿ) . Reflection and transmission factors: r1 =

1−z 1+z

t1 = 1 + r1 =

;

r0 = −r1 2 1+z

t = t0 t1 e−y = e−y

;

1 − e−2y (1 − z2)(1 − e−2y ) = − , 1 − r12 e−2y (1 + z)2 − (1 − z)2 e−2y t0 =

1 − r1 2z2 = , 2 −2y (1 + z)2 − (1 − z)2 e−2y 1 − r1 e

(10)

4z (1 + z)2 − (1 − z)2e−2y

with abbreviations:

y: = a d cos Ÿi = k0d · an · cos Ÿi

;

z: = Zan

cos Ÿ . cos Ÿi

(11)

For a given transmission coefficient ‘(Ÿ) = |t(Ÿ)|2, the sound transmission coefficient with diffuse sound incidence is: Ÿ max

in three dimensions: ‘3D−diff = 2

‘(Ÿ) cos Ÿ sin Ÿ dŸ,

(12)

0 Ÿ max

in two dimensions:

‘2D−diff =

‘(Ÿ) cos Ÿ dŸ.

(13)

0

Transmission loss values for different kinds of excitation (with a = a + j · a ): Plane wave excitation 1 1 Za Z0 . 1+ with normal incidence: R⊥ ≈ 8.68 · a d + 20 log + 2 2 Z0 Za Conphase excitation 1 Z0 with given v0: 1+ Rv0 ≈ 8.68 · a d + 20 log . 2 Za Conphase excitation 1 Za 1+ with given p0 : Rp0 ≈ 8.68 · a d + 20 log 2 Z0 .

(14) (15) (16)

506

I.2

I

Sound Transmission

Sound Transmission through a Slit in a Wall


Let a slit of width 2a be in a hard wall of thickness d. A plane wave pe is incident at the angles indicated in the graph. The slit is possibly filled with a porous material having the normalised characteristic values an , Zan . For air in the slit: an → j; Zan → 1. The slit orifices may be covered with (poro-elastic) foils with surface mass densities m1 , m2 . These can represent plastic sealing masses.

The sound field on the front side has the components: p1 = pe + pr + ps pe = incident plane wave; pr = reflected wave from a hard wall; ps = scattered wave

(1)

Sound Transmission

I

507

with the following formulations: pe (x, y, z) = Pe · e−jk0 (x · sin Ÿi cos ¥i + y · sin Ÿi sin ¥i + z · cos Ÿi ) ,

(2)

pr (x, y, z) = Pe · e−jk0 (x · sin Ÿi cos ¥i + y · sin Ÿi sin ¥i − z · cos Ÿi ) and −j–0 V1 · 2a −jkx x ps (x, y, z) = ·e · 2

+∞ −∞

√ 2 2 sin (a) e−jy+z −‚ d, a 2 − ‚ 2

(3)

where V1 is the average particle velocity in the entrance orifice and kx = k0 · sin Ÿi · cos ¥i ,

(4)

‚ 2 = k02 − kx2 . The waves in the slit are: pi1 (x, z) = Pi1 · e−a (x · sin Ÿ2 + z · cos Ÿ2 ) ,

(5)

pi2 (x, z) = Pi2 · e−a (x · sin Ÿ2 − z · cos Ÿ2 ) with the internal angle Ÿ2 from sin Ÿ2 j k0 cos ¥i j cos ¥i = = sin Ÿi a an

;

The transmitted wave is: j–0 V2 · 2a −jkx x ·e pt (x, y, z ) = 2

+∞ −∞

an cos Ÿ2 =

2 + sin2 Ÿ cos2 ¥ . an i i

(6)

√ 2 2 sin (a) e−jy−z −‚ d a 2 − ‚ 2

(7)

with V2 the average particle velocity amplitude in the exit orifice and shifted z coordinate z (z = z − d). Both ps and pt satisfy the boundary condition at the hard-wall surfaces. Let Z1 = j–m1 + Zr1 and Z2 = j–m2 + Zr2 be the sums of the sealing impedance and radiation impedance Zr of the orifices (see“Radiation Impedance”and End Corrections” in > Ch. F,“Radiation of Sound”). Setting the arbitrary amplitude Pe = 1, the boundary conditions give the following system of equations: ⎛

1 1

⎜ ⎜ − d cos Ÿ 2 ⎝ e a − d cos Ÿ a 2 e

1 −1

−e+a d cos Ÿ2 e+a d cos Ÿ2

Z1 −Za / cos Ÿ2 0 0

⎞ ⎛ ⎞ 0 Pi1 ⎟ ⎜ Pi2 ⎟ 0 ⎟·⎜ ⎟ −Za / cos Ÿ2 ⎠ ⎝ V1 ⎠ V2 −Z2 ⎛ 2 si (k0a sin Ÿi sin ¥i ) ⎜ 0 =⎜ ⎝ 0 0

⎞ ⎟ ⎟ ⎠

(8)

I

508

Sound Transmission

with si(z) = (sinz)/z. The average particle velocity amplitude in the exit orifice V2 is: Za · si (k0 a sin Ÿi sin ¥i ) cos Ÿ2 V2 = . (9) 2 Za Za (Z1 + Z2 ) cosh (a d cos Ÿ2 ) + + Z1 Z2 sinh (a d cos Ÿ2 ) cos Ÿ2 cos Ÿ2 2

The sound transmission coefficient ‘(Ÿi , ¥i ) is then: 2 V2 Z0 ‘(Ÿi , ¥i ) = · Re{Zr2 } · . cos Ÿi Pe

(10)

The sound transmission coefficient ‘dif for diffuse sound incidence follows by integration: 1 ‘dif = =

4

2

/2 d¥ ‘(Ÿ, ¥ ) cos Ÿ sin Ÿ dŸ,

0

0

(11)

/2

/2 d¥ ‘(Ÿ, ¥ ) cos Ÿ sin Ÿ dŸ.

0

0

The normalised radiation impedance Zrn = Zr /Z0 is (with u = 2‚a): 1 (2) 2j (2) (2) (u) + (u) S (u) − H (u) S (u) ] − (u) + Zrn = 2k0a H(2) , [ H H 0 1 0 1 0 2 u 1 u2

(12)

where H(2) n (u) are Hankel functions of the second kind and Sn (u) are Struve functions. The real and imaginary parts of Zrn are: J1 (u) Zrn = 2k0a J0 (u) − + [J1 (u) S0(u) − J0 (u) S1(u)] , u 2 (13) Y1 (u) 2 Zrn = −2k0a Y0 (u) − + (u) S (u) − Y (u) S (u)] , − [Y 1 0 0 1 u u 2 2 and an approximation for small u (with C = 0.577216 Euler’s constant) is: Zrn = k0a 1 − u 2 /24 + u4/960 − u 6/64512 + u8/6635520 j 3 − 19 u 2/144 + 7 u 4/1800 − 353 u 6/5419008 u + 413 u 8/597196800 − ln + C 2 − u 2/12 + u 4/480 − u 6/32256 2 8 + u /3317760 .

+

(14)

Approximations (for ¥i = 0): Use the set of non-dimensional parameters (with m1 = m2 = m): F = f d/c0 = d/Š0

;

A = 2a/d

;

X = ¡d/Z0

;

M = m/0 d.

(15)

Sound Transmission

I

509

For the analytical discussion below, an → j and Zan → 1 are used for an empty slit; in numerical evaluations it is better to use the propagation constant and wave impedance of a flat capillary for a , Za (see > Ch. J,“Duct Acoustics”). Neglecting terms (FA·cosŸ)n with n > 1: ‘(Ÿ) =

M + j 4 − 2C − 2 ln (FA cos Ÿ) + 2 cos Ÿ A cos Ÿ j 1 X + · {1 + sin 2 Ÿ − 22 F2 + j ‰FX } + A cos Ÿ 2 F ‰ −2 − j 42 ‰F2M2 ,

(16)

where ‰ is the adiabatic exponent of air and the porosity of the absorber material. For an empty slit, set X = 0 and ‰ → 1. For a very narrow and empty slit: ‘(Ÿ) =

−2 A 1 + 2M − 42 F2 M (1 + M) cos2 Ÿ . F

(17) ‘(Ÿ) =

A . F

(18)

(or at low frequencies, F 1):

‘(Ÿ) =

A . F(1 + 2M)2

(19)

For a narrow slit with porous material, in a thin wall (or at low frequencies, F 1):

‘(Ÿ) = 4 2 · FA/X2.

(20)

For a very narrow, empty slit without sealing: For an empty, narrow slit with sealing, in a thin wall

The frequency response curves of R(Ÿ) = −lg‘(Ÿ) evidently can have very different shapes. Θi=45° ; Φi=90° ; X=0 ; M=0 ; 2a=0.01 [m]

20 [dB] 15 R 10

A= 0.01 0.05 0.1 0.2

5 0

0.4

-5 -10 -15

0

0.2

0.4

Empty slit, oblique incidence

0.6

0.8

F

1

I

510

Sound Transmission

Θi=0 ; Φi=0 ; X=1 ; M=0

30 [dB] 25 R 20

A= 0.01 0.05 0.1 0.2

15 10

0.4

5 0

0

0.2

0.4

0.6

0.8

F

1

Filled slit, normal incidence

Θi=0 ; Φi=0 ; A=0.1 ; M=0

40 [dB] 30

8

R 20

4 2 1 0.5

10 0 -10

0

X= 0.01 0.2 0.4

0.1 0.6

0.8

F

1

Slits with different absorber material fill; normal incidence

Θi=0 ; Φi=0 ; A=0.1 ; X=2

80 [dB] 70 R 60

8 4

50

2

40

1

30

0.5

20 10 0 0

0.2

0.4

0.6

M= 0.1 0.01 0.8 1 F

Slits with absorber fill and different sealing

Sound Transmission

511

Θi=0 ; Φi=0 ; A=0.1 ; X=0 ; 2a=0.01 [m]

70 [dB] 60 R 50 40

8 4 2

30

1 0.5

20 10

0.1

0 -10

I

0

0.2

0.4

M=0.01 0.6

0.8

F

1

Empty slit with different sealing

I.3

Sound Transmission through a Hole in a Wall


A circular hole with diameter 2a is in a wall of thickness d. The hole is possibly filled with a porous material having a normalised propagation constant an and a normalised wave impedance Zan . Possibly poro-elastic foils with effective surface mass densities m1 , m2 seal the hole orifices. A plane wave pe is incident at a polar angle Ÿi . The graphs show the co-ordinates and wave components used. The sound field p1 on the front side is formulated as p1 = pe + pr + ps , where pr is the plane wave after reflection at a hard wall and ps is the scattered field.

If the hole is “empty”, i.e. filled with air, substitute for analytical discussions an → j and Zan → 1, and for numerical evaluations use the propagation constant and wave impedance for sound propagation in a circular capillary (see > Ch. J,“Duct Acoustics”).

512

I

Sound Transmission

Field formulations: pe (x, y) = Pe · e−j k0 (z cos Ÿi +y sin Ÿi ) , pr (x, y) = Pe · e−j k0 (−z cos Ÿi +y sin Ÿi ) , ps (r1 , Ÿ1 ) = j (k0 a)2 Z0 · V1 ·

(1)

−j k0 r1

J1 (k0a sin Ÿ1 ) e , k0 r1 k0a sin Ÿ1

where V1 is the average particle velocity in the front side orifice. Further, in the hole: pi1 (z) = Pi1 · e−a z

pi2 (z) = Pi2 · e+a z ,

;

(2)

and for the transmitted wave: pt (r3, Ÿ3 ) = j (k0 a)2 Z0 · V2 ·

e−j k0 r3 J1 (k0a sin Ÿ3 ) , k 0 r3 k0a sin Ÿ3

(3)

where V2 is the average particle velocity in the back side orifice. ps and pt satisfy the boundary condition at the wall. The boundary conditions for matching the field components give the following system of equations: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 1 j–m1 + Zr1 0 Pi1 2 ⎜ 1 ⎟ ⎜ Pi2 ⎟ ⎜ 0 ⎟ −1 −Za 0 ⎜ − d ⎟ ⎜ ⎟ ⎜ ⎟ (4) ⎝ e a ⎠ · ⎝ V1 ⎠ = ⎝ 0 ⎠ , −e+a d 0 −Za V2 0 e−a d e+a d 0 −j–m2 − Zr2 where Zr1 , Zr2 are the radiation impedances of circular piston radiators in a baffle wall (see > Ch. F, “Radiation of Sound”). The solutions are, with Z1 = j–m1 + Zr1 , Z2 = j–m2 + Zr2 : Pi1 /Pe = 2 Za (Za + Z2 ) e+a d /D, Pi2 /Pe = 2 Za (Za − Z2 ) e−a d /D, V1 /Pe = 4 (Za cosh (a d) + Z2 sinh (a d) ) /D,

(5)

V2 /Pe = 4 Za /D with the determinant of the matrix D = 2 [ Za (Z1 + Z2 ) cosh (a d) + (Z2a + Z1 Z2 ) sinh (a d) ].

(6)

The transmission loss R of the hole follows from the transmission coefficient ‘, which is the ratio of the transmitted effective power ¢t to the incident effective power ¢e : R = −10 lg ‘ [dB]

;

‘(Ÿi ) =

1 |Pe |2 ¢e (Ÿi ) = S · cos Ÿi · 2 Z0

;

¢t , ¢e (Ÿi ) 1 ¢t = S · Re{Zr2 } · |V2 |2 2

(7)

Sound Transmission

I

513

(S = orifice area). Thus: 2 V2 Z0 ‘(Ÿi ) = · Re{Zr2 } · cos Ÿi Pe 2 Z0 2 Za . = · Re{Zr2 } · 2 cos Ÿi Za (Z1 + Z2 ) cosh (a d) + (Za + Z1 Z2 ) sinh (a d)

(8)

The sound transmission coefficient ‘dif for diffuse sound incidence from the whole halfspace is simply ‘dif = · ‘(0) because of ‘(Ÿi ) = ‘(0)/cosŸi. (9) Approximations and special cases: (normal incidence Ÿi = 0, and m1 = m2 ; index n at impedances means normalisation) Use the set of non-dimensional parameters: F = f d/c0 = d/Š0

;

A = 2a/d

;

X = ¡d/Z0

;

M = m/0 d,

(10)

where ¡ = flow resistivity of the porous fill material (needed for the evaluation of an , Zan ). For 2k0a 1 and |a d| 1: Re{Zrn } ‘(0) ≈ 2 , Z1n + (Z2 + Z2 ) k0d an an 1n 2 Zan

(11)

and with the leading term of the radiation impedance: ⎞2

⎛ ‘(0) ≈

⎟ 1 ⎜ AF ⎟ ≈ 2 ( AF/X)2, ⎜ 2 ⎠ ⎝ 2 2 2 A F + X/(2) 2

(12)

where = porosity of the absorber fill. If additionally the hole is empty (X = 0): ⎛ ‘(0) ≈

⎞2

1 ⎜ A ⎟ ⎠ . ⎝ 4 8 A+M 3

(13)

Empty and unsealed hole (M = 0; X = 0) at low frequencies: ‘(0) ≈

92 = 0.694. 128

(14)

The following diagrams give some examples of R for normal sound incidence and some parameter combinations.

I

514

Sound Transmission

Θi=0 ; Φi=0 ; X=0 ; M=0 ; 2a=0.01 [m]

20 [dB] 15 R 10

A= 0.01 0.05 0.1 0.2

5 0

0.4

-5 -10 -15

0

0.2

0.4

0.6

0.8

F

1

Empty and unsealed holes

Θi=0 ; Φi=0 ; X=1 ; M=0

30 [dB] 25 R

A= 0.01

20 0.05 0.1 0.2

15 10

0.4

5 0

0

0.2

0.4

0.6

0.8

F

1

Filled but unsealed holes

Θi=0 ; Φi=0 ; A=0.1 ; X=0 ; 2a=0.01 [m]

70 [dB] 60 R 50 40

8 4 2

30

1 0.5

20 10

0.1

0 -10

0

0.2

0.4

Sealed but empty holes

M=0.01 0.6

0.8

F

1

Sound Transmission

515

Θi=0 ; Φi=0 ; A=0.1 ; M=0

40 [dB] 30

8

R 20

4 2 1 0.5

10 0 -10

I

0

X= 0.01 0.2 0.4

0.1 0.6

0.8

F

1

Filled but unsealed holes

Θi=0 ; Φi=0 ; A=0.1 ; X=2

80 [dB] 70 R 60 50

8 4 2

40

1

30

0.5

20 10 0 0

0.2

0.4

0.6

M= 0.1 0.01 0.8 1 F

Filled and sealed holes

I.4

Hole Transmission with Equivalent Network


Sound transmission through holes and slits in a wall is derived in the previous > Sects. I.2 and I.3 as a boundary value problem. Equal results are obtained by application of the method of equivalent networks.

The equivalent network of a sealed and filled hole (or slit) in a wall is in the pU-analogy and in the pI-analogy:

I

516

Sound Transmission

The elements are: m1 , F1; m2, F2

surface mass densities and resilience values of the orifice seals; radiation impedances of the orifices;

Zr1 , Zr2 Z3 = Za · sinh (a d); Z5 =

1 · sinh (a d); Za

1 1 cosh (a d) − 1 = ; Z4 Za sinh (a d) cosh (a d) − 1 1 . = Za Z6 sinh (a d)

(1)

Combine Z1 = Zr1 + j–m1 +

1 j–F1

;

Z2 = Zr2 + j–m2 +

1 j–F2

(2)

and ZI = Z1 +

1 1 cosh (a d) − 1 = Zr1 + j–m1 + + Za . Z6 j–F1 sinh (a d)

1 1 cosh (a d) − 1 . ZII = Z2 + = Zr2 + j–m2 + + Za Z6 j–F2 sinh (a d)

(3)

Then: V2 =

2Pe , ZI + ZII + Z5 · ZI · ZII

and the transmission coefficient of the hole is: 2 Z0 2 . · Re{Zr2 } · ‘ (Ÿi ) = cos Ÿi ZI + ZII + Z5 · ZI · ZII Insertion of the abbreviations leads to the result of the boundary value problem.

(4)

(5)

Sound Transmission

I

517

For a slit apply the substitutions: a → a cos Ÿ2

;

Za → Za / cos Ÿ2 ;

Zr,circle → Zr,strip ;

(6)

2Pe → 2Pe si (k0a sin Ÿi sin ¥i ) with si(z) = sin(z)/z. This returns the result of

I.5

>

Sect. I.2.

Sound Transmission through Lined Slits in a Wall


Sometimes joints of wall elements which form slits in the wall cannot be sealed or filled, e.g. the lower gap at a door. The quantities and relations are those of > Sect. I.2,except that a ,Za are the propagation constant and wave impedance of the least attenuated mode in a flat (silencer) duct of height a, which is lined with an absorber having a surface admittance Gy .

The lining is preferably made locally reacting (if necessary with thin partitions). Because k0a 1 holds in general, low-frequency approximations can be used for the determination of a . Channel wave components (with substitutions a → s ; Za → Zs in order to avoid confusion with material data): pi1 (x, y, z) = Pi1 · e−s

(x·sin Ÿ2 + z·cos Ÿ2 )

−s (x·sin Ÿ2 − z·cos Ÿ2 )

pi2 (x, y, z) = Pi2 · e

· cos —y y · cos —y y

;

—2y = s2 + k02.

(1)

—y a is a solution of the characteristic equation for a locally reacting lining: —y a · tan —y a = j · k0 a · Z0 Gy : = j · U. The following low-frequency approximation is applicable: 105 + 45jU ± 11 025 + 5250jU − 1605U2 2 . (—y a) = 20 + 2jU

(2)

(3)

518

I

Sound Transmission

s a = (—y a)2 − (k0 a)2 j Zs = . Z0 s /k0

The sign of the root is chosen so that the real part of is a minimum. The wave impedance Zs of the least attenuated mode is:

(4) (5)

If the lining is a bulk reacting homogeneous, porous absorber layer of thickness t with characteristic values a , Za of the material, the equation to be solved is: a Za −1 t 2 2 (—y a) − (†a) · tan (—y a)2 − (†a)2 , —y a · tan (—y a) = −j k0 Z0 a (6) (†a)2 : = (a a)2 + (k0 a)2 . A continued fraction approximation for low frequencies leads to a polynomial equation: a0 + a1 · (—y a)2 + a2 · (—y a)4 + a3 · (—y a)6 + a4 · (—y a)8 = 0

(7)

with coefficients a0 = −11025 (†a)2 − 1050 (t/a)2(†a)4, a1 = 11025 + (†a)2(4 725 + 2100 (t/a)2) + 450 (t/a)2(†a)4 − A · [11025/(t/a)2 + 4725 (†a)2 + 105 (t/a)2(†a)4 ],

(8)

a2 = − {4725+1050 (t/a)2 +(†a)2 (105 + 900 (t/a)2)+10(t/a)2 (†a)4 + A · [1050 /(t/a)2 +4725+(†a)2 (450+210 (t/a)2)+10 (t/a)2 (†a)4]}, a3 = 105 + 450 (t/a)2 +20 (t/a)2 (†a)2 − A · [450+105 (t/a)2 +20 (t/a)2(†a)2], a4 = 10 (t/a)2(A−1) and the abbreviation

A: = j

t a Za . a k0 Z0

(9) (10)

The least attenuated mode in the slit channel has a cosine profile; the matching to the exterior sound fields has to be performed with the average values of sound pressure and axial particle velocity. The boundary conditions in the orifices lead to the following system of equations: si (—y a) · (Pi1 + Pi2 ) + Zr1 V1 + 0 = 2 Pe si (ky a) j si (—y a) · (Pi1 − Pi2 ) + Z0 V1 + 0 = 0 si (—y a) · (Pi1 e−‚ + Pi2 e+‚ ) + 0 − Zr2 V2 = 0 j si (—y a) · (Pi1 e−‚ − Pi2 e+‚ ) + 0 + Z0 V2 = 0

(11)

with si(z) = sin(z)/z and the abbreviations : = s /k0 · cos Ÿ2 = (—y /k0)2 − 1 + sin2 Ÿi cos2 ¥i , ‚ : = s d · cos Ÿ2 = k0d

(12) )2

2

(—y /k0 − 1 + sin Ÿi

cos2

¥i .

The matrix determinant is: D = 2 si 2 (—y a) · [j Z0 (Zr1 + Zr2 ) · cosh ‚ + (2 Zr1 Zr2 − Z20 ) · sinh ‚ ],

(13)

Sound Transmission

I

519

and the desired average axial particle velocity V2 in the exit orifice is: 2j Z0 si (ky a) V2 . = Pe j Z0 (Zr1 + Zr2 ) · cosh ‚ + (2 Zr1 Zr2 − Z20 ) · sinh ‚

(14)

The coefficient of transmission is, with the normalised radiation impedances Zr1n , Zr2n of the orifices ( > Sect. I.2): ‘(Ÿi , ¥i ) =

2 2j · si (k0a sin Ÿi sin ¥i ) Re{Zr2n } . 2 cos Ÿi j (Zr1n + Zr2n ) · cosh ‚ + ( Zr1n Zr2n − 1) · sinh ‚

(15)

A set of non-dimensional parameters for a slit with a lining consisting of a layer of porous material (flow resistivity ¡, made locally reacting by partitions) covered with a foil of surface mass density m is: F = f d/c0 = d/Š0 ; A = 2a/d ; X = ¡ t/Z0 ; Ms = m/0 d ; T = t/d.

(16)

The frequency response curves of the sound transmission loss R = −10 · lg(‘) have a great variety of forms, depending on the parameter values. A few examples will be given below. 70 [dB] 60 R 50

Θi=45° ; Φi=45° ; T=4 ; X=1 ; M s=0 A= 0.05 0.1

40 0.2

30

0.4 0.8

20 10 0 -10 0

70 [dB] 60 R 50 40 30

0.2

0.4

0.6

0.8

F

1

Θi=45° ; Φi=45° ; A=0.1 ; X=1 ; M s=0

0.4 2 1

20

0.2

T= 0.1

10 0 -10 0

0.2

0.4

0.6

0.8

F

1

I

520

60 [dB] 50 R 40

Sound Transmission

Θi=45° ; Φi=45° ; A=0.1 ; T=2 ; X=1 Ms= 0

30 1

20

2

10

4 8

0 -10 0

I.6

0.2

0.4

0.6

0.8

F

1

Chambered Joint


Some joints between construction elements have a cross section which, in principle, is a sequence of chambers (e.g. joints of facade elements, joints of the window frame, etc.). Let a plane sound wave pe be incident with the wave vector in the plane normal to the wall and the length of the joint, at a polar angle Ÿi .

Let Si ; i = 1, 2, . . ., I; be the cross-section areas of the duct elements into which the joint can be subdivided. The sound transmission coefficient ‘(Ÿi ) will be: SI Re{ZrI /Z0} Z0 VI 2 SI Re{ZrI /Z0} · =4 ‘(Ÿi ) = S1 cos Ÿi Pe S1 cos Ÿi

2 Z0 · SI VI /(2Pe) , SI

(1)

where ZrI is the radiation impedance of the exit orifice SI and Zr1 the radiation impedance of the entrance orifice i = 1. The composed duct can be described with an equivalent chain network, in which the duct sections are represented by ¢-fourpoles separated in the longitudinal branch by mass reactances Zmi,k of an internal orifice i if it enters into a wider duct section k.

Sound Transmission

i=1

2Pe

521

ZmI,I-1

Zm1,2

Zr1

I

i=2

i=I

ZrI

Interpret the network as a pq-network, i.e. with volume flows qi = Si vi instead of particle velocities. Then the ¢-fourpole elements (longitudinal impedances Zi , transversal admittances Gi ) are: Zi =

Zai sinh (ai di ) Si

;

Gi =

Zai tanh (ai di ) Si

−1 −

1 , Zi

(2)

and the longitudinal impedances Zmi,k from the end corrections are: Zmi,k = j

Z0 i,k k0bi , bi bi

(3)

where the end corrections are taken from > Ch. F, or from > Sects. H.4–H.7. The required quantity SI VI /(2Pe ) is evaluated with the iterative method of > Sect. C.5, with SI VI SI pN pN , p0 taken from there, and = . (4) 2Pe ZrI p0

I.7 “Noise Sluice”


Imagine two neighbouring rooms, room A very loud, room B with many working places, and a heavy traffic of fork trucks between the rooms (for material transport), so that sound-insulating doors would be inconvenient. Design an open passageway as a sufficiently wide, lined duct. It is important that the opening of the duct be placed in room A such that sound incidence on it is predominantly under large polar angles Ÿ (see below for additional measures to achieve that). If the opening is near a side wall of room A (as shown),that side wall should be made absorbing on a certain length to produce an unsymmetrical sound incidence. Formulation (in two dimensions) of the directly incident plane wave pe and of the reflected incident wave pr : pe (y, z) = Pe ejk0 y sin Ÿ · e−jk0 z cos Ÿ , pr (y, z) = r0 · pe (−a, z) · e−jk0 y sin Ÿ = r0 Pe e−jk0 a sin Ÿ · e−jk0 y sin Ÿ · e−jk0 z cos Ÿ .

(1)

522

I

Sound Transmission

The sound field p(x, y) in the sluice duct is formulated as a sum of symmetrical and anti-symmetrical silencer modes: cos (—m y) −m z p(y, z) = pm (y, z) = Am e · sin (—m y) m m ; m2 = —2m − k02 . (2) m cos (—m y) −m z j Am e · Z0 vz (y, z) = − k0 sin (—m y) m The transversal wave numbers are solutions of the characteristic equations: —msy a · tan (—msy a) = j k0 a · Z0 G = jU,

(3)

—mas a · cot (—mas a) = −j k0 a · Z0 G = −jU

with U = k0 a · Z0 G, if the lining is locally reacting with a surface admittance G (see > Ch. J,“Duct Acoustics”, for an evaluation of sets of mode wave numbers). Mode norms: Nmsy

1 := 2a

Nmas : =

1 2a

a −a

a

sin (2—msy a) 1 cos (—msy y) dy = 1+ , 2 2—msy a 2

sin2 (—mas y) dy =

−a

(4)

sin (2—mas a) 1 1− . 2 2—mas a

Mode-coupling coefficients with a plane wave: 1 Smsy± : = 2a

Smas± : =

1 2a

+a

e±jk0 y sin Ÿ · cos (—msy y) dy,

−a

+a −a

(5) e±jk0 y sin Ÿ · sin (—mas y) dy.

Sound Transmission

I

523

Because Smsy+ = Smsy− and Smas+ = − Smas− , only the index + is needed further, so we simplify: Smsy = Smsy+ , Smas = Smas+ : 1 Smsy = 2 j Smas = 2

sin (—msy a − k0 a sin Ÿ ) —msy a − k0a sin Ÿ

+

sin (—msy a + k0 a sin Ÿ )

—msy a + k0a sin Ÿ

sin (—mas a + k0a sin Ÿ ) sin (—mas a − k0 a sin Ÿ ) − —mas a − k0a sin Ÿ —mas a + k0 a sin Ÿ

with limit values for —m a → ±k0a · sin Ÿ: and if —m a → 0:

Smsy → Nmsy ; Smas → ±j Nmas , Nmsy → 1; Nmas → 0.

(6)

(7)

Field matching at the sluice entrance gives for the mode amplitudes: Amas Amsy Smsy+ Sm 1 + r0 e−j k0 a sin Ÿ ; = = as+ 1 − r0 e−j k0 a sin Ÿ . Pe Nmsy Pe Nmas Effective sound power of the incident wave pe : Effective modal sound power incident in the duct on the exit orifice:

(8)

a |Pe |2 cos Ÿ. Z0 ⎧ ⎫ (9) a ⎨ ⎬ 1 ∗ ¢m (d) = Re pm (y, d)·vzm (y, d) dy . ⎩ ⎭ (10) 2 ¢e (Ÿ) =

−a

Assuming a good radiation efficiency of the modes at the duct exit (because the duct is wide), the transmission coefficient of the noise sluice is: $

‘(Ÿ) =

¢m (d)

1 m −2m d e ¢e(Ÿ) 2 cos Ÿ m k0 Sm 2 sinh (2—m a) sin 2(—m a) −j k0 a sin Ÿ ± 1 ± r0 e × Nm 2—m a 2—m a m

=

(11)

(with m = m + j · m ; —m = —m + j · —m ). The summation is over the symmetrical (with the upper sign in ±) and anti-symmetrical modes (with the lower sign in ±). In the following examples of the transmission loss R = −lg(‘(Ÿ)) the lining of the duct and of the wall in front of the duct is assumed to be a layer of glass fibre material, if necessary made locally reacting by internal partitions; the flow resistivity of the material is ¡. The mode orders m in the summation are taken in a range Max (1 , m0 − m) ≤ m ≤ m0 + m with an interval width 2m and the central order m0 selected so that |Re{—m a − k0 a · sin Ÿ}| = min (m).

524

I

Sound Transmission

30 [dB] 25 R 20

Δm= 3 5 7

15 10 5 0 50

100

200

500 f [Hz]

1k

2k

4k

Influence of order interval width m on R(Ÿ). Input parameters: Ÿ = 45◦ ; a = 1[m]; d = 3[m]; t = 0.25[m]; t0 = 0.25[m]; ¡ = 10[kPa · s/m2 ] 30 [dB] 25 R 20

Ξ [kPas/m2]= 10

15

20

10 50

5 0 50

100

200

500 f [Hz]

1k

2k

4k

Influence of lining resistivity ¡ on R(Ÿ). Input parameters: Ÿ = 45◦ ; a = 1[m]; d = 3[m]; t = 0.25[m]; t0 = 0.25[m]; m = 5 50 [dB] 40 R 30

75° 60° 45°

20 Θ=30°

10 0 50

100

200

500 f [Hz]

1k

2k

4k

Influence of angle of incidence on R(Ÿ). Input parameters: a = 1[m]; d = 3[m]; t = 0.25[m]; t0 = 0.25[m]; ¡ = 10[kPa · s/m2 ]; m = 5

Sound Transmission

30 [dB] 25 R 20

I

525

45°≤Θ≤85°

15 10

30°≤Θ≤85°

5 0 50

100

200

500 f [Hz]

1k

2k

4k

Sound transmission loss R of a noise sluice for quasi-diffuse sound incidence with two ranges of incidence angle. Input parameters: a = 1[m]; d = 3[m]; t = 0.25[m]; t0 = 0.25[m]; ¡ = 10[kPa · s/m2 ]; m = 5

Measures to avoid sound incidence at small Ÿ:

I.8

Sound Transmissionßindexsound transmission through plates through Plates, Some Fundamentals


Bending-wave equation:

[ − kB4 ] vz =

j– ƒp B

(1)

with = Laplace operator, – = angular frequency, B = bending stiffness, ƒp = pfront − pback = driving sound pressure difference, kB = free bending wave number. % – 2 √ 1 – 4 4 = = – 4 m/B = 12 (1 − 2) = kL d 12 (1 − 2) cB ŠB cL d d % f = k0 fcr /f −−−−−→ 4.515 =0.35 cL d

kB =

(2)

I

526

Sound Transmission

z

y

Fz

x

My vz

u

∂M y dx My+ ∂x

d ∂F Fz + z dx ∂x

∂u , w ϕy = y ∂x

Kinematic and dynamic quantities at a plate section, in Cartesian coordinates. u vz œ wz Fz My

= = = = = =

elongation velocity rotation angle with y as axis rotational velocity transversal force torsional moment with y as axis

with cB = bending wave velocity, ŠB = bending wavelength, m = surface mass density, d = plate thickness, kL = longitudinal wave number, cL = longitudinal wave velocity, = Poisson ratio, f = frequency, fcr = critical coincidence frequency. & % c20 c20 12(1 − 2 ) 60 761 c20 m = = 12(1 − 2 ) −−−−−→ . (3) fcr = =0.35 2 B 2d E 2d cL d cL Coincidence frequency for angle Ÿ of incidence:

fc =

fcr . sin2 Ÿ

(4)

Table 1 Characteristic wave speeds. See Table 4 for S, E, D, B Wave type

p

Shear wave, torsional wave

cs =

Compressional wave in a bar

cL;St =

Compressional wave, longitudinal wave

cD =

p

Speed

p

E=

D=

= c0 = ‚P0 in a gas %

Bar-bending wave Plate-bending wave

Rayleigh wave

cB;St =

4

Remark = plate material density

S=p

–2 BSt m0

cB;PI = cB =

p

m0 = mass per length %

–4

cRayl 0:92 cs

‚ = adiabatic exponent P0 = static pressure

B m

m = mass per area (surface mass density)

Sound Transmission

I

527

If a plate is infinite in its lateral extension (in practice: if its smallest lateral dimension is large compared to the bending wavelength ŠB , so that plate resonances can be neglected in their influence on the transmission loss) the most important quantity of a plate is its partition impedance ZT (see below and > Sect. H.17). If finite plate dimensions must be considered, the boundary conditions at the plate boundaries must be taken into account. “Classical” boundary conditions are given in Table 2.They can be expressed also with force impedances ZF and momentum impedances ZM at the boundary {xR , yR }. These impedances are defined by: F(xR , yR ) = ZK (xR , yR ) · vz (xR , yR ),

(5)

Mi (xR , yR ) = ZM (xR , yR ) · wi (xR , yR ). Additional corner impedances ZE apply at plate corners {xE ,yE }: FE (xE , yE ) = ZE (xE , yE ) · vz (xE , yE ).

(6)

Table 2 Classical boundary conditions for plates Fixation

Simply supported

Clamped

Condition

Symbol

@ 2 v(; 1) =0 @†2 1=ZK = ZM = 0

v(; 1) =

η

@v(; 1) =0 @† 1=ZK = 1=ZM = 0

v(; 1) =

@2 v @2 v + 2 2 = 0 @†2 @ Free

@3 v @3 v + (2 − )2 =0 @†3 @†@ 2

β

η

ZK = ZM = 0 Hinged

ZK = 1=ZM = 0

η

Partition impedance ZT : Let a thin, i.e. incompressible, plate in an orthogonal co-ordinate system {x1 , x2 , x3 } be on the co-ordinate surface x3 = , and let pf (x1 , x2 , 3 ) be the sound pressure at the plate on its front side (side of excitation) and pb (x1 , x2 , 3) the sound pressure at the plate on its back side; further, let p = pf (x1 , x2 , 3 ) − pb (x1 , x2 , 3 ) be the driving pressure

528

I

Sound Transmission

difference, which generates a plate velocity v(x1 , x2 , 3 ) (counted positive in the direction front → back). The boundary conditions at the plate are: v f = vb = v

;

pf − pb = ZT · v,

(7)

where the last condition demands proportionality between p and v with a constant factor ZT if the plate is homogeneous in x1 , x2 (i.e. for example, the plate is large or a closed shell with constant curvature). It is assumed that pf and pb satisfy the wave equation in air (and possible other boundary conditions at surfaces other than the plate). The bending-wave equation for v

j– · p x1 ,x2 x1 ,x2 − kB4 v = B

(8)

and the last boundary condition can be combined into: j – p j – x1 ,x2 x1 ,x2 v − kB4 = · = · ZT . v B v B

(9)

For stationary sound fields and homogeneous plates the function v(x1 , x2 ) is, up to a constant factor, the same as pf (x1 , x2 , 3 ), pb (x1 , x2 , 3 ). So the first term in the last equation is known for given sound field formulations.

Θ Θ d

pr

pe

x

vz

pt z Θ

For a plane plate in the plane z = 0 and a plane wave pe incident in the x,z plane at a polar angle Ÿ, i.e. with a trace wave number kx = k0 · sinŸ, the partition impedance for a plate with surface mass density m is: B 4 p k4 − k 4 ZT = = kx − kB4 = j–m B 4 x = j–m v j– kB 2 f 4 sin Ÿ . = j–m 1 − fcr

2 f 1− fc

(10)

Sound Transmission

I

529

If the plate has bending losses, i.e. B → B(1 + j†) with the loss factor †, then: 2 ZT –m f f 2 4 = sin Ÿ +j 1 − sin4 Ÿ † Z0 Z0 fcr fcr = 2 Zm F †F2 sin4 Ÿ + j 1 − F2 sin4 Ÿ

F: =

f 2 f 2 = 2 Zm F † +j 1−( ) ; fc fc

f fcr m fcr d ; Zm : = = (11) fcr Z0 Z0

where is the plate material mass density. As long as the compressibility of the plate can be neglected, the Timoshenko-Mindlin theory of thick plates gives in an analogous way for the partition impedance: √ k 4 − (k2 − k 2 )(k 2 − k2 ) 12 ; kR = (12) kS ZT = j–m B 4 x 2 L2 x 2 R kB − kR (kx − kL) with kx the trace wave number of the exciting field, kB the free bending wave number, kL the longitudinal wave number and kS the shear wave number.

Table 3 Density and elastic constants of materials Material

Density

E modulus

fcr d

Zm

Loss fact.

[kg=m3 ]

E[MN=m2 ]

[Hz m]

[−]

†[−]

Construction materials Concrete

2100–2300

25–40103

15–18.5

77–104

Lean concrete

2000

15 000

23

112

800–1400

1.5–3103

37–48

72–164

0.015

Porous concrete

600–700

1.4–2103

37–45

54–77

0.01

Cement floor

2200

30 000

17

91

Xylolith floor

1600

6000

32.5

127

0.03

Asphalt floor

2200

6–15103

24–42

129–225

0.03–0.3

Plaster floor

1200

20000

15.5–16

45–47

0.006

Gypsum panel

1000–1200

3.5–7103

24–35

58–102

0.004

Plaster board

1000

3200

31–35

85

0.03

Fibre cement board

2000–2100

20–30103

16.5–20

80–102

0.01

1700–1800

9–25103

16–27

66–118

0.04

2500

60–80103

11–13

67–79

0.001

Light concrete

Brick wall Glass

0.05

530

I

Sound Transmission

Table 3 continued Material

Density

E modulus

fcr d

Zm

Loss fact.

[kg=m3 ]

E[MN=m2 ]

[Hz m]

[−]

†[−]

600–1000

2–5103

23–36

34–88

0.03

Plywood

600–800

5–12103

14–34.5

20–65

0.02

Oak wood

700

200–1000

18–32

31–55

0.01

Pine wood

480

100–500

20–32

23–37

0.01

Hard board

1000

3–4.5103

29.5–36.5

72–89

0.015

Acryl glass

1200

5600

29

85

0.06

Polypropylene

1100

3000

38

102

0.1

Polyester

1200

4500

32.5

95

0.14

PVC, hard

1300

2700

43.5

138

0.04

PVC, 30% softener

1250

48

1220

Polyethylene, hard

950

1700

47

109

0.04

Polyethylene, soft

920

400

95.5

214

0.1

Polystyrene

1070

3000

37.5

98

0.01

Polystyrene+30% glass fibre

1450

8000

27

95

Polyester+glass fibre

2200

11500

27.5

147

0.02

Aluminium

2700

74 000

12

79

710−5

Lead

11400

18000

48.5

1348

0.02

Copper

8900

125000

17

369

Brass

8500

96000

18.6

386

0.001

Steel, cast steel

7800

200000

12.3

234

110−4

Malleable iron

7500

170000

13.2

241

Cast iron with spher. graphite

7250

120000

15.4

272

0.01

Cast iron with lamell. graphite

7250

120000

15.4

272

0.02

Zinc

7130

13000

46.5

809

Tin

7280

4400

81

1438

Chip board

Plastics

Metals

Sound Transmission

I

531

Table 3 above gives the required elastic data of materials. Of special interest are the density and the product fcr · d which is a material constant.

Table 4 Relations of elastic constants Quantity

Symbol, relations Š; ‹

Lame constants

Remark

!

−pii = Š div s +2‹ —ii −pik = 2‹ —ik = −—yy =—xx =

Poisson number

=

Š 2(Š + ‹)

E −1 2S

‹ 1 − 2 = Š 2

0 < 0:5

;

S = ‹ [Pa] Shear modulus =

1 1 E; 21+

E=‹ E modulus,Young’s modulus

3Š + 2‹ Š+‹

Compression modulus

=

incompressible: =0.5

α α

( < 0:4)

[Pa]

= 2‹=(1 + ) = 2S(1 + )

K=Š +

ε

ε

2 ‹ [Pa] 3

2 2 + 3 1 − 2

Δ

ε

S

‰ = compressibility

1 E = = 3(1 − 2) ‰ D = Š + 2‹ [Pa] Dilation modulus

=2

1− 1− S= E 1 − 2 (1 + )(1 − 2)

11− K = 31+

for gas: D = 0 c20 = ‚P

532

I

Sound Transmission

Table 4 continued Quantity

Symbol, relations Bst =

Bar-bending modulus

bd3 E [Pa m4 ] 12

B = Bpl = I E =

Plate-bending modulus

Remark

[Pa m3 ]

d3 ‹(Š + ‹) d3 E = 3 Š + 2‹ 12(1 − 2 )

moment of inertia

d3 S = 6(1 − )

Dpl =

Plate-dilation modulus

I.9

E 1−

I=

d3 12(1 − 2 )

[Pa]

Sound Transmission through a Simple Plate


A plate is “simple” if it is thin (incompressible), homogeneous (except possibly on a micro-scale),isotropic,and unbounded (or at least with large dimensions,so that boundary effects can be neglected).

Transmission as a boundary value problem: Field formulations: pe (x, z) = Pe · e−jkx x · e−jkz z pr (x, z) = r · Pe · e−jkx x · e+jkz z −jkx x

pt (x, z) = Pt · e

−jkz z

·e

kx2 + kz2 = k02, kx = k0 · sin Ÿ

;

kz = k0 · cos Ÿ,

(1)

Plate velocity:

v(x) = V · e−jkx x ,

(2)

Plate driving pressure:

ƒp(x) = pe (x, 0) + pr (x, 0) − pt (x, 0),

(3)

Sound Transmission

From the bending-wave equation: 4 j– ∂ 4 ƒp(x) − k vz (x) = B ∂x4 B (kx4

−

kB4 )

−jkx x

·V·e

j– = (Pe (1 + r) − Pt ) · e−jkx x , B

I

533

(4)

the partition impedance (boundary condition for pressure) follows: ƒp (Pe (1 + r) − Pt ) B 4 k4 − k 4 = = kx − kB4 = j–m B 4 x vz V j– kB 2 2 f f = j–m 1 − = j–m 1 − sin4 Ÿ . fc fcr

ZT =

(5)

With boundary conditions for particle velocity: !

vz = vtz (x, 0) =

kz kz ! Pt · e−jkx x ; vz = vez (x, 0) + vrz (x, 0) = Pe (1 − r) · e−jkx x , (6) k0 Z0 k0 Z0

the system of equations follows: ⎞ ⎛ ⎞ ⎞ ⎛ ⎛ 1 −1 1 V ZT ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ ⎠ ⎝ 1 0 0 − cos Ÿ/Z0 ⎠ · ⎝ rPe ⎠ = ⎝ cos Ÿ/Z0 1 cos Ÿ/Z0 0 Pt Pt 1 ZT cos Ÿ −1 = 1+ . Pe 2 Z0 −2 Pt 2 = 1 + 1 ZT cos Ÿ ‘(Ÿ) = Pe 2 Z0

having the solution: Sound transmission coefficient:

After insertion: 2 2 cos Ÿ k0 1 =1 + –m 1 − sin Ÿ ‘(Ÿ) 2Z0 kB 2 2 cos Ÿ f =1 + –m 1 − 2Z0 fc 2 = 1 + Zm F · cos Ÿ · (1 − F2 sin4 Ÿ)

(7)

(8) (9)

(10)

' (2 = 1 + Zm F(F − y) · (1 − y 2 ) with non-dimensional parameters: y: =

f fc

;

F: =

f fcr

;

Zm : =

fcr m fcr d = . Z0 Z0

(11)

534

I

Sound Transmission

Transmission coefficients for diffuse sound incidence in three and two dimensions: Ÿhi ‘(Ÿ) cos Ÿ sin Ÿ dŸ ‘3−dif =

0

=

Ÿhi cos Ÿ sin Ÿ dŸ

2 sin2 Ÿhi

Ÿhi ‘(Ÿ) cos Ÿ sin Ÿ dŸ 0

0

(12)

Ÿhi ‘(Ÿ) cos Ÿ dŸ ‘2−dif =

1 = sin Ÿhi

0

Ÿhi cos Ÿ dŸ

Ÿhi ‘(Ÿ) cos Ÿ dŸ 0

0

(Ÿhi = upper limit, ≤ /2, of sound incidence angles). Approximations and special cases: Berger’s law for f fcr :

cos Ÿ 2 1 ≈ –m . ‘(Ÿ) 2Z0

f · sin Ÿ 1: fc

1 ≈1+ ‘(Ÿ)

Normal or grazing incidence:

‘(0) =

Diffuse incidence:

‘3−dif =

2

–m cos Ÿ sin2 Ÿ 2Z0

1 1 + ( Zm F)2 2 F sin Ÿhi

2

F sin Ÿhi

0

(13)

1+

2 Z2m

;

2 2 f . fc

(14)

‘(/2) = 1.

(15)

dy . F (F − y)(1 − y 2 )2

(16)

When Rdif = −10 · lg(‘dif ) is plotted over F, it depends on the single parameter Fm (which is a material constant) (besides the parameter Ÿhi of the test conditions). Transmission with equivalent circuit: The equivalent circuit method is adequate for the sound transmission through a simple plate. The immediate result is, as above: −2 Pt 2 1 ZT = 1+ cos Ÿ . ‘(Ÿ) = Pe 2 Z0

(17)

Sound Transmission

I

535

ZT

Z0/cosΘ 2Pe

Z0/cosΘ

Source

Wall

Load

Plate with bending losses: With bending loss factor †, use above: 2 f 2 4 f 4 † sin Ÿ + j 1 − sin Ÿ fcr fcr = 2 Zm F †F2 sin4 Ÿ + j 1 − F2 sin4 Ÿ

ZT –m = Z0 Z0

(18)

2 f 2 f +j 1− = 2 Zm F † . fc fc Transmission coefficient for diffuse sound incidence:

‘3−dif =

2

2 F sin Ÿhi

F sin Ÿhi

dy

2

0

[ 1 + †Zm F(F − y) y 2 ]2 + 2 Z2m F (F − y)(1 − y 2)2

.

(19)

Thick plates: Use in above relations the partition impedance fromßindexTimoshenko-Mindlin theory the Timoshenko-Mindlin theory: √ kB4 − (kx2 − kL2 )(kx2 − kR2 ) 12 kS ZT = j–m ; kR = 4 2 2 2 kB − kR (kx − kL)

(20)

with kx the trace wave number of the exciting field, kB the free bending-wave number, kL the longitudinal wave number and kS the shear wave number. 4 ks Alternatively, use in ZT = j–m 1 − the “corrected” bending-wave number kB: kB 4.43 –2 m d2 + kB2 = 24 B

&

4.43 –2 m d2 24 B

2

m–2 − B

0.26 –2md −1 . E

(21)

I

536

Rdif 80 [dB] 70 60 50

Sound Transmission

Θhi=85°

Z m= 50 100 200 400

40 30 20 10 0 0.1

0.2

0.5

1

2

5 F = f / fcr

10

Transmission loss for diffuse sound incidence of simple plates with different values of Zm Rdif 80 [dB] 70 60

Z m=100 ; Θhi=90°

0.08 0.04 0.02 0.01

50 40 30

η= 0

20 10 0

0.1

0.2

0.5

1

2

5 F = f / fcr

10

Transmission loss for diffuse sound incidence of simple plates with different values of bending loss factor † Rdif Z m=1438 80 [dB] 70 Θhi= 90° 85° 80° 60 50

0.02 0.01 η= 0

40 30 20 10 0

0.1

0.2

0.5

1

2

5 F = f / fcr

10

Highest possible transmission loss values of a simple plate for diffuse sound incidence; the plate consists of tin

Sound Transmission

I

537

Plane wave transmission through unbounded, thin, homogeneous and anisotropic plate between two different fluids: ( See also: Maysenhölder,Acustica 84 (1998)) A thin plate with thickness h, mass density II and generalized bending stiffnesses B‚ƒ [see > Sect. Q.10.3, eq. (35)] separates two fluids with densities I and III and sound speeds cI and cIII . The polar angle of incidence (relative to plate normal) is ŸI ; the azimuthal angle of incidence (in plane of plate) is ¥ . The polar angle Ÿ III of the transmitted wave is given by Snell’s law: sin ŸIII cIII = . sin ŸI cI

(22)

The transmission factor T is: III cIII cos ŸIII I cI + T= 2 III cIII cos ŸI cos ŸIII )−1 sin ŸI 4 3 ˆ + j – II h − – B cI with

Bˆ =

2

B ‚ ƒ œ œ œ‚ œƒ ;

(23)

œ1 = cos ¥ ; œ2 = sin ¥ .

(24)

, , ‚, ƒ = 1

The transmission coefficient is: ‘(ŸI ) =

I cI cos ŸIII | T |2 III cIII cos ŸI

(25)

(if cosŸIII = real, otherwise ‘ = 0), and the transmission loss is R(ŸI ) = −10 · lg ‘. In the special case of an isotropic plate between identical fluids one recovers Cremer’s result (B the usual bending stiffness) with: −1 sin ŸI 4 cos Ÿ1 3 . (26) – II h − – B T = 1 + j 2 I cI cI In the special case where there is no plate between two different fluids: T =

2 . I cI cos ŸIII 1 + III cIII cos ŸI

(27)

I

538

I.10

Sound Transmission

Infinite Double-Shell Wall with Absorber Fill


Two simple plates with thicknesses d1 , d2 form an interspace which is filled with a fraction = ha /h with a bulk reacting porous material having the characteristic values a , Za .

A plane wave pe is incident at a polar angle of incidence Ÿ. The sound transmission is evaluated with the method of equivalent networks. Z0/cosΘ ZT1

2Pe =p0

Source

Ga1

Shell 1

Za1

Zl1

Ga1

Zl1

Gl1

Absorber

Air

ZT2

cosΘ/Z0

Pt = pN

Shell 2 Load

ZTi , i = 1, 2, are partition impedances of the plates; the network elements of the absorber layer are: Za1 Za sinh (a ha cos Ÿa ) = Z0 Z0 cos Ÿa

;

Z0 Ga1 =

cos Ÿa cosh (a ha cos Ÿa ) − 1 , Za /Z0 sinh (a ha cos Ÿa )

(1)

and of the air layer: Z1 1 − cos (k0h cos Ÿ) =j Z0 cos Ÿ sin (k0 h cos Ÿ)

;

Z0 G1 = j cos Ÿ sin (k0h cos Ÿ)

(2)

Sound Transmission

& with internal angle in the absorber layer from

cos Ÿa =

The sound transmission coefficient is:

2 Pt ‘(Ÿ) = Pe

The numerical evaluation may apply the iterative method of with:

I

sin2 Ÿ . (a /k0)2 2 pN = 4 . p0

1+

539

(3) (4)

> Sect. C.5, or analytically

pN Pt 1 = = p0 2Pe ze + (b + ga1 ze ) za

(5)

and the auxiliary quantities: za = zT1 + 1/ cos Ÿ

;

zb = z1 + zT2

zd = zc + a · z1

;

ze = zd + b · za1 ;

a = cos Ÿ + g1 zc

;

b = a + ga1 zd ,

;

zc = 1 + zb cos Ÿ; (6)

in which z, g are normalised (to Z0 ) impedances and admittances. If the interspace is full (with absorber material), then: Pt 1 pN = = , p0 2Pe zc + (a + ga1 zc ) za za = zT1 + 1/ cos Ÿ a

;

zb = 1 + zT2 cos Ÿ

;

zc = zb + a · za1 ,

(7)

= cos Ÿ + ga1 · zb .

If the interspace is empty (i.e. no absorber), then: pN Pt 1 = = , p0 2Pe za + (cos Ÿ + g1 za ) zb za = 1 + (z1 + zT2 ) cos Ÿ

;

(8)

zb = z1 + zT1 + 1/ cos Ÿ,

or after insertion: Pt pN = p0 2Pe

= 2+(2 z1 +zT1 +zT2 ) cos Ÿ+g1

1 . 1 1 + z1 + zT1 + z1 + zT2 cos Ÿ cos Ÿ

(9)

I

540

Sound Transmission

80 [dB] 60 R dif 40 0.25 β=0

0.5

20 0 50

100

200

500

1k

2k 4k f [Hz]

Sound transmission loss for diffuse sound incidence of a double-shell wall of plaster board shells, with different fill factors = ha /h of the absorber layer. Input parameters: Ÿhi = 85◦ ; h = 0.06[m]; d1 = d2 = 0.0125[m]: fcr d1 = fcr d2 = 31 [Hz · m]; 1 = 2 = 1000[kg/m3]; †1 = †2 = 0.03; ¡ = 10000[Pa · s/m2 ] The double-shell resonance for an empty interspace is at: & c0 1 0 1 f0 (Ÿ) = + . 2 cos Ÿ h m1 m2

I.11

(10)

Double-Shell Wall with Thin Air Gap


The present object is formally treated as a double-shell wall, completely filled with absorber material, from the previous > Sect. I.10, but using for a , Za the characteristic values in a flat capillary. Finally, the limit transition a ha → 0 is applied: pN Pt 1 = = , p0 2Pe zc + (a + ga1 zc ) za za = zT1 + 1/ cos Ÿ ; a

zb = 1 + zT2 cos Ÿ ;

zc = zb + a · za1 ;

(1)

= cos Ÿ + ga1 · zb .

a Za 1 1 cos Ÿa ; ga1 → k0ha − = 0, k0 Z0 cos Ÿ za1 za1 and therewith the transmission factor: 1 pN ≈ . a Za p0 2 + (zT1 + zT2 ) cos Ÿ + k0 ha cos Ÿa k0Z0 In the limit

za1 →

(2)

(3)

In the third term of the denominator for very small ha (with † = dynamic viscosity of air): 12 † a Za ≈ . k0 Z0 –0 h2a

(4)

Sound Transmission

60 [dB] 50 R dif 40

I

541

ha = 0.0005 [m] =0.001 [m] =0.002 [m]

30 20 simple plate with sum thickness

10 0 50

100

200

500

1k

2k 4k f [Hz]

Double-glass pane with thin air gap of different thickness. Input parameters: Ÿhi = 85◦ ; d1 = 0.004[m]; d2 = 0.008[m]; fcr d1 = fcr d2 = 11[Hzm]; 1 = 2 = 2500[kg/m3]; †1 = †2 = 0.002

I.12

Plate with Absorber Layer Behind


It is assumed that the structure-borne sound transmission between the plate and the porous absorber layer is negligible (no or only loose contact between them). The evaluation uses the equivalent network method (it produces identical results with the solution of a boundary value problem). a , Za are the characteristic values of the porous material; an = a /k0, Zan = Za /Z0 ; z, g are normalised (with Z0 ) impedances or admittances. Positions of the absorber layer in front of or behind the plate give the same transmission coefficients.

1/cosΘ

2Pe =p0

Source

The network elements are: Zan za = sinh (an k0 da cos Ÿa ), cos Ÿa cos Ÿa cosh (an k0 da cos Ÿa ) − 1 ga = Zan sinh (an k0da cos Ÿa )

za

zT

ga

Plate

ga

Absorber

cosΘ

Pt = pN

Load

(1)

I

542

Sound Transmission

&

ga za = cosh (an k0 da cos Ÿa ) − 1

;

cos Ÿa =

= 2 Zm F [† F2 · sin4 Ÿ + j (1 − F2 · sin4 Ÿ ) ]

zT

1+

sin2 Ÿ 2 an

;

Zm =

fcr dp ; Z0

F=

f fcr

(f cr = critical frequency of the plate ; = its density ; † = its bending loss factor). Transmission loss:

R(Ÿ) = 10 lg(1/‘(Ÿ)) = 10 · lg (0.25 |p0 /pN |2 ) Pt 1 pN = = p0 2Pe z2 + (a + ga · z2 )/ cos Ÿ

with

(2) (3)

z1 = 1 + zT cos Ÿ ; a = cos Ÿ + ga · z1 ; z2 = z1 + a · za , and after insertion: p0 2 Pe = pN Pt = 2 + ga · (2 za + 2 zT + ga za zT ) + (za + zT + ga za zT ) · cos Ÿ

(4)

+ ga · (2 + ga za )/ cos Ÿ.

I.13

Sandwich Panels


Sandwich panels are combinations of sheets with high E and shear modulus G (index 2) with boards having lower E and shear moduli (index 1). The layers are combined with an adhesive layer, either very thin (or at least with no shear) or of thickness ƒ, the adhesive having a shear modulus G (without index). One must distinguish between boards which are tight and boards which are porous. Tight boards: It is sufficient to know the effective bending stiffness B of the sandwich. The required partition impedance ZT is then obtained from > Sect. I.8.

Sound Transmission

I

543

Table 1 Sandwich panels No.

Sandwich

Connection

1

tight joint

2

tight joint

3

connection with shear 3:5 10−3 G − 1:3 10−12 [m] ƒ h1 E1 E2 h2 2 107 [Pa m]

4

connection with shear 0:25 10−3 G ƒ [m] h1 E1 hi in [m]; G, Ei in [Pa]

Table 2 Effective bending moduli B for sandwiches from Table 1 No.

1

B = B2

Effective bending modulus 2 3 2 4 h1 E1 E1 h1 h1 h1 +3 1+2 +2 2 + E2 h2 h2 h2 h2 E2 1+

2

h2 3 B B1 1 + + B2 h1 B B1 + B2 + 3Gƒ

3

+ 2gE2 h1 h2 h1 + h 2 4 E1 h1 + g (E1 h1 + E2 h2 ) (h1 =2)3 B = B2 + 2E1 12(1 − 12 ) − 3Gƒ

4

h1 E1 h2 E2

h21 E1 h1 E2 h2 (h1 =2 + h2 =2)2 g + 4 E1 h1 + g (E1 h1 + E2 h2 ) E1 h21

Remark

ƒ h1 g=

G B=m ƒE1 h1 –

544

I

Sound Transmission

Sandwich with porous board on front side: A porous layer of thickness h propagates both sound waves in the pores (index 2) and dilatational waves in the matrix (index 1). Their coupling with each other leads to two characteristic propagation constants ± . The boundary conditions between board and sheet neglect shear stresses at the boundary II–III.

Field formulations (without common factor e−j kx x ): pe (z) = Pe · e−j kz z ps (z) = Ps · e+j kz z

;

kx2 + kz2 = k02

Ps = r · Pe

kz vez (0) + vsz (0) = Pe (1 − r) k0 Z0

pt (z) = Pt · e−j kz (z−h−d)

;

kx = k0 sin Ÿ

vtz (z) =

;

kz = k0 cos Ÿ,

kz Pt · e−j kz (z−h−d) . k0Z0

(1)

(2)

Sound waves in the matrix (index 1) and in the pores (index 2) of the board: p1 (z) =

z+ z− P12+ [A · e−z+ z + B · e+z+ z ] + P12− [C · e−z− z + D · e+z− z ] + −

Z0 v1z (z) =

z+ V12+ z− V12− [A · e−z+ z − B · e+z+ z ] + [C · e−z− z − D · e+z− z ] + Z22+ − Z22− −z+ z

p2 (z) = A · e Z0 v2z (z) = with

+B·e

+z+ z

−z− z

+C·e

+z− z

+D·e

z+ /+ z− /− [A · e−z+ z − B · e+z+ z ] + [C · e−z− z − D · e+z− z ] Z22+ Z22−

2 2 + z± ±2 = x±

;

x± = j kx

;

2 z± = ±2 + kx2 = ±2 + k02 sin2 Ÿ

(3)

Sound Transmission

I

545

The free field propagation constants ± are solutions of the characteristic equation:

k0

4 ·

K 1 K2 + K 0 K0

k0

2 K 1 K2 1 j K 1 K2 · ” + + ”−1+ − K0 K0 0 2E K0 K0 1 j 1 + = 0 (4) + (” − 1) + ” − 0 2E 0

(± indicates the sign of the root in the solution 2 ), or, in an approximation for ≈ 1; ” ≈ 1; 1 /0 1: 2 4 K1 K2 K1 K2 1 j K 1 K2 1 j · + · ” + − + = 0.(5) + ”− k0 K0 K0 k0 K0 K0 0 2E K0 K0 0 2E The coefficients Z22± , P12± , V12± are: Z22± =

p2 ± K 2 = −j Z0 v2 k 0 K0

2E · (” − ) − j 2 ± K2 − 1 2E · ” − 1 + − j k0 K0 2 ± K2 − j 2E · ” + k0 K0

P12± =

p1 = p2

(6)

2E · (” − ) − j

± 2 K2 2E · ” + − j k0 K0 v1 = V12± = 2 v2 K2 − 1 ± − j 2E · ” − 1 + k0 K0

with the compression modulus of the air in the pores: K2 1 + jE/E0 = K0 ‰ + jE/E0

;

E0 =

0 f0 ¡

;

2f0 · ‘0 = 1.

(7)

See the inset frame for the meaning of other symbols. The velocity V of the sheet is defined with its partition impedance ZT and the driving pressure difference p: V · ZT = p1 (h) + p2 (h) − pt (h + d).

(8)

The seven unknown amplitudes Ps (or r ), Pt , A, B, C, D, V are determined from the boundary conditions (one boundary condition is contained in the definition of V):

at I–II:

p1 (0) = (1 − ) Pe (1 + r), p2 (0) = Pe (1 + r), Z0 [(1 − ) v1z (0) + v2z (0) ] = Z0 (ve (0) + vs (0)) =

at II–III: v1z (h) = V

;

v2z (h) = V,

kz Pe (1 − r), k0

(9)

(10)

I

546

Sound Transmission

at III–IV: vtz (d + h) = V.

(11)

0 = density of air; c0 = adiabatic sound velocity; ‰ = adiabatic exponent of air; k0 = –/cO2 ; K0 = 0 c0 = adiabatic compression modulus of air; 1 = density of matrix material; K1 K2 ”

= = = =

dilation modulus of matrix material; compression modulus of air in the pores; porosity of porous material; tortuosity of porous material;

¡ = flow resistivity of porous material; E = 0 f /¡ = absorber variable of porous material; E0 = value of E at relaxation frequency f0 in the pores The system of equations ⎛

Ps /Pe

⎜ ⎜ Pt /Pe ⎜ ⎜ A/P e ⎜ ⎜ (Matrix) • ⎜ B/Pe ⎜ ⎜ C/P e ⎜ ⎜ ⎝ D/Pe Z0 V/Pe

⎞

⎛

1−

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ k /k ⎟ ⎜ z 0 ⎟ ⎜ ⎟=⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎠ ⎝ 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(12)

has a matrix with the following columns: {Ps /Pe } = {−(1 − ), −, kz/k0, 0, 0, 0, 0} , {Pt /Pe} = {0, 0, 0, 0, 0, kz/k0, 1} , {A/Pe} = {z+ P12+ /+ , 1, z+( + V12+ (1 − ))/(+ Z22+ ),

(13)

z+ V12+ /(ez+ h + Z22+ ), z+/(ez+ h + Z22+ ), 0, −(1 + z+ P12+ /+ )/ez+ h , {B/Pe } = {z+ P12+ /+ , 1, −z+( + V12+ (1 − ))/(+ Z22+ ), −z+ V12+ ez+ h /( + Z22+ ), −z+ ez+ h /( + Z22+ ), 0, − (1 + z+ P12+ /+ ) · ez+ h ,

(14)

Sound Transmission

I

547

{C/Pe} = {z−P12− /− , 1, z−( + V12− (1 − ))/(− Z22− ), z− V12−/(ez− h − Z22− ), z−/(ez− h − Z22− ), 0, − (1 + z− P12− /− )/ez− h , {D/Pe} = {z−P12− /− , 1, −z−( + V12− (1 − ))/(−Z22− ),

(15)

− z− V12− ez− h /( − Z22− ), −z− ez− h /( − Z22− ), 0, −(1 + z− P12− /− ) · ez− h , {Z0 V/Pe} = {0, 0, 0, −1, −1, −1, ZT/Z0 } . The ultimately desired transmission coefficient ‘(Ÿ) follows from * 2 ‘(Ÿ) = Pt Pe .

(16)

The following example is for a sandwich with a front-side glass fibre board, having different bulk densities RG = (1 − ) · 1 (with 1 = 2500 [kg/m3 ]) and flow resistivities ¡, and a back-side plaster board, d = 12.5 [mm] thick. 90 [dB] 80 R(45°) 70 60 50

RG [kg/m3] | Ξ [kPa s/m2] 40 | 16.2

40 30

20 | 5.08

80 | 51.8

20 10 50

100

200

500

1k

4k 2k f [Hz]

Sound transmission loss for oblique sound incidence of a sandwich with a front-side glass fibre board of different bulk densities RG and flow porosities ¡, and a back-side plaster board, d = 12.5 [mm] thick. Parameters: h = 0.1 [m] ; d = 0.0125 [m] ; fcr d = 31 [Hzm] ; p = 1000 [kg/m3 ]; †p = 0.03; ” = 1.35; †a = 0.25 Sandwich with porous board on back side: The field definitions remain as in the previous arrangement. The boundary conditions now are as follows: plate: V · ZT = pe (0) + ps (0) − p1 (d) + p2 (d)

(17)

548

I

Sound Transmission

at I–III:

vez (0) + vsz (0) =

kz ! (Pe − Ps ) = V, k0Z0

(18)

at III–II:

v1z (d) = V

v2z (d) = V,

(19)

;

p1 (d + h) = (1 − ) Pt , at II–IV:

p2 (d + h) = Pt ,

kz Z0 [(1 − ) v1z (d + h) + v2z (d + h) ] = Z0 vtz (d + h) = Pt . k0

The system of equations to be solved is: ⎞ ⎞ ⎛ ⎛ − cos Ÿ Ps /Pe ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ Pt /Pe ⎟ ⎜ 0 ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ A/P ⎟ ⎜ 0 e ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ (Matrix) • ⎜ B/Pe ⎟ = ⎜ ⎟ 0 ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ C/P ⎟ ⎜ 0 e ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎠ ⎝ D/Pe ⎠ ⎝ 0 Z0 V/Pe

(20)

(21)

−1

with the following matrix columns: {Ps /Pe } = {− cos Ÿ, 0, 0, 0, 0, 0, 1} {Pt /Pe} = {0, 0, 0, −(1 − ), −, − cos Ÿ, 0} {A/Pe} = 0, z+ V12+ /(+ Z22+ ez+ d ), z+ /(+ Z22+ ez+ d ), z+P12+ /(+ ez+ (d+h) ), 1/ez+ (d+h) , z+/(+ Z22+ ez+ (d+h) ) · [ + (1 − )V12+ ], −(1 + z+ P12+ /+ )/ez+ d ,

(22)

{B/Pe } = 0, −z+ V12+ ez+ d /(+ Z22+ ), −z+ ez+ d /(+ Z22+ ), z+ P12+ ez+ (d+h) /+ , ez+ (d+h) , −z+ ez+ (d+h) /(+ Z22+ ) · [ + V12+ (1 − )], − ez+ d · (1 + z+ P12+ /+ ) ,

(23)

Sound Transmission

I

{C/Pe} = 0, z− V12−/(− Z22− ez− d ), z− /(− Z22− ez− d ), z− P12− /(− ez− (d+h) ), 1/ez− (d+h) , z− /(− Z22− ez− (d+h) ) · [ + (1 − )V12−], −(1 + z− P12− /− )/ez− d , {D/Pe} = 0, −z− V12− ez− d /(− Z22− ), − z− ez− d /(− Z22− ), z− P12− ez− (d+h) /− , ez− (d+h) , z− (d+h)

− z− e

z− d

/(− Z22− ) · [ + (1 − )V12−], −e

549

(24)

(25)

· (1 + z− P12− /− ) , (26)

{Z0 V/Pe} = {−1, −1, −1, 0, 0, 0, −ZT } . * 2

The ultimately desired transmission coefficient ‘(Ÿ) follows from ‘(Ÿ) = Pt Pe . (27) The following example is for the same object as above. Both the example and the equations show that the position of the absorber board modifies the sound transmission loss. 90 [dB] 80 R(45°) 70 60 50

RG [kg/m3] | Ξ [kPa s/m2] 40 | 16.2

40

80 | 51.8

20 | 5.08

30 20 10 50

100

200

500

1k

4k 2k f [Hz]

Same sandwich as above, but with a reversed arrangement of the glass fibre board and the plaster board Sandwich with a porous layer asßindexsandwich panel!with porous board as core core: The transmitted wave pt is modified as follows: pt (z) = Pt · e−j kz (z−h−d1 −d2 ) vtz (z) =

kz Pt · e−j kz (z−h−d1 −d2 ) . k0 Z0

The equations for the cover plates are: V1 · ZT1 = pe (0) + ps (0) − p1 (d1 ) + p2 (d1 ) V2 · ZT2 = p1 (d1 + h) + p2 (d1 + h) − pt (d1 + d2 + h).

(28)

(29)

550

I

Sound Transmission

The boundary conditions are now as follows: kz ! (Pe − Ps ) = V1 k0Z0 ; v2z (d1 ) = V1

at I–V:

vez (0) + vsz (0) =

(30)

at V–II:

v1z (d1 ) = V1

(31)

at II–III:

v1z (d1 + h) = V2

;

v2z (d1 + h) = V2

kz ! Pt = V 2 . k0Z0 The system of equations to be solved is: ⎞ ⎛ ⎞ ⎛ − cos Ÿ Ps /Pe ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 Pt /Pe ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 A/Pe ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ 0 B/Pe ⎟ ⎜ ⎜ ⎟ (Matrix) • ⎜ ⎟ ⎟=⎜ 0 C/Pe ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ 0 D/Pe ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ −1 ⎠ ⎝ Z0 V1 /Pe ⎠ ⎝ 0 Z0 V2 /Pe

at III–V:

vtz (d1 + d2 + h) =

(32) (33)

(34)

with the following matrix columns: {Ps /Pe } = {− cos Ÿ, 0, 0, 0, 0, 0, 1, 0} {Pt /Pe} = {0, 0, 0, 0, 0, cos Ÿ, 0, −1} {A/Pe} = 0, z+V12+/(+ Z22+ ez+ d1 ), z+/(+ Z22+ ez+ d1 ), z+ V12+/(+ Z22+ ez+ (d1 +h) ), z+ /(+ Z22+ ez+ (d1 +h) ), 0, − (1 + z+P12+ /+ )/ez+ d1 , (1 + z+P12+ /+ )/ez+ (d1 +h) ,

(35)

(36a)

{B/Pe } = 0, −z+V12+ ez+ d1 /(+ Z22+ ), −z+ez+ d1 /(+ Z22+ ), − z+ V12+ ez+ (d1 +h) /(+ Z22+ ), −z+ ez+ (d1 +h) /(+ Z22+ ), 0, − ez+ d1 · (1 + z+ P12+ /+ ), ez+ (d1 +h) · (1 + z+P12+ /+ ) ,

(36b)

Sound Transmission

{C/Pe} = 0, z−V12− /(− Z22− ez− d1 ), z− /(− Z22− ez− d1 ), z− V12− /(− Z22− ez− (d1 +h) ), z− /(− Z22− ez− (d1 +h) ), 0, − (1 + z− P12− /− )/ez− d1 , (1 + z− P12− /− )/ez− (d1 +h) , {D/Pe} = 0, −z−V12− ez− d1 /(− Z22− ), −z− ez− d1 /(− Z22− ), − z− V12− ez− (d1 +h) /(− Z22− ), −z−ez− (d1 +h) /(− Z22− ), 0, − ez− d1 · (1 + z− P12− /− ), ez− (d1 +h) · (1 + z−P12− /− ) , {Z0 V1/Pe } = {−1, −1, −1, 0, 0, 0, −ZT1, 0}

I

551

(37)

(38)

(39)

{Z0 V2/Pe } = {0, 0, 0, −1, −1, −1, 0, −ZT2} .

* 2 The ultimately desired transmission coefficient ‘(Ÿ) follows from ‘(Ÿ) = Pt Pe . (40) The example is for a sandwich with a glass fibre core for different bulk densities RG and flow resistivities ¡ of the glass fibre material. 80 [dB] 70 R(45°) 60 50 40

RG [kg/m3] | Ξ [kPa s/m2] 20 | 5.08

40 | 16.2

30 20 80 | 51.8

10 0 50

100

200

500

1k

2k 4k f [Hz]

Sandwich with a glass fibre core and plaster board cover plates, for different bulk densities RG and flow resistivities ¡ of the core material. Parameters: h = 0.1 [m]; d1 = 0.0125 [m]; d2 = 0.0095 [m]; fcr d1 = 31 [Hzm]; p1 = 1000[kg/m3]; †p1 = 0.03; fcr d2 = 31 [Hzm]; p2 = 1000 [kg/m3 ]; †p2 = 0.03; c = 1.35; ha = 0.25

I.14

Finite-Size Plate


Let a plate be two-dimensional (for ease of writing, a three-dimensional plate is treated similarly), infinite in the y direction, and with supported borders at x = ±h. The nondimensional co-ordinate = x/h will be used. In this section solutions vn ( ) = vn (‚n ) of the homogeneous bending wave equation 4 ∂ 4 − ‚n vn ( ) = 0 (1) ∂ 4

552

I

Sound Transmission

are given which satisfy the boundary conditions of different kinds of boundary support. These solutions are plate modes which will be used to synthesise plate velocity patterns: V( ) = Vn · vn ( ). (2) n

The bending wave equation 4 ∂ j– 4 V( ) = h4 ƒp( ) − (k h) B 4 ∂ B

(3)

then gives: j– ƒp( ), Vn ‚n4 − (kB h)4 vn ( ) = h4 B n which can be written as: Vn ZTn · vn ( ) = ƒp( ),

(4)

(5)

n

thereby defining the modal partition impedances ZTn : ‚n 4 kbn 4 B 4 4 (‚n /h) − kB = j–m 1 − = j–m 1 − . ZTn = j– kB h kB

(6)

In the last expression kbn = ‚/h is the modal bending-wave number. For a plate with bending losses († = bending loss factor) correspondingly: ‚n 4 ‚n 4 +j 1− ZTn = –m † , kB h kB h f fcr m fcr d ; Zm = = . (7) 4 4 F = fcr Z0 Z0 ZTn ‚n ‚n 2 2 , = 2 Zm F † F + j 1−F Z0 k0 h k0 h Depending on the pattern of excitation,symmetrical modes vn(s) ( ) and anti-symmetrical modes vn(a) ( ) will be excited,even for symmetrical support at = ±1 (symmetry defined with respect to = 0). With different supports at both sides, the plate velocity pattern can be written as a mode sum over symmetrical and anti-symmetrical modes. For the “classical” supports ( > Sect. I.8) the plate modes are orthogonal: +1 1 ; m=n () () vm ( ) · vn() ( ) d = ƒmn · NPn ; ƒmn = , 0 ; m = n

(8)

−1

()

where ƒmn is a Kronecker symbol and NPn the norm of the plate mode. Modes with different types of symmetry are evidently always orthogonal to each other. Simply supported plate (the most probable classical support for many technical fixations of construction panels):

Sound Transmission


vn ( ) =

∂ 2 vn ( ) =0 ∂ 2

;

I

= ±1.

553

(9)

Solutions:

vn() ( )

=

⎧ (s) ⎪ ⎨ cos (‚n ) ;

‚n(s) = no

no = 1, 3, 5, . . .

⎪ ⎩ sin (‚ (a) ) ; n

‚n(a)

ne = 2, 4, 6, . . . .

; 2 = ne ; 2

(10)

Polynomial solutions are excluded. Mode norms:

() NPn

+1 =

[vn() ( )]2 d = 1.

(11)

−1

Clamped plate:

= s,a


or

= 0, 1. ∂vn (±1) =0 ∂

vn (±1) =

(12)

General solutions: (s) cos (‚n(s) ) + C(s) () n cosh (‚n ) , vn ( ) = (a) (a) (a) sin (‚n ) + Cn sinh (‚n )

(13)

()

where Cn are solutions of C() n = (−1)

()

sin ‚n

()

sinh ‚n

()

=−

cos ‚n

()

cosh ‚n

.

(14) ()

The second equation gives the characteristic equation for ‚n : tan

‚n()

= ∓ tanh

‚n()

;

=

with approximate solutions:

s

(15)

a ()

‚n ≈ (n ∓ 1/4) ;

n = 1, 2, 3, . . . .

Table 1 Characteristic valuess ‚n (exact and approximations) for a clamped plate n

(s) ‚n;ex

(s) ‚n;apr

(a) ‚n;ex

(a) ‚n;apr

1

2.36502

2.35619

3.92660

3.92699

2

5.49780

5.49779

7.06858

7.06858

3

8.63938

8.63938

10.21018

10.21018

4

11.78097

11.78097

13.35177

13.35177

5

14.92257

‚n(s) (n − 1=4)

16.49336

‚n(a) (n + 1=4)

(16)

I

554

Sound Transmission

Mode norms: () NPn

+1 =

()

[vn() (‚n() )]2 d = 1 ±

−1

() ± [Cn ]2

+2

() Cn () ‚n

sinh (2‚n )

2‚n

()

2‚n

(17) ()

sin ‚n() · cosh ‚n() ± cos ‚n() · sinh ‚n

()

≈1 ±

()

()

1±

sin (2‚n )

sin (2‚n ) ()

2‚n

()

2 ± [C() 1± n ]

sinh (2‚n )

()

2‚n

Table 2 Mode norms (exact and approximate) for a clamped plate N(s) Pn;ex

n

N(s) Pn;apr

N(a) Pn;ex

N(a) Pn;apr

1

1.017651

1.029835

0.9992230

0.9995197

2

1.000034

1.000043

0.9999986

0.9999989

3

1.000000

1.000000

1.000000

1.000000

4

1.000000

1.000000

1.000000

1.000000

Free plate: = s,a or = 0, 1 ∂ 2 vn ( ) ∂ 3 vn ( ) = = 0 ; = ±1. ∂ 2 ∂ 3 ⎧ (s) ⎪ ⎨ cos (‚n(s) ) + C(s) n cosh (‚n ) () vn ( ) = ⎪ ⎩ (a) sin (‚n(a) ) + C(a) n sinh (‚n ),


General solutions:

()

()

()

Cn = −(−1)

where Cn are solutions of: ()

Characteristic equation for the ‚n :

sin ‚n

()

sinh ‚n

()

()

tan ‚n = ∓tanh ‚n

(18)

(19)

()

= ;

cos ‚n . cosh ‚n s = a

(20) (21)

with the same solutions and approximations as for the clamped plate. Additionally a piston-like oscillation (with index n = 0) of the form v0(s) ( ) = 1 + C(s) 0 = const is (s) (s) possible with the characteristic value ‚0 = 0. With the choice C0 = 1, this mode has a norm with unit value. Further, an anti-symmetrical mode is possible with: v0(a) ( ) = C(a) 0 ·

;

‚0(a) = 0

;

C(a) 0 =1

;

n=0

(22)

(the choice C(s) 0 = 1 is arbitrary). These additional modes are called polynomial modes.

Sound Transmission

I

555

The mode norms are the same as for the clamped plate,except for the polynomial modes: 2 (a) 2 2 (s) 2 C (23) N(s) N(a) = . P0 = 2 (1 + C0 ) = 8 ; P0 = 3 0 3 The polynomial modes are orthogonal to the other modes.

I.15

Single Plate across a Flat Duct


A flat (two-dimensional) duct of width a = 2h with hard walls is subdivided by a plate of thickness d; the plate will have different “classical” fixations at the duct walls in x = ±h, i.e. in = x/h = 1. The excitation will first be by a single duct mode of order ‹ and (arbitrary) amplitude Pe . Later “mode mixtures” will be considered.

The field in front of the plate (i.e. in zone I) is formulated as a sum: pI ( , z) = pe ( , z) + pr ( , z) + ps ( , z),

(1)

where pe ( , z) is the incident duct mode, pr ( , z) the hardly reflected duct mode and ps ( , z) the scattered field. The scattered field ps and the transmitted wave pt are formulated as sums of duct modes. The velocity pattern of the plate is formulated as a sum of () plate modes (see previous > Sect. I.14, especially for ‚n and Cn ). ()

pe ( , z) = Pe · q‹ ( ) · e−j k‹z z ()

Z0 vez ( , z) = cos Ÿ‹ Pe · q‹ ( ) · e−j k‹z z

(2)

()

pr ( , z) = Pe · q‹ ( ) · e+j k‹z z ()

Z0 vrz ( , z) = − cos Ÿ‹ Pe · q‹ ( ) · e+j k‹z z () ps ( , z) = PsŒ · qŒ ( ) · e+j kŒz z Œ

Z0 vsz ( , z) = −

Œ

()

cos ŸŒ PsŒ · qŒ ( ) · e+j kŒz z

(3)

(4)

I

556

pt ( , z) =

Sound Transmission

Œ

Z0 vtz ( , z) = V( ) =

()

PtŒ · qŒ ( ) · e−j kŒz (z−d)

()

cos ŸŒ PtŒ · qŒ ( ) · e−j kŒz (z−d)

(5)

Œ

Vn · vn() ( )

n

with the duct mode lateral profiles for symmetrical ( = s) and anti-symmetrical ( = a) modes: ‰Œ(s) = Œe ; Œe = 0, 2, 4, . . . (s) 2 cos (‰Œ ) () qŒ ( ) = (6) sin (‰Œ(a) ) ‰Œ(a) = Œo ; Œo = 1, 3, 5, . . . 2 having the duct mode norms () NŒ

+1 =

() [qŒ ( )]2

()

d = 1 ±

sin (2‰Œ )

−1 ()

() 2‰Œ

2 = ƒŒ

symm. ; ƒi = anti-symm.

()

1; i = 0 . (7) 2;i > 0

()

Setting kvx = ‰v /h, one can introduce duct mode angles ŸŒ , using the wave equation, by: ()

()

()

()

k02 = [kŒx ]2 + [kŒz ]2 = k02 [sin2 ŸŒ + cos2 ŸŒ ], ()

()

kŒx = k0 sin ŸŒ , ()

()

kŒz = k0 cos ŸŒ

() = k0 1 − sin2 ŸŒ

;

(8) √ Re { . . .} ≥ 0 . √ Im { . . .} ≥ 0

They represent the angle with the z axis of plane waves, from which the duct modes can be composed. Coupling coefficients between duct modes and plate modes:

()

SŒn =

,+1 −1

()

()

qŒ ( ) · vn ( ) d .

(9)

The remaining boundary conditions of zone matching are: PsŒ = −PtŒ , $ Œ

2

PtŒ cos ŸŒ · qŒ ( ) =

$ Œ

$ n

Z0 Vn · vn ( )

[ƒ‹Œ Pe − PtŒ ] · qŒ ( ) =

$ n

ZTn Vn · vn ( ).

(10)

Sound Transmission

I

They lead to the following system of equations for Z0 Vn : ⎡ ⎤ ZTm SŒm SŒn ⎦ = 2 S‹m · Pe . Z0 Vn · ⎣ƒmn NPm + ƒŒ Z0 cos ŸŒ n(‹ )

557

(11)

Œ(Œ )

n(‹ ), Œ(‹ ) means: the summation over n and m are in the range of indices belonging to the symmetry type of the incident duct mode, and Œ is in the range of plate modes for the type of plate fixation used. The transmitted duct mode amplitudes PtŒ are evaluated with: ƒŒ SŒn · Z0 Vn (12) PtŒ = 2 cos ŸŒ n(‹ )

and the scattered duct mode amplitudes PsŒ from the first boundary condition. The sound transmission coefficient ‘‹ for the incident ‹-th duct mode is: 2 nob Œgr ƒ‹ ƒŒ Z V 0 n . · SŒn (13) ‘‹ = 4 cos Ÿ‹ cos ŸŒ Pe Œ(‹ ) n(‹ ) The range of duct mode indices used is the range of cut-on modes, with the limit: ncon = 1 − /2 +

k0 a 2

(14)

( = 0 for symmetrical duct modes, = 1 for anti-symmetrical duct modes). Simply supported plate: S(s) Œn

+1 = −1

=

S(a) Œn =

sin( (Œe − no )/2) sin( (Œe + no )/2) 4 no + = (−1)(Œe +no −1)/2 2 , (Œe − no )/2 (Œe + no )/2 no − Œe2 +1 −1

=

' ( ' ( cos Œe · cos no d 2 2

' ( ' ( sin Œo · sin ne d 2 2

sin( (Œo − ne )/2) sin ((Œo + ne )/2) 4 ne − = (−1)(ne +Œo −1)/2 2 . (Œo − ne )/2 (Œo + ne )/2 Œo − n2e

(15)

I

558

Sound Transmission

Clamped plate:

()

SŒn

()

SŒn

⎧ ⎫ +1 ⎪ ⎨ cos (Œe 2 ) ⎪ ⎬ () = (s); Œ = Œe = 0, 2, 4, . . . · vn() (‚n() ) d ; = ⎪ ⎪ () = (a); Œ = Œo = 1, 3, 5, . . . ⎩ sin (Œ ) ⎭ o −1 2 ⎧ ⎪ sinh ‚n(s) sin ‚n(s) ⎪ (s) Œe /2 (s) ⎪ 8 (−1) ‚n ⎪ + Cn ⎪ ⎪ ⎨ 16 (‚n(s) )4 − (Œe )4 16 (‚n(s) )4 + (Œe)4 = ⎪ ⎪ (a) (a) ⎪ cosh ‚ cos ‚ ⎪ n n (a) (Œo −1)/2 (a) ⎪ ‚n − Cn ⎪ ⎩ − 8 (−1) 16 (‚n(a) )4 − (Œo )4 16 (‚n(a) )4 + (Œo )4

=

⎧ (‚n(s) )3 sin ‚n(s) ⎪ ⎪ ⎪ 64 (−1)Œe /2 ⎪ ⎪ ⎨ 16 (‚n(s) )4 − (Œe )4

.

⎪ (a) 3 (a) ⎪ ⎪ (Œo −1)/2 (‚n ) cos ‚n ⎪ ⎪ ⎩ − 64 (−1) (a) 4 16 (‚n ) − (Œo )4

(16)

(17)

Free plate:

()

SŒn =

⎧ ‚n(s) sin ‚n(s) ⎪ Œe /2 2 ⎪ 16 (−1) (Œ ) ⎪ e ⎪ ⎨ 16 (‚n(s) )4 − (Œe )4 ⎪ ⎪ ⎪ (Œ −1)/2 ⎪ (Œo )2 ⎩ − 16 (−1) o

‚n(a) cos ‚n(a)

.

(18)

; Œe = 0, Œe > 0

(19)

(a)

16 (‚n )4 − (Œo )4

Coupling coefficients of polynomial plate modes: sin (Œe /2) 2 (1 + C(s) 0 ) = 4 = = 2 (1 + Œe /2 0; 8 (a) 8 (−1)(Œo −1)/2 = (−1)(Œo −1)/2 . S(a) Œo 0 = C0 Œo Œo

S(s) Œe 0

C(s) 0 )

Relation between coupling coefficients for free and clamped plates: () SŒn, free

2 = 4

Œ () () ‚n

2 ()

· SŒn, clamped ≈

Œ () /2 n ∓ 1/4

2

()

· SŒn, clamped .

(20)

Mixture of incident duct modes (written for a three-dimensional, rectangular duct with mode angles Ÿm,n and mode amplitudes Am,n of incident modes, with an arbitrary reference pressure p0 ):

Sound Transmission

I

559

Sound transmission coefficient ‘ for an arbitrary mixture of incident duct modes, each of which has a modal transmission coefficient ‘(Ÿm,n ): $ ‘(Ÿmn ) Amn 2 · cos Ÿmn · p0 m,n ƒm ƒn 1; m=0 ‘= ; ƒm = (21) Amn 2 $ 1 2 ; m > 0. · · cos Ÿmn ƒ ƒ p m,n

m n

0

In the special case where all incident duct modes have the same energy density (a model that corresponds best to the diffuse sound incidence of room acoustics): Amn 2 ƒm ƒn (22) p = $ Re{cos Ÿ } . 0 mn m,n

Therefore:

$ ‘=

m,n

‘(Ÿmn ) cos Ÿmn $ cos Ÿmn

(23)

mn

(this is, up to the discretisation of the angle of incidence, the relation for diffuse sound incidence). 60 [dB] 50 R 40 30

simply supported

20

clamped

10 0 50

100

200

free

500

1k

4k 2k f [Hz]

Sound transmission loss R of a plaster board across a duct, a = 2 [m] wide, for all propagating duct modes incident with same energy density, and three different boundary fixations of the plate. Parameters: a = 2 [m]; d = 0.0125 [m]; fcrd = 31 [Hzm]; p = 1000 [kg/m3 ]; †p = 0.03

I.16

Single Plate in a Wall Niche


The influence of mounting a wall in a niche of the partition wall (baffle wall) between an emission and a receiving room has played for a while as a “niche effect” some role in the discussion of sound transmission tests.

560

I

Sound Transmission

The object and the sound field formulations here are similar to those in the previous > Sect. I.15, except for the new fields in the niches with depths t1 , t2 , which are formulated as sums of “niche modes”. The plate and the niche have a width b = 2h .

The niche is centred in the baffle wall. The baffle wall is hard and rigid. Only a freely supported plate will be considered below (most parts of the formulas below can be () used for other types of fixation also; then the pertinent values for ‚n and Si,n must be used). The incident wave pe is the ‹-th (propagating) duct mode (symmetrical or anti-symmetrical). Because of the central position of the niche, all other modes have the same symmetry as the incident mode. The field on the front side is composed as pI = pe + pr + ps with: pe = incident duct mode, pr = incident duct mode after hard reflection, ps = scattered field.

(1)

pr , ps and the transmitted wave pt are formulated as sums of duct modes. The plate vibration is formulated as a sum of plate modes. So far the field formulations are similar to those in > Sect. I.15; here additional mode sums of “niche modes” will be used for the fields p1 , p2 in the front and back niche, respectively. Non-dimensional lateral co-ordinates:

= x/h; = x/b = x/(2h )

(2)

Field formulations: Zone I (emission side): pe ( , z) = Pe · q‹ ( ) · e−j k‹z z , Z0 vez ( , z) = cos Ÿ‹ Pe · q‹ ( ) · e−j k‹z z , pr ( , z) = Pe · q‹ ( ) · e+j k‹z z , Z0 vrz ( , z) = − cos Ÿ‹ Pe · q‹ ( ) · e+j k‹z z , PsŒ · qŒ ( ) · e+j kŒz z , ps ( , z) = Œ cos ŸŒ PsŒ · qŒ ( ) · e+j kŒz z . Z0 vsz ( , z) = − Œ

(3) (4)

(5)

Sound Transmission

I

561

Zone III (transmission side): pt ( , z) = PtŒ · qŒ ( ) · e−j kŒz (z−t) , Œ

Z0 vtz ( , z) =

(6)

cos ŸŒ PtŒ · qŒ ( ) · e−j kŒz (z−t) .

Œ

Zone IV (front side niche): Ai e−j giz (z−t1 ) + Bi e+j giz (z−t1 ) · œ(1) p1 ( , z) = i ( ), i

Z0 v1z ( , z) =

(7)

cos ¥i Ai e−j giz (z−t1 ) − Bi e+j giz (z−t1 ) · œ(1) i ( ).

i

Zone V (back side niche, if its width h is different from that of the front side niche; = x/h ): ( ' (2) (2) p2 ( , z) = Ci e−j giz (z−t1 −d) + Di e+j giz (z−t1 −d) · œ(2) i ( ), i

Z0 v2z ( , z) =

(2)

cos ¥i

'

(2)

Ci e−j giz

( (2) (2) − Di e+j giz (z−t1 −d) · œi ( ).

(z−t1 −d)

(8)

i

For h = h is = and the upper niche indices (1),(2) are not needed. Zone II (plate): Z0 Vn · vn ( ). Z0 V( ) =

(9)

n

Mode profiles Duct modes: ⎧ ⎨ cos (‰Œ(s) ) ; symmetrical () qŒ ( ) = ⎩ sin (‰Œ(a) ) ; anti-symmetrical

2 = Œo 2

‰Œ(s) = Œe

;

Œe = 0, 2, 4, . . .

‰Œ(a)

;

Œo = 1, 3, 5, . . .

(10)

having the duct mode norms () NKŒ =

+1

() [qŒ ( )]2

()

d = 1 ±

−1

Niche modes: ⎧ ⎪ ⎨ cos (ie 2 ) ; œi ( ) = ⎪ ⎩ sin (i ) ; o 2

sin (2‰Œ ) () 2‰Œ

2 = ƒŒ

symm. anti-symm.

=s=0

;

ie = 0, 2, 4 . . .

=a=1

;

io = 1, 3, 5 . . .

; ƒn =

1; n = 0 2 ; n > 0.

(11)

(12)

562

I

Sound Transmission

with axial wave numbers and mode angles: √ Re . . . ≥ 0 2 giz = k0 cos ¥i = k0 1 − sin ¥i ; √ Im . . . ≤ 0 ()

having niche mode norms: NNi = Plate modes: ⎧ (s) ⎪ ⎨ cos (‚n ) ; () vn ( ) = ⎪ ⎩ sin (‚ (a) ) ; n having plate mode norms:

;

sin ¥i = i ()

2 . ƒi

2 = ne 2

;

no = 1, 3, 5, . . .

‚n(a)

;

ne = 2, 4, 6, . . .

+1 =

(13) (14)

‚n(s) = no

() NPn

Š0 2b

(15)

[vn() ( )]2 d = 1.

(16)

−1

Auxiliary amplitudes: Xi± = Ai ± Bi ; Yi± = Ai e+j giz t1 ± Bi e−j giz t1

Ui± = Ci ± Di ; Wi± = Ci e−j giz t2 ± Di e+j giz t2

Xi+ =

j j · Xi− − · Yi− , tan (giz t1 ) sin (giz t1 )

j j · Xi− − · Yi− , Yi+ = sin (giz t1 ) tan (giz t1 )

;

Ui+ = ;

−j j · Ui− + · Wi− , tan (giz t2 ) sin (giz t2 )

−j j Wi+ = · Ui− + · Wi− . sin (giz t2 ) tan (giz t2)

(17)

(18)

The boundary conditions of field matching at the zone limits lead to two coupled systems of equations for Y1− , W1− : {M11} ◦ {Yˆ− } + {M12} ◦ {Wˆ− } = 2 Pe · Q‹i ‹ , (19) {M21} ◦ {Yˆ− } + {M22} ◦ {Wˆ− } = 0,

Sound Transmission

I

563

with matrices

{M11} = − j {Iii } ∗ NNi cos(giz t1 ) b cos ¥ˆ t sin(giz t1 ) ∗ {TŒi } ◦ {TŒˆ } ∗ + , a NKŒ cos ŸŒ {M12} = + j {Iii } ∗ NNi cos(giz t2 ) b cos ¥ˆ sin(giz t2 ) ∗ {TŒi }t ◦ {TŒˆ } ∗ , a NKŒ cos ŸŒ NNi , {M21} = j {Iii } ∗ sin (giz t1 ) NNi {M22} = j {Iii } ∗ − {Gii } ◦ {Hi ˆ } sin (giz t2 )

(20)

−

and right-side vector Q‹i ‹ = sin(giz t1 )T‹i ‹ ,

(21)

where ◦ indicates a matrix multiplication; {am } ∗ {cm } = {am cm } indicates a term-wise multiplication of two vectors, and correspondingly {cm xmn } = {cm }∗{xmn } indicates the multiplication of the mth row of the matrix {xmn } with the element cm of the vector {cm }, and {dn xmn } = {dn }∗{xmn } {{dn }∗{xmn }t }t indicates the multiplication of the n-th column of {xmn } with dn ; {xmn }t is the transposed matrix; and {Iii } is the unit matrix. ()

The coupling coefficients TŒi = TŒi between duct modes and niche modes (where the second form indicates the symmetry type = s, a) are:

+1 TŒi = () TŒi

qŒ −1 +1

= −1

b · œi ( ) d , a

cos (Œe /2 · b/a · ) · cos (ie /2 · ) sin (Œo /2 · b/a · ) · sin (io/2 · ) .

sin ((i − Œb/a) /2) sin ((i + Œb/a) /2) = ± (i − Œb/a) /2 (i + Œb/a) /2 Abbreviation used: {Giˆ } : = j {Iii } ∗ NNi with which {Giˆ } ◦ {Uˆ− } = j

d

;

=

;

=

s a

s; Œe , ie . a; Œo, io

ZTn · cos ¥ˆ sin (giz (t1 + t2 )) t , − {Sin } ◦ {Sˆn } ∗ sin (giz t1 ) · sin (giz t2 ) NPn

NNi Wi− NNi Yi− + . sin (giz t1 ) sin (giz t2 )

(22)

(23)

(24)

The symbol ZTn denotes the n-th modal partition impedance of the plate ( > Sect. I.15).

I

564

Sound Transmission

With the solutions Y1− , W1− the other mode amplitudes in the duct are evaluated from: PsŒ =

−b/a b/a cos ¥i TŒi · Yi− ; PtŒ = cos ¥i TŒi · Wi− . NKŒ cos ŸŒ NKŒ cos ŸŒ i

(25)

i

If one is only interested in the PtŒ , a simplified system of equations is: − {M11} ◦ {M21}−1 ◦ {M22 } + {M12} ◦ {Wˆ− } = 2 Pe · Q‹i ‹ . The transmission coefficient ‘‹ for a single incident (propagating) duct mode is: $ 2 ¢tŒ PtŒ 1 Œ = N cos Ÿ ‘‹ = KŒ Œ P b/a · ¢e‹ b/a · NK‹ cos Ÿ‹ Œ e 1 Wi− 2 b/a {TŒi } ◦ cos ¥i = NK‹ cos Ÿ‹ Œ NKŒ cos ŸŒ Pe 2 1 Wi− b/a = TŒi cos ¥i . NK‹ cos Ÿ‹ Œ NKŒ cos ŸŒ Pe

(26)

(27)

i

60 [dB] 50 R 40

t1=0.1 [m] ; t2=0.3 [m]

30 20 t1=t2=0.2 [m]

10 0 50

100

200

500

1k 2k f [Hz]

4k

Transmission loss R through a plaster board plate in a niche in partition wall between test rooms. All propagating modes of the emission side duct are incident with equal energy density. The example shows the singularity of a central position of the test object in a niche

I.17

Strip-Shaped Wall in Infinite Baffle Wall


A wall of thickness d and width a = 2c is placed in a hard baffle wall, also of thickness d. Elliptic-hyperbolic cylindrical systems of co-ordinates , ˜ are used with focus positions at x = ±c.

Sound Transmission

I

565

A plane wave pe is assumed to be incident at a polar angle Ÿ; it becomes pr after hard reflection at the front side surface.An additional scattered wave ps is needed to formulate the front side sound field: pI = pe + pr + ps.

(1)

The transmitted field is pt . Both ps and pt are formulated as sums of Mathieu functions ( See also: Mechel (1997) for notations, formulas and generation of Mathieu functions). The velocity pattern of the plate is formulated as a sum of plate modes in the normalised co-ordinate = x/c. The sound-transmitting wall here is assumed to be a simply supported single plate. Other types of walls are treated correspondingly. Transformation between Cartesian and elliptic-hyperbolic co-ordinates: ⎫ x = c · cosh · cos ˜ ⎬ ; 0 ≤ < ∞ ; − ≤ ˜ ≤ +. ⎭ z = c · sinh · sin ˜

(2)

Field formulations: pe (, ˜ ) + pr (, ˜ ) = 4Pe

∞

(−j)m cem () · Jcm () · cem (˜ ),

(3)

Dm (−j)m cem () · Hc(2) m () · cem (˜ ),

(4)

m=0

pt (, ˜ ) = −ps (, ˜ ) = 2

∞ m=0

566

I

Sound Transmission

Z0 vt (0, ˜ > 0) Z0 vs (0, ˜ < 0)

) =

±j Dm (−j)m cem () Hc(2) m (0) · cem (˜ ), sin ˜ m=0

(5)

where Pe is the amplitude of the incident plane wave, = /2 − Ÿ, = k0c/2, cem (˜ ) are the even azimuthal Mathieu functions, Hc(2) m () are the radial Hankel-Mathieu functions of the second kind (associated with the cem (˜ )), and Dm are the mode amplitudes. Plate velocity: Vn · vn(s) ( ) V( ) =

(6)

n

with (symmetrical, = s, and anti-symmetrical, = a) plate modes: ⎧ ⎪ ⎨ cos (‚n(s) ) ; ‚n(s) = no/2 ; no = 1, 3, 5, . . . ; () = (s) = symmetrical vn() ( ) = ⎪ ⎩ sin (‚ (a) ) ; ‚ (a) = n /2; n = 2, 4, 6, . . . ; () = (a) = anti-symmetrical n

having mode norms:

e

n

NPn =

,+1 −1

(7)

e

vn()2( ) d = 1,

(8)

and modal plate partition impedances: ZTn fcr m fcr d ‚n 4 ‚n 4 f 2 2 = 2Zm F †F +j 1−F = P , (9) ; F = ; Zm = Z0 k0 c k0 c fcr Z0 Z0 where: f = frequency ; f cr = critical frequency ; m = d · P = plate surface mass density; † = plate loss factor. The boundary conditions give a linear system of equations for the plate mode amplitudes Vn :

Z0 Vn ƒn,Œ

n

ZTŒ 8 Hc(2) m (0) Qm,Œ Qm,n · NPŒ + (2) Z0 j m≥0 Hcm (0) = 4 Pe

(−j)m cem () Jcm (0) Qm,Œ

;

Œ = 1, 2, 3, . . . , (10)

m≥0

and with its solutions the mode amplitudes Dm of the transmitted wave: 2 (j)m−1 Qm,n · Z0 Vn. Dm = cem () Hc(2) m (0) n

(11)

Therein: ƒn,Œ = Kronecker symbol, and mode coupling coefficients: +1 Qm,n : = −1

= 0

cem (arccos ; 2 ) · vn() (‚n ) d (12) sin ˜ · cem (˜ ; 2 ) · vn()(‚n cos ˜ ) d˜ ,

Sound Transmission

I

567

which are evaluated for symmetrical plate modes, for which m = 2r, by: &

i s 2 i! 2 Q2r,n = A2s J1/2(‚n ) + (1 − ƒ0,s) (−1)i ‚n s=0 (2i)! ‚n i=1 i−1 1 · Ji+1/2 (‚n ) 4s2 − 4k 2 ,

(13)

k=0

and for anti-symmetrical plate modes, for which m = 2r + 1: Q2r+1,n = &

s i 2 i! 1 A2s+1 J3/2 (‚n ) + (1 − ƒ0,s ) (−1)i (2s + 1)2 − (2k − 1)2 ‚n s=0 (2i)! i=1 k=1 i 2 · · Ji+3/2 (‚n ) . ‚n

(14)

Here Jn (z) are Bessel functions and An are the Fourier series components needed for the evaluation of the azimuthal Mathieu functions. The sound transmission coefficient ‘(Ÿ) for oblique incidence finally is: ‘(Ÿ) =

Dm 2 1 ce2 (/2 − Ÿ), cos Ÿ m≥0 Pe m

(15)

and for diffuse sound incidence (two-dimensional):

‘2−dif

I.18

2 =

+/2

‘(Ÿ) cos Ÿ dŸ.

(16)

−/2

Finite-Size Plate with a Front Side Absorber Layer


A simply supported plate of thickness d with a porous layer of thickness t and characteristic values a , Za of the material on its front side is placed across a flat duct with lateral dimension a = 2h. There is no (or only loose) mechanical contact between the plate and the layer.

568

I

Sound Transmission

The ‹-th (propagating) duct mode is assumed to be the incident wave; the index = s, a indicates whether the incident duct mode is symmetrical or anti-symmetrical. All other fields are of the same symmetry type. The field pI on the front side is composed as pI = pe + pr + ps of the incident wave pe , of its hard reflection pr (at x = 0) and of a scattered wave. The scattered wave ps and the transmitted wave pt , as well as the field pa in the absorber layer, are formulated as duct mode sums. The plate vibration V( ) is a plate mode sum with = x/h. Field formulations: ()

pe ( , z) = Pe · q‹ ( ) · e−j k‹z z , ()

Z0 vez ( , z) = cos Ÿ‹ Pe · q‹ ( ) · e−j k‹z z ,

(1)

()

pr ( , z) = Pe · q‹ ( ) · e+j k‹z z , ()

Z0 vrz ( , z) = − cos Ÿ‹ Pe · q‹ ( ) · e+j k‹z z , ps ( , z) =

(2)

()

PsŒ · qŒ ( ) · e+j kŒz z ,

Œ

Z0 vsz ( , z) = −

()

cos ŸŒ PsŒ · qŒ ( ) · e+j kŒz z ,

(3)

Œ

pt ( , z) =

Œ

Z0 vtz ( , z) =

()

PtŒ · qŒ ( ) · e−j kŒz (z−d) ,

()

cos ŸŒ PtŒ · qŒ ( ) · e−j kŒz (z−d) ,

(4)

Œ

V( ) =

Vn · vn() ( ),

(5)

n

pa ( , z) =

Œ

Z0 vaz ( , z) =

PaŒ · qŒ ( ) · e−Œ z + rŒ e+Œ z ,

Œ PaŒ · qŒ ( ) · e−Œ z − rŒ e+Œ z a Zan Œ

(6)

Sound Transmission

with

() qŒ ( )

=

cos (‰(s) Œ )

;

sin (‰(a) Œ ) &

kvz Œ = cos ŸŒ = 1− k0 2k0 h & 2 a Œ 2 Œ + = k0 k0 2k0 h

and vn() ( ) =

() ‰Œ

=

() kŒx h

2 ;

=

Œe Œo

2

;

;

Œo = 1, 3, 5, . . . ,

(7) ()

NKŒ = 2/ƒŒ ,

⎧ (s) ⎪ ⎨ cos (‚n ) ;

‚n(s) = no

no = 1, 3, 5, . . .

⎪ ⎩ sin (‚ (a) ) ; n

‚n(a)

ne = 2, 4, 6, . . . .

; 2 = ne ; 2

569

Œe = 0, 2, 4 . . .

√ Re{ } ≥ 0 √ Im{ } ≤ 0,

√ Re{ } ≥ 0

;

)

I

(8)

The boundary conditions give a linear system of equations for the plate mode amplitudes Vn : n

ZTm ƒŒ SŒm SŒn (1 + CŒ )2 − (1 − CŒ )2 e−2Œ t Z0 Vn · ƒm,nNPm + Z0 2 cos ŸŒ (1 + CŒ ) − (1 − CŒ ) e−2Œ t Œ =

4 C‹ S‹m e−‹ t · Pe ; m (1 + C‹ ) − (1 − C‹ ) e−2‹ t

(9)

with the plate mode norms NPm = 1, the abbreviations: CŒ : = an Zan cos ŸŒ

k0 Œ

;

an = a /k0

;

Zan = Za /Z0

(10)

and the mode coupling coefficients: S(s) Œn

+1 = −1

=

S(a) Œn

' ( ' ( cos Œe · cos no d 2 2

(11)

sin ((Œe − no )/2) sin ((Œe + no )/2) 4 no + = (−1)(Œe +no −1)/2 2 , (Œe − no )/2 (Œe + no )/2 no − Œe2 +1

= −1

' ( ' ( sin Œo · sin ne d 2 2

(12)

sin ((Œo − ne )/2) sin ((Œo + ne )/2) 4 ne = − = (−1)(ne +Œo −1)/2 2 . (Œo − ne )/2 (Œo + ne )/2 Œo − n2e One computes, with the solutions Z0 Vn , the transmitted duct mode amplitudes PtŒ by: PtŒ =

ƒŒ SŒn · Z0 Vn , 2 cos ŸŒ n

(13)

I

570

Sound Transmission

or, directly, the transmission coefficient: 2 n Œlim hi ƒ‹ ƒŒ Z0 Vn ‘‹ = · SŒn , 4 cos Ÿ‹ cos ŸŒ Pe Œ

(14)

n

where the upper summation limit Œlim is the index limit for propagating duct modes and nhi is the upper index limit for the plate modes used, which is set by the convergence of the system of equations for Z0 Vn . 70 Ξ=10 [kPas/m2]

60 [dB] 50 R(μ) 40

Ξ=5 [kPas/m2]

Ξ=0

30 20 10 0 50

μ=0

100

μ=2

200

μ=4

500

1k

4k 2k f [Hz]

Sound transmission loss R(‹) for the ‹-th duct mode through a plaster board with a front side glass fibre layer (if ¡ > 0) having flow resistivity values ¡. Parameters: a = 2 [m]; dp = 0.0125 [m]; t = 0.1 [m]; fcrd = 31 [Hzm]; p = 1000 [kg/m3 ]; † = 0.03

I.19

Finite-Size Plate with a Back Side Absorber Layer


See the previous sition.

>

Sect. I.18 for the duct, the plate, its mounting and the field compo-

The incident wave is again the ‹-th propagating duct mode.

Sound Transmission

I

The system of equations for the plate mode amplitudes Vn now reads: ZTm ƒŒ 1 + rŒ 1 + CŒ · SŒn SŒm = 2 S‹m · Pe Z0 Vn · ƒm,nNPn + Z0 2 cos ŸŒ 1 − rŒ n Œ

571

(1)

with the modal reflection factors at the back side of the absorber layer: rŒ = e−2Œ t

1 − CŒ . 1 + CŒ

(2)

After solving for Z0 Vn , the transmitted duct mode amplitudes are evaluated from: PtŒ = ƒŒ

CŒ e−Œ t SŒn · Z0 Vn , 1 + CŒ (1 − rŒ ) cos ŸŒ n

(3)

or the sound transmission coefficient directly from: 2 2 nhi Œlim ƒ‹ C Z ƒŒ V Œ 0 n · e−2Œ t S · ‘‹ = Œn cos Ÿ‹ Œ cos ŸŒ (1 − rŒ ) (1 + CŒ ) Pe n

(4)

with = Re{ }; see the previous section for the mode order limits Œlim and nhi . The numerical results are the same as with a front side absorber layer (see previous > Sect. I.18), except for some details around the coincidence frequency of the plate.

I.20

Finite-Size Double Wall with an Absorber Core


A double wall with a porous absorber layer as a core is mounted in a flat, hard duct. The plate borders are simply supported at the duct walls (as an example; other supports only need other plate mode wave numbers ‚n , mode norms NPn , and modal partition impedance ZTn ). There is no (or only a loose) mechanical contact between the plates and the absorber layer. The characteristic values of the porous material are a , Za , or, in normalised forms, an = a /k0, Zan = Za /Z0 .

572

I

Sound Transmission

The incident wave pe is the ‹-th (symmetrical or anti-symmetrical) propagating duct mode. The sound field on the front side is composed as: pI = pe + pr + ps,

(1)

where pr is the incident mode after hard reflection at z = −(t/2 + d1 ) and ps is the scattered field. The scattered wave and the transmitted wave pt are formulated as duct mode sums, as well as the sound field pa in the absorber layer. The plate velocity patterns V(i) ( ); i = 1, 2; are plate mode sums with = x/h. Field formulations: pe ( , z) = Pe · q‹ ( ) · e−j k‹z (z+t/2+d1 ) , Z0 vez ( , z) = cos Ÿ‹ · pe ( , z), pr ( , z) = Pe · q‹ ( ) · e+j k‹z (z+t/2+d1) , Z0 vrz ( , z) = − cos Ÿ‹ · pr ( , z), ps ( , z) =

(2)

(3)

PsŒ · qŒ ( ) · e+j kŒz (z+t/2+d1 ) ,

Œ

Z0 vsz ( , z) = −

cos ŸŒ PsŒ · qŒ ( ) · e+j kŒz (z+t/2+d1 ) ,

(4)

Œ

pt ( , z) =

Œ

Z0 vtz ( , z) =

PtŒ · qŒ ( ) · e−j kŒz (z−t/2−d2 ) ,

cos ŸŒ PtŒ · qŒ ( ) · e−j kŒz (z−t/2−d2) ,

(5)

Œ

pa ( , z) =

Œ

Z0 vaz ( , z) = V(i) ( ) =

n

AŒ e−Œ z + BŒ e+Œ z · qŒ ( ),

Œ AŒ e−Œ z − BŒ e+Œ z · qŒ ( ), a Zan Œ

(6)

(i) V(i) n · vn ( )

(7)

;

i = 1, 2.

Sound Transmission

Duct modes: ' Œ ( ⎧ ⎪ ; Œ = 0, 2, 4, . . . ; symmetrical ⎨ cos 2 qŒ ( ) = ' ( ⎪ ⎩ sin Œ ; Œ = 1, 3, 5, . . . ; anti-symmetrical, 2 & √ kŒz Œ 2 Re{ . . .} ≥ 0 = cos ŸŒ = 1 − ; √ k0 2k0h Im{ . . .} ≤ 0, & Œ Œ 2 √ 2 + = an ; Re{ . . .} ≥ 0 ; an = a /k0. k0 2k0h

I

573

(8)

(9)

Plate modes (for simplyßindexsound transmission!through simly supported plates supported plates; see > Sect. I.14 for other supports): ⎧ (s) (s) ⎪ ⎨ cos (‚n ) ; ‚n = no 2 ; no = 1, 3, 5, . . . vn() ( ) = (10) ⎪ ⎩ sin (‚n(a) ) ; ‚n(a) = ne ; ne = 2, 4, 6, . . . , 2 () NPn

+1 =

[vn() ( )]2 d =

−1

ZTn ZTn Z0

2 = 1, ƒn

(11)

‚n 4 ‚n 4 = –m † +j 1− , kB h kB h f fcr m fcr d 4 4 F = f ; Zm = Z = Z (12) cr 0 0 ‚n ‚n = 2 Zm F † F2 + j 1−F2 ; k0 h k0 h

Mode norms: 1 NKŒ : = −1

2 [qŒ ( )] d = ƒŒ 2

;

() NPn : =

1

[vn() ( )]2 d .

(13)

−1

Abbreviation: CŒ = an Zan cos ŸŒ

k0 . Œ

(14)

Auxiliary amplitudes: XŒ± : = AŒ e+Œ t/2 ± BŒ e−Œ t/2 ,

(15)

YŒ± : = AŒ e−Œ t/2 ± BŒ e+Œ t/2 with intrinsic relations: YŒ− XŒ− − ; XŒ+ = tanh (Œ t) sinh (Œ t)

YŒ+ =

YŒ− XŒ− − . sinh (Œ t) tanh (Œ t)

(16)

I

574

Sound Transmission

The boundary conditions for them give the following coupled systems of linear equations: ) (1) 1 1 + ƒŒ,Œ NKŒ cos ŸŒ YŒ− S(1) Œ n SŒn + − XŒ− (1) CŒ tanh (Œ t) sinh (Œ t) n N(1) Pn ZTn /Z0 Œ = 2 Pe

(1) S(1) Œ n S‹n

, (17) (1) (1) NPn ZTn /Z0 ) (2) −XŒ− 1 1+ƒŒ,Œ NKŒ cos ŸŒ S(2) Œ n SŒn + YŒ− =0 + (2) (2) sinh (Œ t) CŒ tanh (Œ t) Œ n NPn ZTn /Z0 n

with the mode coupling coefficients: S(i) vn : =

1

qv ( ) · vn(i) ( ) d .

(18)

−1

The scattered and transmitted mode amplitudes are: PsŒ =

−1 · XŒ− CŒ

;

PtŒ =

1 · YŒ− , CŒ

(19)

and the transmission coefficient ‘‹ for a single incident mode is: ‘‹ =

Œlim ƒ‹ cos ŸŒ cos Ÿ‹ Œ ƒŒ

YŒ− 2 , · CŒ

(20)

where Œlim is the mode order limit for propagating duct modes.

I.21

Plenum Modes

Mechel, Vol. III, Ch. 19 (1998) This section serves as preparation for the next solutions in the plenum of suspended ceilings.

> Sect. I.22. It deals with

characteristic

Sound Transmission

I

575

The object is a flat (two-dimensional) duct with one hard wall (at z = d + t + h), an air space of height h (d + t ≤ z ≤ d + t + h), a porous absorber layer of thickness t (d ≤ z ≤ d + t), and an elastic plate of thickness d (0 ≤ z ≤ d). The free space z ≤ 0 belongs to the object. The porous absorber material is described by its characteristic propagation constant a and wave impedance Za (or, in normalised form, an = a /k0 , Zan = Za /Z0 ). The elastic plate is described by its partition impedance ZT for a polar angle of incidence ¥ . Elementary solutions are sought which obey the wave equations in air and in the absorber material : + k02 pH = 0 ; + k02 pt = 0 ; − a2 pA = 0 (1) and the bending wave equation of the plate (which is guaranteed by using the partition impedance), as well as the boundary conditions. The solutions are called “plenum modes”, with mode index n. A normalised longitudinal co-ordinate may be used with some reference length a: = x/a. Formulations of the component fields: pHn ( , z) = PHn · e±n a · cos (—n (z − h − t − d)) , Z0 vHnz ( , z) = −j

—n PHn · e±n a · sin (—n (z − h − t − d)) , k0

(2)

ptn ( , z) = Ptn · e±n a · e−n a · ej ‰n z , (3)

‰n ptn ( , z), k0 pAn ( , z) = e±n a Cn · e−‚n (z−d) + Dn · e+‚n (z−d) . Z0 vtnz ( , z) = −

(4)

From the wave equation in the plenum space it follows that: n2 − —2n + k02 = 0, 2 —n n 2 + = 1 = sin2 ¥n + cos2 ¥n j k0 k0 n h = (—n h)2 − (k0 h)2 ; Re{n h} ≥ 0,

;

sin ¥n = n /j k0, cos ¥n = —n /k0

(5)

which defines a modal angle of incidence ¥n . The corresponding equations in the free space z ≤ 0 lead to —n = ±‰n . The wave equation in the absorber material is satisfied with: ‚n √ 2 + 1 − (— /k )2 ; = an Re{ . . .} ≥ 0. (6) n 0 k0 An abbreviation used later is:

Gn : = an Zan

—n tan (—n h). ‚n

(7)

576

I

Sound Transmission

The modal partition impedance of the plate is ( > Sect. I.9): ZTn = 2 Zm F †F2 sin4 ¥n + j 1−F2 sin4 ¥n Z0 2 sin4 ¥n = 1 − (—n /k0)2 .

;

F: =

fcr m fcr d f ; Zm : = = fcr Z0 Z0

(8)

The remaining boundary conditions give the following homogeneous linear system of equations for the mode amplitudes Cn , Dn : —n ‚n /k0 —n ‚n /k0 Cn e−‚n t · j + Dn e+‚n t · j = 0, tan (—n h) − tan (—n h) + k0 an Zan k0 an Zan (9) ‚n /k0 k0 ZTn ‚n /k0 k0 ZTn + Dn · 1 − = 0. Cn · 1 + + + an Zan ‰n Z0 an Zan ‰n Z0 A non-trivial solution exists if the determinant of the coefficient matrix vanishes; this gives the following characteristic equation for the wave numbers of the plenum modes: cosh (‚n t)·

—n ‚n /k0 k0 ZTn j tan (—n h) · tanh (‚n t) + + k0 an Zan ‰n Z0 ‚n /k0 ‚n /k0 k0 ZTn = 0. + · 1+ + an Zan an Zan ‰n Z0

(10)

The leading factor can be assumed to be cosh (‚n t) = 0; thus the expression in the curled brackets must be zero. Taking z = ‰n h as the quantity for which solutions shall be found, the equation reads: j z · tan z · z · tanh

‚n k0 t k0

1 ‚n ZTn + 1+z an Zan k0 Z0 1 ‚n ZTn 1 ‚n z+ 1+z = 0 (11) + an Zan k0 an Zan k0 Z0

(‚n and ZTn are functions of z). A method of solving for a set zn of modes is described in [Mechel, Vol. III, Ch. 19 (1998)].

Im{κnh}

Sound Transmission

I

577

2 1 0

n=-3

n=-2

n=-1

f n=1

n=2

n=3

-1 -2

-7.5

-5

-2.5

0

2.5

5 Re{κnh}

7.5

Example of plenum mode solutions ‰n h for a plaster board as the elastic plate, and with a t = 4 [cm] thick glass fibre mat as absorber layer. Parameters: h = 0.4 [m]; d = 0.0095 [m]; t = 0.04 [m]; fcr d = 31 [Hz · m]; p = 1000 [kg/m3 ]; † = 0.1; ¡ = 10 [kPa s/m2 ]

I.22

Sound Transmission through Suspended Ceilings


A typical set-up of a suspended ceiling is taken from the previous > Sect. I.21, from where notations for the component fields are also used. The next graph shows the arrangement of an emission room of width as (index s on the emission side from the German Sendeseite) and a receiving room of width ae (index e on the receiver side from the German Empfangsseite). The suspended ceiling spans over both rooms; the partition wall between the two rooms is rigid. The target quantity is the flanking transmission loss Rf = −10 · lg(‘) of sound transmission through the suspended ceiling. First, the transmission coefficient ‘‹ for the ‹-th propagating mode of the emission room as incident wave pe will be given; then the transmission loss for all propagating emission room modes (with equal energy densities) follows. The non-dimensional axial coordinates s = x/as ; e = −x/ae will be used. The back walls of the plenum are assumed to have reflection factors rs , re . The sound field in the emission room is composed as pe + pr + ps , where pe is the ‹-th propagating room mode of the emission room, pr is this mode after hard reflection at the lower surface of the suspended ceiling and ps is a (back-)scattered field. Both ps and the transmitted sound pt are composed as mode sums of the room modes in the relevant rooms.

578

I

Sound Transmission

The sound fields in the plenum spaces and in the absorber layer are composed as sums of plenum modes.A lower index = s, e indicates to which side the sound wave belongs; an upper index (±) indicates the direction of a mode. The incident and reflected emission room modes are (‹ = 0, 1, 2, . . .): pe ( s , z) = Pe · cos (‹ s ) · e−j k‹z z

;

Z0 vez ( s , z) = cos Ÿ‹ · pe ( s , z),

pr ( s , z) = Pe · cos (‹ s ) · e+j k‹z z

;

Z0 vrz ( s , z) = − cos Ÿ‹ · pr ( s , z)

(1)

with modal angles of incidence: k‹x ‹ = sin Ÿ‹ = k0 k0as

;

k‹z cos Ÿ‹ = = k0

& 1−

‹ k0as

2 ;

√ Re{ . . .} ≥ 0 (2) √ Im{ . . .} ≤ 0

and similarly modal angle ŸŒ in the receiving room. The transmitted sound field is (Œ = 0, 1, 2, . . .): PtŒ · cos (Œ e ) · ej kŒz z , pt ( e, z) = Œ

Z0 vtz ( e, z) = −

PtŒ cos ŸŒ · cos (Œ e ) · ej kŒz z .

(3)

Œ

The desired sound transmission coefficient is: Œlim ƒ‹ cos ŸŒ PtŒ 2 ‘‹ = cos Ÿ‹ Œ ƒ Œ Pe

(4)

with Œlim the mode order limit for propagating modes, given by the condition

Œ≤

k0ae 2ae = . Š0

The modal components in the plenum space are formulated as ( = s, e): 3 2 +n a −n a · cos (—n (z − h − t − d)) , + P(−) pHn ( , z) = P(+) Hn · e Hn · e

(5)

(6)

Sound Transmission

Z0 vHnz ( , z) = −j

3 —n 2 (+) −n a PHn · e+n a + P(−) · sin (—n (z − h − t − d)) , · e Hn k0

I

579

(7)

and in the absorber layer: −‚n (z−d) +‚n (z−d) + D(+) pAn ( , z) = e+n a C(+) n ·e n ·e −‚n (z−d) +‚n (z−d) . + D(−) + e−n a C(−) n ·e n ·e

(8)

The reflection at the back walls of the plenum with given reflection factors r can serve to eliminate some sets of amplitudes: −2 n a · P(−) P(+) Hn = r · e Hn ,

(9)

and from the matching of fields at the surface between the plenum space and absorber layer: 1 (±) +‚n t (±) Cn = PHn ·e · cos (—n h) · 1 + j Gn ; 2 1 (±) (±) Dn = PHn · e−‚n t · cos (—n h) · 1 − j Gn . 2

(10)

The field matching in the plane x = 0 leads to two coupled systems of linear equations for the P(−) Hn : 2 (−) 3 −2 n ae · Mmn PHsn 1 + rs e−2 n as − P(−) Hen 1 + re e n

= j h S‹m · P(h) Hs‹ −

( t ' (−) (+) · Bs‹ , (11) R‹m · As‹ + R‹m an Zan

n 2 (−) 3 (−) PHsn 1 − rs e−2 n as + PHen 1 − re e−2 n ae · Mmn = 0. k0 n

(12)

(±) (±) With the solutions one evaluates P(+) Hn , and with these Cn , Dn . The amplitudes PtŒ , which are needed for the transmission coefficient, are evaluated from:

PtŒ = −

(+) (−) (−) ƒŒ ‚n /k0 (+) (−) Cen − D(+) en TŒn + Cen − Den TŒn . cos ŸŒ n an Zan

(13)

One has in the above equations: 1 +”‹ t e cos (k‹z h) + j C‹ sin (k‹z h) · PH , 2 1 B‹ = e−”‹ t cos (k‹z h) − j C‹ sin (k‹z h) · PH , 2

A‹ =

(14)

580

I

Sound Transmission

and PH =

4 Pe · e−”‹ t cos (k‹z h) 1 + e−2 ”‹ t + 1/C‹ · 1 + ZT‹ /Z0 · cos Ÿ‹ 1 − e−2 ”‹ t + . . . ... −2 ” t . . . + j sin (k‹z h) C‹ 1 − e ‹ + 1 + ZT‹ /Z0 · cos Ÿ‹ 1 + e−2 ”‹ t

with k‹z h = k0 h · cos Ÿ‹ C‹ : = an Zan

;

”‹ = k0

&

2 + an

‹ k0 a

2 ;

√ Re{ . . .} ≥ 0,

k0 cos Ÿ‹ . ”‹

(15)

(16)

Further, the weight factors Mm,n of the inter-orthogonality of plenum mode factors are used; they are defined by: 1 Mm,n 1 pHm (z) · pHn (z) dz + pAm (z) · pAn (z) dz: = (17) j k0 Z0 a Za k0Z0 H

A

(if the plate of the suspended ceiling is rigid, then Mm,n = ƒm,n = Kronecker symbol) with evaluation by: cos (—m h) cos (—n h) h sin ((—m − —n ) h) sin ((—m + —n ) h) Mm,n = + + t 2j (—m − —n ) h (—m +—n ) h 4an Zan 1 − e−(‚m +‚n ) t 1−e+(‚m+‚n ) t (18) − 1−jGm 1−jGn + 1+jGm 1+jGn (‚m +‚n ) t (‚m +‚n ) t 1 − e−(‚m −‚n ) t 1 − e+(‚m −‚n ) t − 1 − jGm 1 + jGn . + 1+jGm 1−jGn (‚m −‚n ) t (‚m − ‚n ) t Other factors are mode coupling factors, between directly transmitted field and plenum mode field in the absorber layer: (±) R‹n

1 := t

d+t (±) e±”n (z−d) · pAn (z) dz d

cos (—n h) ‚n t e · 1 + j Gn ‚n ± ”n − e−‚n t · 1 − j Gn ‚n ∓ ”n = 2 2 2t (‚n − ”n ) + 2 e±”n t j ‚n Gn ± ”n

(19)

Sound Transmission

Rdif 80 [dB] 70 60

as=2 [m] ae=4.75 [m] rs=1 re=1 as=2 [m] ae=2 [m] rs=1 re=1

40 20 10 0 100

581

as=2 [m] ae=4.75 [m] rs=1 re=0

50 30

I

as=4.2 [m] ae=4.75 [m] rs=1 re=1 200

500

1k

2k

4k f [Hz]

Sound transmission loss for diffuse sound incidence through a suspended ceiling with a plaster board plate, a t = 4 [cm] glass fibre layer on it, a plenum height of h = 39 [cm], for different situations of room sizes as , ae and plenum back wall reflections factors rs , re . Parameters: h = 0.39 [m]; t = 0.04 [m]; d = 0.0095 [m]; 1 ≤ n ≤ 5; fcr d = 31 [Hz · m]; p = 1000 [kg/m3 ]; † = 0.1; ¡ = 10 [kPa s/m2 ], a = 20 [kg/m3 ]

Roo ms

3 dB

-90Emis sion dB

side

3· t

Rec

eivi

ng

side

h

Sound pressure level profile below and in the plenum of a suspended ceiling for the first higher room mode as incident mode. Left rear: emission room; right rear: receiving room; left front: plenum above emission, right front: plenum above receiving room. The space occupied by the absorber layer is drawn with a 3-fold magnification. The suspended ceiling consists of a d = 9.5 [mm] plaster board covered with a t = 8 [cm] glass fibre mat; the plenum is h = 35 [cm] high; the room sizes are as = ae = 4 [m]. The plenum back walls are hard

I

582

Sound Transmission

between the room modes and the absorber field along the common surface: (±) TŒn :=

1

qŒ ( ) · e±n a d =

0

±n a (−1)Œ · e±n a − 1 , (n a)2 + (Œ)2

(20)

and between directly transmitted and plenum mode field in the plenum space: S‹n : = =

1 h 1 2

h+t+d

cos k‹z (z − h − t − d) · cos —n (z − h − t − d) dz

t+d

(21)

sin ((k‹z − —n ) h) sin ((k‹z + —n ) h) + . (k‹z − —n ) h (k‹z + —n ) h

The simulation of diffuse sound incidence assumes that all propagating room modes have the same sound energy density and are incident modes. The sound transmission coefficient for diffuse sound incidence is evaluated by: 4 ‹lim ‹lim ‘ ‹ cos Ÿ‹ cos Ÿ‹ . (22) ‘dif = ‹=0

I.23

‹=0

Office Fences


In a two-dimensional room of height H with a hard floor and an absorbent ceiling is placed a (thin) absorbent and sound transmitting fence with its upper corner at y = h and (possibly) a gap of height b towards the floor. The rooms are anechoic in the ±x direction. The ceiling is assumed (as an approximation) to be locally reacting with an admittance Gc . The surface admittance Gw of the fence may be different on both sides, Gw1 , Gw2.

First, a non-transmissive fence without a lower gap (b = 0) will be considered; the modifications of the results for other conditions will be explained below.

Sound Transmission

I

583

Let the incident wave be the ‹-th mode of the room: pe (x, y) = Pe · cos (—‹ y) · e−‹ x ,

(1)

where —‹ H is a solution of the characteristic equation —‹ H · tan (—‹ H) = j k0H · Z0 Gc ‹2

and

=

—2‹

−

(2)

k02 .

(3)

The field on the emission side (1) is composed as p1 = pe + prs with the backscattered field formulated as a sum of room modes: Bn · cos (—n y) · e+n x . (4) prs (x, y) = n

Similarly the transmitted field pt is a sum of room modes: pt (x, y) = Dn · cos (—n y) · e−n x .

(5)

n

The matching of the fields to the fence admittance and to each other leads to two coupled systems of linear equations for Bn , Dn : $ $ n m Bn · j ƒm,n Nm − Sm,n − Dn · Z0 Gw2 − ƒm,n j Nm k0 k0 n n (6) ‹ = j Pe ƒ‹,mNm − S‹,m , k0 n Bn · j Sm,n − ƒm,n Nm Z0 Gw1 − Z0 Gw1 Dn · Sm,n − ƒm,n Nm k0 n n (7) ‹ = Pe j S‹,m + ƒ‹,m N‹ Z0 Gw1 , k0 where ƒm,n is the Kronecker symbol, Nm the mode norms: 1 H

H cos (—m y) · cos (—n y) dy = ƒm,n · 0

sin (2—m H) 1 1+ 2 2—m H

(8)

= ƒm,n · Nm −−−−−−→ 1 —m =—n =0

and Sm,n the mode-coupling coefficients 1 H

h sin ((—m − —n )h) sin ((—m + —n )h) cos (—m y) · cos (—n y) dy = + 2H (—m − —n )h (—m + —n )h 0 h sin (2—m h) 1+ . =: Sm,n −−−−→ —m =—n 2H 2—m h h

(9)

If the fence is sound transmissive, its surface admittances Gw1 , Gw2 are determined with a free space termination of the fence. If there is a bottom gap (b = 0), the modecoupling coefficients Sm,n are evaluated for the interval (b, h) of integration, i.e. everywhere Sm,n (0, h) is replaced by Sm,n (0, h) → Sm,n (b, h).

I

584

I.24

Sound Transmission

Office Fences, with Second Principle of Superposition


The object is the same as in the previous > Sect. I.23, but it will be treated here with the second principle of superposition (PSP) from > Sect. B.10. The advantage of the PSP is that it halves the size of the system of equations to be solved and it makes the field formulations more plausible. The PSP can be applied if the object has a plane of symmetry S (which is x = 0 in our case). It splits the task into two subtasks: first the sound transmissive parts of S are assumed to be hard (upper index = (h)), and second the sound transmissive parts of S are assumed to be soft (upper index = (w)). The () surface of the fence has the admittances Gw in both subtasks (i.e. with hard or soft termination at x = 0), respectively. The incident wave pe in both subtasks is assumed to be the ‹-th mode of the emission room, which is lined on one side with a locally reacting ceiling: pe (x, y) = Pe · cos (—‹ y) · e−‹ x .

(1)

It is associated on the front side (1) (see graph in > Sect. I.23) with the mode field () pr (x, y) after hard or soft reflection, respectively, at x = 0: +‹ x p() r (x, y) = ±Pe · cos (—‹ y) · e

;

() = (h), (w),

and a scattered wave +n x p() C() , s (x, y) = n · cos (—n y) · e

(2)

(3)

n

which is a sum of duct modes. The sound fields in front of and behind the screen are then: 1 (h) ps (x, y) + p(w) s (x, y) , 2 1 (w) p2 (x > 0, y) = pe (x, y) + p(h) s (−x, y) − ps (−x, y) . 2 p1 (x < 0, y) = pe (x, y) +

(4)

Application of the boundary conditions in the surface plane of the screen gives two () systems of linear equations for the Cn : m (h) (h) C(h) · Z G · S − ƒ · j N 0 w m,n m,n m = −2 Z0 Gw S‹,m · Pe , n k 0 n (5) ‹ n (w) (s) Cn · j · Sm,n − ƒm,n · Z0 Gw Nm = 2 j S‹,m · Pe . k0 k0 n Here the Nm are norms of the duct modes: 1 H

H cos (—m y) · cos (—n y) dy = ƒm,n · 0

sin (2—m H) 1 1+ = ƒm,n · Nm −−−−−−→ 1, (6) —m =—n =0 2 2—m H

Sound Transmission

I

585

and the Sm,n = Sm,n (b, h) are mode coupling coefficients: 1 H

h cos (—m y) · cos (—n y) dy =: Sm,n (b, h).

(7)

b

The method can be applied for any sound source on the emission side if its source profile (either a given pressure or a given velocity) can be synthesised with room modes. Then the above evaluation will be performed mode-wise and the results superimposed. The numerical examples below for the sound pressure level change 20 · lg(|p/pe |) due to the fence assume the following parameter values: frequency f = 500 [Hz]; H = 3.5 [m]; h = 2 [m]; b = 0.1 [m] (if there is a gap); ceiling (from low to high): d= 2 [cm] boards of compressed mineral fibre; bulk density RG = 400 [kg/m3 ]; flow resistance 5 · Z0 ; elastic constant fcr · d = 70; plus a 5 [cm] thick mineral fibre felt with flow resistivity ¡ = 10 [kPa · s/m2 ] and bulk density RG = 15 [kg/m3] and a locally reacting air layer 40 [cm] thick below the hard construction ceiling; fence: a mineral fibre board, 10 [cm] thick, flow resistivity ¡ = 10 [kPa · s/m2 ], covered (on both sides) with a perforated metal sheet, 1.5 [mm] thick, porosity 36%, round perforations 4 [mm] wide (with no, or only loose, mechanical contact between the metal sheet and the mineral fibre board). If the fence is assumed to be non-transmissive, a heavy metal sheet may be assumed in its centre. The incident wave is the fundamental room mode ‹ = 1. f=500 [Hz] ; H=3.5 [m] ; h=2 [m] ; b=0 [m]

20·lg |p/pe| dB 5 0

–10

–20

–30 2 –40

1.5 1 0.8

1

y/H 0.6

x/H

0.5

0.4 0.2 0

0

Sound pressure level change behind the fence; the fence is non-transmissive and has no gap at its foot

586

I

Sound Transmission

f=500 [Hz] ; H=3.5 [m] ; h=2 [m] ; b=0.1 [m] ; μ=1 ; nmax=11 20·lg |p/pe| dB 5 0

–10

–20

–30 2 –40 1

1.5 0.8

1

x/H

y/H 0.6 0.5

0.4 0.2 0

0

As above, but fence has a gap between foot and floor, b = 10 [cm] wide

f=500 [Hz] ; H=3.5 [m] ; h=2 [m] ; b=0.1 [m] ; nmax=10 20·lg |p/pe| dB 5 0

-10

-20

-30 2 -40 1

1.5 0.8

1

x/H

y/H 0.6 0.5

0.4 0.2 0 0

As above (i.e. transmissive fence with gap), but the incident wave is a plane wave

Sound Transmission

I.25

I

587

Infinite Plate Between Two Different Fluids

See also: Alekseev/Dianov (1976)

An infinite plate of thickness d is placed between two fluids with index 0 on the side of incidence of a plane wave pi , and index 1 on the side of the transmitted wave pt .

α0

pr ρ0 ; c0

pi p0

v0

d ; ρ ; E ; σ ; cc ; cs p1

ρ1 ; c1 pt

v1

α1

The fluids are characterised by their densities 0 , 1 and their sound velocities c0 , c1 . The plate material has a density ,Young’s modulus E, Poisson number , compressional wave speed cc and shear wave speed cs . Sound pressures at the plate surfaces:

p0 = pi + pr

;

p1 = pt ,

(1)

v0 , v1 are the corresponding plate surface velocities. Definitions: Normal components of the fluid wave impedances:

Z0n =

Reflection factor:

R=

Transmission factor:

0 c0 1 c1 ; Z1n = . (2) cos 0 cos 1

pr . pi pt T= . pi

(3) (4)

Pressure ratio:

K=

p1 T pt . = = pi + pr p0 1 + R

(5)

Plate input impedance:

Z=

p0 1+R = Z0n . v0 1−R

(6)

Symmetrical impedance (impedance for plate compression):

Zs =

p0 + p1 . v0 − v1

(7)

Za =

p0 − p1 . v0 + v 1

(8)

Anti-symmetrical impedance (impedance for plate bending):

From the relation of the effective powers for a plate without losses Re{p0 v0∗ } = Re{p1 v1∗ }: |K|2 = Z1n · Re{1/Z}.

(9)

I

588

Sound Transmission

In the special case Z0n |Z| & |Zs | |Za | & |Zs | Z1n is K · Z ≈ Z1n . If the medium on the side of incidence is nearly soft, i.e. Z0n is negligible: K(1 ) =

Z1n (Zs − Za ) ≈ D(1 )/2, 2 Zs Za + Z1n (Zs + Za )

Z(1 ) =

2 Zs Za + Z1n (Zs + Za ) 1 . ≈ Z0n 2 Z1n + Zs + Za 1 − R(1)

(10)

The symmetrical and anti-symmetrical impedances Zs , Za follow from: a1 b1 a1 b1 Zs = −j Wc · cot + Ws · cot ; Za = +j Wc · tan + Ws · tan (11) 2 2 2 2 cc cs · cos2 (2) ; Ws = · sin2 (2) ; Wc = cos cos cc cs sin = · sin 1 ; sin = · sin 1 ; c1 c1 (12) with –d –d · cos ; b1 = · cos ; a1 = cc cs & & E (1 − ) E ; cs = , cc = (1 + ) (1 − 2) 2 (1 + ) where is the diffracted angle of the compressional wave in the half-infinite plate material and is the diffracted angle of the shear wave in the half-infinite plate material. The sound transmission can be represented by an equivalent circuit. The transformer has the transformer ratio: Zs + Za N= . (13) Zs − Za 2Z s 2Z a

Z s /2 Z1

2p i Z a /2 1:N

Special cases: a) Normal radiation 1 = 0:

K(0) = cos

–d cc

1 cc –d +j sin 1 c1 cc

;

N = cos

–d . cc

(14)

Sound Transmission

I

589

b) Coincidence of the incident wave with the anti-symmetrical wave in the free plate, i.e. Za = 0: K(1) = 1

;

Z(1 ) =

Z1 · Zs 2Z1 + Zs

;

N = 1.

(15)

c) Coincidence of the incident wave with the symmetrical wave in the free plate, i.e. Zs = 0: K(1) = −1

;

Z(1 ) =

Z1 · Za 2Z1 + Za

;

N = −1.

(16)

d) Inpermeable plate, i.e. Zs = Za : K(1) = 0

;

Z(1 ) = Zs = Za

;

N → ∞.

(17)

e) Incidence at first critical angle, i.e. Zs → ∞: K(1) =

Z1n Z1n + 2Za

;

Z(1 ) = Z1n + 2Za

;

N = 1.

(18)

f) Only near field on the receiver side, i.e. sin1 → ∞ ; cos1 → j∞: K(1) = 0

;

Z(1 ) → −2j cc

cs cc

2

c2c − c2s sin 1 → −j ∞. cc · c1

If, additionally, the plate is thin: * 2 3 cs 2 1 − cs cc . Z(1 ) → −16 j –d

I.26

(19)

(20)

Sandwich Plate with an Elastic Core

See also: Beshenkov (1974)

An elastic layer of thickness 2h and = density, E = Young’s modulus, G = shear modulus, = Poisson ratio of its material when it is covered on both sides with identical thin sheets of thickness d and 1 = density, E1 = Young’s modulus, 1 = Poisson ratio of the sheet material.

I

590

Sound Transmission

A plane wave pi is incident at a polar angle ˜ . The moduli may be complex with loss factors †E , †G for the core, and †E1 for the cover sheets. v1 , w1 and v2 , w2 are the velocities and elongations of the sheets; pr is the reflected wave; pt is the transmitted wave. The trace wave number and normal field impedance are: k˜ = k0 · sin ˜

;

Z˜ = Z0 / cos ˜ .

(1)

Decomposition in compressional (symmetrical, index s) and translational (anti-symmetrical, index a) parts: 1 vs = (v1 − v2 ) 2

;

va =

1 (v1 + v2 ), 2

1 1 (2) ws = (w1 − w2 ) ; wa = (w1 + w2 ), ∗) 2 2 1 1 (pi + pr )z=−(h+d) + (pt )z=+(h+d) ps = ; pa = (pi + pr )z=−(h+d) − (pt )z=+(h+d) . 2 2 Impedances: ps = Zs + j Zs Zs = vs

;

Za =

pa = Za + j Za . va

(3)

Transmission coefficient ‘˜ : (Zs − Za )2 + (Zs − Za )2 . ‘˜ = Z2˜ (Z˜ + Zs )2 + Z2 · (Z˜ + Za )2 + Z2 s a

(4)

The impedances Zs , Za are evaluated from: j k˜2 A1 a7 j k˜2 B1 − – · a6 + − – · c3 , Zs = ; Za = 2 – A2 – 2 – B2 A1 = k˜4 c1 · a4 + a12 − k˜2 (a4 · a5 + 2 a1 · a3 ) + k˜2 –2 (c2 · a4 + c1 · c3 + 2 a1 · a2 ) + –4 c2 · c3 + a22 − –2 (c3 · a5 + 2 a2 · a3 ) + a32 , (5)

A2 = k˜2 · a4 + –2 c3 , B1 = k˜4 c1 · b4 −b21 − k˜2 (b3 · (b4 +c1 +2 b1 )) + k˜2 –2 (c2 · b4 +c1 · b5 −2 b1 · b2 ) + –4 c2 · b5 − b22 − –2 (b3 · (b5 + c2 + 2 b2 )) , B2 = k˜2 · b4 + b3 + –2 b5.

The coefficients ai , bi , ci are evaluated with the dilatation moduli D, D1 of the core and cover sheets, respectively: D=E ∗)

1− (1 + )(1 − 2)

;

see Preface to the 2nd edition.

D1 = E1

1 1 − 12

(6)

Sound Transmission

I

591

from the relations: a1 = D1 · d2

a2 = −1 · d2

;

;

2 a5 = G · h ; 3

a4 = 2D1 · d + 2D · h ;

, 1− 1 a6 = −2 1 d + h 3

a3 = −2D

;

2D a7 = − , h

(7)

b1 = −a1 · h ; b2 = −a2 · h ; b3 = −2Gh, 1 2 ; b5 = −a6 · h2 , b4 = −2 h E1 d + Eh 3 2 c1 = − E1 d3 3

;

c2 =

2 1 d3 3

;

c3 = −2 h + 1 d .

If the core is a viscous fluid with density , sound speed c, and kinematic viscosity Œ, then the equivalent elastic constants are: D → c2

;

G → j–Œ

;

→

–Œ 1 1−j 2 , 2 c

(8)

and the coefficients become (if the mass h of the core can be neglected compared to the surface mass density m1 = 1 d of the cover sheet): a1 = D1 · d

2

;

a2 = −1 · d

a4 = 2D1 · d + 2c2 · h b1 = −a1 · h

;

;

2

;

2

a3 = −2 c

2 a5 = j –Œ · h 3

b2 = −a2 · h

;

;

–Œ 1−j 2 c

5 –Œ 1+j 2 , c

a6 = −2m1

;

a7 = −2

b3 = −2j –Œ h,

c2 , h (9)

2 b4 = −2D1 · h2 − c2 · h2 ; b5 = −a6 · h2 , 3 2 2 c1 = − E1 d3 ; c2 = m1 d2 ; c3 = −2m1 . 3 3

I.27 Wall of Multiple Sheets with Air Interspaces

See also: Sharp/Beauchamps (1969)

N elastic plates i = 1, . . ., N with thicknesses ti and possibly different materials are at mutual distances di . A plane wave pe is incident at a polar angle ˜ .

592

I

Sound Transmission

Incident wave pe : pe (x, z) = px (x) · e−j kz z , px (x) = P · e−j kx x

;

(1)

kx = k0 · sin ˜

;

kz = k0 · cos ˜ .

Sound pressure fields: p0 = px (x) · e−j kz z + R · e+j kz z

;

z < 0,

.. . pi = px (x) · Ai · e−j kz (z−zi ) + Bi · e+j kz (z−zi ) .. . pN = px (x) · T · e−j kz (z−zN )

;

(2)

zi < z < zi+1 ,

;

z > zN ,

where R is the front side reflection factor, T the (total) transmission factor, and Ri and Ti the reflection and transmission factors at the i-th (single) plate. Boundary conditions: i =1: A1 = T1 + B1 · R1

;

R = R1 + B1 · T1 ,

i = 2, . . ., N − 1 : Ai = Ti · Ai−1 · e−j kz di−1 + Ri · Bi i = N: T = TN · AN−1 · e−j kz dN−1

;

(3)

Bi−1 · e+j kz di−1 = Bi · Ti + Ri · Ai−1 · e−j kz di−1 ,

;

BN−1 · e+j kz dN−1 = RN · AN−1 · e−j kz dN−1 .

(4) (5)

From this a matrix equation follows:

T 0

)

=C•

1 R

with coefficients:

) ;

C=

N 1 i=1

1 · C(i) Ti

;

C(i) =

⎧ (i) ⎪ ⎨ c11 ⎪ ⎩

c(i) 21

⎫ ⎪ c(i) 12 ⎬ c(i) 22

⎪ ⎭

(6)

Sound Transmission

i =1: 2 2 c(1) 11 = T1 − R1

c(1) 12 = R1

;

i = 2, . . ., N − 1: 2 2 −j kz di−1 ; c(i) 11 = (Ti − Ri )e i =N: 2 +j kz dN−1 c(N) 11 = TN e

;

c(1) 21 = −R1

;

+j kz di−1 c(i) ; 12 = Ri e

c(N) 12 = 0

;

;

I

c(1) 22 = 1,

(7)

−j kz di−1 c(i) ; 21 = −Ri e

−j kz dN−1 c(N) 21 = −RN e

;

593

+j kz di−1 c(i) , 22 = e

+j kz dN−1 c(N) . 22 = e

(8) (9)

The required single plate reflection and transmission factors R, T (index i is dropped) are: cos2 ˜ cos ˜ (10) R = −F · 1 + ZB · ZE ; T = −F · (ZB + ZE ) 2 2Z0 4Z0 with the auxiliary quantity: F=

cos ˜ 1 + ZB 2Z0

1 cos ˜ 1 − ZE 2Z0

(11)

and the bending-wave impedance ZB and the longitudinal-wave impedance ZE of the plate, which are, for thin plates: B · kx4 4 m · c2s 2kx2 − (1 − ) · –2 /c2s ZB = j – m − ; ZE = j , (12) – – t2 (1 − ) kx2 − –2 /c2d for thick plates: ZB = −j

8 m c4s 2 kx tanh (t/2) − kx2 − –2 /(2c2s) tanh (t/2)/ , 3 t–

ZE = −j

8 m c4s − kx2 coth (t/2) + kx2 − –2 /(2c2s) coth (t/2)/ , 3 t–

2 = kx2 − –2 /c2d

;

(13)

2 = kx2 − –2 /c2s ,

where m = p t is the surface mass density, p the plate material density, E the Young’s modulus, B the bending modulus, the Poisson ratio, and cs , cd the velocities of shear and dilatational waves: & & E t3 E E (1 − ) ; cs = ; cd = . (14) B= 2 12 (1 − ) 2 p (1 + ) p (1 + )(1 − 2) The transmission loss with oblique incidence is R˜ = −10 · log|T|2; the transmission loss for diffuse incidence is: ˜lim |T|2 sin (2˜ ) d˜ R = −10 · log ‘

;

‘=

0

˜lim sin (2˜ ) d˜ 0

.

(15)

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Special case: double sheet T=

T1 T2 e−j kz d . 1 − R1 R2 e−2j kz d

(16)

Special case: triple sheet T=

−T1 T3 e+j kz (d1 +d2 ) +2j kz d2 +2j kz d1 . 1 e e R1 R3 T2 − − R2 − R2 T2 R3 R1

(17)

References Alekseev; Dianov: Sovj. Phys. Acoust. 22, 181–183 (1976) Beshenkov, S.N.: Sovj. Phys. Acoust. 29, 115–117 (1974) Mechel, F.P.: Schallabsorber, Vol. III, Ch. 7: The absorber barrier. Hirzel, Stuttgart (1998) Mechel, F.P.: Schallabsorber, Vol. III, Ch. 8: Sound transmission through gaps and holes (openings) in a wall. Hirzel, Stuttgart (1998)

Mechel, F.P.: Schallabsorber, Vol. III, Ch. 14: Sound transmission through a finite wall in a duct. Hirzel, Stuttgart (1998) Mechel, F.P.: Schallabsorber, Vol. III, Ch. 15: Simple plate in a niche of the baffle wall. Hirzel, Stuttgart (1998) Mechel, F.P.: Schallabsorber, Vol. III, Ch. 16: A strip of a plate in an infinite baffle wall. Hirzel, Stuttgart (1998)

Mechel,F.P.: Schallabsorber,Vol.III,Ch.9: The noise sluice. Hirzel, Stuttgart (1998)

Mechel, F.P.: Schallabsorber, Vol. III, Ch. 17: Sound transmission through a finite plate combined with an absorber layer. Hirzel, Stuttgart (1998)

Mechel, F.P.: Schallabsorber, Vol. III, Ch. 10: Sound transmission through infinite,simple plates.Hirzel, Stuttgart (1998)

Mechel, F.P.: Schallabsorber, Vol. III, Ch. 18: Multilayer finite walls. Hirzel, Stuttgart (1998)

Mechel, F.P.: Schallabsorber, Vol. III, Ch. 11: Sound transmission through infinite, multiple plates. Hirzel, Stuttgart (1998) Mechel, F.P.: Schallabsorber, Vol. III, Ch. 12: Infinite sandwich plate. Hirzel, Stuttgart (1998)

Mechel, F.P.: Schallabsorber, Vol. III, Ch. 19: Flanking sound transmission through suspended ceilings. Hirzel, Stuttgart (1998) Mechel, F.P.: Schallabsorber, Vol. III, Ch. 24: Fences in a room. Hirzel, Stuttgart (1998) Sharp; Beauchamps: J. Sound Vibr. 9, 383 (1969)

J Duct Acoustics This chapter deals with sound propagation in ducts. It begins with hard and smooth ducts in which viscous and thermal losses at the walls are taken into account; this is important in narrow ducts (capillaries). The rest of the chapter deals with lined ducts of different cross sections and different linings. Sometimes the duct is assumed to be infinitely long; sometimes it has a finite length but is still long enough to neglect reflections from the duct exit at the duct entrance. This assumption makes the contents of this chapter different from those of > Ch. K “Acoustic Mufflers”, where the reflections at both ends of duct sections play a dominant role.A section at the end of this chapter will discuss the influence of flow on sound attenuation in lined ducts in an approximation which is precise enough for most technical applications.A more sophisticated discussion of the influence of flow will be given in > Ch. N “Flow Acoustics”.

J.1

Flat Capillary with Isothermal Boundaries


For the fundamental relations and notations used see

> Sect. B.1

The duct has a width of 2 h; x is the axial co-ordinate and y the transversal co-ordinate normal to the walls; the duct is infinite in the z direction (its dimension in this direction is much larger than that in the y direction). The co-ordinate origin is placed in the middle of the height. The specific heat and the heat conduction of the wall material are assumed to be much higher than those of air; therefore the isothermal boundary condition holds; the acoustic temperature fluctuation of the walls is zero. The scalar potentials ¥, for the density wave (index ) and the temperature wave (index ), as well as for the vector wave potential ¦Œ of the viscosity wave (index Œ), are formulated with a common axial propagation constant as: ¥, (x, y) = A,e− x cos(—, y), ¦z (x, y) = AŒ e− x sin(—Œ y) .

(1)

The wave equations for the three types of waves then give the following secular equations: 2 . —2,,Œ = 2 + k,,Œ

The wave number definitions used are: 2 – ‰– – 2 2 = ‰ Pr ·kŒ2 . ; kŒ2 = −j ; k0 = −j k0 = c0 Œ

(2)

(3)

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The boundary conditions at the walls lead to a system of equations (in matrix form) for the following amplitudes: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ h cos(— h) —Œ h cos(—Œ h) h cos(— h) 0 A ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ — h sin(— h) h sin(—Œ h) ⎠ · ⎝A ⎠ = ⎝0⎠ . (4) ⎝— h sin(— h) Ÿ cos(— h) 0 Ÿ cos(— h)0 AŒ See > Sect. B.1 for Ÿ , Ÿ . For the existence of a non-trivial solution the determinant must be zero: Ÿ Ÿ tan(—Œ h) −1 — h · tan(— h) = 0 . (5) ( h)2 + — h · tan(— h) − Ÿ —Œ h Ÿ This is the exact characteristic equation for . A good explicit approximation to the solution is under the condition |— h| 1:

k0

2

tan(k0h) k0 h . tan(kŒ h) 1− kŒ h

1 + (‰ − 1) ≈−

(6)

The characteristic axial wave impedance Z is (with the same degree of approximation): tan(— h) — h p(x, y)y k0 Z = ≈j Z0 Z0 vx (x, y)y tan(— h) tan(kŒ h) − — h kŒ h or, with some transformations:

tan(kŒ h) tan(k0 h) 1− , = j 1 + (‰ − 1) k0 k0 h kŒ h Z = Z0

(7)

(8)

1

. tan(k0h) tan(kŒ h) 1 + (‰ − 1) · 1− k0h kŒ h

The amplitude ratios are as follows: Ÿ cos(— h) Ÿ AŒ h cos(— h) A ; . =− = −1 A Ÿ cos(— h) A Ÿ —Œ h cos(—Œ h)

(9)

(10)

The axial particle velocity profile (relative to the axial velocity of the density wave in the centre) is: Ÿ cos(— y) Ÿ cos(— y) vx (x, y) cos(—Œ y) = cos(— h) · − + −1 vx (x, 0) cos(— h) Ÿ cos(— h) Ÿ cos(—Œ h) (11) Ÿ cos(k0 y) Ÿ cos(kŒ y) cos(kŒ y) ≈1− + ≈1− . −1 Ÿ cos(k0h) Ÿ cos(kŒ h) cos(kŒ h)

Duct Acoustics

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597

The transversal particle velocity profile (relative to the axial velocity of the density wave in the centre) is: vy (x, y) — h = sin(— y) vx (x, 0) h

cos(— h) — h Ÿ cos(— h) h Ÿ sin(— y) + sin(—Œ y) −1 h Ÿ cos(— h) —Œ h Ÿ cos(—Œ h) 2 k0 h y ‰ − 1 sin(k0y) 2 1 sin(kŒ y) ≈ 1+ − . + /k0 h k0 k0h cos(k0h) k0 kŒ h cos(kŒ h) −

Example of particle velocity profiles (with 2 h = 4 × 10−4 m):

(12)

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J.2

Flat Capillary with Adiabatic Boundaries

Duct Acoustics



> Sect. B.1.

No heat exchange takes place between the medium in the capillary and the walls. The field formulation and secular equations are the same as in > Sect. J.1.The boundary conditions are ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ h cos(— h) —Œ h cos(—Œ h) 0 A h cos(— h) ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ — h sin(— h) h sin(—Œ h) ⎠ · ⎝A⎠ = ⎝0⎠ . (1) ⎝— h sin(— h) Ÿ — h sin(— h) 0 Ÿ — h sin(— h)0 AŒ The characteristic equation from det = 0 becomes: Ÿ 2 tan(—Œ h) — h tan(— h) − — h tan(— h) ( h) —Œ h Ÿ Ÿ − − 1 · — h tan(— h) · — h tan(— h) = 0 . Ÿ

(2)

Approximate solutions for the propagation constant are as follows: 2 −1 −1 −1 ≈ ≈ . (3) ≈ tan kŒ h tan kŒ h 1 1 tan kŒ h k0 1+ 1− 1 − Ÿ kŒ h kŒ h (k0 h)2 kŒ h −1 1 + (‰ − 1) Ÿ (k0 h)2 An approximate solution for the wave impedance Zi is: j Zi ≈ 0 c0 /k0

j 1 ≈− ( /k0)2 ≈ −j . tan(kŒ h) /k0 k0 1− kŒ h

(4)

With adiabatic boundary conditions, the normalised wave impedance is approximately the rotated normalised propagation constant. The amplitude ratios of the component waves are: Ÿ — h tan(— h) Ÿ — h sin(— h) AŒ A h cos(— h) ; · −1 . =− = A Ÿ — h sin(— h) A —Œ h cos(—Œ h) Ÿ — h tan(— h)

J.3

Circular Capillary with Isothermal Boundary



> Sect. B.1.

Capillaries have a radius of a. The temperature at the wall is constant.

(5)

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599

The formulation of the scalar potentials ¥, for the density wave (index ) and the temperature wave (index ), as well as for the vector wave potential ¦Œ of the viscosity wave (index Œ) with a common axial propagation constant , is as follows: ¥, (r, z) = A, · e− x · J0 (—, r) ,

(1)

¦Œ (r, z) = AŒ · e− x · J1 (—Œ r) 2 . with Bessel functions Jn (z). The secular equations are —2,,Œ = 2 + k,,Œ

(2)

The boundary conditions at the capillary walls (in matrix form) are: ⎛ · J0 (— a) ⎝— · J1 (— a) Ÿ · J0 (— a)

⎞ ⎛ ⎞ ⎛ ⎞ 0 · J0 (— a) —Œ · J0 (—Œ a) A — · J1 (— a) · J1 (—Œ a) ⎠ · ⎝A ⎠ = ⎝0⎠ . 0 Ÿ · J0 (— a) 0 AŒ

(3)

The characteristic equation for the propagation constant is: ( a)2

J1 (— a) Ÿ Ÿ J1 (— a) J1 (—Œ a) + — a − =0. −1 — a Ÿ —Œ a · J0(—Œ a) J0 (— a) Ÿ J0 (— a)

(4)

An approximate solution is:

k0

2

Ÿ J1 (k0a) 1 · k0 a − Ÿ (k0a)2 J0 (k0a) 2 ≈ Ÿ J1 (kŒ a) 1 −1 + Ÿ kŒ a · J0 (k0 a) 2 J1 (k0 a) 1 + (‰ − 1) · 2 1 + (‰ − 1) · J1,0 (k0 a) k0 a · J0 (k0 a) =− ≈− J1 (kŒ a) 1 − J1,0 (kŒ a) 1−2 kŒ a · J0 (kŒ a)

The wave impedance Z with J1,0 (x) =

.

2 J1 (x) is: x J0 (x)

¢ Ÿ (k a)2 J1,0 (— a) − J1,0 (— a) 2 (k a) ¢ Ÿ Z (k0 a) =j Ÿ Ÿ Z0 k0 a · a (k a)2 J1,0 (— a) − J1,0 (— a) + − 1 J1,0 (—Œ a) 1−‰ Ÿ Ÿ (k0a)2 4‰ Pr (k0 a)2 J1,0(— a) − 1 − J1,0 (— a) (‰ − 1) j 3 (k0 a)2 ≈ (k0 a)2 (k0a)2 /k0 J1,0 (—Œ a) J1,0(— a)−(‰ − 1) J (— a) − 1 + (‰ − 1) 1,0 (k0 a)2 (k0a)2 2

≈

(5)

(6)

1−

J1,0(— a) j 1 j ≈ . /k0 J1,0(— a)−J1,0 (kŒ a) /k0 1 − J1,0 (kŒ a)

(7)

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The combined solutions for numerical applications are:

1 + (‰ − 1)J1,0 (k0 a) =j , k0 1 − J1,0(kŒ a) 1 Z = . Z0 [1 + (‰ − 1)J1,0(k0a)] · [1 − J1,0 (kŒ a)] The effective density eff is:

eff Zi 1 = −j · = . 0 k0 Z0 1 − J1,0 (kŒ a)

The effective compressibility Ceff is:

Ceff Zi = −j / = 1 + (‰ − 1)J1,0(k0 a) C0 k0 Z0

(8)

(9) (10)

(0 , C0 values for air without losses). The amplitude ratios of component waves are: Ÿ J0 (— a) Ÿ AŒ A a J0 (— a) ; . =− = −1 A Ÿ J0 (— a) A Ÿ —Œ a J0(—Œ a)

(11)

Approximations from the literature to the components of the propagation constant in capillaries are shown in the next diagram (for more details, see [Mechel,Vol. I–III (1989, 1995,1998)]); the thick curve represents the solution of the exact characteristic equation. The approximation given above nearly coincides with the exact curve; it agrees with the approximation by Zwikker and Kosten (1949), although it is derived in a different way:

Duct Acoustics

J.4

J

601

Lined Ducts, General

In general, the interior space of lined ducts is prismatic or cylindrical, and the surfaces of the lining can be assumed to lie on co-ordinate surfaces of co-ordinate systems in which the wave equation is separable. In Cartesian co-ordinates, for example, a solution of the wave equation (without flow; see later sections for flow superposition) with the axis in the x direction is: p(x, y, z) = a · cos(—m y)+b · sin(—m y) · · cos(†n z)+ · sin(†n z) (1) · c·e−m,n x +d·e+m,n x with the secular equation

2 = —2m + †n2 − k02 . m,n

(2)

If the x axis is in the duct centre, then the first terms in parentheses (with a, ) describe symmetrical fields and the second terms (with b, ) describe anti-symmetrical fields. The lateral wave numbers —m , †n are solutions of a characteristic equation which follows from the boundary conditions at the lining surfaces. For locally reacting linings (see later sections for other types of linings) with surface admittances Gy , Gz on both sides of the y direction and z direction, respectively, with half-duct heights hy , hz in these directions, the characteristic equations for the symmetrical field components are: —m hy · tan(—m hy ) = jk0hy · Z0 Gy = : jUy , †n hz · tan(†n hz ) = jk0hz · Z0 Gz = : jUz .

(3)

The right-hand sides, and therefore Uy , Uz are known values for given linings. The equations have an infinite number m, n = (0), 1, 2, . . . of solutions because of the

602

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Duct Acoustics

periodicity of the tan function. They are mode solutions. It is important that the task of finding mode solutions be the same regardless of whether the duct is two-dimensional or three-dimensional; in the latter case the same task has to be solved twice, and only in the evaluation of the axial propagation constant from 2 m,n = —2m + †n2 − k02

(4)

does the dimensionality become important.That is why in Cartesian co-ordinates mostly two-dimensional (flat) ducts are considered. The linings on opposite sides of the duct are largely the same and the duct is symmetrical. If in addition the sound excitation is symmetrical, then it is sufficient to consider symmetrical modes only. (It is important in this context that the least attenuated mode is a symmetrical mode.)

A “standard form” of a rectangular lined duct therefore is a flat duct with a hard wall at y = 0 (plane of symmetry) and a lining surface at y = h. The secular equation then becomes: m2 = —2m − k02 .

(5)

It is automatically satisfied if a modal angle Ÿn is introduced by: 1 = (—n /k0)2 − (n /k0)2 = (—n /k0)2 + (n /jk0)2 = sin2 Ÿn + cos2 Ÿn sin Ÿn =

—n ; k0

cos Ÿn =

n . jk0

(6)

With the present choice of association, the modal angle Ÿn is the angle which the wave vector of the plane waves includes with the x axis, which by their superposition form the trigonometric lateral mode profile: 1 +j—y y cos(—y y) (7) ± e−j—y y . e = sin(—y y) 2 The mode angles Ÿn are defined even when they become complex quantities. The most important target quantity of a silencer of finite length L (L sufficiently large, so that end reflections can be neglected; see later sections for end effects) is its transmission loss DL = 8.68 · Re{ L} = L/h · Dh with Dh = 8.68 · Re{ h}. Dh is the preferred quantity for the presentation of the silencer attenuation because it follows immediately from the secular and characteristic equations and permits a better comparison between silencers with different linings.

Duct Acoustics

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603

Suppose we have a set of computed Dh curves, plotted in a double-logarithmic scale over f · h [Hz · m], and a required transmission loss DL plotted in a double-logarithmic scale over f [Hz]. Suppose also that one of the diagrams is on a transparent foil (or the diagrams are plotted in a graphics computer program which permits drawing and moving of graphs in different levels). Then select a suitable silencer with the following procedure.

The required DL over f is plotted as points in the shaded diagram. The computed Dh over f · h are plotted as lines in the other diagram. Move one of the diagrams so that the DL values are just below a computed Dh . Read opposite to f = 1 kHz the value of h in mm and opposite to Dh = 1 the value of L/h. The other values needed are taken from the parameter list The main subtask in the determination of Dh is the solution of the characteristic equation. Important tools are Muller’s procedure for the solution of transcendent complex equations and the continued fraction representation of the tan(z) and cot(z) functions as well as of Bessel function ratios. Muller’s procedure for the solution of the equation f (z) = 0 requires three starting values, zi−2 , zi−1 , zi, , and the associated function values fi−2 = f (zi−2 ), fi−1 = f (zi−1 ), fi = f (zi ). A new approximation of the solution is: zi+1 = zi + Ši+1 · (zi − zi−1 ) ,

(8)

where Ši+1 is a solution of the quadratic equation: 2 Ši+1 · Ši hi + Ši+1 · gi + fi ƒi = 0

(9)

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Duct Acoustics

with the abbreviations: zi − zi−1 , Ši = zi−1 − zi−2 zi − zi−2 ƒi = 1 + Ši = , zi−1 − zi−2

(10)

hi = fi−2 Ši − fi−1 ƒi + fi , gi = fi−2 Ši2 − fi−1 ƒ2i + fi (Ši + ƒi ) . Therefore: 1 −2fi ƒi . −gi ± gi2 − 4fi Ši ƒi hi = Ši+1 = 2Ši hi gi ± gi2 − 4fi Ši ƒi hi

(11)

The sign of the root is selected such that the denominator in the second form has the maximum magnitude. Special cases are: Ši = 0 :

Ši+1 = −

hi = 0 :

Ši+1 = −

gi = 0 :

Ši+1

fi , fi − fi−1

fi ƒ i , gi fi ƒ i = ±j , Ši hi

radicand = 0 : Ši+1 = −

(12)

gi 2fi ƒi =− . 2Ši hi gi

Approximations zi should not coincide. The iteration is terminated if either or both |f (zi+1 )| ≤ ƒ2 and/or |1 − zi /zi+1 | < ƒ

(13)

−8

with a small number ƒ (≈ 10 ). One can, to some degree, influence the direction of the search for a solution by the arrangement of the starters zi−2 , zi−1 , zi . Continued fractions (Cf) may be written in one of the following forms: a1 a1 a2 a3 ... . Cf = b0 + = b0 + a2 b 1 + b2 + b3 + b1 + a3 b2 + b3 + . . .

(14)

The evaluation “from behind” is fast if one knows where to truncate the expansion. An evaluation in the opposite direction uses the following recursion: An a1 a2 a3 an = b0 + , ... Bn b1 + b2 + b3 + bn A−1 ≡ 1 ; A0 = b0 ; B−1 ≡ 0 ; B0 ≡ 1 ,

Cfn =

An = bn An−1 + an An−2 , Bn = bn Bn−1 + an Bn−2 . Tests of convergence can be performed repeatedly after a certain number of steps.

(15)

Duct Acoustics

J.5

J

605

Modes in Rectangular Ducts with Locally Reacting Lining


Let the axial co-ordinate x be in the centre of the duct with heights 2hy and 2hz . Let the linings on opposite walls be equal and have the surface admittances Gy and Gz on the walls normal to the y and z axes, respectively. Modes (i. e. solutions to the wave equation and of the boundary conditions) have the following form: p(x, y, z) = qy (y) · qz (z) · e− x with lateral profiles: cos(—y y) ; qy (y) = sin(—y y) ; cos(—z z) ; qz (z) = sin(—z z) ;

symmetrical mode, anti-symmetrical mode; symmetrical mode, anti-symmetrical mode.

(1)

(2)

606

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Duct Acoustics

The wave equation is satisfied if the following secular equation holds: 2 = —2y + —2z − k02 ;

Re{ } ≥ 0 ;

Im{ } ≥ 0

(3)

(the first sign convention has priority; the second convention holds if Re{. . .} = 0). The boundary conditions at the lining surfaces give the following characteristic equations: —y hy · tan(—y hy ) = jk0 hy · Z0 Gy = : jUy ,

symmetrical modes

—z hz · tan(—z hz ) = jk0 hz · Z0 Gz = : jUz ; anti-symmetrical modes

—y hy · cot(—y hy ) = −jk0 hy · Z0 Gy = : − jUy , —z hz · cot(—z hz ) = −jk0 hz · Z0 Gz = : − jUz .

(4)

(5)

Uy and Uz are known quantities for a given lining. If two opposite walls are hard, e. g. Gz = 0, then: —z hz =

n ; (n + 1/2) ;

n = 0, 1, 2, . . . ; symmetrical modes . n = 0, 1, 2, . . . ; anti-symmetrical modes

(6)

Duct Acoustics

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607

Modes in locally lined ducts are orthogonal to each other over the duct height: 1 2hy

hy qym (y) · qyn (y) dy = ƒm,n · Nyn = −hy

sin(2—m hy ) ƒm,n ; 1± 2 2—m hy

symm. (7) anti-symm.

with ƒm,n = 1 for m = n and ƒm,m = 0; Nyn denotes the mode norms. The characteristic equations for locally reacting linings have the form, with —h → z: z · tan z = jU ;

symm. ;

z/ tan z = −jU ;

anti-symm.

(8)

with U a known (in a general complex) number with positive real part. These equations induce a transformation between z and U. If one plots for given real or imaginary parts of z = z + j · z

with a running second part being the evaluated value of U in the complex plane, then one gets a type of “Morse chart” (see diagrams at beginning of this section). If one plots the above charts over the complex U plane and introduces a third dimension Re{z}, one gets more instructive three-dimensional charts, for example for symmetrical modes:

Evidently there are two types of modes: one set of modes with curved chart lines, and a single mode with nearly rectilinear chart lines. The modes with curved lines have correspondences in a hard duct; the mode with the rectilinear lines is a surface wave; there is no corresponding solution in the hard duct

J.6

Least Attenuated Mode in Rectangular, Locally Lined Ducts


Designing a silencer with the attenuation of the least attenuated mode is a “safe” design because the least attenuated mode is one of the modes to excite easily (see later section

608

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Duct Acoustics

about excitation efficiency of modes), and for a sufficiently long silencer other possibly excited modes will have decayed at the silencer exit, so that the least attenuated mode determines the exit sound pressure level. The least attenuated mode is among the two lowest symmetrical modes. A number of methods have been described for its evaluation. The lateral wave number z = —h is a solution of the characteristic equation z · tan z = jU

(1)

with U = k0h · Z0 G .

(2)

The characteristic equation is even in z, so most approximate solutions described are for z2 .

+ j · zex An approximate solution zap can be tested as follows: let the parts of zex = zex run and evaluate the associated U from the characteristic equation; then solve with this value of U for the approximation zap ; plot the lines for zex in the complex plane of zap . If the approximation is good, it reproduces (approximately) the co-ordinate grid of that plane.

First approximation: Expand tan(z) as a power series around z = 0 and retain the first term; this gives: z2 ≈ j · U.

(3)

Second approximation: The power series expansion up to terms z4 gives the following approximation: z2 ≈ 3/2 · −1 + 1 + 4jU/3 .

(4)

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Third approximation: Perform first the identical transformation, z · tan z = z2 /z cot z ,

(5)

and then expand: z · cot z = 1 − z2/3 − z4 /45 − . . . .

(6)

This gives: z2 ≈

3 15 + 5jU − 2

225 + 150jU − 45U2 . jU

(7)

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Fourth approximation: This approximation is of some systematic interest because it does not use the characteristic equation. It starts from the general admittance equation: Z0 G =

1 −grad arg(p) + jgrad ln |p| k0

(8)

and uses the expansion for the sound pressure profile q(y) = cos(—y): ln q(y) = ln cos(—y) =

∞

ln 1 −

n=1

4(—y)2 2 (2n − 1)2

.

(9)

Simple transformations give: U = −8jz2

∞ n=1

1 . 2 (2n − 1)2 − 4z2

(10)

The solution with an upper summation limit n = 1 yields: z2 ≈

2 jU , 4 2 + jU

(11)

and summation up to n = 2 yields: 2 10 + 5jU − 100 + 64jU − 16U2 2 . z ≈ 4 4 + jU

(12)

An approximation 2

z ≈

27.40 + 12.34jU −

750.56 + 322.47jU − 97.409U2 6.452 + jU

(13)

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is obtained if one neglects 4z2 in the denominator of the expression for U for n ≥ 3 and applies the numerical series: ∞ n=1

1 2 , = 2 (2n − 1) 8

(14)

which has the value 1/8 − 10/(92) if it begins with n = 3.

Fifth approximation: A new class of precision is attained if one applies in the characteristic equations for symmetrical and anti-symmetrical modes z · tan z = jU ;

z/ tan z = −jU

(15)

the continued fraction expansions: z2 z2 z 2 z2 z 2 z2 z 2 . . . ; z · cot z = 1 − ... . 1− 3− 5− 7− 3− 5− 7− They begin to converge with the partial fraction for which z · tan z =

(16)

|z|2 ; (2n)2 − 1 > |z|2 . (17) 2n + 1 If one truncates the continued fraction with increasing depth, then one gets polynomial equations of higher and higher degrees. The polynomials and explicit solutions (where they exist) are, for symmetrical modes: 2n − 1 >

z2 = jU , jU , z2 = 1 + jU/3

(18)

z 4 − (15 + 6jU)z2 + 15jU = 0, 1 15 + 6jU ± 225 + 120jU − 36U2 , z2 = 2

(19)

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Duct Acoustics

(10 + jU)z4 − (105 + 45jU)z2 + 105jU = 0, 2

z =

105 + 45jU ±

11025 + 5250jU − 1605U2 , 20 + 2jU

(20) ∗)

z 6 − (105 + 15jU)z4 + (945 + 420jU)z2 − 945jU = 0, (21 + jU)z6 − (1260 + 210jU)z4 + (10395 + 4725jU)z2 − 10395jU = 0,

(21)

(22)

z8 − (378 + 28jU)z6 + (17325 + 3150jU)z4 − (135135 + 62370jU)z2 + 135135jU = 0,

(23)

(36 + jU)z 8 − (6930 + 630jU)z6 + (270270 + 51975jU)z4 − (2027025 + 945945jU)z2 + 2027025jU = 0 .

(24)

The precision test of the approximation (21) is shown in the graph on the next side. In the grey area the sign of the root was chosen so as to make the real part of the root positive and in the other range negative. The limit line passes through the first branch point (where the lines are curved). Evidently this approximation can be used also for parts of the second “Morse chart”, i. e. for the second mode. Higher-degree polynomials of the continued fraction expansion give more than just one solution for z2 . They belong (with different precision) to the lower-order modes. To find the solution for the least attenuated mode, exclude all approximations z which are not in the first quadrant and take from the remaining approximations that which makes Re{ h} a minimum. Frommhold has modified the coefficients of the continued fraction approximations to move the range of application more towards the range of technical values of U; he proposes the following: If 0 ≤ Re{Z0 G} ≤ 3 ; z2 ≈

−1.5 ≤ Im{Z0 G} ≤ 1.5:

(2.74 − 0.52 · j)jU . 2.88 − 0.55 · j + jU

If 2 ≤ Re{Z0 G} ≤ 5 ;

3 ≤ Im{Z0 G} ≤ 6, then:

√ (78.94 − 5.43 · j) + (34.47 − 2.2 · j)jU ± . . . z2 ≈ , (16.1 − 1.11 · j) + 2jU √ . . . = (6203 − 857 · j) + (2887.3 − 372 · j)jU − (867.4 − 130 · j)U2 . The sign of the root is determined with the criterion Re{ h} = minimum. ∗)

(25)


(26)

Duct Acoustics

J.7

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613

Sets of Mode Solutions in Rectangular, Locally Lined Ducts


The charts of the transformation z → U which is induced by the characteristic equation show branch points. The evaluation of sets of mode solutions begins with the determination of these branch points zb, and the associated values Ub follow from the characteristic equations z · tan z = jU ; z/ tan z = −jU ;

symmetrical modes, anti-symmetrical modes.

(1)

The branch points are solutions of tan z +

z =0; cos2 z

symmetrical ;

cot z −

z sin2 z

=0;

anti-symmetrical.

(2)

Approximations of the functions zb

= f (zb ) ; U

b = g(U b ) ; zb (m) ; U b (m) (with zb = zb + j · zb

; Ub = U b + j · U

b and m = 0, 1, 2, . . . the mode order) are: zb

= 0.702568 · (zb )1/3 + 0.216438 · (zb )1/2 − 0.036625 · zb

+ 0.000143119 · zb 2 ; symm. zb

= 0.0232164 + 0.829796 · (zb )1/2 − 0.0827732 · z b + 0.000351925 · z 2b ; anti-symm. U

b = 2.02599 · (U b )1/3 − 0.655631 · (U b )1/2 + 0.00631631 · U b + 0.00000250827 · U b 2 ; symm. U

b = 1.0 + 0.553673 · (U b )1/2 − 0.0412894 · U b + 0.000120216 · U b 2 ;

anti-symm.

(3)

(4)

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√ zb (m) = −1.4403 m + 3.76029 · m − 0.0284415 · m2 + 0.000620241 · m3 ; symm. zb (m) = 3.39478 · m − 0.023865 · m2 + 0.000669072 · m3 ; anti-symm. √ U b (m) = −1.50237 m + 3.76029 · m − 0.0284415 · m2 + 0.000620241 · m3 ; symm. U b (m) = 3.39042 · m − 0.023312 · m2 + 0.000651624 · m3 ; anti-symm.

(5)

(6)

The characteristic equations transform the U plane (with Re{U} ≥ 0) into a strip in the first quadrant of the z plane. A one-to-one correspondence of a z-strip and the U plane, with limit curves which can be evaluated (!), is shown in the graph below. The z-strip is limited by lower limit curves zgl (m) (g = Grenze in German), which are vertical lines from the branch point zb (m) to the Re{z} axis, and by upper limit curves zgu (m), which

are quarter ellipses between zb (m) and zg0 (m) on the Im{zb } axis. The associated limit curves in the U plane are shown in the second graph. In the shaded range of the U plane, the surface wave mode is evaluated.

A strip in the z plane into which the U plane (with Re{U} ≥ 0) is transformed. It is limited by vertical lower curves through the branch points and by upper elliptic arcs

Transformation of the limits of the z-strip into the U plane. If U is in the shaded range, the surface wave mode is evaluated

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Table 1 Branch points zb and associated Ub for symmetrical and anti-symmetrical modes in flat ducts with locally reacting lining Symmetrical m

zb

Anti-symmetrical Ub

zb

Ub

0

0.+ j 0.

0.+ j 0.

0.+ j 0.

0.+ j 1.

1

2.1062 + j 1.12536

2.05998 + j 1.65061

3.74884 + j 1.38434

3.71944 + j 1.89528

2

5.35627 + j 1.55157

5.33471 + j 2.05785

6.94998 + j 1.6761

6.93297+ j 2.18022

3

8.53668 + j 1.77554

8.52264 + j 2.27847

10.1193 + j 1.85838

10.1073 + j 2.36058

4

11.6992 + j 1.9294

11.6888 + j 2.43112

13.2773 + j 1.99157

13.2681+ j 2.49295

5

14.8541 + j 2.04685

14.8458 + j 2.54799

16.4299 + j 2.09663

16.4224 + j 2.59758

6

18.0049 + j 2.14189

17.9981 + j 2.64271

19.5794 + j 2.1834

19.5731 + j 2.6841

7

21.1534 + j 2.22172

21.1476 + j 2.72234

22.7270 + j 2.25732

22.7216 + j 2.75786

8

24.3003 + j 2.29055

24.2952 + j 2.79103

25.8734 + j 2.32171

25.8686 + j 2.82214

9

27.4462 + j 2.35105

27.4417 + j 2.85144

29.0188 + j 2.37876

29.0146 + j 2.87911

10

30.5913 + j 2.40501

30.5872 + j 2.90533

32.1636 + j 2.42996

32.1598 + j 2.93025

11

33.7358 + j 2.45372

33.7321 + j 2.95399

35.3079 + j 2.4764

35.3044 + j 2.97665

12

36.8799 + j 2.4981

36.8765 + j 2.99833

38.4518 + j 2.5189

38.4486 + j 3.01911

13

40.0236 + j 2.53887

40.0205 + j 3.03906

41.5954 + j 2.55807

41.5924 + j 3.05825

14

43.1671 + j 2.57656

43.1642 + j 3.07673

44.7387 + j 2.59439

44.7359 + j 3.09455

15

46.3103 + j 2.61161

46.3076 + j 3.11176

47.8819 + j 2.62825

47.8793 + j 3.1284

16

49.4534 + j 2.64436

49.4509 + j 3.1445

51.0248 + j 2.65997

51.0224 + j 3.1601

17

52.5963 + j 2.6751

52.5939 + j 3.17522

54.1677 + j 2.68979

54.1654 + j 3.18991

18

55.7390 + j 2.70407

55.7368 + j 3.20417

57.3104 + j 2.71794

57.3082 + j 3.21804

19

58.8817 + j 2.73144

58.8796 + j 3.23154

60.4530 + j 2.74459

60.4509 + j 3.24468

20

62.0242 + j 2.7574

62.0222 + j 3.25748

63.5955 + j 2.76988

63.5935 + j 3.26997

The equation for the limit curve in the U plane in the form U g (m) = f U

g (m) for both Ugu and Ugl is: ⎛ ⎞2 U g (m) = U b(m)

⎜ ⎟ U

g (m) − U

b (m) ⎜ ⎟ 1−⎜

⎟ . ⎝ zb (m) · sin(2zb (m)) ⎠ !

+ U (m) b 1 + cos(2zb (m))

(7)

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The equation for the elliptic limit curve in the z plane in the form zgu (m) = f zgu (m) is:

2 zgu (m) − zb

(m) !

; zb

(m) ≤ zgu (m) ≤ zg0 (m) , (8) zgu (m) = zb (m) 1 −

zg0 (m) − zb

(m)

and in the form zgu (m) = f zgu (m) : 2

(m) zgu !

zgu (m) = zb (m) + zg0 (m) − zb (m) 1− ; zb (m)

0 ≤ zgu (m) < zb (m) .

(9)

(m) ; U

g0(m) are contained in the table below. Required values of zg0

It should be noticed that the branch points, range limit curves and endpoints of the elliptic arcs coincide with the origin for m = 0. The modes are counted as m = 1, 2, 3, . . . in the procedure for the evaluation of a mode solution z = —h for given values of U and m (usually counting is m = 0, 1, 2, . . .). The procedure works with the following steps: Step 1: special case U = 0? (m − 1); Take z = —h = (m − 1/2) ;

symmetrical . anti-symmetrical

(10)

Step 2: U in the surface wave range? That is, U is in the range limited by (a) the curve U

b = g(U b ) which connects the branch point images, (b) the imaginary axis and (c) the curves U g (n) = f U

g (n) for n = m − 1 and n = m. In that case use the fast converging (because z ≈ U) iteration i = 0, 1, 2, . . . zi+1 = jU/ tan(zi ) ;

symm. ;

zi+1 = −jU/ cot(zi ) ;

anti-symm.

(11)

with z0 = U. Step 3: Else: Expand the characteristic equation in a continued fraction, using the periodicity of tan(z), cot(z): z(z − m) (z − m)2 (z − m)2 . . . = jU; 1− 3− 5− z (z − m)2 (z − m)2 z · cot(z − m) = 1− . . . = −jU; z − m 3− 5− z · tan(z − m) =

symm. (12) anti-symm.

and derive a polynomial equation in z 2 by truncation. Take the solution for which zb

(m) ≤ z

< zb

(m+1).Produce two start solutions for Muller’s procedure by truncating the continued fraction at different depths, and take as the third starter the mean value between them. Then solve the characteristic equation (in its original form) with these

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Table 2 Values of endpoints of elliptic arcs on imaginary axis m

Symmetrical

Anti-symmetrical

z00g0 (m) & U00g0 (m)

z00g0(m) & U00g0 (m)

0

0.00000

0.00000

1

3.55637

5.39512

2

7.13605

8.83041

3

10.4985

12.1485

4

13.788

15.4182

5

17.0412

18.6597

6

20.276

21.8853

7

23.4928

25.0985

8

26.7013

28.2986

9

29.8984

31.4972

10

33.0907

34.6867

11

36.2800

37.8715

12

39.4623

41.0533

13

42.6456

44.2313

14

45.8196

47.4114

15

48.9979

50.5792

16

52.1657

53.7582

17

55.3359

56.9203

18

58.5124

60.0893

19

61.6746

63.2566

20

64.8482

66.4240

starters and Muller’s procedure (for many applications the solution of the polynomial equation, if the degree of the polynomial is not too low, is already sufficiently precise).

Because the curves U

gu (m) = f U gu (m) ; zgu (m) = f zgu (m) are not exact transforms of each other, it may happen that no solution of the polynomial equation has zb

(m) ≤ z

< zb

(m + 1). Then the desired solution is in or near the range of the surface wave mode. Take in this (rare) instance the solution from the iteration above. The (important) advantage of the procedure is that no mode in a mode set is missed or returned twice; the disadvantage is that the surface wave mode (if any exists) is subdivided and the “pieces” are attributed to different modes, so mode solution curves

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Duct Acoustics

in the z plane do not look “nice”; but this feature does not disturb modal analysis computations. In the numerical example shown below, the surface wave solution the arc which approximately agrees with the curve U(k0 h) is subdivided; a mode solution jumps whenever this solution crosses a limit of a z-strip.

The graph shows (in the U plane) the curve U(k0h) (full line with dots); the curve zb (m) connecting the branch points (thin full line); the limit curves zg (m) of the z-strips (thin dashed curves); and the mode solutions for the modes m = 0, 1, . . ., 5. The direction of increasing frequency is indicated by arrows A procedure for a set of mode solutions and a list of k0 h values, which avoids the subdivision of the surface wave mode, which returns continuous mode solutions (for a variation of k0 h) and which is relatively robust against “mode jumping”, proceeds as follows.It is assumed that the list {k0 h} begins with low values (if necessary,prepend such values; you may drop them later), and the difference k0 h is not too large (k0 h ≤ 0.1; maximum ≤ 0.2). The procedure is first described for symmetrical modes. Begin to work through the list (i = 1, 2, . . .) of k0h with starting values zs1 = m; zs2 = m + 0.01 · j; zs3 = m + 0.01 + 0.01 · j for Muller’s procedure and find zi=1 . Write the characteristic equation as: ⎡

⎤

⎢ (z − m)2 (z − m)2 ⎢ ... z2 ≈ m · z+j · k0h · Z0 G · ⎢1− ⎣ 3− 5−. . .

(z − m)2 (z − m)2 (nhi − 2)2 − n2hi 2 2 −−−−−→ (m) + j · k0 h · Z0 G −−−−−−−−−−−−−−−→ (m) , z→m

⎥ ⎥ ⎥ ⎦ (13)

k0 h→0 and/or G→0

and use on the right-hand side z → zi−1 to find a starting approximation zs3 for zi . The other starters for Muller’s procedure are zs2 = zi−2 ; zs1 = zi−3 (in the second step i = 2, take the previous solution as zs2 and the mean value of zs2 , zs3 as zs1 ). The order

Duct Acoustics

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619

of the starters is important; do not use the Newton-Raphson method; otherwise mode jumping will happen. The next graph shows again the mode solutions from above,but now with this method; in the second graph the range over which the surface wave mode spans has been enlarged; the mode index m = 0 is attributed to the surface wave mode.

Mode solutions as above, but with a method returning steady curves for the modes

Mode solutions as above, but with an extended span of the surface wave mode (m = 0) In the case of anti-symmetrical modes, proceed as above, but replace m → (m + 1/2) because of cot(z) = −tan(z − (m + 1/2)) .

J.8

Flat Duct with a Bulk Reacting Lining


A flat duct with axial co-ordinate x and lateral co-ordinate y has a hard wall at y = 0 and is lined with a bulk reacting absorber at y = h. The lining is a layer of thickness d of

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Duct Acoustics

a porous material with characteristic values a , Za (or, in normalised form, an = a /k0 , Zan = Za /Z0 ), possibly covered with a poro-elastic foil having a partition impedance Zs (“s” from series impedance). Fundamental relations are shown in Table 1.

Table 1 Relations in the free rectangular duct and in the absorber layer Relation

Free duct + k02 p(x; y) = 0

Wave equation Field formulation ( symm. Profile in y anti-symm. ( Profile z

Absorber layer − a2 pa (x; y) = 0

p(x; y; z) = P0 q(y) s(z) e− x ( cos(—y y) q(y) = sin(—y y) (

symm. anti-symm.

in

s(z) =

Secular equation

—2y = + k02 − —2z

Velocity in y direction

vy =

Lateral admittance y = h symmetrical

Gy = −j

Lateral admittance y = h

Gy = j

anti-symmetrical

pa (x; y; z) = Pa qa (y) s(z) e− x qa (y) = cos —ay (˙y − h − d)

cos(—z z) sin(—z z)

—2ay = 2 − a2 − —2z

j @p k0 Z0 @y

vay =

—y h tan(—y h) k0 hZ0

−1 @pa a Za @y

Gay = −

—h cot(—h) k0 hZ0

—ay d tan(—ay d) a dZa

Ga y = −

—ay d tan(—ay d) a dZa

The boundary condition is the agreement of the lateral admittances on both sides of y = h: !

Gy = 1

)

Zs + 1/Gay

.

(1)

This gives the following characteristic equations:

for symmetrical modes:

—y h · tan(—y h) = jk0 h

for anti-symmetrical modes: —y h · cot(—y h) = −jk0 h secular equation:

a Za Zs − cot(—ay d) = jU; (2) Z0 —ay Z0 a Za Zs − cot(—ay d) Z0 —ay Z0

(—ay h)2 = (—y h)2 − (a h)2 − (k0 h)2 .

= −jU; (3)

(4)

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Function U now contains the solution —y h (in contrast to locally reacting linings),and the form of the characteristic equation and the method of its solution thus depend on the structure of the lining. Without a cover of the absorber layer (i. e. Zs = 0), function U is: U=−

h k0 Z0 —a d · tan(—a d) . d a Za

(5)

If the absorber layer is made locally reacting (e. g. either by a high flow resistivity or by internal partition walls normal to the surface), the characteristic equation for symmetrical modes becomes: —h · tan(—h) = jk0 h

tanh (a d) . Za /Z0

(6)

If, on the other hand, the term Zs /Z0 is large compared with the second term in the k0 h parentheses of the characteristic equation then U → , i. e. the lining behaves like Zs /Z0 a locally reacting lining. The characteristic equation is even in —h (as for locally reacting absorbers), but now Re{U} < 0 is possible (in contrast to locally reacting absorbers); therefore solutions —h are no longer necessarily in the first quadrant. “Morse charts” for the solutions cannot be drawn. Modes in a duct with a bulk reacting lining (terminated with a hard wall towards the outer space) are orthogonal to each other if the lateral field profile within the outer walls is written as q(y) = q(1) (y) + q(2) (y), where (i) = (1) stands for the free duct and (i) = (2) for the absorber layer. The orthogonality relation is: 1 1 (1) (1) (2) qm · qn dA1 + qm · qn(2) dA2 = ƒm,n · Nm , (7) jk0 Z0 a Za A1

A2

where ƒm,n is the Kronecker symbol and Nm the mode norm: (1) 2 (2) 2 1 1 Nm = qm dA1 + qm dA2 . jk0 Z0 a Za A1

(8)

A2

For multi-layer absorbers an additional term i > 2 will appear on the left-hand side of (7) for each layer.

J.9

Flat Duct with an Anisotropic, Bulk Reacting Lining


The object is the same as in the previous > Sect. J.8, but the porous material layer is now assumed to be anisotropic, i. e. with characteristic values ax , Zax and ay , Zay in the x, y co-ordinate directions. The relations in the free duct and in the absorber layer are presented in Table 1.

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Table 1 Relations in the free duct and in the anisotropic absorber layer Relation

Free duct

Absorber layer

Wave equation

+ k02 p(x; y) = 0

2 @ =(ax @x)2 + @2 =(ay @y)2 − 1 pa (x; y) = 0

Field formulation

p(x; y) = P0 q(y) e− x

pa (x; y) = Pa qa (y) e− x

Profile

symm. anti-symm.

q(y) =

qa (y) = cos —ay (˙y − h − d)

cos (—y y) sin (—y y)

Secular equation

—2y = 2 + k02

Velocity in y direction

vy =

Lateral admittance symmetrical

Gy = −j

Lateral admittance anti-symmetrical

Gy = j

2 = 2 = 2 − 1 —2ay =ay ax

j @p k0 Z0 @y

vay =

—y h tan (—y h) k0 hZ0

—h cot (—h) k0 hZ0

−1 @pa ay Zay @y

Gay = −

—ay d tan (—ay d) ay dZay

Gay = −

—ay d tan (—ay d) ay dZay

The characteristic equations for duct modes are now as follows:

Zs ay Zay − cot (—ay d) symmetrical mode: —y h · tan (—y h) = jk0 h Z0 —ay Z0 = jU ;

anti-symmetrical mode:

—y h · cot (—y h) = −jk0 h = −jU ;

secular equation:

Zs ay Zay − cot (—ay d) Z0 —ay Z0

(1)

(2)

—2ay = 2 · ay2 /ax2 − ay2 = —2y − k02 · ay2 /ax2 − ay2 ;

axial propagation constant: h =

(—y h)2 − (k0h)2 .

(3) (4)

No principally new features are introduced by the anisotropy, only the amount of computation is somewhat increased.

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623

Mode Solutions in a Flat Duct with Bulk Reacting Lining


Because no transformation between z = —h and a meaningful known variable, like U with locally reacting linings, can be defined drawing Morse charts is no way to solutions for the solutions for characteristic equation. Continued fraction expansion, symmetrical mode (flat duct, —z = 0; applicable for the least attenuated mode only): Transform the characteristic equation into:

d a Za Zs · —ay d · tan (—ay d) − —y h · tan (—y h) = j—ay d · tan (—ay d) k0h · Z0 h k0 Z0 with: —ay d = (d/h)2 (—y h)2 − (k0h)2 − (a d)2 ,

(1) (2)

and apply the continued fraction expansion to z · tan z (with z = —y h and a = (1 + 2 an )(k0 h)2 ; Zsn = Zs /Z0). Even with the rather low precision of expansion up to the second fraction and the special case Zs = 0 (i. e. no cover foil on the porous layer), the polynomial equation becomes somewhat lengthy: 2 2 Zan ) − (z2 )6 (d/h)4 (k0h)2 + (z2 )5 (d/h)4 (k0h)2 (6 + 4a − an 2 2 2 2 Zan + (d/h)2 9 + 6a2 − 2a(−12 + an Zan ) − (z2 )4 (d/h)2 (k0h)2 −6an 2 2 2 2 Zan − 6(d/h)2 a · an Zan + (d/h)4 a · 36 + 4a2 + (z2 )3 (k0 h)2 −9an 2 2 − a · (−36 + an Zan ) −(z2 )2 (d/h)4 a · (k0h)2 (54 + 24a + a2 )

(3)

+ (z2 ) · 6(d/h)4 a3 (k0 h)2 (6 + a) − 9(d/h)4 a4 (k0h)2 = 0 . With a cover foil (i. e. Zs = 0) having the same depth of expansion, the equation becomes, for z = —h , z14 · 9(d/h)4 Z2sn + z13 · 6j(d/h)4 k0hZsn − z12 · (d/h)4 ((k0h)2 + 36aZ2sn ) − z11 · 6j(d/h)4 k0h(3 + 4a)Zsn 2 2 + z10 · (d/h)4 (k0h)2 6 − an Zan + 2a 2 + 27aZ2sn /(k0h)2 + z9 · 36j(d/h)4 ak0 hZsn (2 + a) 2 2 2 2 − z8 · (d/h)2 (k0h)2 (−6an Zan + (d/h)2 (9 − 2a(−12 + an Zan )

+ 6a2 (1 + 6aZ2sn /(k0h)2 )) − z7 · 12j(d/h)4 a2 k0hZsn (9 + 2a) 2 2 2 2 2 2 Zan − 6(d/h)2 aan Zan + (d/h)4 a 36 − a(−36 + an Zan ) + z6 · (k0 h)2 −9an + a2 (4 + 9aZ2sn /(k0h)2 ) + z5 · 6j(d/h)4 a3 k0hZsn (12 + a) − z4 · (d/h)4 a2 (k0h)2 (54 + 24a + a2 ) − z3 · 18j(d/h)4 a4 k0 hZsn + z2 · 6(d/h)4 a3 (k0h)2 (6 + a) − 9(d/h)4 a4 (k0 h)2 = 0 .

(4)

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The problem is finding the right solution among the roots. Root z in the first quadrant and with the smallest magnitude is often the right choice. Iteration through a list of {k0 h}: At very low k0h the admittance of all linings becomes small. Then the mode solution with the corresponding absorber, in which all layers are assumed to be locally reacting, are suitable starters for Muller’s procedure. For later entries of the k0h list, take previous solutions as starters. It may be necessary to make the steps k0 h very small. It is helpful (and permits larger k0 h values) to use as the third starter for the solution zi an extrapolated value. Let zi−1 , zi−2 , zi−3 be the previous solutions for the previous k0h values.

The extrapolated zi (which is the third start value together with zi−1 and zi−2 ) is evaluated from (with ∠z the argument of the complex z): zi = |z3 | · ej∠z3 ;

|z3 | = c · |z2 | = c · |zi−2 − zi−1 | ;

∠z3 = ∠z2 + œ = 2 · ∠z2 − ∠z1 ; −œ = ∠z1 − ∠z2 ; z2 = zi−1 − zi−2 ;

(5)

z1 = zi−2 − zi−3 .

Start the numerical solution with mode values for the locally reacting absorber: Sometimes it is proposed to take the mode solution for the absorber, with all layers made locally reacting (and some values nearby as the two other starters) as start values for Muller’s method, not only for the lowest entries of a k0h list, as above, but for all k0 h values. This method fails, except in very harmless cases (because the numerical procedure may pass on its way from the starters to the true value through apparent resonances of the lining, which do not exist). Iteration through the modal angle: Define modal angles from the secular equation (below for three dimensions) in the following form: 1 = (—y /k0)2 + (—z /k0)2 − ( /k0)2 (6) = cos2 œ + sin2 œ · sin2 • + cos2 • .

Duct Acoustics

If one associates the terms as cos œ = —y /k0 ;

sin œ =

1 − (—y /k0)2 ,

J

625

(7)

then œ is the angle of incidence on the absorber.The surface admittance of a bulk reacting lining can be written as a function of cosœ,sinœ (which may be complex).The admittance of the locally reacting absorber is obtained for normal incidence. Start the evaluation by finding the mode solution for the locally reacting absorber with the method described in > Sect. J.7. Evaluate with it the mode angle œ as above. Insert this into the admittance formula of Z0 Gay of the lining, and solve for the next approximation with the method for locally reacting absorbers (i. e. œ is kept constant during the performance of Muller’s procedure). Repeat until the solution becomes stationary. The advantage of this method of œ-iteration is its robustness against mode jumping. It can also be applied for multilayer absorbers, where other methods mostly run into problems. After about eight iterations the result is mostly stationary in its first four to five decimals.

J.11

Flat Duct with Unsymmetrical, Locally Reacting Lining


The idea behind silencers with unsymmetrical lining is explained with the diagram below.It combines modal attenuations Dh for the first symmetrical and anti-symmetrical modes in two ducts with symmetrical linings; both ducts are 2h wide, and the lining is a simple, locally reacting layer of glass fibres.

The parameters in the ducts shown, i = 1, 2, are: d1 /h = 1 ;

¡1 d1 /Z0 = 1 ;

d2 /h = 2 ;

¡2d2 /Z0 = 2 ,

where di = layer thickness and ¡i = flow resistivity.

626

J

Duct Acoustics

The “idea” expects that in a duct with an unsymmetrical lining the anti-symmetrical mode is used, together with the symmetrical mode, to make up the sound field, and due to its higher attenuation it will increase the attenuation of the least attenuated mode in such ducts. The field formulation in a flat duct with an unsymmetrical lining is: p(x, y) = A · cos(—y) + B · sin(—y) · e− x Z0 vy (x, y) =

j— −A · sin(—y) + B · cos(—y) k0

(1)

2 = —2 − k02.

The boundary conditions are * Z0 vy ** j— −A · sin(—h) + B · cos(—h) ! = = Z0 G1 , p *y=+h k0 A · cos(—y) + B · sin(—y) * Z0 vy ** j— A · sin(—h) + B · cos(—h) ! = = −Z0 G2 . * p y=−h k0 A · cos(—h) − B · sin(—h)

(2) (3)

First, they give the amplitude ratio: B —h · tan(—h) − jk0h · Z0 G2 ∗) . = − cot(—h) A —h · cot(—h) + jk0 h · Z0 G2 Second, they lead to a characteristic equation for —h, with Ui = k0h · Z0 Gi : —h · cot(—h) + jU2 —h · tan(—h) − jU1 + —h · cot(—h) + jU1 —h · tan(—h) − jU2 = 0 .

(4)

(5)

1 1 (6) (U1 + U2) ; Ua = (U1 − U2) 2 2 (where the sides i = 1, 2 are selected so that Re{Ua } ≥ 0) the equation transforms into: —h · tan(—h) − jUs · —h · cot(—h) + jUs = U2a . (7)

Setting:

∗)

Us =


Duct Acoustics

J

627

If Ua = 0, it represents the product of the characteristic equations for symmetrical and anti-symmetrical modes in a symmetrically lined duct. By continued fraction expansions with increasing depth one gets the polynomial equations for z 2 = (—h)2 : (8) 5jUs − 3 U2s − U2a − −3 − 4jUs + U2s − U2a · z2 = 0, − 45jUs + 45 U2s − U2a + 45 + 78jUs − 18 U2s − U2a · z2 (9) + −18 − 9jUs + U2s − U2a · z4 = 0, − 99225jUs + 99225 U2s − U2a + 99225 + 185850jUs − 53550 U2s − U2a · z2 + −53550 − 41895jUs + 5775 U2s − U2a · z4 (10) + 5775 + 2250jUs − 150 U2s − U2a · z6 + −150 − 25jUs + U2s − U2a · z8 = 0 . The modes in the unsymmetrical duct with solutions of the above characteristic equation are orthogonal to each other over the duct height. The numerical example compares the attenuation Dh = 8.6858 · Re{ h} [dB] of the least attenuated modes in an unsymmetrically lined duct (full line) with those in the two ducts having each of the linings as a symmetrical lining (dashed lines). The attenuation in the unsymmetrical duct generally lies between the attenuations in the symmetrical ducts, but nearer to the higher attenuation.

Attenuation Dh of the least attenuated mode in duct with unsymmetrical, locally reacting lining (full line), compared with Dh in ducts with symmetrical linings (dashed lines)

628

J.12

J

Duct Acoustics

Flat Duct with an Unsymmetrical, Bulk Reacting Lining


The object is as in the previous > Sect. J.11,but the lining consists of bulk reacting layers of a porous material (mineral fibre in the numerical examples) having the normalised characteristic values an,i , Zan,i on the duct sides i = 1, 2, possibly covered with a foil having the partition impedance Zsi . The secular and characteristic equations for the lateral wave number z = —h are: h = z2 − (k0h)2 ;

z · tan z − jUs · z · cot z + jUs = U2a ;

Us =

1 (U1 + U2 ) ; 2

Ua =

(1)

1 (U1 − U2) 2

with: yi · tan yi −yi · tan yi ; −−−−→ di Zsi →0 di · an,i Zan,i Zsi /Z0 · yi · tan yi − k0 h · an,i Zan,i h h 2 yi2 = (di /h)2 z2 − (k0h)2 (1 + an,i ) ; i = 1, 2 . Ui = k0 h

(2)

Attenuation Dh of the least attenuated mode in a flat duct with unsymmetrical, bulk reacting lining (full line), and in ducts with symmetrical lining (dashed lines)

Duct Acoustics

J.13

J

629

Round Duct with a Locally Reacting Lining


The lining is defined by a surface admittance G. The form of a mode is: p(r, ‡, x) = P · q(r, ‡) · e− x .

(1)

The lateral profile q(r, ‡) must satisfy the Bessel differential equation: 2 ∂ 1 ∂2 1 ∂ 2 2 + + + + k0 q(r, ‡) = 0 . ∂r2 r ∂r r2 ∂r2

(2)

Solutions have the general form: q(r, ‡) = cos(m‡) [Jm (—m r) + b · Ym (—m r)] ;

m = 0, 1, 2, . . .

(3)

with Bessel functions Jm (z) and Neumann functions Ym (z). If the origin r = 0 belongs to the field area, the Neumann function must be excluded because it is singular there. Thus the mode profiles in round ducts are: q(r, ‡) = cos(m‡) · Jm (—m r) ;

m = 0, 1, 2, . . . .

The Bessel differential equation requires (secular equation):

(4) —2m = 2 + k02

(5)

with → m . The radial particle velocity is: vr =

j—m j ∂p = P cos(m‡) · J m (—m r) · e− x . k0 Z0 ∂r k0Z0

(6)

With it the boundary condition gives the characteristic equation for —m h: (—m h)

J m (—m h) = −jk0 h · Z0 G = : − j · U Jm (—m h)

(7)

630

J

or, with (—m h)

Duct Acoustics

J m (z) = Jm−1 (z) −

m Jm (z): z

Jm−1 (—m h) = m − jU . Jm (—m h)

The function Fm (z): = z Fm (z) = 2m −

Jm−1 (z) can be expanded into a continued fraction: Jm (z)

z2 z2 z2 ... . 2(m + 1)− 2(m + 2)− 2(m + 3)−

(8) (9) (10) (11)

Therewith the characteristic equation can be written as (with the abbreviations z = —m h) z2 z2 z2 . . . = jU + m . 2(m + 1)− 2(m + 2)− 2(m + 3)− It becomes for the fundamental azimuthal mode m = 0: z 2 z2 z2 z2 . . . = jU . 2− 4− 6− 8− The solution with the indicated length of the fraction is: 96 + 36jU ± 9216 + 2304jU − 912U2 . (—0 h)2 ≈ 12 + jU

(12)

(13)

(14)

Its precision test is contained in the diagram. The root is evaluated with a negative real part in the range above the dash-dotted limit curve and with a positive real part below that curve. The coefficients ai for polynomial approximations a0 + a1 · z2 + a2 · z4 + . . . + ai · z2i = 0

(15)

of the characteristic equation with increasing depth of the expansion are given in Table 1. Mode charts can be plotted as lines of Re{z} = const and Im{z} = const in the complex U plane for azimuthal mode orders m = 0, 1, 2, . . . and radial mode orders n = 0, 1, 2, . . . .

4

3

2

1

m 0

a0 8jU −384jU 46080jU −10321920jU 3715891200jU 24(1 + jU) 1920j(j − U) 322560(1 + jU) 92897280j(j − U) 40874803200(1 + jU) 48(2 + jU) 5760j(2j − U) 1290240(2 + jU) 464486400j(2j − U) 245248819200(2 + jU) 80(3 + jU) 13440j(3j − U) 3870720(3 + jU) 1703116800j(3j − U) 1062744883200(3 + jU) 120(4 + jU) 26880j(4j − U) 9676800(4 + jU) 5109350400j(4j − U) 3719607091200(4 + jU)

a1 −(4 + jU) 24(8 + 3jU) 1920j(12j − 5U) 322560(16 + 7jU) 92897280j(20j − 9U) −(7 + jU) 48(13 + 3jU) 5760j(19j − 5U) 1290240(25 + 7jU) 464486400j(31j − 9U) −(10 + jU) 240(6 + jU) 13440j(26j − 5U) 3870720(34 + 7jU) 5109350400j(14j − 3U) −(13 + jU) 120(23 + 3jU) 26880j(33j − 5U) 9676800(43 + 7jU) 5109350400j(53j − 9U) −(16 + jU) 168(28 + 3jU) 241920j(8j − U) 21288960(52 + 7jU) 13284311040j(64j − 9U)

a2 − −(12 + jU) 96(20 + 3jU) 17280j(28j − 5U) 5160960(36 + 7jU) − −(17 + jU) 480(9 + jU) 40320j(37j − 5U) 15482880(47 + 7jU) − −(22 + jU) 240(34 + 3jU) 80640j(46j − 5U) 38707200(58 + 7jU) − −(27 + jU) 336(41 + 3jU) 725760j(11j − U) 85155840(69 + 7jU) − −(32 + jU) 1344(16 + jU) 241920j(64j − 5U) 170311680(80 + 7jU)

a3 − − −(24 + jU) 800(12 + jU) 94080j(48j − 5U) − − −(31 + jU) 1200(15 + jU) 188160j(59j − 5U)4 − − −(38 + jU) 1680(18 + jU) 1693440j(14j − U) − − −(45 + jU) 2240(21 + jU) 564480j(81j − 5U) − − −(52 + jU) 2880(24 + jU) 887040j(92j − 5U)

a4 − − − −(40 + jU) 600(56 + 3jU) − − − −(49 + jU) 840(67 + 3jU) − − − −(58 + jU) 3360(26 + jU) − − − −(67 + jU) 1440(89 + 3jU) − − − −(76 + jU) 1800(100 + 3jU)

a5 − − − − −(60 + jU) − − − − −(71 + jU) − − − − −(82 + jU) − − − − −(93 + jU) − − − − −(104 + jU)

Table 1 Coefficients of the polynomial approximations to the characteristic equation for azimuthal modes of orders m = 0; 1; 2; 3; 4

Duct Acoustics

J 631

632

J

Duct Acoustics

Duct Acoustics

J

633

Three-dimensional mode charts are created if one plots the mesh points at a height Re{z} above the U plane for any azimuthal order m.

3D mode chart for the azimuthal mode m = 0 and some radial modes The evaluation of sets of mode solutions z = —h is similar to the task in > Sect. J.7. It needs the branch points zb and their transforms Ub . The zb are solutions of the equation Jm−1 (z) Jm−1 (z) ! 2m − z − z = 0 ; z = 0 . (16) Jm (z) Jm (z) The following tables contain branch points zb (m, n) and associated Ub (m, n) for azimuthal orders m = 0, 1, . . ., 10 and radial orders n = 0, 1, . . ., 20. The entries of the tables are the real and imaginary parts. Further, the coefficients are given (again with their real and imaginary parts) in tables for the representation of zb (m, n), Ub (m, n) as functions of n for given m: zb (m, n) = am + bm · n1/2 + cm · n + dm · n2 + em · n3 ,

(17) Ub (m, n) = Am + Bm · n1/2 + Cm · n + Dm · n2 + Em · n3 , and coefficients for the functions zb

(m, n) = f zb (m, n) ; U

b (m, n) = f U b (m, n) as:

1/2

3/2 zb

= a¯ m + b¯ m · zb + c¯m · zb + d¯ m · zb + e¯ m · zb 2 + f¯m · zb 3 ,

1/2 ¯ m · U 3/2 + E¯ m · U 2 + F¯ m · U 3 . U

b = A¯ m + B¯ m · Ub + C¯ m · U b + D b b b

(18)

0: 0:

0: 0:

0: 0:

0: 0:

0: 0:

0: 0:

0: 0:

0: 0:

0: 0:

0: 0:

0: 0:

0

1

2

3

4

5

6

7

8

9

10

15:386730 2:1923528

14:243760 2:1381313

13:091486 2:0804554

11:928184 2:0186875

10:751546 1:9519719

9:5583600 1:8791146

8:3439388 1:7983610

7:1009646 1:7069432

5:8168507 1:6000118

19:2764083 2:27401119

18:0694113 2:22939479

16:8505539 2:18235758

15:6179031 2:13251552

14:3689547 2:07937182

13:1003688 2:02226408

11:8075223 1:96027390

10:4837004 1:89206035

9:11848718 1:81551915

7:69410395 1:72697154

6:17515307 1:61871738

zb(m; 2)

22:861283 2:3378991

21:610265 2:2996848

20:346310 2:2596692

19:067527 2:2176033

17:771528 2:1731742

16:455227 2:1259798

15:114516 2:0754868

13:743724 2:0209608

12:334640 1:9613351

10:874574 1:8949433

9:3419610 1:8188728

zb (m; 3)

26:308655 2:39326958

25:0243271 2:35975104

23:7266496 2:32484276

22:4138509 2:28837560

21:0837394 2:25014251

19:7335515 2:20988517

18:3597199 2:16727350

16:9575046 2:12187295

15:5203741 2:07308877

14:0388913 2:02006280

12:4985071 1:96145954

zb (m; 4)

29:677677 2:4427332

28:366891 2:4128473

27:042689 2:3818611

25:703439 2:3496560

24:347158 2:3160901

22:971391 2:2809907

21:573052 2:2441437

20:148156 2:2052780

18:691419 2:1640392

17:195565 2:1199462

15:650104 2:0723098

zb(m; 5)

32:9965422 2:48760786

31:6640769 2:46063214

30:3183214 2:43276785

28:9577821 2:40392982

27:5806698 2:37401800

26:1848108 2:34291347

24:7675211 2:31047269

23:3254221 2:27651937

21:8541613 2:24083200

20:3479682 2:20312418

18:7989117 2:16301098

zb(m; 6)

36:280895 2:5287392

34:930267 2:5041544

33:566582 2:4788405

32:188476 2:4527351

30:794337 2:4257663

29:382231 2:3978501

27:949815 2:3688878

26:494193 2:3387612

25:011725 2:3073267

23:497724 2:2744062

21:945980 2:2397725

zb (m; 7)

39:5402853 2:56673216

38:1741788 2:54415004

36:7953083 2:52096170

35:4024288 2:49712076

33:9940832 2:47257456

32:5685480 2:44726287

31:1237603 2:42111618

29:6572168 2:39405346

28:1658298 2:36597916

26:6457175 2:33677883

25:0918858 2:30631281

zb (m; 8)

zb(m; 10)

42:7809382 46:007117 2:60204547 2:6350391

41:4014603 44:615958 2:58116683 2:6156280

40:0095430 43:212698 2:55977848 2:5957841

38:6040478 41:796294 2:53784550 2:5754810

37:1836547 40:365546 2:51532886 2:5546894

35:7468196 38:919064 2:49218466 2:5333769

34:2917166 37:455223 2:46836325 2:5115077

32:8161611 35:972101 2:44380805 2:4890417

31:3175030 34:467400 2:41845412 2:4659342

29:7924754 32:938331 2:39222622 2:4421349

28:2369731 31:381461 2:36503612 2:4175870

zb(m; 9)

J

4:4662985 1:4674704

2:9803824 1:2796025

zb (m; 0) zb(m; 1)

m

Table 2a Branch points zb (m; n) for azimuthal orders m and radial orders n in round ducts with a locally reacting lining

634 Duct Acoustics

zb (m; 11)

34:525496 2:4651401

36:083507 2:4875121

37:615957 2:5092906

39:125763 2:5305124

40:615330 2:5512115

42:086671 2:5714189

43:541487 2:5911633

44:981236 2:6104708

46:407174 2:6293655

47:820395 2:6478694

49:221856 2:6660028

m

0

1

2

3

4

5

6

7

8

9

10

52:4273767 2:69517382

51:0167569 2:67816304

49:5947157 2:66083276

48:1603706 2:64316737

46:7127217 2:62515010

45:2506280 2:60676298

43:7727783 2:58798682

42:2776539 2:56880112

40:7634791 2:54918414

39:2281563 2:52911295

37:6691784 2:50856370

zb (m; 12)

55:625353 2:7227494

54:206533 2:7067329

52:776622 2:6904391

51:334805 2:6738558

49:880164 2:6569703

48:411660 2:6397692

46:928107 2:6222383

45:428140 2:6043630

43:910183 2:5861281

42:372390 2:5675181

40:812583 2:5485179

zb (m; 13)

58:8170688 2:74889562

57:3908614 2:73376582

55:9538830 2:71839384

54:5053795 2:70277034

53:0445061 2:68688559

51:5703113 2:67072952

50:0817171 2:65429176

48:5774940 2:63756183

47:0562300 2:62052925

45:5162895 2:60318383

43:9557618 2:58551614

zb (m; 14)

62:003528 2:7737536

60:570629 2:7594194

59:127268 2:7448727

57:672744 2:7301061

56:206278 2:7151124

54:726996 2:6998839

53:233915 2:6844132

51:725920 2:6686927

50:201740 2:6527155

48:659915 2:6364750

47:098757 2:6199657

zb(m; 15)

65:1855288 2:79744428

63:7465385 2:78382802

62:2973818 2:77002428

60:8374082 2:75602747

59:3658956 2:74183199

57:8820393 2:72743228

56:3849379 2:71282291

54:8735766 2:69799875

53:3468057 2:68295513

51:8033147 2:66768817

50:2415986 2:65219513

zb (m; 16)

68:363712 2:8200725

66:919153 2:8071071

65:464711 2:7939758

63:999779 2:7806740

62:523689 2:7671977

61:035696 2:7535428

59:534972 2:7397055

58:020588 2:7256823

56:491500 2:7114703

54:946524 2:6970676

53:384312 2:6824733

zb (m; 17)

71:5386020 2:84172925

70:0889297 2:82935680

68:6296464 2:81683668

67:1601857 2:80416564

65:6799238 2:79134056

64:1881718 2:77835854

62:6841652 2:76521697

61:1670524 2:75191372

59:6358794 2:73844721

58:0895726 2:72481678

56:5269161 2:71102288

zb(m; 18)

74:7106281 2:86249447

73:2562454 2:85066423

71:7925113 2:83870220

70:3188954 2:82660595

68:8348159 2:81437322

67:3396329 2:80200202

65:8326389 2:78949068

64:3130496 2:77683800

62:7799907 2:76404342

61:2324837 2:75110719

59:6694268 2:73803069

zb (m; 19)

77:880149 2:8824385

76:421412 2:8711060

74:953571 2:8596555

73:476130 2:8480852

71:988543 2:8363935

70:490216 2:8245792

68:980491 2:8126413

67:458644 2:8005793

65:923871 2:7883934

64:375276 2:7760845

62:811857 2:7636547

zb (m; 20)

Table 2b Branch points zb (m; n) for azimuthal orders m and radial orders n in round ducts with a locally reacting lining (continued)

Duct Acoustics

J 635

0: 0:

0: 1:

0: 2:

0: 3:

0: 4:

0: 5:

0: 6:

0: 7:

0: 8

0: 9:

0: 10:

0

1

2

3

4

5

6

7

8

9

10

11:835078 2:8502677

11:169070 2:7267290

10:479213 2:5990743

9:7617169 2:4667050

9:0116093 2:3288533

8:2221186 2:1845043

7:3835846 2:0322668

6:4813156 1:8701363

5:4907943 1:6950243

16:5358998 2:65088497

15:7194383 2:56267754

14:8758052 2:47206344

14:0010662 2:37877745

13:0901833 2:28250429

12:1365161 2:18286741

11:1309878 2:07941812

10:0605876 1:97163375

8:90541529 1:85895745

7:63203406 1:74101667

6:17515307 1:61871738

Ub (m; 2)

20:589105 2:5959057

19:674737 2:5259193

18:732066 2:4543972

17:757338 2:3812247

16:745849 2:3062807

15:691562 2:2294453

14:586504 2:1506167

13:419744 2:0697508

12:175571 1:9869593

10:829870 1:9027655

9:3419610 1:8188728

Ub (m; 3)

24:3536822 2:58538746

23:3673841 2:52707712

22:3526020 2:46775429

21:3058567 2:40738075

20:2228660 2:34592951

19:0982555 2:28339614

17:9251176 2:21981999

16:6943117 2:15532518

15:3932865 2:09020428

14:0039585 2:02510184

12:4985071 1:96145954

Ub (m; 4)

27:955719 2:5931956

26:913312 2:5431644

25:842719 2:4924595

24:740752 2:4410835

23:603546 2:3890567

22:426342 2:3364279

21:203156 2:2832936

19:926292 2:2298322

18:585549 2:1763663

17:166901 2:1234860

15:650104 2:0723098

Ub (m; 5)

33:757471 2:5911533

32:604957 2:5519495

31:422838 2:5124977

30:207976 2:4728523

28:956619 2:4330944

27:664212 2:3933440

26:325136 2:3537805

24:932316 2:3146755

23:476634 2:2764495

21:945980 2:2397725

Ub (m; 7)

31:45462389 34:883035 2:6095514 2:6300728

30:3667933 2:56575149

29:2513290 2:52150723

28:1053201 2:47684338

26:9252866 2:43180353

25:7070081 2:38645997

24:4452811 2:34092956

23:1335608 2:29539999

21:7634172 2:25017531

20:3236666 2:20575852

18:7989117 2:16301098

Ub (m; 6)

38:2607134 2:65257265

37:1031938 2:61758702

35:9194713 2:58243119

34:7071219 2:54714695

33:4633108 2:51179287

32:1846863 2:47645099

30:8672365 2:44123699

29:5060904 2:40631550

28:0952333 2:37192429

26:6270897 2:33841359

25:0918858 2:30631281

Ub (m; 8)

41:6004613 2:67588250

40:4154669 2:64413811

39:2050406 2:61230611

37:9669654 2:58043032

36:6986717 2:54856962

35:3971529 2:51680343

34:0588524 2:48523973

32:6795116 2:45402684

31:2539566 2:42337139

29:7757956 2:39356630

28:2369731 2:36503612

Ub (m; 9)

44:911000 2:6993511

43:702092 2:6703241

42:468523 2:6412699

41:208261 2:6122325

39:918969 2:5832688

38:597938 2:5544540

37:241995 2:5258873

35:847388 2:4977011

34:409622 2:4700748

32:923231 2:4432550

31:381461 2:4175870

Ub (m; 10)

J

4:3645604 1:5016772

2:9803824 1:2796025

Ub (m; 0) Ub (m; 1)

m

Table 3a Branch points Ub (m; n) for modes with azimuthal order m and radial order n in a round duct with a locally reacting lining

636 Duct Acoustics

Ub (m; 11)

34:525496 2:4651401

36:069713 2:4884634

37:562987 2:5128291

39:011063 2:5379526

40:418666 2:5636249

41:789737 2:5896899

43:127608 2:6160297

44:435123 2:6425538

45:714733 2:6691925

46:968572 2:6958912

48:198507 2:7226073

m

0

1

2

3

4

5

6

7

8

9

10

51:4674976 2:74543936

50:2189128 2:72071188

48:9471592 2:69603480

47:6505153 2:67144898

46:3270288 2:64700563

44:9744709 2:62276930

43:5902778 2:59882201

42:1714748 2:57526884

40:7145778 2:55224590

39:2154611 2:52993170

37:6691784 2:50856370

Ub (m; 12)

54:721357 2:7677292

53:456107 2:7447305

52:168396 2:7218066

50:856634 2:6989962

49:519028 2:6763473

48:153544 2:6539191

46:757859 2:6317861

45:329301 2:6100418

43:864770 2:5888055

42:360632 2:5682308

40:812583 2:5485179

Ub (m; 13)

57:9626776 2:78941536

56:6824395 2:76793265

55:3804194 2:74654278

54:0551440 2:72528223

52:7049613 2:70419550

51:3280090 2:68333713

49:9221743 2:66277443

48:4850439 2:64259107

47:0138402 2:62289203

45:5053390 2:60381027

43:9557618 2:58551614

Ub (m; 14)

61:193485 2:8104709

59:899688 2:7903279

58:584759 2:7702909

57:247328 2:7503941

55:885870 2:7306788

54:498673 2:7111952

53:083807 2:6920041

51:639082 2:6731805

50:161996 2:6548173

48:649669 2:6370303

47:098757 2:6199657

Ub (m; 15)

64:4153818 2:83089039

63:1092582 2:81193925

61:7826203 2:79310361

60:4341950 2:77441551

59:0625683 2:75591320

57:6661623 2:73764260

56:2432072 2:71965912

54:7917072 2:70203008

53:3093968 2:68483785

51:7936874 2:66818403

50:2415986 2:65219513

Ub (m; 16)

67:629659 2:8506816

66:312276 2:8327973

64:974968 2:8150351

63:616548 2:7974250

62:235703 2:7800025

60:830973 2:7628097

59:400731 2:7458970

57:943149 2:7293250

56:456167 2:7131673

54:937445 2:6975133

53:384312 2:6824733

Ub (m; 17)

70:8373648 2:86986025

69:5096543 2:85293593

68:1625812 2:83613827

66:7950354 2:81949525

65:4057936 2:80303968

63:9935030 2:78681032

62:5566605 2:77085312

61:0935882 2:75522285

59:6024038 2:73998526

58:0809835 2:72521973

56:5269161 2:71102288

Ub (m; 18)

74:0393625 2:88844680

72:7021446 2:87239063

71:3460989 2:85646395

69:9701856 2:84069288

68:5732632 2:82510783

67:1540736 2:80974448

65:7112253 2:79464478

64:2431718 2:77985839

62:7481871 2:76544436

61:2243340 2:75147339

59:6694268 2:73803069

Ub (m; 19)

77:236369 2:9064642

75:890368 2:8911966

74:526049 2:8760600

73:142435 2:8610789

71:738459 2:8462813

70:312949 2:8317003

68:864612 2:8173741

67:392019 2:8033480

65:893580 2:7896752

64:367523 2:7764189

62:811857 2:7636547

Ub (m; 20)

Table 3b Branch points Ub (m; n) for modes with azimuthal order m and radial order n in a round duct with a locally reacting lining (continued)

Duct Acoustics

J 637

−0:368205765 0:1469519108

0:9879138803 0:6540921622

2:0521650936 0:9577515405

3:0330022534 1:1785983848

3:9850498138 1:3540592956

4:9274383020 1:5007561805

5:8677962944 1:6275460840

6:8093247657 1:7397127229

7:7533394623 1:8406628730

8:7003048861 1:9327222066

9:6502912998 2:0175497001

1

2

3

4

5

6

7

8

9

10

am

0

m

3:22448082853 0:15833007251

3:00519948413 0:19417363167

2:76875711191 0:23475325470

2:51247480112 0:28141377516

2:23322708646 0:33611110602

1:92754209316 0:40182259263

1:59208568134 0:48335506807

1:22534943787 0:58911235561

0:83345985094 0:73558052419

2:503386726405 0:017288702225

2:530009158087 0:011967067444

2:561568904705 0:005668071753

2:599110725057 −0:00193820478

2:643937503965 −0:01134703166

2:697648032647 −0:02333700567

2:762104033654 −0:03920700769

2:839114176362 −0:06130501868

2:929014637243 −0:09439767730

3:024301642897 −0:15014414119

3:071531336840 −0:26965854932

cm

0:0131273172995 −0:000650474842

0:0129972690267 −0:000593571691

0:0127481558357 −0:000518443632

0:0123445094137 −0:000417131079

0:0117393813727 −0:000277093023

0:0108708424789 −0:000077651008

0:0096590276873 0:0002173543069

0:0080091011338 0:0006764141953

0:0058440123552 0:0014452291919

0:0032892893190 0:0028975202864

0:0019178022618 0:0064183572442

dm

−0:0001891806589 8:90808346238 10−6

−0:0001902652854 8:518568157 10−6

−0:0001896588790 7:902688174 10−6

−0:0001867537156 6:942356734 10−6

−0:0001807192236 5:445314093 10−6

−0:0001704204923 3:0844587418 10−6

−0:0001543296062 −7:29673305 10−7

−0:0001305099148 −7:14670392 10−6

−0:0000970698817 −0:0000186834770

−0:0000552435395 −0:0000419843633

−0:0000317773239 −0:0001024222785

em

J

0:45297970214 0:96167589506

0:27627919216 1:39952627045

bm

Table 4 Coefficients for zb (m; n) = f(n) for n 1 638 Duct Acoustics

−0:368205759 0:1469519014 0:7282201609 0:7715863448 1:2844860115 1:2656576981 1:6588783877 1:6855162704 1:9527849797 2:0573696067 2:2049928625 2:3955102277 2:4329866599 2:7085693213 2:6455580966 3:0021502783 2:8475035353 3:2800950094 3:0416180708 3:5451532755 3:2296254730 3:7993627839

1

2

3

4

5

6

7

8

9

10

Am

0

m

6:83134065850 −1:3728521687

6:26736142134 −1:2029842154

5:67793946463 −1:0242412011

5:05948744153 −0:8346524594

4:40793395345 −0:6314780024

3:71899899875 −0:4107674316

2:98917578557 −0:1665669779

2:21884022363 0:11061198552

1:42244565712 0:43704401766

0:66715771889 0:84366874624

0:27627918188 1:39952628507

Bm

1:755509313657 0:428274604376

1:842563478661 0:388662803041

1:937478158243 0:346875537688

2:041808793319 0:302371835308

2:157483380377 0:254380707608

2:286842692153 0:201761528312

2:432549760658 0:142748671627

2:597005607418 0:074449811796

2:779892426397 −0:00823731455

2:967519068383 −0:11529468602

3:071531341247 −0:26965855525

Cm

Table 5 Coefficients for Ub (m; n) = f(n) for n 1

0:0295043406055 −0:012455254269

0:0282880484561 −0:011456954116

0:0268771251663 −0:010405619456

0:0252172122947 −0:009286287303

0:0232363066293 −0:008076867207

0:0208394175184 −0:006743458033

0:0179043227384 −0:005231565146

0:0142874040691 −0:003448411178

0:0098790246486 −0:001223792151

0:0049072430267 0:0017887755870

0:0019178020235 0:0064183575291

Dm

−0:0004332708985 0:00020873028611

−0:0004195460889 0:00019269335736

−0:0004030388285 0:00017585217640

−0:0003828178735 0:00015796641044

−0:0003576065424 0:00013867336588

−0:0003256545970 0:00011740354356

−0:0002845999367 0:00009321965254

−0:0002314514703 0:00006448705901

−0:0001633416420 0:00002813268974

−0:0000825306369 −0:0000222567162

−0:0000317773180 −0:0001024222849

Em

Duct Acoustics

J 639

10

9

8

7

6

5

4

3

2

0: 0:

0: 1:

0: 2:

0: 3:

0: 4:

0: 5;

0: 6:

0: 7:

0: 8:

0: 9:

0: 10:

z00 U00

z00 U00

z00 U00

z00 U00

z00 U00

z00 U00

z00 U00

z00 U00

z00 U00

z00 U00

am &Am

z00 U00

to

1:35289583687 −4:7889854598

1:30152387453 −4:3721534847

1:25324561024 −3:9416611884

1:20830030824 −3:4943027473

1:16700545337 −3:02571125991

1:12979055127 −2:5297526272

1:09724952002 −1:9974527414

1:07021984409 −1:4149143941

1:04988152225 −0:7586897835

1:03768613394 0:01695130841

1:03255472049 1:03255471568

bm &Bm

−0:351763534989 1:1676373868072

−0:330472119525 1:1096650362265

−0:310074608988 1:0474900263916

−0:290694642818 0:9796199915083

−0:272501726616 0:9038948758619

−0:255733684141 0:8170922590251

−0:240732875512 0:7142045954165

−0:228003625757 0:5869873503596

−0:218290124515 0:4206222861074

−0:212546972815 0:1842150467024

−0:209891280205 −0:209891275705

cm &Cm

0:04919089721 −0:1315676758

0:04602637876 −0:1297334263

0:04293030500 −0:1275745816

0:03992283817 −0:1248598502

0:03703335580 −0:1212266312

0:03430556310 −0:1160930652

0:03180605385 −0:1084914003

0:02963847580 −0:0967221457

0:02796403550 −0:0775339878

0:02699886920 −0:0437173781

0:02650507820 0:02650507684

dm &Dm

−0:002908009942 0:0062138410826

−0:002732925622 0:0063654705151

−0:002557583186 0:0065223770294

−0:002383059269 0:0066724572248

−0:002211096670 0:0067942928935

−0:002044526252 0:0068500426717

−0:001887999641 0:0067714996070

−0:001749250152 0:0064302118295

−0:001641020525 0:0055645102340

−0:001581192667 0:0035595268032

−0:001547034821 −0:001547034677

em &Em

4:591708949 10−6 −6:40447321 10−6

4:394274371 10−6 −7:14831915 10−6

4:184329860 10−6 −7:99458282 10−6

3:962241644 10−6 −8:93938211 10−6

3:729591742 10−6 −9:96235424 10−6

3:490226516 10−6 −0:0000110096253

3:252218427 10−6 −0:0000119578544

3:031500895 10−6 −0:0000125320673

2:857957129 10−6 −0:0000120934311

2:777630939 10−6 −8:96778277 10−6

2:717121859 10−6 2:717121329 10−6

f m &Fm

U00b (m; n) = f U0 (m; n) of the curve connecting the branch points

J

1

0

m

Table 6 Coefficients for zb00 (m; n) = f z 0 (m; n) ; 640 Duct Acoustics

Duct Acoustics

J

641

Above the curve U

b (m, n) = f U b (m, n) connecting the branch points in the U plane for the surface wave mode it is evident that z ≈ U. This permits an iterative approximation to a solution with the following iteration scheme: (m − jU)U z ≈ z1 = Fm (U) ≈ z2 = (m − jU)

(m − jU)z1 Jm (z1) = Jm−1 (z1 ) Fm (z1 )

≈ z3 = (m − jU)

(m − jU)z2 Jm (z2) . = Jm−1 (z2 ) Fm (z2 )

(19)

The radial modes n to a given azimuthal mode order m subdivide the z plane into “mode strips”. The limits pass through the branch points zb (m, n). The lower-limit branches zgl (m, n) are vertical lines down to the real axis Re{z}; the upper branches of the limits zgu (m, n) are quarter ellipses. The transforms of these limit branches are nearly coincident quarter elliptic arcs with the following forms (prime and double prime indicate real and imaginary parts):

2 1 − U 2 (20) U

g (m, n) = U

b (m, n) + U

g0(m, n) − U

b (m, n) g (m, n)/Ub (m, n) , where U

g0(m, n) are the endpoints of the arcs on the imaginary axis Im{U}; they have

the same values as zg0 (m, n): + ,

(m, n) = Im j · Fm Re{zb (m, n)} −m . (21) U

g0(m, n) = zg0

642

J

Duct Acoustics

A set of mode solutions in a round duct with locally reacting lining, evaluated with the method described. A surface wave mode, if it exists, is distributed over some mode orders. The solution with the large arc is a surface wave mode The next graph shows a negative example of what may happen if either imprecise start values and/or an unfavourable method of numerical solution is used; the returned solutions are “hopping” between modes (mode hopping). This would have bad (if not catastrophic) consequences in a modal field analysis attempted with such results (some solutions are missing, others are returned several times).

An example of “mode hopping” In the following procedure for evaluation of a value z(m, n) = —m,n h of a desired mode set for a given value U, the radial mode counting is n = 0, 1, 2, . . . and the azimuthal mode counting is m = 0, 1, 2, . . . . A solution of radial mode order n is sought.

Duct Acoustics

J

643

Step 1: special case U = 0? Take the (n + 1)-th (non-zero) root of J m (z) = 0 .

(22)

Step 2: U in the surface wave range? That is, U is in the range limited by • the curve U

b = g(U b ) which connects the branch point images Ub , • the imaginary axis, • the curves U g (m, n) = f U

g (m, n) for n and n + 1. Then evaluate z with the above iteration. Step 3: Else: The solution belongs to the lower part of the z strip,where continued fraction expansions converge quickly. Take as starters zsi ; i = 1, 2, 3 for Muller’s procedure 4(1 + m)(2 + m)(m + jU) , 4 + 3m + jU 2 2 = (2 + m)(3 + m)(8 + 5m + 3jU) zs3 12 + 5m + jU ± (2 + m)(3 + m) · (384 + 608m + 294m2 + 57m3 + 5m4 1/2 . + 6j(2 + m)(8 + 3m + m2 )U − (38 + 25m + 5m2 )U2 ) ,

2 zs1 =

(23)

zs2 = (zs1 + zs3 )/2 . Select the sign of the root in zs3 so that it lies in the lower part of the z strip (i. e.

zb (m, n) < zs3 ≤ zb (m, n + 1) and 0 ≤ zs3 ≤ f zb (m, n) , with which the curve connecting the branch points zb (m,n) is indicated.

J.14

Admittance of Annular Absorbers Approximated with Flat Absorbers


The evaluation of the surface admittance G of annular absorbers may be tedious. This is illustrated with a simple porous layer of thickness d having characteristic values a , Za . Let its interior surface be at the radius r = h. If the layer is made locally reacting by cellular partitions (i. e. locally reacting in all directions), its surface admittance is: Z0 G =

jk0 J1 (−ja h) · Y1 (−ja (h + d)) − Y1 (−ja h) · J1 (−ja (h + d)) . Zan J0 (−ja h) · Y1 (−ja (h + d)) − Y0 (−ja h) · J1 (−ja (h + d))

(1)

If the layer has ring-shaped partitions, its surface admittance for the mth azimuthal mode is: Z0 G =

jk0 J m (−ja h) · Y m (−ja (h + d)) − Y m (−ja h) · J m (−ja (h + d)) . Zan Jm (−ja h) · Y m (−ja (h + d)) − Ym (−ja h) · J m (−ja (h + d))

(2)

644

J

Duct Acoustics

The next graph shows sound attenuation curves Dh (for the least attenuated mode) in a round duct with a simple glass fibre layer with cellular partitions, first evaluated with the admittance of ring-shaped absorbers, then with the admittance of the same, but plane, absorber.

Attenuation Dh in a round duct with a locally reacting porous layer, evaluated either as a cylindrical layer or as a plane layer

As above, but the thickness of the plane absorber increased in the evaluation

Duct Acoustics

J

645

The rule which can be taken from this example (checked with other examples, too) is as follows: • For frequencies up to the first maximum in the Dh curve use the admittance of a plane absorber after increasing the thickness of all air or porous layers in the absorber by 1 d/h → d/h · (1 + d/h) . 3 • For higher frequencies use the plane absorber with the original thicknesses.

J.15

Round Duct with a Bulk Reacting Lining


A round duct, 2 h wide, is lined with a layer of thickness d of porous material having characteristic values a , Za , or, in normalised form, an = a /k0, Zan = Za /Z0 . The layer is possibly covered with a foil having a partition impedance Zs . The analysis proceeds somewhat in parallel with the analysis in surface admittance G of the lining now becomes field dependent.

>

Sect. J.13, but the

The field formulation in the free duct for the m-th azimuthal mode is: p(r, ‡, x) = P · cos(m‡) · Jm (—m r) · e−m x ; m = 0, 1, 2, . . . , j ∂p j—m vr = P cos(m‡) · J m (—m r) · e−m x , = k0 Z0 ∂r k0Z0 —2m = m2 + k02 .

(1)

The characteristic equation for —m h is: (—m h)

J m (—m h) = −jk0 h · Z0 G Jm (—m h)

(2)

or (—m h)

Jm−1 (—m h) = m − jU ; Jm (—m h)

U =: k0h · Z0 G .

(3)

J

646

Duct Acoustics

So far the analysis is as in > Sect. J.13. To obtain the surface admittance G, the field in the absorber layer is formulated as: pa (r, ‡, x) = [BJm (—am r) + CYm (—am r)] · cos(m‡) · e−m x , −—am

var (r, ‡, x) = BJm (—am r) + CY m (—am r) · cos(m‡) · e−m x a Za 2 with —2am = 2 − a2 = —2m − k02 1 + an .

(4) (5)

The boundary condition at the hard outer duct wall gives: C = −B

J m (—am (h + d)) . Y m (—am (h + d))

(6)

The surface admittance of the lining (without Zs ) becomes: Z0 G =

−—am J m (—am h) · Y m (—am (h + d)) − Y m (—am h) · J m (—am (h + d)) , a Zan Jm (—am h) · Y m (—am (h + d)) − Ym (—am h) · J m (—am (h + d))

and with Zs

) Z0 G → 1 (Zs /Z0 + 1/Z0G) .

(7) (8)

With the derivatives of the Bessel and Neumann functions substituted, function U becomes for Zs = 0 and y = —am h (for abbreviation): U=

1 an Zan

mJm (y) − yJm−1 (y) · mYm y(1 + d/h) − y(1 + d/h)Ym−1 y(1 + d/h) − . . . · (9) m · Jm (y) · Ym y(1 + d/h) − Ym (y) · Jm y(1 + d/h) + . . . . . . − mYm (y) − yYm−1 (y) · mJm y(1 + d/h) − y(1 + d/h)Jm−1 y(1 + d/h) , . . . + y(1 + d/h) · Ym (y) · Jm−1 y(1 + d/h) − Jm (y) · Ym−1 y(1 + d/h) and for Zs = 0 and m = 0:

y J1 (y) · Y1 y(1 + d/h) − Y1 (y) · J1 y(1 + d/h) . U = k0hZ0 G = an Zan J0 (y) · Y1 y(1 + d/h) − Y0 (y) · J1 y(1 + d/h)

(10)

Function U of the characteristic equation contains in a complicated manner the desired solution z = —m h of that equation. It is assumed for the following solution methods that modes will be determined for a list of k0h values which begins at low values (if a mode for a single k0 h value is needed, a list might have to be prepended). First method: iteration of layer resistance: This method makes use of the fact that a mode-safe method for locally reacting linings exists and that at low frequency and/or high flow resistance values R = ¡ · d/Z0 (¡ = layer flow resistivity) the bulk reacting absorber becomes nearly locally reacting. So start the iteration i = 1, 2, . . . through the list k0 h for i = 1, 2, 3 and begin for each i the evaluation of the mode for a locally reacting absorber with a high value Rk (approx.Rk > 8); evaluate three solutions for Rk=1 , Rk=2 , Rk=3 with the tendency of Rk → R for

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increasing k. Take these values as start values in Muller’s procedure for the solution of the characteristic equation with the bulk reacting absorber, but with Rk=3 . Then iterate with this task through Rk , taking the previous solutions zk as the new starters. For values i > 2 take the previous solutions zi−2 , zi−1 as two starters for the new zi , and take the approximation from the iteration through Rk as the third starter (it helps to avoid mode hopping, which happens when only previous zi are used, even with small steps k0 h). Second method: start with approximations for low frequencies: The iteration through Rk in the above method may be time consuming. If the absorber is locally reacting, then U is independent of z. The characteristic equation reads as follows: z · J m (z) + jU · Jm (z) = 0 ;

m>0,

z · J1 (z) − jU · J0 (z) = 0 ;

m=0.

(11)

The method makes use of the fact that for k0 h → 0 the function for every absorber decreases at least with the square of k0h. Thus starter solutions at low frequency (for the radial modes n) are the solutions zm,n of J m (z) = 0 for m > 0 and of J1 (z) = 0 for m = 0. If n ≥ 1, this value and nearby values with small shifts −z and +j · z

are used as starters for Muller’s procedure. If n = 0 (i. e. the fundamental radial mode), then these approximations are not precise enough; in that case approximations from the continued fraction expansion of Eq. 11 should be applied. From the solution i = 4 on (of the list of k0 h) use previous solutions together with an extrapolated estimate of the new solution (See > Sect. J.10). If the absorber is bulk reacting, the statement U −−−−−→ O (k0h)i ; i > 2 still holds at k0 h→0

low frequencies. For n > 0 the same starters can be used as above, but for n = 0 they must be more precise. Either use in this case the “iteration through R” from above or derive continued fraction approximations for the characteristic equation. For the least attenuated mode one determines solutions for n = 0 and n = 1 and takes the solution with minimum Re{n h}.

J.16

Annular Ducts


Such ducts are in principle different from round ducts because now the Neumann functions appear. We distinguish with indices i, a radii hi , ha and absorber functions Ui = k0 hi · Z0 Gi and Ua = k0 ha · Z0 Ga on the interior or outer side, respectively, of the free duct, which spans over hi ≤ r ≤ ha . The field formulation for the m-th azimuthal mode is: p(r, ‡, x) = [A · Jm (—r) + B · Ym (—r)] · cos(m‡) · e− x , j— vr (r, ‡, x) = A · J m (—r) + B · Y m (—r) · cos(m‡) · e− x k0 Z0

(1)

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(we neglect for the moment the indexing of —, with m). The boundary conditions give the following system of equations: A · j— · J m (—ha ) − k0 Z0 Ga · Jm (—ha ) + B · j— · Y m (—ha ) − k0 Z0 Ga · Ym (—ha ) = 0 , (2) A · j— · J m (—hi ) + k0 Z0 Gi · Jm (—hi ) + B · j— · Y m (—hi ) + k0Z0 Gi · Ym (—hi ) = 0 . A solution must nullify the determinant: j— · J m (—ha ) − k0Z0 Ga · Jm (—ha ) · j— · Y m (—hi ) + k0 Z0 Gi · Ym (—hi ) − j— · J m (—hi ) + k0Z0 Gi · Jm (—hi ) · j— · Y m (—ha ) − k0 Z0 Ga · Ym (—ha ) = 0 .

(3)

With the recurrence relations for derivatives of Bessel and Neumann functions, and using the abbreviations z = —ha ; = hi /ha, a different form of the equation is: z · Jm−1 (z) + (jUa − m) · Jm (z) · z · Ym−1 (z) − (jUi + m) · Ym (z) (4) − z · Jm−1 (z) − (jUi + m) · Jm (z) · z · Ym−1 (z) + (jUa − m) · Ym (z) = 0 . In the special case m = 0 with J 0 (z) = −J1 (z); Y 0 (z) = −Y1 (z) it becomes: z · J1 (z) − jUa · J0 (z) · z · Y1 (z) + jUi · Y0 (z) − z · J1 (z) + jUi · J0 (z) · z · Y1 (z) − jUa · Y0 (z) = 0 ,

(5)

and after multiplication: z2 · [J1 (z) · Y1 (z) − J1(z) · Y1 (z)] − jUa z · [J0 (z) · Y1 (z) − J1 (z) · Y0 (z)]

(6)

+j Ui z · [J1 (z) · Y0 (z) − J0 (z) · Y1 (z)] + Ua Ui · [J0 (z) · Y0(z) − J0 (z) · Y0 (z)] = 0. An interior porous layer of thickness di < hi has the following surface admittance (i , Zi the characteristic values of the material): −jk0 J1 (ji hi ) · Y1 ji (hi − di ) −Y1 (ji hi ) · J1 ji (hi − di ) , (7) Z0 Gi = Zi J0 (ji hi ) · Y1 ji (hi − di ) −Y0 (ji hi ) · J1 ji (hi − di ) and if di = hi , i. e. a central absorber (locally reacting): Z0 Gi =

−jk0 J1 (ji hi ) . Zi J0 (ji hi )

(8)

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An outer porous layer has the admittance: jk0 J1 (ja ha ) · Y1 ja (ha + da ) −Y1 (ja ha ) · J1 ja (ha + da ) . Z0 Ga = Za J0 (ja ha ) · Y1 ja (ha + da ) −Y0 (ja ha ) · J1 ja (ha + da )

649

(9)

For the numerical solution, an iterative scheme over a list of k0 h which begins at low values is recommended. For m = 0 and locally reacting absorbers: For low k0 h the last three terms in the last form of the characteristic equation disappear with the highest order in k0h; therefore an approximate solution is found from: [J1 (z) · Y1 (z) − J1 (z) · Y1(z)] = 0 .

(10)

Such solutions are tabulated in the literature.A regression for zm=0,n=1 over 0 ≤ ≤ 0.8 is: z0,1 ≈ 3.8050757 + 2.65957569 · − 21.7646292042 + 135.27002785 · 3 − 249.58953616 · 4 + 172.92523266 · 5

(11)

(use small shift −z and +j·z

for the two other starters of Muller’s procedure).A starter at low k0h for the case n = 0 (i. e. lowest radial mode) is taken from the following power series expansion of the characteristic equation: 2 j (Ua + Ui ) + Ua Ui ln 0 z2 / − 1 − 2 2 + j (Ua − Ui ) + Ua Ui + ln · Ua − 2j Ui + 2 Ui + 2j Ua 2 z4 / 1 − 2 8 + 2jUa − 10jUi + 3Ua Ui + 2 8 + 10jUa − 2jUi + 3Ua Ui + 64 0 + 2 ln · Ua − 4j Ui + 42 Ua − 2j Ui + 2j + 4 Ua Ui + 4j =0.

(12)

Take in the solution for z2 the root with a negative real part. Further, take the solution of this equation with the fourth power term z4 neglected as a second starter, and the mean value of both as the third starter. If the absorber is bulk reacting and m = 0: Find initial starters (at the lower end of the k0h list) with the assumption that the absorber is approximately locally reacting. Take these solutions as starters with the equation for the bulk reacting absorber, and for later k0h values take previous solutions and an extrapolated new solution (see > Sect. J.10). If the absorber is bulk reacting and m > 0: For n > 0 take the solution of the term without Ua , Ui in the expanded characteristic equation as starter. For n = 0 solve the power series expansion of the characteristic equation, now with expansion of Ua , Ui , for a starter.

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]

Attenuation curves Dhi for ring-shaped silencers with locally reacting glass fibre layers for different thickness ratios of the outer layer. Input parameters: hi/ha = 0.5; di /hi = 1; ¡a da /Z0 = ¡i di /Z0 = 1.5; Zsa = Zsi = 0

J.17

Duct with a Cross-Layered Lining


Often silencers need low values of the (normalised) flow resistance R = ¡ · d/Z0 of porous layers to obtain high attenuation values Dh , but the layer thickness d must not be small, otherwise the lower-limit frequency of attenuation would be high. The necessarily low flow resistance values ¡ lead to low bulk densities or to coarse fibres; both measures reduce the mechanical stability of the absorber. A remedy can be to place layers side by side, one of the layers just being a “placeholder” made out of (e. g.) scrambled wire mats. If characteristic values , Z , = a, b, are normalised with k0, Z0 , respectively, this is indicated with an additional index n. In case (a),in which both layers are separated from each other,if both layer thicknesses a, b are small compared to the wavelength, the effective lining admittance is the weighted average Z0 Gy =

a/T b/T tanh (a d) + tanh (b d) . Zan Zbn

(1)

In case (b) a couple of layers form a lined cross-directed duct, and one of the layer heads is covered towards the main duct. In case (c) layer couples form cross-directed, lined ducts, and both layer heads are open towards the main duct. In cases (b) and (c) first the propagation constant s in the side ducts is determined. For arrangement (b) the average wall admittance seen from the main duct is approximately: Z0 Gy =

b/T s tanh (s d) . Zbn b

(2)

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For arrangement (c) the average admittance of the cross-layered lining of the main duct is: b/T a/T Z0 Gy = · s tanh (s d) . + (3) a Zan b Zbn Average admittance values Gy are sufficient only for small T/Š0.Otherwise the main duct should be treated as a duct with an axially periodic lining. Formulas will be given below, first with the assumption of only the fundamental mode in the side duct, second with higher modes in the side duct. Sometimes the layer indices will be collected as = a, b. Only the fundamental mode in the side duct: The wave and impulse equations in the layers are:

− 2 p (x, y) = 0 ;

v,x (x, y) =

−1 ∂p . Z ∂x

(4)

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Field formulations in the layers are as follows ( may assume the values of a, b): cosh s (y − d) a; x∈a · cos (— (x ∓ )) ; = , p (x, y) = P · b ; x∈b cosh (s d) cosh s (y − d) — 1 vx (x, y) = P · sin (— (x ∓ )) , · Z cosh (s d) −1 s sinh s (y − d) vy (x, y) = · cos (— (x ∓ )) , P · Z cosh (s d) and from the wave equation —2 = s2 − 2 .

(5)

(6)

If the main duct carries higher modes in the z direction, this changes to: —2 = s2 − 2 − —2z .

(7)

(This will be ignored below.) The characteristic equation of the side duct is: −—a —b tan (—a a) = tan (—b b) a Za b Zb

(8)

or, with the abbreviations: zb = —b b ;

A=

b b Zb ; a a Za

B = (b b)2 − (a b)2 ,

(9)

it is:

a 2 a 2 z + B · tan z +B =0. zb · tan(zb ) + A · b b b b

(10)

An approximate equation C0 + C1 · zb2 − C2 · zb4 + C3 · zb6 − C4 · zb8 = 0

(11)

is obtained by continued fraction expansion with the following coefficients: C0 = 525(a/b)2A · B 21 − 2(a/b)2B , / C1 = 15 5(a/b)2A 147 − 7 9 + 4(a/b)2 B + 6(a/b)2B2 0 + 7 105 − 45(a/b)2B + (a/b)4B2 , / C2 = 5 210 + 3(a/b)2 [315 + 7A(45 − B) − 30B)] 0 − 2(a/b)4 (21 − B)B − A(105 − 90B + B2 ) , / 0 C3 = 5(a/b)2 90 + A 21 + (a/b)2(90 − 4B) + (a/b)2(21 − 4B) , C4 = 10(1 + A)(a/b)4 . One obtains with its solution: d 2 2 z + (k0 b)2 bn s d = b b

(12)

(13)

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or, in the case of a higher mode in the z direction: d 2 2 z + (k0 b)2 bn + (—z b)2 . s d = b b

653

(14)

Select from the polynomial solutions the solution with minimum Re{s d} > 0. The normalised surface admittances of the layer heads are: Z0 Gy =

s d tanh (s d) . k0dn Zn

(15)

The field formulation in the main duct with spatial harmonics is: +∞ cos —n (y + h) Pn · p(x, y) = · e−jn x , cos(— h) n n=−∞ —n sin —n (y + h) · e−jn x Z0 vv (x, y) = −j Pn · k cos(— h) 0 n n

(16)

with axial wave numbers n = 0 + n

2 ; T

Im{0 } ≤ 0 ;

—2n = k02 − 2n .

(17)

The wall admittance at y = 0 is periodic with period T = a + b and has the axial profile: Z0 Gby ; −b ≤ x < 0 . (18) Z0 G(x) = Z0 Gay ; 0 < x ≤ a When written as a Fourier series it is: 1 +j2Œ·x/T gŒ · e ; gŒ = Z0 G(x) · e−j2Œ·x/T dx Z0 G(x) = T Œ

(19)

T

with coefficients 1 sin (Œa/T) −jŒa/T sin (Œb/T) +jŒb/T a · Z0 Gay ·e ·e , gŒ = + b · Z0 Gby a+b Œa/T Œb/T 1 g0 = a · Z0 Gay + b · Z0 Gby = Z0 Gy a+b

(20)

(notice in general g−Œ = g+Œ ). One splits the main duct field into a periodic factor and a propagation factor: p(x, y) = P(x, y) · e−j0 x ;

vy (x, y) = Vy (x, y) · e−j0 x .

(21)

The boundary conditions at y = 0 give the following linear, homogeneous system of equations: —n (22) Pn · ƒm,n · j tan(—n h) + gn−m = 0 ; m, n = 0, ±1, ±2, . . . . k0 n

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The determinant set to zero represents the characteristic equation for 0 . A simplified boundary condition requires the matching of the periodic factor Vy (x, 0) with the particle velocity Vabs (x, 0) in the surface of the lining: −jPn

—n 1 tan (—n h) = k0 T

This leads to: ƒm,n − jk0 h n

Z0 Vabs (x, 0) · e+j2nx/T dx .

(23)

T

1 gn−m · Z0 Vabs (x, 0) · e+j2nx/T dx = 0 . —n h · tan (—n h) T

(24)

T

If only one mode exists in the side duct, and this is supposed to have a plane velocity profile, so that Vabs = G · P for an exciting pressure P, then this equation becomes: gn−m · g−n = 0 ; ƒm,n − jk0 h —n h · tan (—n h) n gn · g−n g0 − jk0h =0; m=0. —n h · tan (—n h) n

m = 0 , (25)

The equation for m = 0 is the characteristic equation for 0 h, which is contained in the —n h (the leading term plus the term with n = 0 in the sum form the characteristic equation for a duct with a homogeneous lining). One can take the mode wave numbers —n h in a duct with the average admittance from Z0 Gay , Z0 Gby as starting approximations in the numerical solution with Muller’s procedure for (0 h)2 = (k0 h)2 − (—0 h)2 .

[

]

Mineral fibre boards, a = 4 [cm] thick, ¡ = 30 [kPas/m2 ], with mutual distance b = 6 [cm], form a cross-layered lining of d = 20 [cm] thickness (air in the layers = a) of a h = 10 [cm] wide duct.Points:measured; Full line:periodic duct with spatial harmonics; Dashed: homogeneous duct with average admittance of the layers treated as side ducts

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The assumption of a plane mode profile, made above, can be dropped. One obtains with the true mode profile: ⎧ s tanh (s d) cos (—a (x − a)) ⎪ Pa cos(—a a) ; x∈a ⎪ ⎨Z0 vay (x, 0) = a Zan cos(—a a) Z0 Vabs (x, 0) = , (26) ⎪ cos tanh ( d) (x + b)) (— ⎪ s s b ⎩Z0 vby (x, 0) = ; x∈b Pb cos(—b b) b Zbn cos(—b b) and (notice the definition of the abbreviation ‚n ) 1 Z0 Vabs (x, 0) · e+j2nx/T dx : = P · ‚n T T 1 a ej2na/T a/T = P· j2n − 1 + —a a · tan (—a a) a d Zan (—a a)2 − (2na/T)2 T cos (—a a) 1 e−j2nb/T b b/T b · tan (— b) . + 1 − + — j2n b b b d Zbn (—b b)2 − (2nb/T)2 T cos (—b b) Therewith the equation to be solved is: gn · ‚n =0. ‚0 − jk0 h — h n · tan (—n h) n

(27)

(28)

The changes in the result due to the mode profile (26) as compared with a plane profile often are not worth the larger amount of computations. Higher modes in the side duct (mode index in the side ducts): The field formulation in the side ducts = a, b is: cosh s (y − d) cos (— (x ∓ )) a; · ; = P · p (x, y) = b ; cosh (s d) cos (— ) cosh s (y − d) — sin — (x ∓ ) 1 · , P · vx (x, y) = Z cosh (s d) cos (— ) −1 s sinh s (y − d) cos (— (x ∓ )) vy (x, y) = P · · Z cosh (s d) cos (— )

x∈a , x∈b (29)

with —2 = s2 − 2 and the characteristic equation: −—a —b tan(—a a) = tan(—b b) . a Za b Zb With the transversal mode profiles:

(30) q (x) =

cos (— (x ∓ )) , cos (— )

(31)

the orthogonality relation in the side ducts 1 an Zan

a 0

1 qa (x) · qa‘ (x) dx + bn Zbn

0 qb (x) · qb‘ (x) dx = ƒ,‘ · TN −b

(32)

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gives the mode norms N : T · N =

a/2 sin (2—a a) b/2 sin (2—b b) 1+ + 1 + .(33) an Zan cos2 (—a a) 2—a a bn Zbn cos2 (—b b) 2—b b

The field formulation in the main duct remains as above. Matching the sound fields in the plane y = 0 leads to the homogeneous linear system of equations for the amplitudes Pn of the space harmonics: h Rn, R−m, Pn ƒm,n · —m h · tan (—m h) − j s d tanh (s d) · =0 (34) d N n with the coupling coefficients Rn, between side duct modes and main duct spatial harmonics: 1 T · Rn, : = an Zan

a

e−jßn x ·

0

1 + bn Zbn

0

cos (—a (x − a)) dx cos (—a a)

cos (—b (x + b)) dx cos (—b b) −b j/2 2 2—2a − ß2n −jßn a −1 · = + e an Zan cos (—a a) ßn ßn 4—2a − ß2n e−jßn x ·

1 ßn cos (2—a a) − 2j—a sin (2—a a) 4—2a − ß2n j/2 1 2 2—2b − ß2n +jßn b · + − e bn Zbn cos (—b b) ßn ßn 4—2b − ß2n

+

(35)

1 − 2 ßn cos (2—b b) + 2j—b sin (2—b b) . 4—b − ß2n The determinant of the above system of equations set to zero is the characteristic equation of the system. It can be simplified if only the periodic factor of the main duct field is matched to the field in the side ducts: h Sn, S−m, =0, (36) Pn ƒm,n · —m h · tan (—m h) − j s d tanh (s d) · d N n where the Sn, are obtained from the Rn, by the substitution n → 2n/T, especially for n = 0: sin (2—a a) b/2 sin (2—b b) a/2 T · S0, = 1+ + 1+ an Zan cos (—a a) 2—a a bn Zbn cos (—b b) 2—b b (37) S0, · S0, b/T a/T + . −−−−−−−−−−−→ |—a a|,|—b b|1 an Zan N bn Zbn

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The determinant equation of the second system can be approximated by: h b/T a/T —0 h · tan(—0 h) − j (s0 d) · tanh (s0 d) =0, + d an Zan bn Zbn which is just the characteristic equation with the average head admittance.

657

(38)

A special form of ducts with cross-layered linings is the “pine-tree silencer”. These silencers are used in air flows with heavy dust load. Their axis is vertical. The intention is that dust deposits on the branches of the trees will slide down due to the vibrations of the structure. The pine-tree baffles mostly belong to type (b) of the initial graph of this section. Because of the inclination of the branches, the layer thicknesses change to a → a · cosœ, b → b · cosœ and the admittances Gy → Gy · cosœ. Sometimes the air-filled channels (layers with b in the sketch) are terminated near the “trunk” with an absorber layer having a reflection factor r for the incident fundamental side duct mode. The surface admittance at y = 0 then becomes approximately: Z0 Gby =

s 1 − r · e−2s s cos œ . b Zbn 1 + r · e−2s s

(39)

Frommhold derived a correction formula to be applied to the reflection factor r for normal incidence to obtain approximately the reflection factor reff for the incident mode (claimed to be applicable for f > 70 [Hz]): √ 1 − cos œ √ reff = r · . (40) cos œ + 1 − (f [Hz]/70)3

658

J.18

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Duct Acoustics

Single Step of Duct Height and/or Duct Lining


The steps are isolated, i. e. the ducts on both sides are infinitely long, in principle; in practice it will be sufficient if the next step is far enough to neglect reflections from there at the original step.

One must distinguish: • expanding (a, c) or contracting (b, d) ducts (as seen in the direction of sound); • locally (a, b) or bulk reacting linings; • the same (a, b) or different (c, d) types (local or bulk) on both sides of the step; • if the lining in the narrow duct is bulk reacting, whether its head is open or covered with a hard sheet. There are more combinations than the few examples shown above. Vertical hatching will be used in the sketches below to indicate a locally reacting lining; crossed hatching indicates bulk reacting absorbers. An index = i, e will indicate the duct (and its parameters) on the side of the incident sound and on the exit side. The incident sound pi may be a sum of modes of the entrance duct or, as a special case, one mode of order ‹. Both the reflected wave pr and the transmitted wave pt are formulated as sums of modes in their ducts. Fundamental relations in a duct with a locally reacting lining: Field formulations: pi (x, y) = Pim · cos(—im y) · e−‚im x , m

vix (x, y) =

1 Pim · ‚im cos(—im y) · e−‚im x . jk0Z0 m

In the special case of a single incident mode, Pim → ƒm,‹ · Pi‹ . Prm · cos(—im y) · e+‚im x , pr (x, y) = m

vrx (x, y) =

−1 Prm · ‚im cos(—im y) · e+‚im x , jk0 Z0 m

(1) (2)

(3)

Duct Acoustics

pt (x, y) =

J

Ptn · cos(—en y) · e−‚en x ,

n

vtx (x, y) =

659

(4)

1 Ptn · ‚en cos(—en y) · e−‚en x . jk0Z0 n

2 The secular equation is: —2k = k02 + ‚k ;

= i, e ;

k = ‹, m, n .

If there is a higher mode in the z direction, then it becomes: = i, e ; k = ‹, m, n .

2 —2k = k02 + ‚k − —2z ;

(5) (6)

The characteristic equation for the modes is: (—k h ) · tan(—k h ) = jk0h · Z0 G , and if G = 0, then —k = (k − 1) ;

(7) k = 0, 1, 2, . . . .

(8)

The mode norms Nk are: 1 h

h cos(—k y) · cos(—k y) dy = ƒk,k · Nk , 0

Nk : =

1 h

h

cos2 (—k y) dy =

0

(9)

sin(2—k h ) 1 1+ . 2 2—k h

The mode coupling coefficients of the modes of both ducts are: 1 Rm,n (h, , ) : = h

=

1 2

h cos(—m y) · cos(— n y) dy 0

sin (—m − — n )h sin (—m + — n )h + . (—m − — n )h (—m + — n )h

(10)

Evidently Rm,n (h, , ) = Rn,m (h, , ). Other coupling coefficients over the range h of the height difference are: 1 Sm,n (h, , ): = h

h cos(—m y) · cos(— n y) dy h

(11)

h = Rm,n (h , , ) − Rm,n (h, , ) −−−−→ 0 . h→h h In ducts with a locally reacting lining: Sm,n (h, , ): = ƒm,n · Nm −

h Rm,n (h, , ) . h

(12)

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Fundamental relations in a duct with a bulk reacting lining: In order to write the mode field in a single line, a “switch function” is introduced by: 1; a≤y 1 ⎪ ⎩ k=1

The incident wave is a sum of modes of the duct i = 0 with a list of amplitudes {Pim (1)} or a single mode with amplitude Pi‹ (1) (note that the index i in field components and amplitudes counts the duct steps). The field formulations of the incident wave pi (i), reflected wave pr (i), and transmitted wave pt (i) at the ith duct step are: pi (i) = pi (i, i, y) = pim (i, i, y) = Pim (i) · qm (i − 1, y) · e−‚m (i−1)· i , m m pim (1, 1, y) = Pim (1) · qm (0, y) · e−‚m (0)· 1 , pi (1) = pi (1, 1, y) = m m (2) prm (i, i, y) = Prm (i) · qm (i − 1, y) · e+‚m (i−1)· i , pr (i) = pr (i, i , y) = m m pt (i) = pt (i, i , y) = ptn (i, i, y) = Ptn (i) · qn (i, y) · e−‚n (i)· i . n

n

The pk (i, i, y), = i, r, t, are the mode components; qm (i, y) are their lateral profiles; and Pk (i) are their amplitudes. The secular equations between lateral mode wave number —k (i) and axial propagation constant ‚k (i) are:

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‚k2 (i) = —2k (i) − k02 ‚k2 (i)

=

—2k (i) −

k02

− —2z (i)

if the field is constant in the z direction ,

(3)

if there is a mode in the z direction .

(4)

The lateral wave numbers are solutions of the characteristic equation in duct section i. The lateral mode pressure profiles are: ⎧ ; locally reacting cos —n (i)y ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ cos —n (i)y sy (0, hi ) (5) qn (i, y) = cos (—n (i)hi ) ⎪ ⎪ ⎪ ⎪ cos ‰n (i)(y − hi − di ) ⎪ ⎩ ; bulk reacting + sy (hi , hi + di ) cos (‰n (i)di ) the lateral profiles of the axial particle velocity are: ⎧ cos —n (i)y ; locally reacting ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ cos —n (i)y sy (0, hi ) (6) qvn (i, y) = cos (—n (i)hi ) ⎪ ⎪ ⎪ ⎪ jsy (hi , hi + di ) cos ‰n (i)(y − hi − di ) ⎪ ⎩ ; bulk reacting − in Zin cos (‰n (i)di ) 1; a≤y Sect. J.18. Begin with the given Pim (1) and evaluate with those equations the transmitted mode amplitudes Ptm (1) at the first duct step. With the last relation from above, they give the incident mode amplitudes Pim (i + 1) at the next step. Repeat the evaluation until the last step.

J.20

A Section with Feedback Between Sections Without Feedback


A duct section is said to have feedback if the reflected waves from its exit influence the boundary conditions at its entrance. Neglecting feedback (because the section is long and/or its attenuation is high) simplifies the field evaluation in cascades to a straightforward computation. Feedback in all sections, on the other hand, leads to chains of systems of equations. The amount of computational work is still reasonably low if only one section is assumed to have feedback between sections without feedback.

Duct Acoustics

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673

These sketches show two examples of a duct section i with feedback between duct sections i − 1 and i + 1 without feedback. —n (i), ‚n (i), Nn (i) are, respectively, the lateral mode wave numbers, axial mode propagation constants, and mode norms in section i. The (half) duct height are hi ; i are the axial co-ordinates of the sections ( > Sect. J.18). Sound pressure and axial particle velocity conditions at the entrance i = 0 of the i-th section are: (Pim (i) + Prm (i)) · cos —n (i − 1)y m (1) Ptn (i) + Prn (i + 1) · e−‚n (i)·Li · cos —n (i)y , = n

(Pim (i) − Prm (i)) ‚n (i − 1) · cos —n (i − 1)y m = Ptn (i) − Prn (i + 1) · e−‚n (i)·Li ‚n (i) · cos —n (i)y .

(2)

n

Sound pressure and axial particle velocity conditions at the exit i = Li of the i-th section are: Ptn (i) · e−‚n (i)·Li + Prn (i + 1) · cos —n (i)y n (3) = Ptn (i + 1) · cos —n (i + 1)y , n

Ptn (i) · e−‚n (i)·Li − Prn (i + 1) ‚n (i) · cos —n (i)y n = Ptn (i + 1)‚n (i + 1) · cos —n (i + 1)y .

(4)

n

The special case of a wide section i with a locally reacting lining between narrow sections i − 1 and i + 1 with locally reacting linings (see sketch (a) above) is as follows.

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There exist two coupled linear systems of equations for the double vector of amplitudes {Ptn (i), Prn (i + 1)} (the mode coupling coefficients Rm,n (h, , ) are defined in > Sect. J.18): Rm,k (hi−1 , i−1, i)·Rm,n(hi−1 , i − 1, i) Ptn (i) ƒk,n‚k (i)hi Nk (i)+ ‚m (i−1)hi−1 Nm (i − 1) n m − Prn (i + 1)e−‚n (i)·Li n · ƒk,n ‚k (i)hi Nk (i) − =2

m

Rm,k (hi−1 , i − 1, i) · Rm,n (hi−1 , i − 1, i) ‚m (i − 1)hi−1 Nm (i − 1)

(5)

Pim (i)‚m(i − 1)hi−1 Rm,k (hi−1 , i − 1, i)

m

and n

Ptn (i)e−‚n (i)·Li

Rm,k (hi+1 , i + 1, i) · Rm,n (hi+1 , i + 1, i) ‚m (i + 1)hi+1 Nm (i + 1)

· ƒk,n ‚k (i)hi Nk (i) − m Prn (i + 1) − n Rm,k (hi+1 , i + 1, i)·Rm,n(hi+1 , i + 1, i) = 0. · ƒk,n ‚k (i)hi Nk (i)+ ‚m (i + 1)hi+1 Nm (i + 1) m

(6)

Using the solutions evaluate: Ptm (i + 1) =

1 Ptn (i) · e−‚n (i)·Li + Prn (i + 1) · Rm,n (hi+1 , i + 1, i) , Nm (i + 1) n

Prm (i) = −Pim (i) +

1 Ptn (i) + Prn (i + 1)e−‚n (i)·Li · Rm,n (hi−1 , i − 1, i) . Nm (i − 1) n

(7)

(8)

The special case of a narrow section i between wider sections i − 1 and i + 1, all with locally reacting lining (see sketch (b) above), is as follows: There exist two coupled linear systems of equations for the double vector of amplitudes {Ptn (i), Prn(i + 1)}: Rk,m (hi , i, i − 1) · Rn,m (hi , i, i − 1) Ptn (i) ƒk,nNk (i) + ‚n (i)hi ‚m (i − 1)hi−1 Nm (i − 1) n m Rk,m (hi , i, i − 1) · Rn,m (hi , i, i − 1) (9) Prn (i+1)e−‚n (i)·Li ƒk,nNk (i) − ‚n (i)hi + ‚m (i − 1)hi−1 Nm (i − 1) n m =2 Pim (i)Rk,m(hi , i, i − 1) m

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Duct Acoustics

and

Ptn (i)e−‚n (i)·Li ƒk,nNk (i) − ‚n (i)hi

n

+

Rk,m (hi , i, i + 1) · Rn,m (hi , i, i + 1) ‚m (i + 1)hi+1 Nm (i + 1)

m

Prn (i + 1) ƒk,nNk (i) + ‚n (i)hi

n

Rk,m (hi , i, i + 1) · Rn,m (hi , i, i + 1) ‚m (i + 1)hi+1 Nm (i + 1)

m

675

(10) = 0.

Using the solutions evaluate: Ptm (i + 1) =

1 ‚m (i + 1)hi+1 Nm (i + 1) Ptn (i)e−‚n (i)·Li − Prn (i + 1) ‚n (i)hi Rn,m (hi , i, i + 1), · n

Prm (i)

·

(11)

1

= Pim (i) −

‚m (i − 1)hi−1 Nm (i − 1) Ptn (i) − Prn (i + 1)e−‚n (i)·Li ‚n (i)hi Rn,m (hi , i, i − 1) .

n

Other step configurations, as in > Sect. J.18, can be treated as follows. The equations at the exit of section i do not change with feedback. The equations at the entrance are modified by: Ptn (i) → Ptn (i) + Prn (i + 1)e−‚n (i)·Li ; pressure , (12) Ptn (i)‚n (i) → Ptn (i) − Prn (i + 1)e−‚n (i)·Li ‚n (i) ; axial velocity. The systems of equations from the boundary conditions at the entrance will have the form: Ptn (i) ƒk,n · Ak + Bk,n ∓ Prn (i + 1)e−‚n (i)·Li ƒk,n · Ak − Bk,n n

=

m

( Pim (i) · Ck,m ;

n

expanding contracting

(13) .

The Ak , Bk,n , Ck,m can be taken from the corresponding systems for sections without feedback.

J.21

Concentrated Absorber in an Otherwise Homogeneous Lining


Sometimes the attenuation produced by a homogeneous lining (say, locally reacting with an admittance G) shall be improved around some (preferably low) frequency. The idea is to place some isolated resonators into the lining; the resonator orifice is the “concentrated absorber”.

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The task is treated with the method of a fictitious volume source in the orifice, having the particle velocity Va . The sound field p = pr + ps + pt which the source produces in the duct satisfies the inhomogeneous wave equation: + k02 p(x, y) = −jk0 Z0 · Va (x0 )dx0 · ƒ(x − x0 ) · ƒ(y − y0 ) (1) with y0 = h; 0 ≤ x0 ≤ a. The Dirac delta function ƒ(y − y0 ) is synthesised with modes of the homogeneously lined duct having the lateral profiles qn (y) and axial propagation constants ‚n , and the mode norms Nhn : ƒ(y − y0 ) =

cn · qn (y) ;

cn =

n

qn (y0) ; y0 · Nhn

Nhn =

1 h

h

qn2 (y) dy .

(2)

0

The sound pressure contribution dp(x, y) of the elementary source Va (x0 ) · dx0 is: jk0 Z0 Va (x0 )dx0 qn (h) · qn (y) −‚n |x−x0 | e . 2h ‚n Nhn n

dp(x, y) =

(3)

The source contributions pr , ps , pt ahead of, in front of, and after the orifice are: pr (x, y) =

jk0a qn (h) · qn (y)e+‚n x · Irn (0) ; 2 n ‚n hNhn

ps (x, y) =

jk0a qn (h) · qn (y) e−‚n x · Itn (x) + e+‚n x · Irn (x) ; 2 n ‚n hNhn

pt (x, y) =

jk0a qn (h) · qn (y)e−‚n x · Itn (a) ; 2 n ‚n hNhn

x Sect. J.22. The range of the index ns for radiating spatial harmonics can now be formulated as follows: 2a ‹ T ‹ 2a T < ns < or: − + − + ‹ < 2ns K < −‹. (5) − Š0 2K Š0 2K Š0 Š0 The incident effective sound power is: 2a 1; 2 cos Ÿ‹ |P1 | ; ƒ‹ = ¢i‹ = 2; ƒ‹ Z0

‹=0 . ‹>0

The transmitted effective sound power is:

2 2 2a Š0 2 |Dns | 1 − ‹ + 2ns K . ¢t‹ = ƒ‹ Z0 n 2a

(6)

(7)

s

The effective sound power reflected at the front side of the splitters is: ⎡ ⎤

2 2a Š0 2 2 ⎣|A0| cos Ÿ‹ + |Ans | 1 − (‹ + 2ns K)2 ⎦ . ¢r‹ = ƒ‹ Z0 2a n =0

(8)

s

Special cases: The transmission loss of the entrance plane of the splitters can be studied using L → ∞. Then Cm → 0,Dm → 0; Xm → Bm ,Ym → 0.In the system of equations of > Sect. J.22 go Fm → 0, and Am,m = −j

‚m h k0 Sm,n Sm ,n ; k 0 T n ‰n

bm = 2P1 Sm,0 .

(9)

Special case of incident plane wave ‹ = 0: Consequences: ‹=0;

Ÿ = Ÿ‹ = 0 ;

0 = 0 ;

n =

‰0 = k0 ; −n = −n ;

2n ; T 2n 2 ) ; ‰n = k0 1 − ( k0 T

‰−n = ‰n ;

(10)

Duct Acoustics

Sm,n =

Sm,0

⎧ (jk hZ G) cos(2nh/T) − (2nh/T) sin(2nh/T) 0 0 ⎪ 2 cos(—m h) ; ⎪ ⎪ ⎪ (—m h)2 − (2nh/T)2 ⎪ ⎪ ⎪ ⎨m = 0, 2, 4, . . .

⎪ ⎪ j(2nh/T) cos(2nh/T) − (k0 hZ0 G) sin(2nh/T) ⎪ ⎪ 2 sin(—m h) ; ⎪ ⎪ ⎪ (—m h)2 − (2nh/T)2 ⎩ m = 1, 3, 5, . . . ⎧ cos(—m h) ⎪ ⎪ ; m = 0, 2, 4, . . . ⎨2jk0hZ0 G · —m h = ; ⎪ ; m = 1, 3, 5, . . . ⎪ ⎩0

;

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685

(11)

(12)

Sm,−n = (−1)m · Sm,n . ∞ k0 Sm,n Sm ,n disappear in the coefficients Am,m ; therefore anti-symme‰ n=−∞ n trical waves play no role. The system of equations for the auxiliary amplitudes Xm , Ym has the coefficients 1 + e−2‚m L ‚m h k0 ƒn Sm,n Sm ,n + 2ƒm ,m (−1)m Nm , Am,m = −j k0 T n≥0 ‰n 1 − e−2‚m L

The sums

Fm

= 4(−1)m Nm

bm

= 2P1 Sm,0 .

e−‚m L , 1 − e−2‚m L

Using its solution the other amplitudes are evaluated from: j Xm · ‚m h · Sm,n ; A−n = An = ƒ0,nP1 + ‰n T m −j Ym · ‚m h · Sm,n ; D−n = Dn = ‰n T m Xm − Ym e−‚m L Xm e−‚m L − Ym −‚m L ; C = e . m 1 − e−2‚m L 1 − e−2‚m L The sound fields in the zones then follow from: ∞ √ 2 2n pI (x, y) = 2P1 e−jk0 x + 2 ƒn · An ejk0 x 1−(nŠ0 /T) · cos y , T n=0 ∞ √ 2n

2 ƒn · Dn e−jk0 x 1−(nŠ0 /T) · cos pIII (x , y) = 2 y , T n=0 pII (x, yŒ ) = 2 [Bm e−‚m x + Cm e+‚m x ] · cos —m yŒ .

(13)

(14)

Bm =

(15)

m

(The number K of the splitters in the main duct has disappeared, as expected.) The range of radiating spatial harmonics is: T T − < ns < . (16) Š0 Š0

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The expressions for the effective sound powers simplify to: ¢i = 2a|P21 |/Z0 , 2a ¢r = ƒns · |Ans |2 · 1 − (ns Š0 /T)2 , Z0 n s

(17)

2a ƒns · |Dns |2 · 1 − (ns Š0 /L)2 . ¢t = Z0 n s

Approximations:

Neglect the reflections at the splitter duct exit, i. e. set everywhere Cm = 0. The consequence is Xm = Bm ;

Ym = Bm · e−‚m L = Xm · e−‚m L .

This simplifies the system of equations with the following coefficients: ‚m h k0 Am,m = −j Sm,n Sm ,n + 2ƒm,m (−1)m Nm , k 0 T n ‰n

(18)

(19)

bm = 2P1 Sm,0 . The amplitudes An are evaluated as before, and the Dn follow from: ∞

k0 h ‚m Xm e−‚m L Sm,n . Dn = −j ‰n T m=0 k0

(20)

Assume only a single mode in the splitter duct (usually the least attenuated mode): The Xm , Ym follow from the two equations: Am,m · Xm + Fm · Ym = bm ,

(21)

Fm · Xm + Am,m · Ym = 0 with the coefficients: ‚m h k0 2 S + 2(−1)m Nm coth (‚m L) , Am,m = −j k0 T n ‰n m,n 1 , Fm = 2(−1)m+1 Nm sinh (‚m L)

(22)

bm = 2P1 Sm,0 , or, as explicit solutions: Xm =

Am,m bm ; A2m,m − F2m

Ym = −

Fm bm , A2m,m − F2m

(23)

Duct Acoustics

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687

and the amplitudes in the main duct from: An = ƒ0,nP1 + j

h ‚m Sm,n · Xm ; T ‰n

Dn = −j

h ‚m Sm,n · Ym . T ‰n

(24)

Assume only a single mode in the splitter duct and neglect its reflection at the splitter duct exit: Then, using the coefficients from above: Xm = Bm =

bm ; Amm + Fm e−‚m L

An = ƒ0,nP1 + j

h ‚m Sm,n · Xm ; T ‰n

(25) Dn = −j

h ‚m Sm,n · Xm · e−‚m L . T ‰n

(26)

Assume an incident plane wave (‹ = 0), and neglect reflections at the splitter duct exit. The simplified system of equations for the Xm from above has the coefficients: Am,m = 2ƒm,m (−1)m Nm − j

∞

‚m h k0 ƒn Sm,n Sm ,n ; k0 T n=0 ‰n

bm = 2P1 Sm,0 ,

(27)

and the amplitudes in the main duct are: j Xm · ‚m h · Sm,n , ‰n T m ‚m k0 h D−n = Dn = −j Xm e−‚m L Sm,n . ‰n T m k0

A−n = An = ƒ0,nP1 +

[

(28)

]

Transmission loss of a splitter silencer; points: measured; solid: theory; dash-dotted: propagation loss of the least attenuated mode; dashed: loss by reflection at the splitter duct entrance

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Sound pressure level in front of, within, and behind a splitter silencer with two splitters in a main duct

The splitters in the numerical example shown consist of layers of mineral fibres with a flow resistivity of ¡ = 11 [kPa · s/m2 ], covered with a porous foil of surface mass density m

= 0.2 [kg/m2 ] and a flow resistance Zser = 1 · Z0 . The points are measured transmission loss values for plane wave incidence; the full curve is evaluated as explained above; the dash-dotted curve represents the propagation loss of the least attenuated mode of the splitter duct; the dashed curve shows the loss by reflection at the entrance of the splitter ducts.

J.24

Splitter Type Silencer with Simple Porous Layers as Bulk Reacting Splitters


Splitters consist of simple porous layers with characteristic values of the material a , Za , or, in normalised form, an = a /k0, Zan = Za /Z0 . The heads of splitters are open. Splitters are not sound transmissive from one splitter duct to the neighbouring splitter duct, either due to a sufficiently high flow resistance of the splitter or due to a central partition (the condition is not necessary for plane wave incidence parallel to the x axis). H = 2h; D = 2d. The incident wave is a mode of the hard main duct; it is composed of two plane waves p1 + p2 with mirror-reflected incidence under the modal angle Ÿ → Ÿ‹ : p1 (x, y) = P1 · e−jkx x · e−jky y ; ky → ky‹ = k0 sin Ÿ‹ ; sin Ÿ‹ =

ky‹ ‹Š0 . = k0 2a

p2 (x, y) = P1 · e−jkx x · e+jky y ;

kx → kx‹

‹ = k0 cos Ÿ‹ = k02 − ( )2 ; a

(1)

Duct Acoustics

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689

The backscattered wave ps in Zone I and the transmitted wave pt in Zone III are sums of spatial harmonics; thus the fields in these zones are formulated as follows: −jk0 x cos Ÿ‹

pI‹ (x, y) = 2P1 e

· cos(k0y sin Ÿ‹ ) + 2

+∞

An ej‰n x · cos(n y) ,

(2)

n=−∞

pIII‹ (x , y) = 2

+∞

Dn e−j‰n x · cos(n y)

(3)

n=−∞

with n = 0 +

2n ; T

n = 0, ±1, ±2, . . . ;

‰0 = kx = k0 cos Ÿ‹ ;

‰n =

0 = ky = k0 sin Ÿ‹ ;

k02 − 2n = k0 1 − (sin Ÿ‹ + nŠ0 /T)2 .

(4) (5)

The field in the Œ-th splitter duct is a sum of splitter duct modes: pII‹ (x, yŒ ) = 2 cos(Œ0 T)

∞

[Bm e−‚m x + Cm e+‚m x ] · qm (yŒ )

(6)

m=0

with lateral mode profiles: ⎧ cos(—m y) ⎪ ⎪ + s|y| (h, h + d) ⎪s|y| (0, h) · ⎪ ⎪ cos(—m h) ⎪ ⎪ ⎪ ⎪ ⎪ cos(m (y − h − d)) ⎪ ⎪ ; m = 0, 2, 4 . . . ⎪ ⎨· cos(m d) qm (y) = ⎪ sin(—m y) ⎪ ⎪ + s|y| (h, h + d) s|y| (0, h) · ⎪ ⎪ ⎪ sin(—m h) ⎪ ⎪ ⎪ ⎪ ⎪ y cos(m (y − h − d)) ⎪ ⎪ · ; m = 1, 3, 5 . . . ⎩· |y| cos(m d)

(7)

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690

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1; which use the “switch function” sy (a, b) = 0;

a≤y

Sect. J.23,

Splitter-Type Silencer with Splitters of Porous Layers Covered with a Foil


The arrangement and sound incidence are as in the previous > Sect. J.24, but the splitters are covered with a (poro-elastic) foil having a partition impedance Zs . The field formulations in Zones I and III remain as in duct and in the porous layer is formulated as: pII‹ (x, yŒ ) = 2 cos(Œ0 T)

∞

> Sect. J.24. The field in a splitter

[Bm e−‚m x + Cm e+‚m (x−L) ] · qm (yŒ )

(1)

m=0

with the following mode profiles: qm (y) = ⎧ s (0, h) · cos(—m y) + s|y| (h, h + d) · bm cos(m (y − h − d)); m = 0, 2, 4 . . . ⎪ ⎨ |y| = y ⎪ ⎩s|y| (0, h)·sin(—m y)+s|y| (h, h + d)· ·bm cos(m (y − h − d)); m = 1, 3, 5 . . . |y|

(2)

(sy (a,b) is the switch function as in > Sect. J.24). The wave equations in the splitter duct and in the porous layer imply: 2 , (3) (m h)2 = (—m h)2 − (k0 h)2 1 + an and either the —m h or the m h are solutions of: * * jm h · sin(m d) * —m h · sin(—m h) * an Zan * * * * k0h · cos(—m h) + jZs —m h · sin(—m h) −k0 h · cos(m d) * Z0

* * * * * *=0; (4) * * * m = 0, 2, 4, . . . ,

Duct Acoustics

* * * —m h · cos(—m h) * * * * * k0h · sin(—m h) − jZs —m h · cos(—m h) * Z

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693

* * * * * * = 0 ; m = 1, 3, . . . . (5) * −k0 h · cos(m d) **

−jm h · sin(m d) an Zan

0

The amplitudes Bm , Cm are solutions of the following coupled systems of linear equations: ‚m j Bm Tm,n + Qm,n − j Qm,n Tm,n + ‰n Z m an an (6) ‚ j m −‚m L + Cm e Tm,n + Qm,n + j Tm,n + = 2ƒ0,n · P1 , Qm,n ‰n an Zan ‚m j −‚m L Bm e Qm,n Tm,n + Qm,n + j Tm,n + ‰ an Zan m n (7) ‚m j Qm,n + Cm Tm,n + Qm,n − j Tm,n + =0 ‰n an Zan with mode coupling coefficients: ( Sm,n cos(—m h) ; m = 0, 2, 4 . . . Tm,n = , Sm,n sin(—m h) ; m = 1, 3, 5 . . . Qm,n = Rm,n bm cos(md) ;

m = 0, 1, 2 . . . ,

where ⎧ —m sin(—m h) ⎪ ⎪ ; m = 0, 2, 4, . . . ⎨jan Zan m sin(m d) bm = . ⎪ — cos(—m h) ⎪ ⎩−jan Zan m ; m = 1, 3, 5, . . . m sin(m d)

(8) (9)

(10)

Using the solutions Bm , Cm the other amplitudes follow as: Bm + Cm e−‚m L Tm,n + Qm,n , An = −ƒ0,n · P1 + m

Dn =

Bm e−‚m L + Cm Tm,n + Qm,n .

(11)

m

J.26

Lined Duct Corners and Junctions


See also

> Sect. J.41 about TV splitters.

Two lined ducts i = 1, 2 form a corner. The corner walls i = 3, 4 opposite the ducts are lined, too. All linings are supposed to be locally reacting (for ease of formulation, mainly) with surface admittances Gi .

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Let the incident wave pi be the ‹-th mode of the duct i = 1. Each of the corner linings with G3, G4 , when mirror-reflected at the y axis and x axis, respectively, will form a fictitious lined duct i = 3, 4. The reflected wave pr in the duct i = 1 is formulated as a mode sum of that duct, and the transmitted wave pt in the duct i = 2 is formulated as a mode sum of the duct i = 2. The scattered waves ps3 , ps4 in the corner area are written as mode sums of the fictitious ducts i = 3, 4. The inlet duct i = 1, the exit duct i = 2, and the corner area form Zones I, II, and III, respectively, of the sound field. The sound fields in the zones are: pI (x, y) = pi (x, y) + pr (x, y) , pII (x, y) = pt (x, y) ,

(1)

pIII (x, y) = ps3 (x, y) + ps4 (x, y) with the formulations of the component fields: pi (x, y) = Pi · q1‹ (x) · e−‚1‹ (y+h2 ) , Am · q1m (x) · e+‚1‹ (y+h2 ) , pr (x, y) = m

pt (x, y) =

n

ps3 (x, y) =

ps4 (x, y) =

Dn · q2n (y) · e−‚2n (x−h1 ) , B · q3 (x) · e−‚3 (y+h2 ) + R · e+‚3 (y+h2 ) ,

(2)

C · q4 (y) · e+‚4 (x−h1 ) + R · e−‚4 (x−h1 ) ,

where the qik (z) are symmetrical and anti-symmetrical mode profiles: cos(—ik z) ; k = 0, 2, 4, . . . ; symm. qik (z) = sin(—ik z) ; k = 1, 3, 5, . . . ; anti-symm.

(3)

Duct Acoustics

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695

with the axial propagation constants ‚ik from the wave equation ‚ik2 = —ik2 − k02 and the lateral wave numbers —ik hi solutions of the characteristic equation:

(4)

—ik hi · qik (—ik hi ) = −jk0 hi · Z0 Gi · qik (—ik hi ) .

(5)

The mode norms are:

1 Nik : = 2hi

hi −hi

qik2 (†i ) d†i =

1 sin(2—ik hi ) . 1 + (−1)k 2 2—ik hi

(6)

R , R are modal reflection factors of the corner linings G4, G3 , respectively,“measured” in the orifice planes of the ducts i = 1, 2; they follow from the reflection factors r , r at the lining surfaces by: R = r · e−4‚3 h2 ; −4‚4 h1

R = r · e

;

r =

j‚3 /k0 + Z0 G4 j‚3 h2 + U4 = , j‚3 /k0 − Z0 G4 j‚3 h2 − U4

j‚4/k0 + Z0 G3 j‚4 h1 + U3 r = = . j‚4/k0 − Z0 G3 j‚4 h1 − U3

(7)

Matching of the sound fields at the zone limits leads to a coupled system of linear equations for the amplitudes B , C : B · S,k ‚3 h2 (1 − R ) + ‚1k h2 (1 + R ) + (8) C · (−1) q4 (h2 ) ‚1k h2 − jU4 e−‚4 h1 · Ia,k + R e+‚4 h1 · Ib,k

= 2ƒ‹,k · Pi · ‚1‹ h2 N1‹ , B · q3 h1 ‚2k h1 − jU3 e−‚3 h2 · IB,k + R e+‚3 h2 · IA,k + C · T,k ‚4h1 1 − R + ‚2k h1 1 + R = 0 .

(9)

These contain the following mode coupling coefficients: h1 1 S,k : = q3 (x) · q1k (x) dx = 0 ; + k = odd 2h1 −h1 sin (—3 + —1k )h1 1 sin (—3 − —1k )h1 (+k)/2 + (−1) · ; + k = even = 2 (—3 − —1k )h1 (—3 + —1k )h1

(10)

h2 1 T,k : = q4 (y) · q2k (y) dy = 0 ; + k = odd 2h2 −h2 sin (—4 + —2k )h2 1 sin (—4 − —2k )h2 (+k)/2 ; + k = even + (−1) · = 2 (—4 − —2k )h2 (—4 + —2k )h2

(11)

with Sk → ƒk · N1k if the lining i = 3 agrees with the lining i = 1, and T,k → ƒ,k · N2k if the lining i = 4 agrees with the lining i = 2.

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Further needed are the integrals: Ia,k

1 := 2h1

IA,k : =

1 2h2

h1

+‚4 x

e −h1 h2

· q1k (x) dx ;

e+‚3 y · q2k (y) dy ;

−h2

1 Ib,k : = 2h1 IB,k : =

h1 −h1 h2

1 2h2

e−‚4 x · q1k (x) dx ; (12) e−‚3 y · q2k (y) dy ,

−h2

which are easily evaluated with the mode profile functions qik (z).With the solutions B , C the other amplitudes follow as: 1 Ak = − ƒ‹,k · Pi · N1‹ N1k ⎤ (13) −‚4 h1 +‚4 h1 + B (1+R )·S,k + C · q4 (−h2 ) e · Ia,k + R e · Ib,k ⎦ ,

Dk =

1 B q3 (h1 ) e−‚3 h2 · IB,k + R e+‚3 h2 · IA,k N2k + C · 1 + R · T,k .

(14)

Sound pressure level in two ducts and their corner. The entrance duct i = 1 and the corner walls are hard; only the exit duct i = 2 is lined. Because the standing wave pattern in the corner agrees well with the first higher mode pattern in the exit duct, this higher mode is predominantly excited, and a high extra corner transmission loss is produced (as compared with the least attenuated mode propagation loss)

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The linings in the numerical examples shown are simple glass fibre layers of thickness di with flow resistivity ¡i , made locally absorbing (if a duct or corner wall is hard, then di = 0 and no ¡ value is given).

Sound pressure level in two lined ducts and their corner. Both ducts and the corner walls are equally lined. Input parameters: f = 2000[Hz]; ‹ = 0; mhi = 8; h1 = h2 = 0.2[m]; d1 = d2 = d3 = d4 = 0.1[m]; ¡1 = ¡2 = ¡3 = ¡4 = 10[kPas/m2 ] T-joints and cross-joints of ducts can be approximately evaluated with the present method if one or both corner wall linings are given a lining admittance Gi = 1/Z0. See references for a more precise method.

J.27

Sound Radiation from a Lined Duct Orifice

See also: Mechel, Vol. III, Ch. 39 (1998); Mechel, Mathieu Functions (1997)

A two-dimensional, flat duct of width 2 h with locally reacting lining of surface admittance G has its orifice in a hard baffle wall. The ‹-th duct mode p‹ is incident on the orifice. Cartesian co-ordinates x, y are used inside the duct; outside the duct an elliptic–hyperbolic system of co-ordinates , ˜ is applied. The orifice is in the plane x = 0 and = 0. The reflected wave pr inside the duct is composed of duct modes; the radiated field ps is formulated as a sum of azimuthal and radial Mathieu functions cem (˜ ), Hc(2) m (). Field formulations: j‚‹ p‹ (x, y) = P‹ · cos(—‹ y) · e−‚‹ x ; v‹x = −P‹ cos(—‹ y) · e−‚‹ x ; k0 Z0 (1) ‚‹2 = —2‹ − k02 ;

698

J

pr (x, y) =

Duct Acoustics

An · cos(—n y) · e+‚n x ;

vnx (0, y) =

n

ps (, ˜ ) =

Dm · Hc(2) m () · cem (˜ ) ;

j‚n h An · cos(—n y) ; k0hZ0

(2)

m

* j ∂ps,m ** vm (0, ˜ ) = vmx (0, ˜ ) = k0Z0 · h sin ˜ ∂ *=0 j Dm · Hc (2) = m (0) · cem (˜ ) . k0Z0 · h sin ˜

(3)

Both the duct modes and the azimuthal Mathieu functions are orthogonal with norms: 1 Nn : = h 0

h

qn2 (y) dy = 1 +

−h

ce2m (˜ ) d˜ = . 2

sin(2—n h) ;, 2—n h

(4)

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699

Coupling coefficients: Rm,n : =

cem (˜ ) · cos(—n h · cos ˜ ) d˜ =

A2s · (−1)s J2s (—n h) ,

(5)

s≥0

0

where Jk (z) is a Bessel function and A2s are Fourier coefficients for the representation of cem (˜ ) ( > Sect. J.15). The field matching in the orifice gives a system of linear equations for the An : (0) 2 Hc (2) m Rm,k · Rm,n An · ƒn,k · ‚k hNk − (2) Hc m (0) n m 2 Hc (2) m (0) = P‹ · ƒ‹,k · ‚‹ hN‹ + Rm,k · Rm,‹ m Hc(2) m (0)

(6)

(a prime indicates the derivative). Using its solutions the Dm can be determined from: 2 (7) Dm = ƒ‹,nP‹ + An · Rm,n . (2) Hcm (0) n A “radiation loss”can be defined by L = −10·lg(‘‹ ) with the transmission coefficient ‘‹ being the ratio of the radiated effective power ¢ s to the effective incident power ¢ ‹ of the ‹-th duct mode: h ∗ 1 j‚‹ 2 p‹ (0, y) · = |P‹ | | cos(—‹ y)|2 dy 2 k0 Z0 −h −h ∗

sin(2—‹ h) sinh(2—‹ h) 1 j‚‹ h = |P‹ |2 + , 2 k0Z0 2— ‹ h 2—

‹ h

1 ¢‹ = 2

¢s =

=

1 2 h 2

h

h −h

∗ v‹x (0, y) dy

(8)

∗ ps (0, y) · vsx (0, y) dy

∗ ps (0, ˜ ) · vs (0, ˜ ) sin ˜ d˜

(9)

0

= |Dm |2 Yc m (0) · Jcm (0) − jYcm (0) , 4k0Z0 m where Jcm (z), Ycm (z) are Mathieu–Bessel and Mathieu–Neumann functions associated with cem (˜ ). The transmission coefficient is thus (writing —n = — n +j—

n ; ‚n = ‚n +j‚n

; ‚n∗ = ‚n − j‚

): |Dm |2 Yc m (0) · Jcm (0) 2 |P | ‹ ¢ s m . (10) ‘‹ = =

¢‹ 2‚‹ h sin(2— ‹ h) sinh(2—

‹ h) + 2— ‹ h 2—

‹ h

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Duct Acoustics

In the special case of a hard duct: ƒ‹

‘‹ = 4 (k0h)2 − (‹)2 (with ‚‹ h =

|Dm |2 m

|P‹

Yc (0) |2 m

· Jcm (0) ;

ƒ‹ =

1; 2;

‹=0 ‹>0

(11)

(‹)2 − (k0 h)2 ).

3D plot of sound pressure level inside and behind a lined duct

Approximate determination of radiation loss L: The radiation loss L of the orifice (in a baffle wall) of a lined duct can be evaluated approximately by: L = −10 · lg(1 − |R|2 ) ; Z‹ = jk0 Z0 /‚‹ ,

R=

Zr − Z‹ ; Zr + Z‹

(12)

where Z‹ is the axial wave impedance of the incident ‹th mode and Zr is the radiation impedance either of a piston radiator with an area equal to the orifice area (if ‹ belongs to the lowest or least attenuated mode) or of a cylindrical radiator with radius a = 2 h/ (see > Sects. F.4, F.7 for radiation impedances).

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701

Radiation loss L of duct orifice, lined with a layer of glass fibres, covered with a resistive foil having a flow resistance Zf . solid line: with a piston radiator; long dash: with a cylindrical radiator; short dash: with elliptic co-ordinates

Radiation loss L of duct orifices of a hard duct (solid) and of a lined duct (dashed) for different radiation angles §. Evaluated with the radiation impedance Zr of a hemi√ spherical radiator with radius a = S/§ (S = orifice area)

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Duct Acoustics

The radiation loss depends on the volume angle § into which the orifice radiates (§ = 4: free space; § = 2: orifice in a baffle wall; § = : orifice in the corner of two walls; § = /2: orifice in the corner of three walls).

J.28

Conical Duct Transitions; Special Case: Hard Walls

See also: Mechel, J. Sound Vibr. 216, pp. 649–671 (1998)

It seems appropriate to describe the sound field in conical (wedge-shaped) duct transitions in cylindrical co-ordinates r, ˜ because the flanks would then be on co-ordinate surfaces and to separate modes in them as p(r, ˜ ) = T(˜ ) · R(r) with azimuthal profiles: cos(†˜ ) ; symmetrical T(˜ ) = (1) sin(†˜ ) ; anti-symmetrical for symmetrical or anti-symmetrical distributions relative to the x axis. Such modes would be orthogonal over ˜ and, therefore, would be suitable for modal analysis of the fields. If the lining of the cone is locally reacting with a surface admittance G, the azimuthal wave numbers † have to be solutions of the equation: (†Ÿ) · tan(†Ÿ) ; symmetrical (2) jk0 r · ŸZ0G = −(†Ÿ) · cot(†Ÿ) ; anti-symmetrical . In general, † = †(r), and this prevents a separation of the mode into factors depending on only one co-ordinate.

Special cases with separation are: • G = 0 (i. e. hard flank): m ; symmetrical †Ÿ = (m + 1/2) ; anti-symmetrical

; m = 0, 1, 2, . . . ,

(3)

Duct Acoustics

• G = ∞ (i. e. soft flank): (m + 1/2) ; symmetrical †Ÿ = m ; anti-symmetrical

;

J

703

m = 0, 1, 2, . . . ,

(4)

• G = G(r) ∼ 1/r: †Ÿ = const(r), • parallel walls (in distance 2 h) with characteristic equation: (†h) · tan(†h) ; symmetrical jk0h · Z0 G = . −(†h) · cot(†h) ; anti-symmetrical

(5)

(6)

In these cases the radial part of the wave equation becomes the Bessel differential equation:

d2 1 †2 d 2 R(k0r) = 0 ; ‰2 = 1 − (kz /k0)2 , + ‰ + − d(k0 r)2 k0 r d(k0 r) (k0 r)2

(7)

where kz = 0 if in the z direction a field variation with cos(kz z), sin(kz z), e±jkz z or a linear combination thereof exists. Thus, if ‰ = 1, the radial factors R(k0 r) are Bessel, Neumann, and Hankel functions of order †. This section further deals with the first special case of hard flanks; the next sections will present methods for the evaluation of sound fields in lined cones. Both the entrance duct with height 2 h1 and the exit duct with height 2 h2 are assumed to be hard also (if they are lined, mainly the lateral mode wave numbers in the ducts —i,n have to be solutions of the characteristic equations in these ducts). The terminating ducts are infinite; the ‹th mode of the entrance duct is the incident wave pi ; it is assumed to be a symmetrical mode. The reflected wave pr in the entrance duct, the transmitted wave pt in the exit duct, and the field pc in the cone are formulated as mode sums. The fields are matched with respect to their pressures and radial particle velocities vr = vx · cos ˜ + vy · sin ˜ at the arcs ri ; i = 1, 2; with hi = ri · sin Ÿ. On these arcs is: x = ri · cos ˜ ; y = ri · sin ˜ . Field formulations: pi (x, y) = Pi · cos (—1‹ y) · e−‚1‹ x ; pr (x, y) =

An · cos (—1n y) · e+‚1n x ;

—in hi = n · ;

‚in2 = —2in − k02 ;

n≥0

pt (x, y) =

n≥0

pc (x, y) =

(8)

Dn · cos(—2n y) · e−‚2n x ; cos(†m ˜ ) · Bm · J†m (k0r) + Cm · Y†m (k0 r) ;

†m Ÿ = m · ;

m≥0

1 Ncm : = 2Ÿ

( +Ÿ 1; m=0 sin (2†mŸ) 1 1 2 1+ = cos (†m ˜ ) d˜ = ; ƒm = . 2 2†mŸ ƒm 2; m>0

−Ÿ

(9)

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Duct Acoustics

Introduce the integrals (i = 1, 2): (±) Iin,k (ri )

(±) (ri ) Jin,k

1 := 2Ÿ 1 := 2Ÿ

(±) Kin,k (ri ) : =

1 2Ÿ

+Ÿ cos(—in ri sin ˜ ) · e±‚in ri cos ˜ · cos(†k ˜ ) d˜ , −Ÿ

+Ÿ cos(—in ri sin ˜ ) · e±‚in ri cos ˜ · cos(†k ˜ ) · cos ˜ d˜ ,

(10)

−Ÿ

+Ÿ sin(—in ri sin ˜ ) · e±‚in ri cos ˜ · cos(†k ˜ ) · sin ˜ d˜ . −Ÿ

The integrals must be evaluated by numerical integration. (Because the integrands are even in ˜ , they can be evaluated as 1 2Ÿ

+Ÿ +Ÿ 1 . . . d˜ = . . . d˜ .) Ÿ

−Ÿ

0

Application of the operator

1 2Ÿ

+Ÿ . . . · cos(†k ˜ ) d˜

(11)

−Ÿ

on the boundary condition for the sound pressure pi (r1, ˜ ) + pr (r1 , ˜ ) = pc (r1 , ˜ ) on the arc with r1 gives the following system of equations: (−) (+) Pi · I1‹,k (r1 ) + An · I1n,k (r1 ) = Nck Bk · J†k (k0r1 ) + Ck · Y†k (k0 r1 ) . (12) n≥0

The same operator applied to the boundary condition for the radial particle velocity at r1 leads to (the prime indicates the derivative): —1‹ (−) −‚1‹ (−) −—1n (+) ‚1n (+) J1 (r1 ) − K1 (r1 ) + An · J1 (r1 ) − K1n,k (r1 ) Pi · k0 ‹,k k0 ‹,k k0 n,k k0 n≥0 . (13) An · an,k = Nck Bk · J †k (k0r1 ) + Ck · Y †k (k0 r1 ) . :=Pi · b‹,k + n≥0

On the arc with r2 drop terms with Pi as factor; substitute r1 → r2 ; An → Dn ; —1n → —2n ; ‚1n → −‚2n ; and substitute the integrals correspondingly; this gives: Dn · I2(−) n,k (r2 ) = Nck Bk · J†k (k0 r2 ) + Ck · Y†k (k0 r2 ) , (14) n≥0

−—2n (−) −‚2n (−) J2n,k (r2 ) − K2n,k (r2 ) k0 k0 n≥0 . Dn · dn,k = Nck Bk · J †k (k0 r2 ) + Ck · Y †k (k0 r2) . :=

Dn ·

n≥0

(15)

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Solve these two equations at r2 (with fixed but arbitrary integer k ≥ 0) for Bk , Ck : k0 r2

Dn · I2(−) n,k (r2 ) · Y†k (k0 r2 ) − dn,k · Y†k (k0 r2 ) , 2Nck n≥0 k0 r2

Ck = − Dn · I2(−) (r ) · J (k r ) − d · J (k r ) . 2 0 2 n,k † 0 2 k †k n,k 2Nck n≥0

Bk =

(16)

This inserted into the equations at r1 leads to two coupled systems of linear equations for the sets An , Dn of amplitudes:

(+)

An · I1n,k (r1 ) +

n≥0

k0 r2 Dn 2 n≥0

· dn,k · J†k (k0 r1)

· Y†k (k0r2 ) − J†k (k0r2 ) · Y†k (k0r1 )

(17)

.

+ I2(−) = −Pi · I1(−) n,k (r2 ) · J†k (k0 r2 ) · Y†k (k0 r1 ) − J†k (k0 r1 ) · Y†k (k0 r2 ) ‹,k (r1 ) , k0 r2 Dn · dn,k · J †k (k0r1 ) · Y†k (k0r2 ) − J†k (k0 r2 ) · Y †k (k0 r1 ) 2 n≥0 n≥0 (18) . (−)

+ I2n,k (r2 ) · J†k (k0 r2 ) · Y†k (k0 r1) − J†k (k0r1 ) · Y†k (k0r2 ) = −Pi · b‹,k .

An · an,k +

The Bk , Ck can be evaluated with the solutions Dn . The sound field is determined.

J.29

Lined Conical Duct Transition, Evaluated with Stepping Duct Sections

See also: Mechel, J. Sound Vibr. 216, pp. 673–696 (1998)

This section makes use of the last special case in > Sect. J.28, i. e. it composes a lined (locally reacting) duct cone with stepping duct sections having parallel walls. All duct sections may have equal linings (for ease of representation) with surface admittance G. The duct section i = 0 is the entrance duct (infinitely long); it sends a mode mix to the stepping sections. The last section i = I is terminated with an admittance Gt . The heads of the steps are assumed to be hard. In each section the forward wave pe,i and the backward wave pr,i are written as sums of modes of that section. The fields are matched at the section limits with their sound pressure and axial particle velocity. Field formulation: pe,i (x, y) = Ai,m · qi,m (y) · e−‚i,m (x−xi−1 ) , m≥0

pr,i (x, y) =

m≥0

Bi,m · qi,m (y) · e+‚i,m (x−xi )

(1)

J

706

Duct Acoustics

with lateral mode profiles: ( cos(—i,m y) ; symmetrical qi,m (y) = sin(—i,m y) ; anti-symmetrical.

(2)

The axial propagation constants ‚i,m are obtained by: 2 = —2i,m − k02 ; ‚i,m

Re{‚i,m } ≥ 0

(3)

from the lateral wave numbers, and these in turn are solutions of the characteristic equations: (—i,m hi ) · tan(—i,m hi ) = jk0 hi · G ; (—i,m hi ) · cot(—i,m hi ) = −jk0 hi · G ;

symmetrical , anti-symmetrical .

(4)

The mode norms are: 1 hi

hi qi,m (y) · qi,n (y) dy = ƒm,n · Ni,m ; 0

Ni,m =

sin(2—i,m hi ) 1 , 1± 2 2—i,m hi

(5)

and the mode coupling coefficients C(i, m; k, n) of the mode of order m in section i with the mode of order n in section k are: 1 hi

hi qi,m (y) · qk,n (y) dy = C(i, m; k, n) , 0

1 C(i, m; k, n) = 2

sin (—i,m − —k,n )hi sin (—i,m + —k,n )hi ± (—i,m − —k,n )hi (—i,m + —k,n )hi

(6)

Duct Acoustics

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707

(± for symmetrical or anti-symmetrical modes,respectively; cross-coupling coefficients between both types of symmetry are zero). By the special termination of the wedge with an admittance Gt one has: 0 0 / 0 / 0 / / (7) BI,m = {Mt } ◦ AI,m = rm · e−‚I,m x ◦ AI,m , where {Mt } is a general coupling matrix (◦ is the symbol for matrix multiplication), which in the present case is a diagonal matrix with the values rm · e−‚I,m x on the main diagonal, rm being the modal reflection factors at the exit of the last section i = I: gm − Gt j‚I,m /k0 + Gt = , (8) rm = gm + Gt j‚I,m /k0 − Gt and gm the normalised axial modal admittances of the modes of pe,I (x, y): ‚I,m vxeI,m (xI ) = −j . (9) gm = Z0 peI,m (xI ) k0 Converging cone: The boundary condition for the sound pressure at the entrance x = xi−1 of the i-th section (i ≥ 1) is: !

pe,i−1 (xi−1 , y) + pr,i−1 (xi−1 , y) = pe,i (xi−1 , y) + pr,i (xi−1 , y) ; 0 ≤ y ≤ hi ,

! Ai−1,m · e−‚i−1,m x + Bi−1,m · qi−1,m (y) = Ai,m + Bi,m · e−‚i,m x · qi,m (y) .

m

(10)

m

Application of the operation

1 hi

hi . . . · qi,m (y) dy on both sides gives the following sys0

tem of equations: Ai−1,n · e−‚i−1,n x + Bi−1,n · C(i, m; i − 1, n) . Ai,m + Bi,m · e−‚i,m x · Ni,m =

(11)

n

This is an upward iteration scheme. The boundary condition for the axial particle velocity at x = xi−1 is: ! 0 ; hi ≤ y ≤ hi−1 , vxe,i−1 (xi−1 , y) + vxr,i−1 (xi−1 , y) = vxe,i(xi−1 , y) + vxr,i (xi−1 , y) ; 0 ≤ y ≤ hi ! Ai−1,m · e−‚i−1,m x − Bi−1,m · ‚i−1,m · qi−1,m (y) = (12) m ⎧ ; hi ≤ y ≤ hi−1 ⎨0 −‚i,m x . Ai,m − Bi,m · e · ‚i,m · qi,m (y) ; 0 ≤ y ≤ hi ⎩ m

1 Application of the operators hi

hi . . . · qi,m (y) dy left ; 0

1 hi

hi−1 . . . · qi,m (y) dy right 0

produces the following system of equations: hi−1 Ai,m − Bi,m · e−‚i,m x · ‚i,m · Ni,m = Ai−1,n · e−‚i−1,n x − Bi−1,n hi n · ‚i−1,n · C(i − 1, n; i, m) .

(13)

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Duct Acoustics

One gets, by combination of both systems of equations: 1 ‚i−1,nhi−1 −‚i−1,n x Ai,m = Ai−1,n · e · C(i, m; i − 1, n) + C(i − 1, n; i, m) 2Ni,m n ‚i,m hi ‚i−1,nhi−1 + Bi−1,n · C(i, m; i − 1, n) − C(i − 1, n; i, m) , ‚i,m hi 1 Bi,m = Ai−1,n ·e−‚i−1,n x 2Ni,m ·e−‚i,m x n ‚i−1,nhi−1 · C(i, m; i − 1, n) − C(i − 1, n; i, m) ‚i,m hi ‚i−1,nhi−1 + Bi−1,n · C(i, m; i − 1, n) + C(i − 1, n; i, m) . ‚i,m hi

(14)

The upward iterations begin at i = 1, where Ai−1,m = A0,m have given numerical values and B0,m are unknown symbols. At any step i of the iteration one will have systems of equations of the form: ai,n + bi,n · B0,n ; Bi,m = i,n + i,n · B0,n (15) Ai,m = n

n

with numerical ai,n , bi,n , i,n , i,n . The iteration ends with i = I, where on the left-hand sides of the iterative equations stand {AI,m } and {BI,m }, which, with the above relation of reflection, reduce to only the {AI,m } as yet unknown amplitudes. Thus the equations for i = I are two linear systems of equations in the two sets of amplitudes {AI,m }, {B0,n }, and they are inhomogeneous systems of equations because of the numerical terms aI,n , I,n . After they are solved for {B0,n }, all amplitudes {Ai,m }, {Bi,m } can be evaluated by insertion. The described iteration with mixed numerical and symbolic expressions can easily be performed with Mathematica or other computer programs for both numerical and symbolic mathematics. Diverging cone: One gets in a similar way the following two downward iterative systems of equations: 1 ‚i,n hi · Ai,n · C(i − 1, m; i, n) + · C(i, n; i − 1, m) Ai−1,m = 2Ni−1,m · e−‚i−1,m x n ‚i−1,m hi−1 ‚i,n hi + Bi,n · e−‚i,n x · C(i − 1, m; i, n) − · C(i, n; i − 1, m) , ‚i−1,m hi−1 1 ‚i,n hi Bi−1,m = · Ai,n · C(i − 1, m; i, n) − · C(i, n; i − 1, m) 2Ni−1,m n ‚i−1,m hi−1 ‚i,n hi · C(i, n; i − 1, m) . + Bi,n · e−‚i,n x · C(i − 1, m; i, n) + ‚i−1,m hi−1 (16)

Duct Acoustics

If one begins the iteration with i = I, the equations have the form: AI−1,m = bI,n · AI,n ; BI−1,m = I,n · AI,n n

709

(17)

n

with still unknown amplitudes {AI,n }, and in the general step i: Ai−1,m = bi,n · AI,n ; Bi−1,m = i,n · AI,n n

J

(18)

n

with numerical values of bi,n , i,n . At the end, with i = 1, one has the known amplitudes {A0,m } of the incident modes on the left-hand side of the first equation. Thus it can be solved for the {AI,n }, and with these all other amplitudes {Ai,m },{Bi,m } are computed by insertion. The numerical examples show 3D plots of the sound pressure level; the spatial coordinates are k0 x, k0 y.

Sound pressure level in a converging cone, with the fundamental duct mode incident

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As above, but with the first higher duct mode incident

Sound pressure level in a diverging cone, with the fundamental duct mode incident

Duct Acoustics

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711

As in the previous graph, but now with the first higher mode incident

Sound pressure level in a converging, nearly hard cone, with the first higher mode incident. It becomes cut off inside the cone

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J.30

Duct Acoustics

Lined Conical Duct Transition, Evaluated with Stepping Admittance Sections

See also: Mechel, J. Sound Vibr. 219 (1999)

This section applies the third special case of > Sect. J.28: †Ÿ = const(r) if the lining admittance G = G(r) ∼ 1/r. This condition is not satisfied over the entire radial range, but in radial sections i = 1, 2, . . . , such that Gi (r) ∼ 1/r, and the average admittance in the sections equals a given value G: Gi = G.

(1)

The computational sectional admittances are: Gi (r) =

G ri − ri−1 . ) r ln ri ri−1

(2)

If the section width r is small compared to the wavelength, and if the variation of Gi (r) is not too strong, the sectored lining will approximately produce a sound field like that for a homogeneous lining with admittance G (the conditions mentioned exclude a lining reaching to the origin r = 0). The sound fields in the sections can be written as sums of modes, which are orthogonal over 0 ≤ ˜ ≤ Ÿ: R† (kr) · T(†˜ ) · Z(kz z) (3) pi (r, ˜ , z) = † 2

k02 − kz2

with k = if the variation Z(kz z) in the z direction is like cos(kz z), sin(kz z), e±jkz z or a linear combination thereof, with cos(†˜ ) ; symmetrical modes T(†˜ ) = , (4) sin(†˜ ) ; anti-symmetrical modes where the radial functions R† (kr) are Bessel, Neumann, or Hankel functions of order †, and with †Ÿ solutions of: (†Ÿ0 ) · tan(†Ÿ0 ) = jkr · Ÿ0 G ; (†Ÿ0 ) · cot(†Ÿ0) = −jkr · Ÿ0 G ;

symmetrical modes , anti-symmetrical modes

(5)

Duct Acoustics

or, more definitely (with the mode counter n in the i-th section): k0r rigid tan G (†i,n Ÿ) · flank at ˜ = 0 . (†i,n Ÿ) = ±jŸ ) soft cot ln ri ri−1

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713

(6)

The mode norms are Ni,n

1 = Ÿ

sin(2†i,n Ÿ) 1 cos2 1± . (†i,n ˜ ) d˜ = sin2 2 2†i,nŸ

Ÿ 0

(7)

The sound field in zone i is formulated as (henceforth only symmetrical modes are assumed; a possible variation in the z direction with Z(kz z) will be dropped): . (2) Ai,m · H(1) pi (r, ˜ ) = (kr) + B · H (kr) · cos(†i,m ˜ ) , i,m †i,m †i,m m≥0

Z0 · vr,i =

. jk (2) Ai,m · H (1) †i,m (kr) + Bi,m · H†i,m (kr) · cos(†i,m ˜ ) k0 m≥0

(8)

(a prime indicates the derivative). The amplitudes Ai,m , Bi,m are determined by field matching at the section limits. In a simple example of application of the method assume the following: 1. A given radial particle velocity distribution V0(˜ ) on the arc r = a. (Some incident duct mode in the duct in front of r = a would lead to a method as described in the previous > Sect. J.29.) 2. The cone is infinitely long. (In practice it is sufficient if it is so long that the admittance step at the outer zone limit becomes small, so that reflections at the cone can be neglected, and the cone has an anechoic termination; other terminations are handled as in > Sect. J.29, see also below.) One needs coupling coefficients between modes of adjacent zones given by the integrals (Ti,m (˜ ) are the azimuthal mode functions): (i) Xm,n

Y(i) m,n

1 = Ÿ 1 = Ÿ

Ÿ Ti,m (˜ ) · Ti+1,n (˜ ) d˜ , 0

(9)

Ÿ Ti,m (˜ ) · Ti−1,n (˜ ) d˜ =

(i−1) Xn,m

.

0

They assume the following values if the flank at ˜ = 0 is rigid: sin (†i,m + †i+1,n )Ÿ 1 sin (†i,m − †i+1,n )Ÿ (i) Xm,n = + , 2 (†i,m − †i+1,n )Ÿ (†i,m + †i+1,n )Ÿ Y(i) m,n

1 = 2

sin (†i,m − †i−1,n )Ÿ sin (†i,m + †i−1,n )Ÿ + , (†i,m − †i−1,n )Ÿ (†i,m + †i−1,n )Ÿ

(10)

(11)

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and if that flank at ˜ = 0 is soft: sin (†i,m + †i+1,n )Ÿ 1 sin (†i,m − †i+1,n )Ÿ (i) Xm,n = − , 2 (†i,m − †i+1,n )Ÿ (†i,m + †i+1,n )Ÿ sin (†i,m + †i−1,n )Ÿ 1 sin (†i,m − †i−1,n )Ÿ (i) Ym,n = − . 2 (†i,m − †i−1,n )Ÿ (†i,m + †i−1,n )Ÿ The boundary condition (source condition) at r = a is: . jk !

(2) Z0 · vr,1 = A1,m · H (1) †1,m (ka) + B1,m · H†1,m (ka) · cos(†1,m˜ ) = Z0 V0 (˜ ) k0 m≥0

(12)

(13)

leading to: . 1 Ÿ jkN1,m (1)

(2) Z0 V0 (˜ ) · cos(†1,m˜ ) d˜ A1,m · H†1,m (ka) + B1,m · H†1,m (ka) = k0 Ÿ

(14)

0

with known Fourier coefficients on the right-hand side. 1 Applying the integral operation Ÿ

Ÿ . . .·cos(†i+1,m ˜ ) d˜ on both sides of the boundary 0

condition for the sound pressure at the zone limit ri between two zones i and i + 1 gives: . (2) Ai+1,m · H(1) †i+1,m (kri ) + Bi+1,m · H†i+1,m (kri ) · Ni+1,m =

-

. (2) (i) Ai,n · H(1) †i,n (kri ) + Bi,n · H†i,n (kri ) · Xn,m ,

(15)

n≥0

and for the radial particle velocity: .

(2) Ai+1,m · H (1) †i+1,m (kri ) + Bi+1,m · H†i+1,m (kri ) · Ni+1,m =

-

.

(2) (i) Ai,n · H (1) (kr ) + B · H (kr ) · Xn,m . i i,n i †i,n †i,n

(16)

n≥0

Elimination of the Bi+1,m and use of the Wronski determinant for Hankel functions returns the upward iterative systems of equations: kri (2)

(1) (2) Ai+1,m = j Ai,n · H(1) †i,n (kri ) · H†i+1,m (kri ) − H†i,n (kri ) · H†i+1,m (kri ) 4Ni+1,m n≥0 .

(2)

(2) (2) (i) + Bi,n · H(2) (kr ) · H (kr ) − H (kr ) · H (kr ) · Xn,m , i i i i †i,n †i+1,m †i,n †i+1,m (17) kri (1)

(1) (1)

(1) Ai,n · H†i,n (kri ) · H†i+1,m (kri ) − H†i+1,m (kri ) · H†i,n (kri ) Bi+1,m = −j 4Ni+1,m n≥0 .

(1) (1)

(2) (i) + Bi,n · H(2) · Xn,m . †i,n (kri ) · H†i+1,m (kri ) − H†i+1,m (kri ) · H†i,n (kri )

Duct Acoustics

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715

If we begin with i = 1, the B1,n on the right-hand sides can be expressed by the (numerical) Fourier coefficients of the particle velocity distribution Z0 V0 (˜ ) at r0 = a and the symbolic A1,n . The equations will have the following general form during the iteration: kri ai,n · A1,n + bi,n ; 4Ni+1,m n≥0 kri = −j i,n · A1,n + i,n , 4Ni+1,m n≥0

Ai+1,m = j Bi+1,m

(18)

where ai,n , bi,n , i,n , i,n are numerical quantities. When the iteration has proceeded up to a value i = I, for which the admittance step at r = rI is small enough to neglect the inward reflection, i. e. Bi≥I,m = 0, the equations will have the form: AI+1,m = j

krI krI aI,n · A1,n + bI,n ; 0 = −j I,n · A1,n + I,n . 4NI+1,m n≥0 4NI+1,m n≥0

(19)

Then they are a coupled system of equations for the amplitude sets A1,n , AI+1,n.After the solution of this system, all other amplitudes follow by insertion. If the cone is not infinitely long but ends with some termination at r = rI , this termination will give a prescription of how to express the BI+1,m by the AI+1,m ,and the procedure remains the same, in principle. In general, the upper limit nhi of the required mode orders will not be high, except if V0 (˜ ) has many details.

J.31

Mode Mixtures


Modes are elementary solutions of a wave equation and of boundary conditions. For some kinds of boundaries they are orthogonal over the duct cross section and therefore suited for a synthesis of sound fields in the duct. Like the “science fiction” of a diffuse sound field in room acoustics, it may be useful to define in duct acoustics mode mixtures in which the modes obey some rules of mixing but may have random phases. Consider a rectangular hard duct with the duct axis in the z direction and the origin of the transversal co-ordinates x, y in a duct corner. A sound wave propagating in the z direction may be described by: pm,n (x, y, z) = Am,n · qm,n (x, y) · e−jkm,n z (1) p(x, y, z) = m,n

m,n

with mode profiles (containing both symmetrical and anti-symmetrical modes with respect to the duct central axis): qm,n (x, y) = cos(—m x) · cos(†n y) ; —m a = m · ; †n b = n · ; m, n = 0, 1, 2, . . .

(2)

and 2 k02 = —2m + †n2 + km,n ;

2 ‰m,n = —2m + †n2 = (m/a)2 + (n/b)2

(3)

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with mode norms Nm,n (S = duct cross-section area): 77 0; m, n = ‹Œ qm,n · q‹,Œ dx dy = ; S · N ; m, n = ‹, Œ m,n S 1 1; k=0 ; ƒk = . Nm,n = 2; k>0 ƒm · ƒn

(4)

The modal angles (relative to the duct axis) are: ‰m,n km,n = arcsin ; cos ¥m,n = 1 − (m/k0a)2 − (n/k0 b)2 .(5) ¥m,n = arccos k0 k0 Thus modes have the form: n m x · cos y · e−jkm,n z , pm,n (x, y, z) = Am,n · cos a b km,n pm,n (x, y, z) = Gzm,n · pm,n (x, y, z) , vzm,n (x, y, z) = k0Z0 where Gzm,n are the modal axial field admittances: km,n 1 Gzm,n = = 1 − (‰m,n /k0)2 k0Z0 Z0 1 cos ¥m,n = 1 − (m/k0 a)2 − (n/k0 b)2 = . Z0 Z0

(6)

(7)

The modal axial effective intensity at a point x is: 0 1 1 / ∗ Izm,n (x) = Re pm,n (x) · vzm,n (x) = |pm,n (x)|2 · Re{Gzm,n } . 2 2 The axial effective intensity of the sound wave (a mode mixture) is: ( 8 1 1 ∗ pm,n (x) · vzm,n(x) = |pm,n (x)|2 · Re{Gzm,n } . Iz (x) = Re 2 2 m,n m,n The effective sound power through a duct cross section is: ( 8 km,n 1 1 ∗ 2 ¢= Re{p · vz } dS = S · Re · |Am,n | · Nm,n 2 2 kZ m,n 0 0 S

1 1 · |Am,n |2 · Re{Gzm,n } = ¢m,n , = S 2 m,n ƒm ƒn m,n

(8)

(9)

(10)

where ¢m,n are the modal effective powers. A mode can transport effective power only if it is cut on (propagating); the condition for cut-on is (the summations in ¢ extend up to these limits): (‰m,n /k0)2 < 1

or:

(m/a)2 + (n/b)2 < (k0 /)2 = 4/Š02 = (2f /c0)2 .

(11)

Below,the sound power ¢ sometimes will be referred to as the sound power ¢0 of a plane S p20 = 1. (12) wave with sound pressure p0 such that ¢0 = 2 Z0

Duct Acoustics

The condition ¢/¢0 = 1 is equivalent to: 1 1 ** Am,n **2 * * · Re{km,n /k0} = * * ƒ ƒ p0 ƒ ƒ m,n m n m,n m n

* * * Am,n *2 * * * p * · Re{Z0 Gzm,n } = 1 . 0

Mode mixture with equal modal amplitudes Am,n : −1 * * * Am,n *2 ! Re{Z0 Gzm,n } * * = = const(m, n) . * p * ƒm ƒn 0 m,n

J

717

(13)

(14)

Mode mixture with equal modal sound powers (or intensities): ¢m,n m,n

¢0

=1;

¢m,n 1 , = const(m, n) = ¢0 N

(15)

where N is the total number of cut-on modes. This leads to the mode amplitudes: * * * Am,n *2 ! ƒm ƒn 1 ƒ m ƒn 1 * * . (16) * p * = N Re{Z G } = N cos ¥ 0 0 zm,n m,n If a mode approaches cut-off, then cos ¥m,n → 0, i. e. this mode-mixing model would require large mode amplitudes near cut-off, and also for large mode orders, because then ¥m,n → /2. Mode mixture with equal mode energy density Em,n : The mode energy density averaged over the duct cross section follows from the mode (17) power: ¢m,n = cgm,n · S · Em,n with the modal group velocity: cph 1 , = cg = – ∂cph dk/d– 1− cph ∂– (18) 1 dkm,n 1 2 km,n = (–/c0)2 − ‰m,n ; , = d– c0 1 − (‰m,n /k0)2 cgm,n = c0 1 − (‰m,n /k0)2 = c0 · Z0 Gzm,n . Therefore the averaged modal energy density is: *2 Re{Gzm,n } ¢m,n 1 1 ** Em,n = Am,n * = . S · cgm,n 2 ƒm ƒn c0 Z0 Gzm,n

(19)

The model of equal modal energy density demands (with restriction to propagating modes, for which Re{Gzm,n } = Gzm,n) : * *2 1 1 *Am,n * Em,n = = const(m, n) , (20) 2 c0 Z0 ƒm ƒn or, with the above power normalisation: * * * Am,n *2 ƒm ƒn ƒm ƒ n * * = . (21) * p * = 0 Re{Z0 Gzm,n } cos ¥m,n m,n

m,n

This is the most plausible model for the simulation of random sound fields.

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J.32

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Duct Acoustics

Mode Excitation Coefficients


Sometimes one is interested in exciting predominantly higher modes (because they are easier to attenuate than lower modes); sometimes one would like to avoid the excitation of higher modes (e. g. in experiments with low modes). It is reasonable to introduce a coefficient which describes the excitation probability of a mode under some standard conditions. Consider a flat lined duct extending over −h ≤ y ≤ +h (other duct geometries are treated similarly) with a sound field formulated as a mode sum, with mode norms Nn : p(x, y) =

−‚n x

An · qn (y) · e

n

;

1 Nn = 2h

h

qn2 (y) dy .

(1)

−h

If the excitation is performed by a given sound pressure profile pi (0, y) in the plane x = 0, then the mode amplitudes are: 1 1 · An = Nn 2 h

h pi (0, y) · qn (y) dy .

(2)

−h

If the excitation is done by a given axial particle velocity vix (0, y), then the mode amplitudes are: j 1 An = · Nn · ‚n /k0 2 h

h Z0 vix (0, y) · qn (y) dy .

(3)

−h

One plausible standard excitation is the excitation by a plane wave pressure profile pi (0, y) = 1 and to introduce mode excitation coefficients Fn for that excitation: 1 1 Fn = · Nn 2 h

h qn (y) dy . −h

For modes with symmetrical (relative to y = 0) profiles qn (y) = cos(—n y): Fn = 2

(4)

sin(—n h)/(—n h) −−−−→ 1 . 1 + sin(2—n h)/(2—n h) —n h=0

(5) (6)

For anti-symmetrical modes with qn (y) = sin(—n y): Fn = 2

(1 − cos(—n h)) /(—n h) . 1 − sin(2—n h)/(2—n h)

(7)

Duct Acoustics

J

719

Magnitude of the mode norm in the range of the first mode in a flat, lined (locally reacting) duct over the plane U = k0h · Z0 G

Magnitude of mode excitation coefficient Fn(U) for n = 1 in a flat lined (locally reacting) duct over the plane U = k0 h · Z0 G. The low values are in the range of the surface wave mode; the peak maximum is at the branch point between the first and second modes.

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Magnitude of mode excitation coefficient Fn(U) for n = 2 in a flat lined (locally reacting) duct over the plane U = k0h · Z0 G. Low values are in the range of the surface wave mode; the peak maxima are at the branch points between the first and second and the second and third modes

J.33

Cremer’s Admittance

See also: Cremer (1953); Mechel, Vol. III, Ch. 41 (1998)

(The author wondered whether he should include this section because its topic requires more words than formulas; but the use of Cremer’s admittance is a modern design of silencers if powerful computing programs for sound absorbers are available. Duct linings are assumed to be locally reacting.) Cremer’s question: Under what condition will the least attenuated mode in a lined duct have its maximum attenuation ? Answer (for a flat duct): !

(1) When U: = k0h · Z0 G = Ub,1 = 2.05998 + j · 1.65061 , where G is the lining admittance and Ub,1 is the value of U in the first branch point of symmetrical modes. !

In circular ducts U: = k0 h · Z0 G = Ub,1 = 2.9803824 + j · 1.2796025 .

(2)

The attenuation Dh in the duct is the sound pressure level decrease per half duct height h (or radius) travel distance: Dh = 8.6858 · Re{ h} ( h)2 = (—h)2 − (k0h)2 , where z =

Duct Acoustics

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721

—h is the solution of the characteristic equation for the least attenuated mode. If the admittance is in the branch point, then: 2.1062 + j · 1.12536 flat duct —h = zb,1 = . (3) 2.9803824 + j · 1.2796025 round duct

Attenuation Dh of least attenuated mode in a flat duct if the lining admittance G has Cremer’s admittance value at all frequencies Facts: 1. U = k0 h· Z0 G = Ub,1 is the condition for maximum attenuation of the least attenuated mode. 2. G has the sign of a spring-like reaction but the frequency dependence of a mass-like reaction. 3. tan ƒ = Im{Ub,1 }/Re{Ub,1 } = Im{G}/Re{G} = 0.801 is a value which is typical at frequencies in the lower half-value point of resonances. 4. Ub,1 is at a coincidence of —0 h and —1 h, i. e. at the cut-on of the first higher mode. 5. Onset of the first higher mode is the criterion for the beginning of “ray formation” in the duct, where the slope Dh ∼ 1/f 2 begins. 6. From the second and third facts it follows that a lining with Cremer’s admittance typically is a narrow-band lining. 7. From the fourth and fifth facts it follows that a lining with Cremer’s admittance is no low-frequency lining (i. e. with small h/Š0 or small f · h). 8. The least attenuated mode for Cremer’s admittance has a large mode excitation coefficient (see previous > Sect. J.32).

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Extension of Cremer’s rule: The condition U = Ub,1 is unnecessarily restrictive. The following extension is more flexible in its application. Every absorber with a characteristic length t and a function Ut = k0 t · Z0 G which crosses the straight line (0, Ub,1 ) has a Cremer admittance at the crossing point if it is applied in a duct with the (half) height h for which t/h = |Ut |/|Ub,1|. The crossing point t/h·Ub,1 on the straight line (0, Ub,1) will be called the“design point”. Design points with t/h > 1 in general are reached more easily, but the maximum of their attenuation curve Dh (f · h) is often in the range f · h of ray formation, so that the attenuation is of little practical use. Therefore design points with t/h < 1 are of greater interest. In order to extend the frequency range of Cremer’s admittance, one must combine resonators which are tuned differently, so that the curve of U for the combination in the complex U plane forms narrow double loops around Ub,1.This combination can be made by resonators in series (one behind the other) or in parallel (resonators side by side). Series combinations must be found by trial and error; > Sect. J.33 below will describe an algorithm for finding parallel combinations. The following graphs are examples for series combinations (more examples in [Mechel, Vol. III, Ch. 41 (1998)]). Pairs of graphs will be shown; the first graph contains the curve of Ut in the complex plane with indications of possible design points, while the second graph shows the Dh (f · h) curve for the indicated design point, together with the (dashed) curve of maximum possible attenuation. The characteristic length t in all examples is the sum of the layer thicknesses of the absorber. The first example is for two Helmholtz resonators (with slit-shaped necks) in series; in each orifice is a resistance foil with normalised flow resistance Rf .

Function Ut in the complex plane for two Helmholtz resonators in series

Duct Acoustics

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723

Attenuation Dh for the above absorber arrangement with a design ratio t/h = 1.2136. The two maxima belong to the two crossings of the line (0, Ub) at about the same design point The next example, for a triple Helmholtz resonator with a front side porous layer, shows the influence of the selection of the design point on the attenuation curve.

Function Ut in the complex plane for three Helmholtz resonators in series with a porous front layer. Three design points can be selected

724

J

t/h = 0.4461

t/h = 0.7573

Duct Acoustics

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725

t/h = 1.2573 It is possible to construct design points with multiple cross-overs with resonators in series, but the individual crossings are separated by full resonance circles and, therefore, in the Dh curves by wide frequency steps.This can be avoided with resonators in a parallel arrangement; see next section.

J.34

Cremer’s Admittance with Parallel Resonators

See also: Mechel, Vol. III, Ch. 41 (1998); Press et al., (1989)

The lining consists of repeated couples of absorbers. If the dimensions of the absorbers (in the direction of the duct axis) are small compared to the wavelength, the weighted (with the absorber surface areas) average of their admittances will determine the attenuation. One of the absorbers will be called the primary absorber (with index p), the other the adjoint absorber (with index a). Let Fp, Fa be the surface areas of the absorbers, Gp and Ga their surface admittances, = Fa /Fp the surface ratio, and t a common characteristic length of both absorbers. Then the extended principle of Cremer’s admittance (see > Sect. J.33) demands that: Ut =

Fp Ut,p + Fa Ut,a Ut,p + Ut,a ! t = = Ub ; F p + Fa 1+ h

Ut, = k0 t · Z0 G ;

= p, a .

(1)

ˆ t,a of a fictitious adjoint absorber: This conditional equation defines the U function U ˆ t,a =! U

1 (1 + ) · t/h · Ub − Ut,p .

(2)

It is called “fictitious” because it is not clear whether and how it can be realised. If, for example,the real part of the brackets is negative,then the associate admittance Gˆ a should have a negative real part, which cannot be realised with passive absorber elements. The ˆ t,a in the U plane: conditional equation gives the “rule of construction” for U

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Duct Acoustics

. ˆ t,a =! − t/h · Ub − Ut,p . · t/h · Ub − U

(3)

ˆ t,a belonging to a value Ut,p is on a straight line According to this condition, the point U through Ut,p and the design point t/h · Ub on the opposite side (with respect to t/h · Ub ) at a distance times the distance between Ut,p and t/h · Ub . Thus, if Ut,p is above ˆ t,a is below that line; if Ut,p is above (0, Ub) and rightthe line (0, Ub ), the point for U ˆ turning (which is normal), Ut,a is below (0, Ub) and right-turning also. All curves for functions U of passive absorbers begin at sufficiently low frequencies near the origin ˆ t,a begins near the line (0, Ub ) beyond Ub , which physically is not of the U plane; thus U ˆ t,a is possible at very low frequencies. possible. Therefore no realisation of U First the steps of the procedure for finding a lining with parallel absorbers with an effective Cremer admittance will be described (with a concrete example), then an algorithm for finding a suitable adjoint absorber will be derived. The first example simply consists of two porous absorbers with different flow resistivity values ¡ and thicknesses tp , ta , arranged side by side. The characteristic length is t = tp . 1. Find a suitable primary absorber (i. e. an absorber with the U function on an arc above the line (0, Ub ), possibly crossing that line beyond Ub ). ˆ t,a in some fre2. Conceive an adjoint absorber whose function Ut,a approximates U ˆ quency interval of interest (Ut,a can be drawn with the above rule of construction). Thus Ut,p , Ut,a are known as functions of frequency and Ut is their average function. Plot these curves in the U plane. The diagram below shows these curves together with the straight line (0, Ub ) and the curve Ut for three surface ratios = Fa /Fp . A design point at t/h = 0.2694 is marked.

Branch point Ub , curves Ut,p, Ut,a for the component absorbers, and average Ut in the U plane. A design point t/h = 0.2694 is marked at which two resonances have contracted to a dent

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Maximum possible attenuation (dashed) and attenuation curve of the least attenuated mode in a duct with the lining from above at a surface ratio = Fa /Fp = 0.2694 The next example is for a similar arrangement, but now with equal thicknesses t = tp = ta = 8 [cm[. The adjoint absorber is covered with a tight, limp foil with surface mass density mf = 0.06 [kg/m2 ]. Two design points are of interest, one at t/h = 0.5971, the other at t/h = 0.7330.

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A surface area ratio = Fa /Fp = 0.3 is selected for the attenuation curves at the two design points.

t=h = 0:5972; = 0:3

t=h = 0:7330; = 0:3

The main problem is the determination of suitable parameters of the adjoint absorber. This task can be performed with a computation algorithm. The initial step is the same as above: 1. Find a suitable primary absorber (i. e. an absorber with the U function on an arc above the straight line (0, Ub), possibly crossing that line beyond Ub ). 2. Establish the structure of an adjoint absorber. The lining in the example used below consists of a primary absorber, which is a porous layer of thickness d = t and a flow resistance (normalised) Ra = ¡ · d/Z0 ; ¡ = 1000 [Pa · s/m2 ] is fixed. an adjoint absorber, which is a Helmholtz resonator with (normalised) flow resistances Rrv , Rrh in the orifices.

Experience shows that if both component absorbers have resonances at frequencies fp , fa , respectively, the resonance of the adjoint absorber should be at about fa ≈ fp /2. The

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resonances of the component absorbers in the example are at about (d = resonator neck length): c0 1 · , 4 t

Fa c0 c0 = fa ≈

2 (Fa + Fp) · d · ta 2 (1 + ) · d · ta

c0 2 = fp · . − −− −→ d →t 2t (1 + ) · ta /t (1 + ) · ta /t fp ≈

(4)

Thus the condition fa ≈ fp /2 can be satisfied. The U functions of the component absorbers are evaluated from (an , Zan are normalised characteristic values of the porous material of the primary absorber; E is the variable needed for their evaluation): Primary absorber: Ut,p = k0 t · Gp = k0t · tanh(k0 t · an )/Zan ;

E=

1 k0 t 0 f = ; ¡ 2 Ra

Ra =

¡·t . Z0

(5)

Adjoint absorber: l0 Ut,a = k0 t · Ga = k0 t/Za ; Za = jk0 a + Rrv + Zsv ; a Zsh + j tan(k0t) ; Zsv = 1 + j · Zsh · tan(k0t) Zsh = Rrh + jk0 a

a/L li −j ; a tan(k0ta )

(6)

li l0 (x) = 1 + f (y) · 1 + g(z) ; a a

; y = lg(d/a) = − lg(a/t) ; 1+ z = lg(ta /L) = x + y + lg(ta /t) ,

x = lg(a/L) = lg

(7)

(8)

where li is the interior end correction of the neck; the functions l0 (x), f (y), g(z) are taken from > Sect. H.4. The parameters to be optimised are = a/L; t/h; ta /t; a/t; Rrv ; Rrh . The conditional equation for the Cremer admittance can be written as: !

ˆ t,a . Ut,p = (1 + ) · t/h · Ub − · U

(9)

ˆ t,a → Ut,a , the task is to find a good approximaIf one replaces on the right-hand side U tion to the known Ut,p by variation of parameters. It can be formulated as a task to find a minimum by parameter variation. Find a minimum of the squared-error sum: 9 ! 2 wn · |zn − f (xn ; a1 , a2 , . . . ; b1 , b2 , . . .)| wn = Min , (10) q(a1 , a2 , . . .) = n

n

where xn = (k0 t)n are discrete values of the frequency variable in a range for which the approximation should be found, zn = (Ut,p )n = zn (xn ) are the known values to be

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approximated, fn (xn ; . . .) are the values of the right-hand side of the above conditional ˆ t,a → Ut,a ), ak are parameters to be varied, bk are paramequation (after substitution U eters to remain fixed, and wn (xn ) is a weight function which may be centred at some point x e. g. with the form of a sine arc w(x) = sin ((x − xlo )/(xhi − xlo ) · ) for easier location of a parameter set in a preliminary run, or it is wn (xn ) = const in later runs. Use a program for minimum seeking which accepts start values for the parameters ak and which accepts limits for the parameter ranges (in order to avoid parameter variation to negative or even complex values of the ak which should be real positive, see [Press et al., (1989)]). The following diagram shows in the U plane the curve for Ut,p with sampling points ˆ t,a in its range used (also with sampling in the k0t interval used, the fictitious adjoint U points), the final Ut,a after optimisation of the parameters (sampling points in the k0 t interval used), and the average Ut (thick line). The reference dimension t was set to t = 0.1 [m]; the ratio a/t was kept on a fixed value a/t = 0.1; the starters for the other parameters ak were: = 0.5¸ ;

t/h = 1.0 ;

ta /t = 0.5 ;

a/t = 0.1 ;

Rrv = 0.05 ;

Rrh = 0.05 .

The optimised values were: = 0.5832 ; Rrv = 0.3579 ;

t/h = 0.4585 ; Rrh = 0.0 .

ta /t = 0.4267 ;

a/t = 0.1 ;

The weight function was w(x) = 1. The (computed) design point is indicated as a point on the straight line (0, Ub).

The design point is on the straight line (0, Ub) near the small loop in the curve of Ut

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Attenuation curve Dh for the combination of a porous layer primary absorber with a Helmholtz resonator as adjoint absorber with the optimised values of the free parameters listed above

J.35

Influence of Flow on Attenuation

See also: Mechel, Vol. III, Ch. 25.4 (1998); Mechel, Vol. III, Ch. 42 (1998)

Consider a duct with axial co-ordinate x and transversal co-ordinate y. A stationary flow with velocity profile V(y) is in the +x direction if V > 0 and in the −x direction if V < 0. Sound waves are assumed to propagate in the +x direction. In this section the simplifying assumption V(y) = const is made; for more details see the chapter “Flow Acoustics”. The presence of flow will modify the fundamental equations mainly by the replacement of the partial time derivative ∂/∂t by the “substantial derivative” D/Dt: ∂ D = 0−−−→0 div v + =0, V=0 ∂t Dt ∂v Dv + grad p = 0−−−→0 + grad p = 0 , 0 V=0 ∂t Dt 2 1 ∂ 1 D2 − 2 2 p = 0−−−→ − 2 2 p = 0 . V=0 c0 ∂t c0 Dt 0 div v +

(1)

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The substitution for the time derivative can be written for an assumed time factor ej–t and a sound wave of the form p(x, y) = P0 · q(y) · e− x as: (2) ∂ ∂ ∂ D ∂ = j– = jc0 k0 −−−→ = +V = c0 jk0 + M V = 0 ∂t Dt ∂t ∂x ∂x (3) = jc0 k0 1 + jM k0 (with M = V/c0 the Mach number). So this effect of the flow can be taken into account by the substitution: M ∂ . (4) = k0 1 + jM k0 −−−→k0 1 − j V=0 k0 ∂x k0 Using the abbreviation w one can write: M ∂ w = 1−j = 1 + jM , k0 ∂x k0 k0 −−−→ k0 · w .

(5)

V=0

The wave equation, for example, becomes + k02w 2 p(x, y) = 0.

(6)

If the lateral sound wave profile q(y) is (for example) q(y) = cos(—y), the characteristic equation for the determination of —h in a duct of (half) width h and with a locally reacting lining with surface admittance G changes to: —h · tan(—h) = jk0 hZ0 G −−−→ —h · tan(—h) = jk0 hZ0 G · w .

(7)

V=0

This form assumes that the boundary conditions at the lining surface are the continuity of sound pressure and normal particle velocity vy . Some authors claim that not the particle velocity should be continuous, but the elongation ey with vy = ∂ey /∂t. This time derivative introduces a new factor w wherever G appears: —h · tan(—h) = jk0 hZ0 G −−−→ —h · tan(—h) = jk0 hZ0 G · w 2 .

(8)

V=0

One can combine both theories of the boundary condition to give: —h · tan(—h) = jk0 hZ0 G −−−→ —h · tan(—h) = jk0 hZ0 G · w ; V=0

= 1, 2 .

The secular equation, which follows from the wave equation, changes to: ( /k0)2 + 1 − (—/k0 )2 = 0 −−−→ ( /k0)2 1 − M2 + 2jM · /k0 + 1 − (—/k0)2 = 0 . V=0

Because the solution shall be without flow: 1 (—h)2 − (k0h)2 , −−−−→ k0 M→0 k0h

(9)

(10)

(11)

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the solution with flow is: −j j 2 )(—h)2 − (k h)2 = (1 − M M + . 0 k0 1 − M2 k0 h Thus w has the form: 1 M 2 (1 − M2 ) − (k h)2 w= (—h) 1 + j . 0 1 − M2 k0 h

733

(12)

(13)

The appearance of —h in w modifies the characteristic equation significantly,and also the method for its numerical solution. In general, one will solve the characteristic equation first for M = 0, i. e. w = 1, and then increase M iteratively to its final value, taking solutions for the earlier M as starting solutions zsi in the numerical procedure: zs1 = —h(Mk−3 ) ,

zs2 = —h(Mk−2 ) ,

zs3 = —h(Mk−1 ) .

(14)

One must take very small steps M,especially at the beginning of the iteration through M. A better choice of starting solutions is at the beginning of the iteration: zs1 = —h(0) ,

zs2 = (zs1 + zs3 )/2 ,

zs3 = —h(0) + M · d(—h)/dM|M=0 ,

(15)

zs3 = —h(Mk−1 ) + M · d(—h)/dM|M(k−1) .

(16)

and at later steps: zs1 = —h(Mk−2 ) ,

zs2 = —h(Mk−1 ) ,

The required derivatives d(—h)/dM are for symmetrical modes, for both exponents = 1, 2, with the abbreviation qw = (—h)2 (1 − M2) − (k0h)2 (17) = 1: −(—h)2 (1 − M2) + (k0h)2 (1 + M2 ) + 2jk0h · M · qw d(—h) ; = Z0 G —h —h dM 2 qw · (1 − M ) M Z0 G + + tan(—h) qw cos2 (—h)

(18)

= 2: 2(k0h + jM · qw) −(—h)2 (1 − M2 ) + (k0h)2 (1 + M2 ) + 2jk0 h · M · qw d(—h) = Z0 G . (19) dM M —h —h M2 —h Z Z +tan(—h) qw·(1−M2 )3 2j G+2 G+ 0 0 1−M2 k0h 1−M2 qw cos2 (—h) For anti-symmetrical modes replace 1/cos2(—h) → −1/sin2(—h), tan(—h) → cot(—h) and Z0 G → −Z0 G.

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The first two mode solutions z = —h in a flat duct with a locally reacting lining, consisting of a simple layer of glass fibres with thickness t and normalised flow resistance Ra = ¡t/Z0. For Mach numbers M = 0 and M = 0.2 with the boundary condition form = 1. The numerical solutions jump at curve sections with short dashes

Same as above, but for the boundary condition form = 2

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Attenuation curves Dh for the least attenuated mode in a duct as above, for Mach numbers M = 0 and M = ±0.2 with the boundary condition form = 1

As above, but for the boundary condition form = 2

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One needs,in a number of applications,the branch points of the complex transformation which is induced by the characteristic equation. We write this equation for symmetrical modes in a flat duct of (half) width h with a locally reacting lining having a surface admittance G in the form: f (z; M) = −j

z · tan z ! =U; w

= 1, 2 ;

z = —h ;

U = k0h · Z0 G .

(20)

The branch points zb,n (M) are determined as solutions of the equation: w (z; M) f (z; 0) ! =· f (z; 0) w(z; M)

(21)

with 1 f (z; 0) 1 = + , f (z; 0) z sin z · cos z M(1 − M2) w (z; M) =j w(z; M) k0 h

z2 (1 − M2 ) − (k0 h)2

z

. M 2 2 2 z (1 − M ) − (k0h) 1+j k0 h

For small Mach numbers the branch points can be approximated by: * dzb ** zb,n (M) ≈ zb,n (0) + M · dM *M=0

(22)

(23)

with the derivative dzb = 2jM(1 − M2) + k0h(1 − M2)/qw · zb2 + k0h · qw + jM · qw 2 dM . + k0 h · qw + jM −(k0 h)2 + (1 − )(1 − M2 ) · zb2 · cos2 (zb ) . + 2j(1 − )(1 − M2)M · zb + k0h(1 − M2) · zb /qw · cos(zb ) · sin(zb ) . − k0 h · qw + jM −(k0 h)2 + (1 − )(1 − M2 ) · zb2 · sin2 (zb ) · +

(24)

2jM2 + k0 hM/qw · zb3 − jzb · qw 2

-

. 2jM2(1 − ) + k0 hM/qw · zb2 − j −(k0 h)2 + (1 − )(1 − M2) · zb2 −1

· cos(zb ) · sin(zb )

,

which for M = 0 becomes with the abbreviation zb = zb,n (0) * 4zb2 − 2(k0h)2 + 2(zb2 − (k0 h)2 ) cos(2zb ) + zb sin(2zb ) dzb ** = jk h 0 . dM *M=0 zb2 − (k0 h)2 2zb (zb2 − (k0 h)2 ) + (1 − )zb2 − (k0 h)2 sin(2zb )

(25)

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One gets for the first branch point n = 1 with zb = 2.1062 + 1.12536 · j: for k0 h = 2 ; for k0 h = 2 ; for k0 h = 1 ; for k0 h = 1 ;

= 1: = 2: = 1: = 2:

zb,1 (M) ≈ zb,1(0) + (−1.33228 + j · 0.403591)M , zb,1 (M) ≈ zb,1(0) + (−0.66614 + j · 0.201793)M , zb,1 (M) ≈ zb,1(0) + (−0.599752 + j · 0.406326)M , zb,1 (M) ≈ zb,1(0) + (−0.299878 + j · 0.203163)M .

The images Ub,n (M) of the branch points zb,n (M) can be approximated by: * dUb ** Ub,n (M) ≈ Ub,n (0) + M · dM *

(26) (27)

(28)

M=0

with the derivative * dUb ** = dM *M=0

(29) (k0 h)2 (2zb + sin(2zb )) 4zb2 − 2(k0h)2 + 2(zb2 − (k0h)2 ) cos(2zb ) + zb sin(2zb ) =j . 2 2zb (zb2 − (k0 h)2 ) + (1 − )zb2 − (k0 h)2 sin(2zb )

One gets for the first branch point n = 1 with Ub = 2.05998 + j · 1.65061: for k0 h = 2 ; for k0 h = 2 ; for k0 h = 1 ; for k0 h = 1 ;

= 1: = 2: = 1: = 2:

Ub,1(M) ≈ Ub,1(0) + (−7.206913·10−6 + j·0.000152866)M , Ub,1(M) ≈ Ub,1(0) + (−3.603788·10−6 + j·0.0000764332)M , Ub,1(M) ≈ Ub,1(0) + (0.0000218572 + j·0.0000352125)M , (30) Ub,1(M) ≈ Ub,1(0) + (0.0000109286 + j·0.0000176063)M .

Influence of the flow on the attenuation for a lining with U = Ub,1 (0):

Influence of flow on the attenuation Dh of the least attenuated mode in a duct whose lining has a U function U = Ub,1(0) which for M = 0 is in the first branch point If, however, the lining has a U function U = Ub,1(M), the attenuation remains high.

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Attenuation Dh of the least attenuated mode in a flat duct having a locally reacting lining with U = Ub,1(M), for M ≥ 0 and boundary condition form = 1

As above, i. e. for = 1, but with M ≤ 0

Duct Acoustics

As above, but for the boundary condition form = 2 and with M ≥ 0

As above, i. e. for = 2, but with M ≤ 0

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J.36

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Influence of Temperature on Attenuation


Silencers are used in gas flows with a wide variety of temperatures. The question is how to take the operation temperature into consideration in the design of a silencer. If the attenuation is evaluated as a non-dimensional quantity (like Dh = 8.6858 · Re{ h}) using only non-dimensional parameters,and if it is plotted over a non-dimensional variable, then the result is valid for all fluids like air, and also for air at different temperatures. Some different influences of the temperature on the attenuation may be distinguished. Below, T is the operation temperature (in Kelvin) and T0 is the standard temperature. (1) Influence of representation: In a plot of Dh over f · h the abscissa comes from f · h = c0 /(2) · k0h. The (linear) abscissa should be multiplied with c0 (T)/c0 (T0). (2) Temperature-dependent input parameters: Some parameters,like frequency f ,geometrical dimensions,porosities,shape factors etc., are not changed by temperature. Other parameters, like bulk densities of porous materials, surface mass densities m of foils and plates, remain virtually unchanged. If, however, a non-dimensional parameter M = m/(0 d) is used, with air density 0 and some thickness d, M becomes M(T) = 0 (T0 )/(T)· M0. Often impedances or admittances are made non-dimensional (normalised) with the free field wave impedance Z0 = 0 c0 . This reference impedance changes as Z0 (T) = 0 (T)c0 (T)/(0c0 ) · Z0 (T0 ). If the impedance which is normalised with Z0 is a mass reactance Zm of a solid element (e. g. foil or plate), it is not modified by the temperature; thus the variation in Zm /Z0 comes from Z0 . Commonly used resistances include the flow resistance ¡ · d of a porous layer (d its thickness; ¡ the material flow resistivity) or the flow resistance Zf of a porous foil or plate.One can always write ¡ · a2 /† = f (d), where a and d are characteristic lengths (e. g. a = fibre radius, d = fibre distance) and † is the dynamic viscosity of air. Thus ¡(T) = †(T)/†(T0) · ¡0. This transformation holds for all other resistances based on the friction of air. (3) Temperatur-dependent non-dimensional material data of air: Theories for the characteristic propagation constant a and wave impedance Za of porous materials take into consideration not only the flow resistivity ¡, but also material data of air, such as the adiabatic exponent ‰ and the Prandtl number Pr. The best procedure is to evaluate a and Za with a physical model theory and to use material data of air at the operation temperature. An important parameter is the product fcr · d of the critical frequency fcr and thickness d of an elastic plate. From the relation fcr d = fd

kb k0

2 = c0

m , B

(1)

and with the assumption that the surface mass density m and the bending modulus B do not (or only slightly) change with temperature, the parameter fcr · d changes as c0(T).

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> Section L.2 contains material data for air and relations for their temperature dependence. For some approximations it may be sufficient to use the ideal gas relations:

0 (T) = 0 (T0 ) · T0 /T , √ c0 (T) = ‰(T)0(T0 )/‰(T0)0 (T) · c0 (T0) ≈ c0 (T0) · T/T0 , ‰(T) ≈ ‰(T0 ) ,

Pr(T) ≈ Pr(T0 ) , )√ )√ T/T0 , k0(T) = k0 (T0 ) T/T0 , Z0 (T) = Z0 (T0 ) √ √ ¡(T) = ¡(T0 ) · T/T0 , †(T) = †(T0 ) · T/T0 , R(T) = R(T0 ) · T/T0 ,

(2)

E(T) = E(T0 ) · (T/T0)−3/2 ,

where R is the gas constant and E = 0 f /¡ is a non-dimensional input parameter for some porous material model theories.

Attenuation Dh of least attenuated mode in a flat duct with a locally reacting glass fibre layer as lining, for three operation temperatures

J.37

Stationary Flow Resistance of Splitter Silencers


The acoustic design of silencers is often in conflict with the static pressure loss of the stationary flow, especially in splitter silencers. The stationary flow resistance is usually described by the … value of the silencer: …=

Pwith − Pno , 0 V2 /2

(1)

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where Pwith is the static pressure drop over the silencer, Pno is the static pressure drop in the empty duct over the same distance and V is the average flow velocity in the duct in front of the splitters. If, alternatively, the average flow velocity Vs is determined in a splitter duct, the corresponding … value is: …s = …/(1 + D/H)2 .

(2)

Many measurements with splitter silencers (the splitters having rectangular corners) can be summarised by: 0.004 L D …s = 0.53 + 0.66 · lg + 0.027 − · . (3) H D/H H Rounding the splitter heads reduces … by about … ≈ 0.5 − 1.5.

J.38

Non-linearities by Amplitude and/or Flow

See also: Mechel, Vol. II, Ch. 28 (1995); Ronneberger (1967/68); Cummings (1975)

High sound amplitudes and stationary flow produce non-linearities in some absorber components, especially in fences and perforated sheets. The references combine their own measurements with a survey of the literature. I. Amplitude non-linearity of fences: Let ps be the sound pressure drop across a fence, vs the particle velocity in the fence orifice (both averaged over the orifice) and the porosity of the fence; then the nonlinear contributionto the normalised partition impedance Zs = (ps /vs )/Z0 of the fence opening can be written as: vs 1 ps = 2 K() . (1) Zs = Z0 vs c0 The following diagram gives values of the factor K() over the porosity .

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Values of factor K() in Zs (1): With the stationary flow resistance coefficient … defined by P = … · 0 /2 · U2 , where P = static pressure drop and U/ =average flow velocity through the fence opening, the relation is K() = 0.42 · … . (2) (2): Slit-shaped orifice with sharp corners: K() = 0.675/ 2 .

(3)

(3): Slit-shaped orifice with rounded corners: K() = 0.119/ 2 .

(4)

(4): Thin perforated sheet:

K() = 0.58/ 2 .

(5)

(5), (6): Some other values for perforated sheets are taken from the literature. II. Non-linearity by flow over orifices: A flow with velocity U is past the orifice with diameter d = 2a of a neck in the duct wall. Experimental results by Cummings for the real part Z of the orifice input impedance and for the orifice end correction can be represented by the following relations (f = frequency; = neck length): Z

= [12.52 · (/d)−0.32 − 2.44] · (U∗ /fd) − 3.2 , 0 fd 1 ; U∗ /f ≤ 0.12d/ = ∗ (1 + 0.6/d) · e−(U /f +0.12d/)/(0.25+/d) − 0.6/d ; 0

(6) U∗ /f > 0.12d/

,

∗ where 0 is the orifice end correction without flow and U is the flow shear velocity. It ∗ −1/4 , (7) is evaluated from U = Š/8 · U ; Š = 0.306 · Re

where U is the average velocity in the duct, Š is the coefficient of flow resistance by viscous shear and Re is the Reynold’s number of the flow using the duct diameter (the

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relation between Š and Re is for square ducts; the corresponding relation in circular ducts with diameter 2R is Š = 0.316 · Re−1/4 ). III. Non-linearity by flow through an orifice: Consider an orifice generated by a step in a hard duct. The flow velocity profile will have shape (a). Ronneberger in his analysis uses shape (b); Cummings applies shape (c), which is also used in the analysis of [Mechel (1995)] presented below.

The sketch below shows the co-ordinates x, r, the duct areas Si , the field zones I, II, III and the component sound fields. M is the Mach number with the average flow velocity. The grey area with limits near x = 0 may contain near fields, which will not be considered in detail.

Integrals of conservation of the mass, the impulse and the energy can be assumed to exist in this transition volume: v · dA = 0 ; (v)v · dA + p · dA = 0 ; Hv · dA = 0 , (8) A

A

A

A

where A is the surface of the volume; , p, v are density, pressure and velocity, respectively; and H is the stagnation enthalpy. The approximation pi3 = pi1 + pr1 will be used at the step. The density variations are = (p + ƒ)/c20 , where ƒ is the pressure produced by variations of the enthalpy S: S=

−ƒ 0 T0 (‰ − 1)

(9)

(T0 = stationary temperature; ‰ = adiabatic exponent). The stagnation enthalpy is: H =T·S+

p 1 2 + |v| . 2

(10)

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Let the x factors of the sound fields pi2 , pi3 be e−jKk0 x with a correction factor K for the free field wave number k0 . The axial particle velocity in zone II is: vi2 =

Kpi2 . 0 c0 (1 − MK)

(11)

At the limit between zones II and III let pi3 = pi2 ; let the fields within a zone be (approximately) constant in the radial direction. Then, with the porosity = S1 /S2, the integrals give: K KS3 pi2 + Mƒ , +M+ (1 + M)pi1 − (1 − M)pr1 = S1 1 − MK 1 2MK 1 1 (12) + 2M + M2 pi1 + − 2M + M2 pr1 = + M2 + pi2 + M2ƒ , 1 − MK MK ƒ pi2 − (1 + M)pi1 + (1 − M)pr1 = 1 + 1 − MK ‰−1 with the Bessel function J0 (z) and the Neumann function Y0 (z); as radial functions the amplitude A follows from the condition of zero radial particle velocity at the outer radius of zone III.The boundary conditions are pi2 (R1 ) = pi3 (R1 ) and 1 ∂pi2 (R1 ) ∂pi3 (R1 ) = . 2 ∂r (1 − MK) ∂r They lead to a characteristic equation for K: √ √ √ √ J1 (k0 R1 1 − K2 ) · Y1 (k0R2 1 − K2 ) − J1 (k0R2 1 − K2 ) · Y1 (k0R1 1 − K2 ) √ √ √ √ J0 (k0 R1 1 − K2 ) · Y1 (k0R2 1 − K2 ) − J1 (k0R2 1 − K2 ) · Y0 (k0R1 1 − K2 ) (1 − MK)2 − K2 · J1 (k0R1 (1 − MK)2 − K2 ) − =0. √ (1 − MK)2 1 − K2 · J0 (k0 R1 (1 − MK)2 − K2 )

(13)

(14)

A start value for its numerical solution is K ≈ 1/(1 + M). The following diagrams show the magnitude of the reflection factor rM = pr1 (x = 0)/pi1 (x = 0).

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Magnitude of reflection factor rM over Mach number M for different porosities = S1 /S2 . Points: measured by Ronneberger; Dashed: computed by Ronneberger; Solid: present computation

Magnitude of the reflection factor rM over the frequency for porosity = 0.5 and different Mach numbers M. Points: measured; Curves: present evaluation IV. Non-linearity by flow along mineral fibre absorbers: The flow resistivity ¡(U) of fibrous absorbers with flow along their surface from measurements with flow velocities up to U = 80 [m/s] can be represented by: ¡(U) = (1 − Af U)−4 ; ¡(0)

0.085 Af [s/m] ≈ . f[Hz]

(15)

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747

V. Non-linearity by flow through porous absorbers: The following representation of the characteristic propagation constant a and wave impedance Za with flow through the porous material does not include the possibility that the material is compressed by the flow! ”. , a = j–Ceff F† ¡ + 2 t · |U| + j–0 (16) 1 0 ”/ 2 + (F† ¡ + 2 t |U|) ! j– Za = Ceff with

1 tan(k0 h) , Ceff = 1 + (‰ − 1) k0h 0 c20 Ÿ − 1 kŒ h · tan(kŒ h) 1 Ÿ , F† = Ÿ 1 Ÿ 3 1− √ tan(k0 h) + − 1 tan(kŒ h) Ÿ ‰ Pr Ÿ √ k0 h = ‰ Pr · kŒ h ; kŒ h = −j6E ; E = 0 f /¡ ,

(17)

(18)

and t from a P − U record of stationary flow with velocity U through a material layer with thickness z according to: −P = ¡ + t · U z · U

(19)

(make a quadratic regression through measured P − U values; the coefficient of the linear term in U gives ¡; the coefficient of the quadratic term gives t ). f= –= 0 = c0 = U= ‰= Pr = = ”= ¡= t = kŒ = k0 = Ÿ , Ÿ

frequency; angular frequency; air density; sound velocity in air; stationary flow velocity; adiabatic exponent; Prandtl number; material porosity; 1.362 = structure factor; measured flow resistivity; quadratic term of flow resistivity; viscosity wave number; thermal wave number; see > Sect. B.1

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The following diagrams show measured points and computed curves for the characteristic values a , Za in a polyurethane foam for velocities U = 0; 0.82; 1.96 [m/s].

Real and imaginary parts of propagation constant a in a PU foam with three flow velocities

J.39

Real and imaginary parts of normalised wave impedance Za =Z0 in PU foam, with three flow velocities

Flow-Induced Non-linearity of Perforated Sheets

See also: Mechel, Vol. II, Ch. 28 (1995); Coelho (1983)

The following table gives partition impedances Zm = Z m + j · Z

m for perforated sheets, which are mostly based on experimental data.The perforations are circular with radius a at mutual distances b; the porosity is = a2 /b2 ; the sheet thickness is t. The Mach number M0 for high sound levels is M0 = v/c0, with v the particle velocity in the exit orifice; in the Mach number M∞ = U∞/c0 the flow velocity U∞ belongs to the undisturbed flow parallel to the sheet. Ranges of the sound pressure level Lp (relative to 20 [‹Pa]) are given by L0l , L0h without flow, and LUl , LUh with flow; Œ is the kinematic viscosity. Other symbols are explained in the table below.

0 p 8Œ–(1 + t=2a)

Z0m = RM ;

(U1 10 m=s)

= 0:30 c0

R2M + R2h ;

1 − 2 (M1 − 0:025) − 40R0 (M1 − 0:05) ;

Z0m =

LUl Lp LUh

œ1 (M0 ) =

Z00m = X0 (ƒ)

Z00m = X0 (ƒ)

1 + 5 103 M20 1 + 104 M20

ƒ = ƒ0 œ1 (M0 )

Z0m = Rh ;

Lp > LUh LUh = 193 + 40 lg M1

ƒ = ƒ0 œ1 (M0 )

M1 0:05

10−2:25+0:025Lp M0 = 0:50 c20 (1 − 2 )

Rh =

1 20 (1 − 2 ) 10−2:25+0:185Lp . –0 - X0 = 8Œ=–(1 + t=2a) + t + ƒ

1 − 2 M1 ; M1 > 0:05 ƒ = ƒ0 œ2 (M1 ) ) œ2 (M1 ) = 1 1 + 305M21

RM = 0:60 c0

Z00m = X0 (ƒ)

Lp < LUl LUl = 175 + 40 lg M1

p p œ0 () = 1 − 1:47 + 0:47 3

ƒ = ƒ0 = 0:85 2a œ0 ()

R0 =

With flow M1 0:025

(U1 8 m=s)

No flow M1 < 0:025

Medium level High level Lp < L0l L0l Lp L0h Lp > L0h L0l = 107 + 27 lg 4(1 − 2 )–0 Œ(1 + t=2a)2 L0h = 137 + 27 lg 4(1 − 2 )–0 Œ(1 + t=2a)2 Z0m = R0 ; Z00m = X0(ƒ) Z0m = jR2h − R20 j ; Z00m = X0 (ƒ) Z0m = Rh ; Z00m = X0 (ƒ)

Low level

Table 1 Formulas for the partition impedance Zm of perforated sheets for different sound pressure levels, with or without parallel flow

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750

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Reciprocity at Duct Joints

See also: Cho (1980); Mechel, Vol. III, Ch. 33 (1998)

Consider two ducts, = a, b, each with constant width and lining, possibly different in the ducts. The ducts are anechoic and be connected with some transition duct (step, corner, cone etc.). In the ducts there are modes with axial propagation constants ‚n and mode norms Nn . In a first “experiment” the m-th mode of duct a with amplitude Paim is incident on the joint; it produces modes n in duct b with amplitudes Pbtn (m). In a second “experiment” the ‹-th mode of duct b with amplitude Pbi‹ is incident on the joint; it produces modes in duct a with amplitudes Patn (‹).A relation of reciprocity holds for the transmitted modes: b Patm (‹) ‚b‹ Nam Pt‹ (m) = . ‚am Nb‹ Paim Pbi‹

(1)

The corresponding relation for the reflected modes in two“experiments”in which modes (e. g. of duct a) of orders m, n are incident on the joint is: Parn (m) ‚am Nan Paim = . Parm (n) ‚an Nam Pain

J.41

(2)

Mode Sets in Flat Ducts with Unsymmetrical, Locally Reacting Lining

See also: Mechel (2006), Grigoryan (1969)

The object is a flat duct with different locally reacting linings on opposite sides.A similar object is treated in > Sect. J.11, where the least attenuated mode was sought. Complete sets of mode solutions shall be determined here. In contrast to > Sect. J.11, the origin of the co-ordinates here is in a wall (with index I, the opposite wall with index a).

Mode formulation (constant in z direction) of the mode with index ‹: p‹ (x, y) = cos(—‹ y) + sin(—‹ y) e−‹ x ; ‹2 = —2‹ − k02, ; —‹

—‹ ∂p‹

=j cos (—‹ y) + sin (—‹ y) e−‹ x Z0 v‹y (x, y) = j k0 ∂(—‹ y) k0

(1)

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satisfying the transversal Helmholtz equation 2 2 ∂ ∂ 2 2 2 + k + (y) = + — p ‹ 0 ‹ ‹ p‹ (y) = 0, ∂y 2 ∂y 2

751

(2a)

and the boundary conditions at y = 0 and y = 2h: Z0 v‹y (x, yi,a ) = ∓Z0 Gi,a · p‹ (x, yi,a ) .

(3a)

They assume, with the non-dimensional quantities —‹ y → † ;

—‹ yi,a → †i,a ;

yi = 0 ;

ya = 2h ;

k0 h · Z0 Gi,a → Ui,a ,

(4)

the forms ) ) ∂ 2 p‹ (—y) ∂(—y)2 + p‹ (—y) = 0 → ∂ 2 p‹ (†) ∂† 2 + p‹ (†) = 0 ,

(2b)

†a cos (†i,a ) + sin (†i,a ) = ±2jUi,a · cos(†i,a ) + sin(†i,a ) .

(3b)

The amplitude ratio of the terms in (1) will be at the walls → i,a :

i = −

2jUi cos †i − †a cos †i

2jUi sin †i − †a sin †i

−−−→ †i =0

2jUi , †a

2jUa cos †a + †a cos †a

2jUa cos †a − †a sin †a =− . a = −

2jUa sin †a + †a cos †a 2jUa sin †a + †a sin †a

(5)

Requiring i = a leads to the characteristic equation for the lateral wave number †a :

2jUi sin †i − †a sin †i · 2jUa cos †a + †a cos †a (6a)

− 2jUi cos †i − †a cos †i · 2jUa sin †a + †a sin †a = 0 . Expansion in continued fractions: Insertion of the derivatives in (6a) will give: (4Ui Ua + †a2) sin †i cos †a − cos †i sin †a + 2j†a (Ui + Ua ) sin †i sin †a + cos †i cos †a = 0 .

(6b)

And, after application of the addition theoreme: −(4Ui Ua + †a2) sin(†a − †i ) + 2j†a (Ui + Ua ) cos(†a − †i ) = 0,

(6c)

and of the special value †i = 0: −(4Ui Ua + †a2) sin †a + 2j†a (Ui + Ua ) cos †a = 0 .

(6d)

Division by sin †a (which is possible for †a = n, n = integer; these values are assumed only in ducts with hard walls) returns: 2j(Ui + Ua ) · †a cot †a − (4Ui Ua + †a2) = 0 ,

(6e)

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where the continued fraction expansion can be applied on z · cot z, resulting in the following form of the characteristic equation fo †a : †a2 †a2 †a2 (7) . . . − (4Ui Ua + †a2 ) = 0 . 2j(Ui + Ua ) · 1 − 3− 5− 7− On truncation and putting everything in one fraction, the numerator becomes a polynomial equation in (†a )2 whose solutions, with Im((†a )2 ) ≥ 0 and |char.eq(†a )| < limit (a value of limit ≈ 80 may be taken), are approximations to a set of mode solutions. They can be taken as starters for Muller’s procedure of a numerical solution (see > Sect. J.4) of the characteristic equation (6d). This is a mode-safe and fast computing procedure for a set of mode eigenvalues †a . Grigoryan’s expansion of the characteristic equation: Grigoryan has applied his method for the expansion of the characteristic equation in bent ducts, [Grigoryan (1969)]. It can be generalised and then applied on asymmetrical flat ducts as well; see [Mechel (2006)]. Equation (6a) may be written as: * * *

*

*sin †i sin †a * * sin †a ** 2 *sin †i * * + †a *

4Ui Ua *

cos †i cos †a * †a * * * cos †i * cos *

*sin †i sin †a * *sin †i sin †a * * * * * = 0, − 2jUi †a * + 2jUa †a *

cos †i cos †a * cos †i cos †a * or with the abbreviations for the determinants: * * (n) *sin †i sin(m) †a * * ; (n), (m) ∈ (0), (1) , * Dn,m (†i , †a ) = * (n) cos †i cos(m) †a *

(6f)

(8)

in which (n), (m) ∈ (0), (1) are degrees of derivatives, in the form: 4Ui Ua D0,0(†i , †a ) + †a2 D1,1 (†i , †a )

(6g)

− 2jUi †a D0,1 (†i , †a ) + 2jUa †a D1,0 (†i , †a ) = 0 .

According to Grigoryan, a Taylor expansion is applied on the second column of Dn,m (†i , †) around † = †i = 0, i. e. as a series in ” = († − †i ): * * * * ”k m+k) *sin(n) † †i ** sin i * k! * * k≥0 * Dn,m (†i , †) = ** * ”k (m+k) *cos(n) † cos †i ** i * (9) k! * * k≥0 =

* ”k *sin(n) † i * k! *cos(n) †i k≥0

* sin(m+k) †i ** . cos(m+k) †i *

In the special case † = †a , i. e. for ” = †a = 2—h, one gets for the determinats in (6g): * * (†a )k (†a )k *sin(n) † sin(m+k) † * i i* * : = Bn,m+k (†i ) , (10) Dn,m (†i , †a ) = (n) (m+k) * * †i k! cos †i cos k! k≥0

k≥0

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which may be interpreted as definitions of the coefficient determinants: * * (n) * * *sin z sin(Œ) z * *f (n) (z) f (Œ) (z) * * , * * * Bn,Œ (z) = * (n) = cos z cos(Œ) z* *g(n) (z) g(Œ) (z)*

753

(11)

where f (z), g(z) stand for independent solutions of the non-dimensional Helmholtz equation f (2) (z) + f (z) = 0 ;

g (2) (z) + g(z) = 0 .

(12)

Evidently, the trivial relations BŒ,Œ = 0; Bn,Œ = −BŒ,n hold, and therefore a recursive evaluation is possible: n=0 : B0,0 (z) = 0 ;

B0,1 (z) = W(f (z), g(z)) ;

n=1;: B1,0 (z) = −W(f (z), g(z)) ;

B0,Œ (z) = −B0,Œ−2 (z) ;

B1,1 (z) = 0 ;

(13a)

B1,Œ (z) = −B1,Œ−2 (z) ,

where W(f (z), g(z)) is the Wronski determinant of the pair of solutions f (z), g(z). In the special case f (z) = sin z; g(z) = cos z, with W(f (z), g(z)) = −1, follows: n=0 : B0,0 (z) = 0 ; B0,1 (z) = −1 ; B0,2 (z) = 0 ; B0,Œ (z) = −B0,Œ−2 (z) = − sin(Œ/2) ;

B0,3 (z) = 1 ;

n=1 : B1,0 (z) = 1 ; B1,1 (z) = 0 ; B1,2 (z) = −1 ; B1,Œ (z) = −B1,Œ−2 (z) = cos(Œ/2) .

B1,3 (z) = 0 ;

B0,4 (z) = 0 ; (13b) B1,4 (z) = 1 ;

Thus, the determinants in (10) can be evaluated by: D0,m (†i , †a ) =

(†a )k k≥0

D1,m (†i , †a ) =

k! k!

(†a )k k!

k≥0

(†a )k k≥0

B0,m+k (†i ) = − B1,m+k (†i ) =

(†a )k k≥0

k!

sin((m + k)/2) , (14a)

cos((m + k)/2) ,

and in special cases of the indices n, m: D1,1 (†i , †a ) =

(†a )k (†a )k cos((1 + k)/2) = , (−1)(1+k)/2 k! k!

kodd ≥1

kodd ≥1

(†a ) (†a )k D1,0 (†i , †a ) = cos(k/2) = , (−1)k/2 k! k! k

keven ≥0

D0,0 (†i , †a ) = −

(14b)

keven ≥0

(†a )k (†a )k sin(k/2) = , (−1)(1+k)/2 k! k!

kodd ≥1

kodd ≥1

(†a )k (†a )k sin((1 + k)/2) = − . D0,1 (†i , †a ) = − (−1)k/2 k! k! keven ≥0

keven ≥0

(14c)

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The characteristic equation (6g) will be, after these transformations: (†a )k (†a )k 4Ui Ua + †a2 (−1)(1+k)/2 (−1)k/2 + 2j†a (Ui + Ua ) =0. k! k! kodd ≥1

(15a)

keven ≥0

On truncation at k = khi this returns an odd polynomial in †a without a constant term (in †a ). Since †a = 0 can be excluded as solution, one may divide (15a) by †a , leading to: (†a )k−1 (†a )k + 2j (Ui + Ua ) =0, (15b) (−1)(1+k)/2 (−1)k/2 4Ui Ua + †a2 k! k! kodd ≥1

keven ≥0

and with the abbreviations keven → 2‰ ; kodd → 2‰ + 1 ; ‰ = 0, 1, 2, . . . ; to: (†a )2‰ (†a )2‰ + 2j (Ui + Ua ) = 0, (−1)‰ (−1)‰ − 4Ui Ua + †a2 (2‰ + 1)! (2‰)! ‰≥0 ‰≥0 or, with collected sums, finally: 2 2 ‰ U + † 4U († ) i a a 2j (Ui + Ua ) − =0. (−1)‰ a (2‰)! (2‰ + 1) ‰≥0

(15c)

(15d)

The degree of the polynomial in (†a2) on truncation at ‰hi is ‰hi + 1. The polynomial solutions †a may be taken as approximations to mode solutions and as starters in Muller’s procedure, after solutions with Im((†a )2 ) < 0 and/or |char.eq| > limit (limit ≈ 80 is feasible) are rejected. The number of unusable polynomial solutions (for equal limits khi of truncation) is higher in Grigoryan’s expansion than with the continuedfraction expansion, and the set of usable mode solutions is less “compact” (i. e. more solutions missing) than with the continued-fraction method.

J.42

Mode Sets in Annular Ducts with Unsymmetrical, Locally Reacting Lining

See also: Mechel (2006); Grigoryan (1969)

The object is a ring-shaped duct with different locally reacting linings on opposite sides. A similar object is treated in > Sect. J.16, where the least attenuated mode was sought. Complete sets of mode solutions shall be determined here. Some methods will be displayed for the numerical evaluation of mode eigenvalues (mode solutions).They all transform the transcendental characteristic equation to a polynomial equation whose solutions shall serve as starters in Muller’s procedure of numerical solution of the exact characteristic equation (see > Sect. J.4). (More methods and details can be found in [Mechel (2006)].) Formulations of modes p‹ (r, œ, z) in an annular duct preferably use the Bessel and Neumann functions, J‹ (kr r), Y‹ (kr r), for the radial factor, with the unknown radial wave number kr (k0 , Z0 = free field wave number and wave impedance); the azimuthal mode index ‹ is mostly given by the angular distribution of the excitation: p‹ (r, œ, z) = J‹ (kr r) + Y‹ (kr r) e−j‹œ e− z ; 2 = kr2 − k02 , (1) kr

kr ∂p =j J‹ (kr r) + Y ‹ (kr r) e−j‹œ e− z . Z0 v‹r (r, œ, z) = j k0 ∂(kr r) k0

Duct Acoustics

The boundary conditions at the walls with wall admittances Gi,a at r = ri,a are: kr ri,a J ‹ (kr ri,a ) + Y ‹ (kr ri,a ) = ±jk0 ri,a Z0 Gi,a J‹ (kr ri,a ) + Y‹ (kr ri,a )

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755

(2a)

(primes indicate derivatives). Introduce non-dimensional quantities: kr r → ;

kr ri,a → i,a ;

k0 ri,a · Z0 Gi,a → Ui,a ;

Š = ri /ra = i /a .

By elimination of the radial term factor from both boundary conditions: i,a J ‹ (i,a ) + Y ‹ (i,a ) = ±jUi,a J‹ (i,a ) + Y‹ (i,a ) ,

(2b)

one gets the characteristic equation for the mode eigenvalues a ; i = Š·a with given Š: Ui J‹ (i ) + ji J ‹ (i ) · Ua Y‹ (a ) − ja Y ‹ (a ) (3a) − Ui Y‹ (i ) + ji Y ‹ (i ) · Ua J‹ (a ) − ja J ‹ (a ) = 0 , or grouped differently with determinants as factors: * *

* *

*J ( ) J‹ (a ) * * * * + i · a *J‹ (i ) J‹ (a ) * Ui · Ua ** ‹ i * * Y‹ (i ) Y‹ (a ) Y‹ (i ) Y‹ (a )* * * * * *J‹ (i ) J ‹ (a ) * *J ‹ (i ) J‹ (a ) * * * = 0. * + j U − ja Ui ** i a*

Y‹ (i ) Y ‹ (a )* Y‹ (i ) Y‹ (a )*

(3b)

With the determinant symbols in which (n), (m) ∈ (0), (1) are orders of derivatives: * * (n) * *J‹ (i ) J(m) ‹ (a ) * * Dn,m (‹, i, a ): = * (4) * ; (n), (m) ∈ (0), (1) , (m) * *Y(n) ‹ (i ) Y‹ (a ) the characteristic equation reads: i a D1,1 (‹, i , a ) − ja Ui D0,1 (‹, i, a ) + ji Ua D1,0 (‹, i, a ) + Ui · Ua D0,0 (‹, i , a ) = 0.

(3c)

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Expansion of characteristic equation with Grigoryan’s method: We generalise the mode formulation (1) by symbolic writing f‹ (kr r), g‹ (kr r) for the radial functions, instead of Bessel and Neumann functions as in (1). These functions may also be the Hankel functions of both kinds or Bessel functions with positive and negative non-integer orders ‹: = f‹ (kr r) + g‹ (kr r) e−j‹œ e− z ; 2 = kr2 − k02 , kr

kr ∂p f‹ (kr r) + g‹ (kr r) e−j‹œ e− z . Z0 v‹r (r, œ, z) = j =j k0 ∂(kr r) k0

p‹ (r, œ, z)

(5)

The radial functions f‹ (kr r) = f‹ () and g‹ (kr r) = g‹ () satisfy the Bessel differential equations: 2 f‹

+ f‹ + (2 − ‹ 2)f‹ ≡ 0 ,

(6)

2 g‹

+ g‹ + (2 − ‹ 2)g‹ ≡ 0 ,

and have a non-zero Wronski determinant W(f‹ , g‹ ) = f‹ g‹ − f‹ g‹ . The boundary conditions at the locally absorbing walls at r = ri,a : f‹ (kr ri,a ) + g‹ (kr ri,a ) ! k0 Z0 v‹r (ri,a , œ, z) =j = ∓ Z0 Gi,a p‹ (ri,a , œ, z) f‹ (kr ri,a ) + g‹ (kr ri,a ) kr

(7)

lead with the abbreviations kr ri,a = i,a ; k0 ri,a Z0 Gi,a = Ui,a ; Š = ri /ra to the characteristic equation (3c) in which the determinants are now: * (n) *f (i ) Dn,m (‹, i, a ): = ** ‹(n) g‹ (i )

* f‹(m) (a ) ** ; g‹(m) (a )*

(n), (m) ∈ (0), (1) .

(8)

We perform a Taylor series expansion of the first column around a in the variable = ( − a ) having the special value x = (a − i ) = a (1 − Š): * * * * (−1)k (n+k) k * (m) f‹ (a ) · x f‹ (a ) ** * * * k≥0 k! * Dn,m (‹, i, a ) = ** k * (−1) (n+k) k (m) * g‹ (a ) · x g‹ (a )** * * * k≥0 k! * * (9) (−1)k *f (n+k) (a ) f (m) (a ) * ‹ * · xk *‹ = k! *g(n+k) (a ) g(m) (a )* ‹

k≥0

=

(−1)k k≥0

k!

‹

Bn+k,m (‹, a ) · xk .

This defines the coefficient determinants BŒ,m : * (Œ) *f () BŒ,m (‹, ) = ** ‹(Œ) g‹ ()

* f‹(m) () ** . g‹(m) ()*

(10)

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757

For their evaluation, one writes down the Œ-th derivative of the Bessel differential equation (6), e. g. for f (): ∂ Œ 2 f

+ f + (2 − ‹ 2)f /∂Œ = Œ · (Œ − 1) f (Œ−2) () + 2Œ f (Œ−1) () + (Œ 2 − ‹ 2 + 2 ) f (Œ) () + (2Œ + 1) f (Œ+1) () + 2 f (Œ+2) () −−→ (−‹ 2 + 2) f (0) () + f (1) () + 2 f (2) () Œ=0

−−→ 2 f (0) () + (1 − ‹ 2 + 2 ) f (1) () + 3 f (2) () + 2 f (3) ()

(11)

Œ=1

−−→ 2 f (0) () + 4 f (1) () + (4 − ‹ 2 + 2 ) f (2) () + 5 f (3) () + 2 f (4) () Œ=2

−−→ 6 f (1) () + 6 f (2) () + (9 − ‹ 2 + 2 ) f (3) () + 7 f (4) () + 2 f (5) () Œ=3

−−→ 12 f (2) () + 8 f (3) () + (16 − ‹ 2 + 2 ) f (4) () + 9 f (5) () + 2 f (6) (). Œ=4

These expressions are used for the cross-product: g(m) · ∂ Œ 2 f

+ f + (2 − ‹ 2)f /∂Œ − f (m) · ∂ Œ 2 g

+ g + (2 − ‹ 2)g /∂Œ (12a) together with the coefficient determinants BŒ,m (‹, ): g(m) · ∂ Œ 2 f

+ f + (2 − ‹ 2)f /∂Œ − f (m) · ∂ Œ 2 g

+ g + (2 − ‹ 2)g /∂Œ = Œ · (Œ − 1)BŒ−2,m (‹, ) + 2Œ BŒ−1,m (‹, ) + (Œ 2 − ‹ 2 + 2) BŒ,m (‹, )

(12b)

+ (2Œ + 1) BŒ+1,m (‹, ) + 2 BŒ+2,m (‹, ) . For Œ = 0 both leading terms vanish; for Œ = 1 the first term is zero. The sum (12b) must vanish identically; therefore (12b) represents an iteration for BŒ,m (‹, ) ; Œ = 2, 3, 4, . . .: (Œ − 2)2 + 2 − ‹ 2 2Œ − 3 BŒ−2,m BŒ−1,m + BŒ,m (‹, ) = − 2 2(Œ − 2) (Œ − 2)(Œ − 3) Œ = 2, 3, . . . + BŒ−4,m ; BŒ−3,m + m = 0, 1 2 2 2 −‹ 1 (13) B1,m + B0,m −−→ B2,m (‹, ) = − Œ=2 2 z 1 + 2 − ‹ 2 2 3 B2,m + B B + −−→ B3,m (‹, ) = − 1,m 0,m Œ=3 2 z 4 + 2 − ‹ 2 4 2 5 B3,m + B B + + B −−→ B4,m (‹, ) = − 2,m 1,m 0,m Œ=4 2 2 and so on. The required starting values B0,m , B1,m of the recursion follow from the identities BŒ,Œ = 0 ; BŒ,m = −Bm,Œ and from the Wronski determinant W(f‹ (), g‹ ()): 0 ; m = 0 B0,m (‹, ) = ; W f‹ (), g‹ () ; m = 1 (14) −W f‹ (), g‹ () ; m = 0 . B1,m (‹, ) = 0 ; m = 1

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The Wronski determinant is the point where the choice of the pair of functions f (), g() enters into the evaluation: 2 4j (2) W J‹ (), Y‹ () = ; W H(1) . (15) ‹ (), H‹ () = − In case of the function pair f‹ () = J‹ (); g‹ () = Y‹ (): the initial terms of the iteration are: m=0 :

2 2 B10 = − ; B20 = ; 2 2 2+‹ 2 4 3 + 3‹2 1− = ; B40 = −1 + ; 2 2 2

B00 = 0 ; B30

m=1 : 2 2 ‹2 ; B11 = 0 ; B21 = −1 + 2 ; B01 = 2 2 3 + 2‹ 2 q4 + 11‹ 2 3‹ 2 + 1− B31 = 1 − 2 ; B41 = . 2 2 4

(16a)

(16b)

Although the coefficients BŒ,m (‹, a ) also contain odd powers of a , the characteristic equation is made up of even powers (a2 )k ; so it is advisable to solve for (a2 ). Then polynomial solutions with Im(a2 ) < 0 can be rejected. Some of the polynomial solutions may produce large magnitudes |char.eq| > lim of the characteristic equation. They should be rejected if a limit of about lim ≈ 80 is exceeded if the polynomial solutions are used as starters for Muller’s procedure (see > Sect. J.4) when solving the characteristic equation; a direct use of polynomial solutions as mode solutions in further field evaluations may be possible for lim ≤ 0.001. In the latter case the summation limit khi must be sufficiently high: about khi ≈ (3 to 4) · k0 ra with larger values of k0ra (> 4), and khi ≈ (12 to 20) with small values of k0ra . It is a principal disadvantage of Grigoryan’s method that it is based on a Taylor series approximation (in (9)) which assumes small values of x = (a − i ). This drawback is avoided with the next method. Expansion of characteristic equation with theorem of multiplication of cylindrical functions: Let the couple of radial functions again be f‹ () = J‹ () ; g‹ () = Y‹ (). We start with the characteristic equation (3c) with the determinants in (4). For any of the basic cylindrical functions C‹ () (Bessel, Neumann, Hankel functions) and a factor Š in the argument with |Š2 − 1| < 1 the theorem of multiplication reads: C‹ (Š) = Š‹

(−1)k

k≥0

(Š2 − 1)k k C‹+k () , 2k k!

(17)

and from this the derivative ∂/∂(Š) = (1/Š) · ∂/∂: C ‹ (Š) = Š‹−1

k≥0

(−1)k

. (Š2 − 1)k - k−1 k

k C () + C () . ‹+k ‹+k 2k k!

(18)

Duct Acoustics

This, when inserted in the determinant (4), gives: * * *J‹ (Ša ) J(m) * ‹ (a ) * D0,m (‹, i , a ) : = ** (m) Y‹ (Ša ) Y‹ (a )* * * 2 k k * * J(m) ‹ (a ) * k (Š − 1) a *J‹+k (a ) ‹ (−1) = Š *Y‹+k (a ) Y(m) (a )* ; 2k k! ‹ k≥0 * * (1) (m) *J (Ša ) J‹ (a ) * * D1,m (‹, i , a ) : = ** ‹(1) * Y‹ (Ša ) Y(m) ‹ (a ) * * 2 k * * (Š − 1) J(m) ‹ (a ) * k k−1 *J‹+k (a ) ‹−1 = Š (−1) ka * * Y‹+k (a ) Y(m) ( ) 2k k! a ‹ k≥0 * * *J ( ) J(m) ( ) * a a * * ‹ + ak * ‹+k * . ( *Y ‹+k (a ) Y(m) a )* ‹

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(19a,b)

We introduce the “subdeterminants” * * *J(n) () J(m) () * * ‹+k * ‹ * * =: dn,m (‹, k, ) , * (n) * *Y‹+k () Y(m) ‹ ()*

(20)

and obtain for the special cases (with the Wronski determinant W(J‹ (), Y‹ ()) ): n = 0 ;

k = 0 : 0 ; m = 0 d0,m (‹, 0, ) = W(J‹ (), Y‹ ()) ; n = 1 ;

k = 0 : −W(J‹ (), Y‹ ()) ; d1,m (‹, 0, ) = 0 ; m = 1

m = 1

(21a)

;

m = 0

.

(21b)

From there one gets, with the recursions for derivatives of cylindrical functions C‹ (): n = 0 ;

k = 1 : ‹ C‹+1 () = C‹ () − C ‹ () ; z ‹ d0,m (‹, 1, ) = d0,m (‹, 0, ) − d1,m (‹, 0, ) ; n = 1 ;

(21c)

k = 1 :

‹+1 C‹+1 () ; ‹+1 d0,m (‹, 1, ) ; d1,m (‹, 1, ) = d0,m (‹, 0, ) −

C ‹+1 () = C‹ () −

k + 1 : ‹+k C‹+k () − C ‹+k () ; C‹+k+1 () = ‹+k d0,m (‹, k, ) − d1,m (‹, k, ) ; d0,m (‹, k + 1, ) =

(21d)

n = 0 ;

(21e)

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k + 1 :

‹+k+1 C‹+k+1 () ; ‹+k+1 d0,m (‹, k + 1, ) . d1,m (‹, k + 1, ) = d0,m (‹, k, ) − C ‹+k+1 () = C‹+k () −

(21f)

Relations (21) show that the subdeterminants dn,m (‹, k, ) can be evaluated by recursion. ⎧ ⎪ ⎪ ‹ + k d0,m (‹, k, ) − d1,m (‹, k, ) ; n = 0 ⎨ dn,m (‹, k + 1, ) = ; ‹+k+1 ⎪ (22) ⎪ d0,m (‹, k + 1, ) ; n = 1 ⎩d0,m (‹, k, ) − k = 0, 1, 2, 3, . . . beginning at k = 1 with: ⎧‹ ⎪ ⎨ d0,m (‹, 0, ) − d1,m (‹, 0, ) ; n = 0 ; dn,m (‹, 1, ) = ‹+1 ⎪ ⎩d0,m (‹, 0, ) − d0,m (‹, 1, ) ; n = 1 and at k = 0 with: ⎧ 0 ; m = 0 ⎪ ⎪ ⎪ ⎨ W(J‹ (), Y‹ ()) ; m = 1 ; n = 0 dn,m (‹, 0, ) = −W(J‹ (), Y‹ ()) ; m = 0 ⎪ ⎪ ; n = 1 ⎪ ⎩ 0 ; m = 1 ⎧ 0 ; m = 0 ⎪ ⎪ ; n = 0 ⎨ 2/() ; m = 1 . = −2/() ; m = 0 ⎪ ⎪ ; n = 1 ⎩ 0 ; m = 1

(23)

(24)

The main determinants (19a,b) then become: D0,m (‹, i , a ) = Š‹

(Š2 − 1)k ak d0,m (‹, k, a ) ; (−1)k 2k k! k≥0

(Š2 − 1)k k−1 ka d0,m (‹, k, a ) 2k k! k≥0 + ak d1,m (‹, k, a ) .

D1,m (‹, i , a ) = Š‹−1

(−1)k

(25a,b)

With these the characteristic equation finally is:

(Š2 − 1)k ak / Ui · Ua + jUa k d0,0 (‹, k, a ) k 2 k! k≥0 0 + a k − jUi d0,1 (‹, k, a ) + jUa a d1,0 (‹, k, a ) + a2 d1,1 (‹, k, a ) = 0.

Š‹

(−1)k

(26)

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The summation over k and the iteration of the dn,m (‹, k, ) in k may run in parallel. Special cases for different summation limits khi are: khi = 0 : 2 −j {Ui + Ua } = 0 ; khi = 1 : 2 −j {Ui + Ua } +

(27a)

(27b)

0 (1 − Š ) / Ui · Ua + jUa + ‹(1 − jUi ) + a2 − jUa (‹ + 1) − ‹(‹ + 1) = 0 ; 2

khi = 4 : b c a − · a4 + · a2 − = 0

(27c)

with factor terms: a=

(1 − Š2 )3 (1 − Š2 )4 + (4 + 4‹ + j(Ui + Ua )) , 24 192

b = (1 − Š2 ) +

(28a)

(1 − Š2 )2 (2‹ + j(Ui + Ua )) 4

+

(1 − Š2 )3 2 5‹ + 4jUi − Ua (Ui − 2j) + 3j‹(Ua + Ui − 2j) 24

+

(1 − Š2 )4 (2 + ‹) 3‹ 2 + 3jUi − Ua (Ui − j) + 2j‹(Ua + Ui − 2j) , 48

(28b)

c = 2j(Ui + Ua ) + (1 − Š2 )(‹ + jUi )(‹ + jUa ) (1 − Š2 )2 (1 + ‹)(‹ + jUi )(‹ + jUa ) 2 (1 − Š2 )3 + (2 + 3‹ + ‹ 2)(‹ + jUi )(‹ + jUa ) 6 (1 − Š2 )4 (6 + 11‹ + 6‹ 2 + ‹ 3 )(‹ + jUi )(‹ + jUa ) . + 24 +

(28c)

One obtains even-order polynomials in (a2 ); the rank is (a2 )khi/2 . The odd powers which appear in the subdeterminants cancel each other during the combination to the characteristic equation. This procedure requires no transcendental functions. The coefficients of the powers of (a2 ) in the polynomial approximation of the characteristic equation are of moderate magnitudes. The polynomial solutions can be used as starters for Muller’s procedure up to about Re(a ) = khi . This procedure computes slower than Grigoryan’s procedure; however, it returns for equal khi more useable approximations, and it is not so sensitive to large a as Grigoryan’s procedure. A reasonable point of transition from Grigoryan’s procedure to the present procedure may be at about |a | ≈ 2.

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Mode Sets in Annular Ducts via Mode Sets in Flat Ducts with Unsymmetrical Lining


The problem in the numerical evaluation of mode sets in annular ducts consists in finding reliable starters for Muller’s procedure in solving the characteristic equation. Under the conditions of narrow ring width, i. e. Š = ri /ra > 0.75, and not too small k0ra , i. e. k0ra > 10, a fast computing method for finding reliable starters makes use of the similarity of transversal mode profiles in “equivalent” annular and flat ducts.

This sketch makes plausible that an annular duct defines an equivalent flat duct; however, several annular ducts (with different radii) may have the same equivalent flat duct. Therefore the equivalence will be best for large radii and small gap widths. Geometrical equivalences are, with the notations from the previous sections for annular ducts: h = (ra − ri )/2 = ra (1 − Š)/2 ;

k0 h = k0 ra (1 − Š)/2 ;

k0 ra =

2k0 h , (1 − Š)

(1)

from which follow the correspondences of modal quantities in annular ducts and in unsymmetrical flat ducts: ∧

kr ra − kr ri = kr (ra − ri ) = a − i = a (1 − Š) = —(ra − ri ) = —2h = †a , ∧

a =

†a . 1−Š

(2)

The procedure works well under the mentioned conditions.

J.44

Bent, Flat Ducts with Locally Reacting Lining

See also: Mechel (2006); Grigoryan (1969)

The objects of this section are modes in circularly bent, lined ducts running in the azimuthal direction (bow modes). The free duct cross-section is between the inner and outer radii ri , ra ; it is unlimited in the z direction (or the mode fields are constant in the

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z direction). The linings at the duct walls are defined by (radial) wall admittances Gi , Ga . Generally the admittances are different from each other (due to the different radii), though the construction of the linings may be similar. The duct is anechoic in the œ direction.

A general formulation of a mode in the cylindrical co-ordinate system {r, œ, z} is composed of factor functions: p(r, œ, z, t) = R(r) · Z(z) · ¥ (œ) · e+j–t

(1)

with radial functions like: (2) R(r) = H(1) ‹ (kr r) + H‹ (kr r)

solutions of

∂ 2 R 1 ∂R ‹2 2 + k + − R = 0 , (2) r ∂r2 r ∂r r2

solutions of

∂2Z + kz2 Z = 0 . ∂z2

(3)

Both parts are connected by the secular equation: k02 = kr2 + kz2.

(4)

= J‹ (kr r) + Y‹ (kr r) = J‹ (kr r) + J−‹ (kr r) and lateral functions like: Z(z) = ‚1 e+jkz z + ‚2 e−jkz z = ‚1 cos(kz z) + ‚2 sin(kz z)

The mode also has azimuthal functions like: ¥ (œ) = —1 e+j‹œ + —2 e−j‹œ = —1 sin(‹œ) + —2 cos(‹œ)

solutions of

∂ 2¥ + ‹2¥ = 0 . ∂œ2

(5)

The assumedly constant profile in the z direction is obtained in the special case kz = 0; ‚1 = 1; ‚2 = 0 in (3),and the anechoic terminations in the œ direction may be represented by —1 = 0; —2 = 1 in (5), and with Re(‹) ≥ 0, Im(‹) ≤ 0 the mode, propagating in +œ direction, obeys the far field condition. The modes p‹ (r, œ) considered further have the form: p‹ (r, œ) = R(r) · ¥ (œ) = J‹ (kr r) + Y‹ (kr r) e−j‹œ , (6) kr

kr ∂p =j J‹ (kr r) + Y ‹ (kr r) e−j‹œ . Z0 v‹r (r, œ) = j k0 ∂(kr r) k0

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The boundary conditions at the walls at ri , ra are for given (radial) admittance values Gi , Ga of the locally reacting walls: J ‹ (kr ri,a ) + Y ‹ (kr ri,a ) ! Z0 v‹r (ri,a , œ, z) k0 =j = ∓ Z0 Gi,a . p‹ (ri,a , œ, z) J‹ (kr ri,a ) + Y‹ (kr ri,a ) kr

(7)

The coefficient must be identical in both of them; thus,eliminating gives the following characteristic equation for the indices ‹ of the modes: k0 k0

Z0 Gi J‹ (kr ri ) + jJ‹ (kr ri ) · Z0 Ga Y‹ (kr ra ) − jY‹ (kr ra ) kr kr (8a) k0 k0

− Z0 Gi Y‹ (kr ri ) + jY‹ (kr ri ) · Z0 Ga J‹ (kr ra ) − jJ‹ (kr ra ) = 0 . kr kr It has a formal similarity with the characteristic equation for modes propagating in the z direction in annular ducts (see previous sections), so some transformations may be similar also, but the unknown mode number here is ‹, whereas it is kr in annular ducts. Define the non-dimensional quantities: kr ri,a → i,a ; k0ri,a ·Z0 Gi,a → Ui,a ; Š = ri /ra < 1 with i = Š · a . The characteristic equation (8a) can be transformed into: * * *

* * J‹ (i ) J ‹ (a ) * * J‹ (i ) J ‹ (a ) * * * * * i · a *

− ja Ui * Y‹ (i ) Y ‹ (a ) * Y‹ (i ) Y ‹ (a ) * * *

* * J‹ (i ) J‹ (a ) * * J ( ) J‹ (a ) * * + Ui · Ua ** ‹ i + ji Ua *

* Y‹ (i ) Y‹ (a ) Y‹ (i ) Y‹ (a ) or, with the determinants in it, written: * * (n) * * J (i ) J(m) ‹ (a ) * ; Dn,m (‹, i, a ): = ** ‹(n) (m) Y‹ (i ) Y‹ (a ) *

* * * = 0, *

(n), (m) ∈ (0), (1) ,

(8b)

(9)

in which (n), (m) ∈ (0), (1) are orders of derivatives; the characteristic equation may then be written as i a D1,1 (‹, i , a ) − ja Ui D0,1(‹, i , a )

(8c)

+ ji Ua D1,0 (‹, i, a ) + Ui · Ua D0,0 (‹, i , a ) = 0 .

The coefficient determinants in this equation, when nullified individually, represent the characteristic equations for the special cases of the bow duct with ideally reflecting walls: • • • •

first determinant = 0: second determinant = 0: third determinant = 0: fourth determinant = 0:

inner and outer walls hard; inner wall soft, outer wall hard; inner wall hard, outer wall soft; inner and outer walls soft;

denoted h-h; denoted w-h; denoted h-w; denoted w-w.

Thus, these special cases are singularities of the general equation. This may become important in solution methods which start with known solutions in hard bow ducts and then proceed iteratively to absorbing walls because the evaluation may approach such singularities during the iteration.

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Of some practical interest may be cases of bow ducts with one wall hard and the other wall absorbing. If the outer wall at ra is hard, i. e., Ua = 0, then the characteristic equation is: i · a J ‹ (i ) · Y ‹ (a ) − Y ‹ (i ) · J ‹ (a ) (9a) − ja Ui J‹ (i ) · Y ‹ (a ) − Y‹ (i ) · J ‹ (a ) = 0 . If the inner wall at ri is hard, i. e., Ui = 0, then: i · a J ‹ (i ) · Y ‹ (a ) − Y ‹ (i ) · J ‹ (a ) + ji Ua J ‹ (i ) · Y‹ (a ) − Y ‹ (i ) · J‹ (a ) = 0 .

(9b)

One sees that a procedure for arriving at a solution of these equations which starts an iteration with real or imaginary solutions ‹ in a double-sided hard duct will produce dramatic changes in ‹ in its first step. So the procedure may fail. Because, on the other hand, one may expect that the eigenvalues ‹ of the “ideal" cases will bracket the eigenvalues of the lined bow duct anyway, and because some published approximations for mode solutions ‹ in hard-walled or soft-walled ducts proved to be erroneous, we first consider some “ideal cases". Special case h-h: hard walls at ri and ra : With Ui = Ua = 0 and Š = ri /ra < 1; i = Š · a , the equation for the eigenvalues ‹ is: J ‹ (Ša )Y ‹ (a ) − J ‹ (a )Y ‹ (Ša ) = 0 , or, with the recursions for the derivatives of Bessel and Neumann functions: ‹ ‹ J‹ (Ša ) − J1+‹ (Ša ) Y‹ (a ) − Y1+‹ (a ) Ša a ‹ ‹ J‹ (a ) − J1+‹ (a ) Y‹ (Ša ) − Y1+‹ (Ša ) = 0 , − a Ša and the terms sorted for ‹ and ‹2 are: ‹ 2 J‹ (a ) Y‹ (Ša ) − J‹ (Ša ) Y‹ (a ) + ‹a J‹ (Ša ) Y1+‹ (a ) − J1+‹ (a ) Y‹ (Ša ) + Š J1+‹ (Ša ) Y‹ (a ) − J‹ (a ) Y1+‹ (Ša ) + Ša2 J1+‹ (a ) Y1+‹ (Ša ) − J1+‹ (Ša ) Y1+‹ (a ) = 0 .

(10a)

(10b)

(10c)

The bracketed factor of ‹ 2 is just the characteristic equation for the double-sided softwalled bow duct (w-w); thus, the solutions of the case w-w are pole positions of the characteristic equation for the hard-hard duct (h-h). An unambiguous survey of the positions of solutions may be obtained by plotting in a 3D plot the values of -lg |char.eq.| over a and ‹, where ‹ is either real or imaginary (both possibilities exist for solutions).

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3D plot of −lg|char.eq.| of characteristic equation for a bow duct with hard walls on both sides over a and real ‹ for Š = ri /ra = 0.5. The positions of the maxima, which belong to solutions, are collected from the plot list in the floor surface of the enclosing cube; the ceiling surface contains the points after improvement by Muller’s procedure The next graph is similar to the plot above, but now for negative-imaginary ‹ = −j|‹|.

Similar plot as above, but now over negative imaginary values of ‹

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The next diagram combines both branches of solutions for real and negative imaginary values of ‹. The curves are from approximations (see below), the points are solutions after improvement by Muller’s procedure.

Real and negative imaginary mode solutions ‹ in a bow duct with hard walls on both sides, for Š = ri /ra = 0.5. Solid lines: real ‹; dashed: imaginary ‹. curves: from approximations; points: after improvement with Muller’s procedure The next graph is similar to the previous graph, but it combines, for Š = 0, 5, mode solutions ‹ for bow ducts with two hard walls (h-h; solid lines), inner hard wall and outer soft wall (h-w; long dashes), and both walls soft (w-w; short dashes). The solutions for the w-w duct partly coincide with solutions for other configurations.

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Real and negative imaginary mode solutions ‹ in a bow duct with hard walls on both sides (solid), hard inner and soft outer wall (long dashes), soft walls on both sides (short dashes), for Š = ri /ra = 0.5. The lowest-order solution in the h-h duct has only a real branch The parabolic branches of real solutions ‹ in the above diagram belong to propagating modes; they will be numbered with a mode order index s = (0), 1, 2, . . . (left to right). The branch aiming at (a , ‹) = (0, 0) with the mode order s = 0 exists only in the duct h-h with two hard walls; it represents there the fundamental mode. The elliptic branches of imaginary solutions ‹ = −j|‹| are numbered s = −1, −2, . . . (left to right); they belong to cut-off modes. The parameters a , Š must exceed some limit values (the values at the dips at ‹ → 0 in the diagram) for the existence of higher propagating modes. Approximations to mode solutions ‹ in bow ducts with ideally reflecting walls: Duct h-h, both walls hard: Characteristic equation (10). Suitable Muller starters ‹ ≈ {z1 , z2, z3 } for the real (parabolic) branches are: / 0 s = 0: {z1 , z2, z3 } = x , 2x /(1 + Š) + 0.001, 2x /(1 + Š) ; x = 0.00001, (11) / 0 s > 0: {z1 , z2, z3 } = x + 0.1, x + 0.05 , x ; x = s/(1 − Š). The imaginary (elliptic) branches are approximated by quarter ellipses. The axes |‹0| = |‹|(a → 0, Š; s)| are obtained from the development of the characteristic equation at a →0: char.eq. −→ a →0

−j|‹|2 (Š2j|‹| − 1) Š1+j|‹| sinh(|‹|) (1 + j|‹|) (1 − j|‹|) 2|‹| sin |‹| ln(Š) . = Š

(12)

Duct Acoustics

Solutions thereof with |‹0| = 0 are

|‹0 | = −s/ ln(Š) ;

s = 1, 2, . . . .

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(13)

The other axes a0 at ‹ → 0 are the solutions of J1 (Š)Y1 () − J1 ()Y1(Š) = 0 from the literature or from mathematical computer programs, or they may be approximated by: ⎧ ƒ ⎪ −3 ⎪ ⎨ 1 + Š ; ƒ ≈ 10 ; s = 0 . (14) a0 ≈ ⎪ ⎪ ⎩ s + 3(1 − Š) ; s>0 1−Š 8sŠ Thus, the quarter elliptic arcs for ‹ = −j · |‹| are given by: ⎧ ⎨|‹0| = |‹0 |(Š, s) = −s/ ln(Š) 2 ; |‹| = |‹0| 1 − a /a0 ; ⎩a0 = a0 (Š, s) (15) ( s = 1, 2, 3, . . . . 0 < Š < 1; 0 < a ≤ a0 For small a (< 4) and not too small Š = i /a = ri /ra the parabolic branches can be approximated with the positive real approximations: 2 ‹ ≈ (77 − 10a2) − Š(137 − 26a2) + Š2 (97 − 22a2) − Š3 (25 − 6a2 ) − 4(1 − Š)2 (7 − 5Š)a2 (16) · (30 − 3a2 ) − Š(36 − 7a2 ) + Š2 (24 − 5a2) − Š3 (6 − a2 ) 2.1/2 + (77 − 10a2) − Š(137 − 26a2) + Š2 (97 − 22a2 ) − Š3 (25 − 6a2 ) / 2(5Š − 7)(Š − 1)2 . The approximations should be used as "Muller starters" for the numerical solution of the characteristic equation (10). Duct h-w, inner wall hard, outer wall soft: Characteristic equation: J ‹ (Ša )Y‹ (a ) − J‹ (a )Y ‹ (Ša ) = 0 ;

Š < 1 .

(17)

The limit values a0 = a (‹ → 0; Š, s) (for the dip points on the a axis) may be approximated by: a0 ≈

(s − 1/2) 3+Š + ; 1−Š 4Š(2s − 1)

s = 1, 2, . . . ,

(18)

and the limit values |‹0| = |‹|(a →0, Š; s) (on the ‹ axis) for the branches of cut-off modes are obtained by series expansion in a of Eq. (17) and nullifying the a -free term: char. eq. ≈

(1 + Š2j|‹| ) (Šj|‹| + Š−j|‹| ) = Šj|‹| ! 2 2 |‹| ln(Š2 ) = cos(|‹| ln(Š)) ≈ 0 = cos 2

(19)

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with solutions

|‹0| =

−(2s − 1)/2 ; ln(Š)

s = 1, 2, 3, . . . .

(20)

The branches of the cut-off modes with imaginary ‹ are approximated by quarter ellipses: 2 |‹| = |‹0| 1 − a /a0 ;

s = 1, 2, 3, . . . . 0 < Š < 1 ; 0 < a ≤ a0

(21)

The parabolic branches of the propagating modes may be approximated for low a (< 5) and not too small Š by positive real values of: + ‹ 2 ≈ (580 − 28a2) − Š(1145 − 74a2) + Š2 (850 − 64a2) − 9Š3 (25 − 2a2) − 16(17425 − 1190a2 + 9a4) − 8Š(142625 − 12035a2 + 108a4) + 15Š2 (133135 − 13400a2 + 144a4 ) − 20Š3 (95975 − 11092a2 + 144a4) + 10Š

4

(107215 − 13672a2

+

216a4)

5

3730a2

− 12Š (27475 − , 1/2 + 9Š6 (4825 − 680a2 + 16a4) / 10(3Š − 4)(Š − 1)2 .

(22)

+ 72a4 )

Duct w-w, both walls soft: The characteristic equation is: ; Š < 1 . J‹ (Ša )Y‹ (a ) − J‹ (a )Y‹ (Ša ) = 0

(23)

The limit values a0 = a (‹ →0, Š; s) on the a axis may be obtained from published solutions of Eq. (23) in the special case ‹ = 0, or from the approximation (they get better the larger are s and Š): ⎧ s ⎪ ; Š ≤ 0.05 ⎨ 1−Š a0 (s) ≈ ; s = 1, 2, . . . . 1−Š ⎪ s ⎩ − 1 − Š 8sŠ

(24)

The limit solutions ‹0 for a → 0 are obtained from the a -free term of the series expansion of Eq. (23): char.eq. ≈ j

sin(|‹| ln Š) ! Šj|‹| − Š−j|‹| −1 + Š2j|‹| = −2 = j = 0 |‹|Šj|‹| |‹| |‹|

(25)

with solutions |‹0 | = −s/ ln Š ;

s = 1, 2, . . . ;

0 < Š < 1 .

(26)

Duct Acoustics

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771

The elliptic branches of the cut-off modes with imaginary ‹ can be approximated with elliptic arcs (21) using (24) and (26). The parabolic branches for propagating modes can be approximated for small a (< 6) and not too small Š by positive real values of: + ‹ 2 ≈ (245 − 10a2 ) − Š(425 − 26a2) + Š2 (295 − 22a2) − Š3 (75 − 6a2) − (43561 − 2520a2 + 16a4 ) − 2Š(86317 − 6256a2 + 48a4) + Š2 (281591 − 25528a2 + 240a4) − 4Š3 (62571 − 6848a2 + 80a4 ) 4

+ Š (130791 −

16328a2

+ 240a4)

5

5152a2

− Š (38170 − , 1/2 + Š6 (4825 − 680a2 + 16a4 ) / 2(5Š − 7)(Š − 1)2 .

(27)

+ 96a4 )

Sets of mode solutions ‹ in bow ducts with locally reacting wall linings: Grigoryan’s method: The modes are formulated as in Eq. (6). They satisfy the boundary conditions at the walls in (7). We still use the non-dimensional quantities kr ri,a → i,a ; k0ri,a · Z0 Gi,a → Ui,a ; Š = ri /ra < 1 with i = Š · a . The characteristic equation for mode eigenvalues ‹ then assumes the form of (8c) with the determinants Dn,m (‹, i , a ) defined in (9), where (n), (m) ∈ (0), (1) are orders of derivatives. A Taylor series expansion with the centre at a and the new variable x = a −i = a (1−Š) is applied on the first column (with variable i ) of the Dn,m (‹, i , a ), which defines coefficient determinants BŒ,m (‹, a ); Œ = n + k: * * * (−1)k ** J(n+k) (a ) J(m) ‹ ‹ (a ) * Dn,m (‹, i, a ) = * (n+k) * · xk * k! * Y‹ (a ) Y(m) ( ) a ‹ k≥0 (28) (−1)k Bn+k,m (‹, a ) · xk . = k! k≥0

The coefficient determinants can be evaluated by recursion, starting with B0,m and B1,m . One of the starters, B0,0 = 0 or B1,1 = 0, is an identity; the other starters B0,1 or B1,0 are obtained from the Wronski determinant. The recursion is (with Œ = n + k ; k = 0, 1, 2, . . .): (Œ − 2)2 + a2 − ‹2 2Œ − 3 (29) BŒ,m (‹, a ) = − BŒ−1,m + BŒ−2,m a a2 2(Œ − 2) (Œ − 2)(Œ − 3) + BŒ−3,m + BŒ−4,m a a2 a2 − ‹ 2 1 → B (‹, ) = − B + B −− 2,m a 1,m 0,m Œ=2 a a2 1 + a2 − ‹ 2 2 3 B2,m + B + B −−→ B3,m (‹, a ) = − 1,m 0,m Œ=3 a a2 a 2 2 4 + a − ‹ 4 2 5 B3,m + B2,m + B1,m + 2 B0,m . −−→ B4,m (‹, a ) = − Œ=4 a a2 a a

772

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Duct Acoustics

The initial members of the recursions are as follows: for m = 0: B00 = 0 ; 4 B40 = 2 a

B10 = −

2 ; a

3 + 3‹ 2 −1 + a2

B20 =

2 ; a2

2 ; a

2 B31 = 2 a

B11 = 0 ;

2 a

1−

2 + ‹2 ; a2

(30a)

;

for m = 1: B01 =

B30 =

B21 =

2 a

−1 +

‹2 ; a2

3‹ 2 3 + 2‹ 2 ‹ 4 + 11‹ 2 2 1 − 2 ; B41 = 1− . + a a a2 a4

(30b)

After truncation of the sum in Eq. (28) at khi the characteristic equation (8c) becomes a polynomial equation in ‹ 2. Numerical tests show that the summation limit should be khi > a ,more precisely khi ≈ 3a /2 for large a but not smaller than about khi ≈ 12.The degree of the polynomial in ‹ 2 will be near khi /2. If the limit khi is too low, solutions will be missed in the mode set; if khi is unnecessarily high, Muller’s procedure will furnish duplicates. Polynomial approximations with Re(‹) < 0 and/or Im(‹) > 0 should be rejected. A principal drawback of the method comes from the Taylor series expansion with the variable x = a − i = a (1 − Š), which prefers narrow ducts and/or low frequencies.This will be avoided in the next method of transformation into a polynomial equation which applies asymptotic expansions of the cylindrical functions. Transformation with Hankel asymptotics: The mode form is again: p‹ (r, œ) = J‹ (kr r) + Y‹ (kr r) e−j‹œ .

(31)

Abbreviations: = kr r; kr ri,a = i,a ; k0 ri,a · Z0 Gi,a = Ui,a ; Š = ri /ra = i /a . The modes satisfy two boundary conditions: (32) i,a J ‹ (i,a ) + Y ‹ (i,a ) = ±jUi,a J‹ (i,a ) + Y‹ (i,a ) , four recursions for derivatives of cylindrical functions: J ‹ (i,a ) = J‹−1 (i,a ) −

‹ J‹ (i,a ) ; i,a

Y ‹ (i,a ) = −Y‹+1 (i,a ) +

‹ Y‹ (i,a ) , i,a

(33)

and two Wronski determinants: J‹ (i,a )Y ‹ (i,a ) − J ‹ (i,a ) Y‹ (i,a ) = 2/(i,a ) .

(34)

Eliminate from this system of eight equations the seven quantities: ;

J ‹ (i ) ;

J ‹ (a ) ;

Y ‹ (i ) ;

Y ‹ (a ) ;

Y‹+1 (i ) ;

Y‹+1 (a ) .

Duct Acoustics

J

This gives the characteristic equation in the form: J‹ (a ) a J‹−1 (a ) − (‹ − jUa )J‹ (a ) − J‹ (i ) i J‹−1 (i ) − (‹ + jUi )J‹ (i ) + J‹ (a )Y‹ (i ) − J‹ (i )Y‹ (a ) 2 · i J‹−1 (i ) − (‹ + jUi )J‹ (i ) a J‹−1 (a ) − (‹ − jUa )J‹ (a ) = 0.

773

(35a)

Divide with J2‹ (a ) = 0 and introduce the well convergent continued fractions for ratios of Bessel functions: F‹ (z) = z

J‹−1 (z) J‹ (z)

z2 z2 z2 z2 = 2‹ − ... 2(‹ + 1) − 2(‹ + 2) − 2(‹ + 3) − 2(‹ + ihi ) resulting in the form of the characteristic equation: 2 F‹ (a ) − (‹ − jUa ) − J‹ (i )/J‹ (a ) F‹ (i ) − (‹ + jUi ) J‹ (i )/J‹ (a ) · J‹ (a )Y‹ (i ) − J‹ (i )Y‹ (a ) + 2 · F‹ (a ) − (‹ − jUa ) F‹ (i ) − (‹ + jUi ) = 0.

(36)

(35b)

The remaining Bessel and Neumann functions are substituted by their asymptotic series: J‹ (z) = 2/(z) P‹ (z) · cos z − (‹/2 + 1/4) −Q‹ (z) · sin z − (‹/2+1/4) , (37) Y‹ (z) = 2/(z) P‹ (z) · sin z − (‹/2 + 1/4) +Q‹ (z) · cos z − (‹/2+1/4) with the component series P(‹, z) = P‹ (z); Q(‹, z) = Q‹ (z): P(‹, z) =

K k=0

Q(‹, z) =

K k=0

(−1)k

K (‹, 2k) = (−1)k/2 · tk , (2z)2k keven =0

(‹, 2k + 1) (−1)k (2z)2k+1

=

K

(38) (k−1)/2

(−1)

· tk ,

kodd =1

where the Hankel symbols (‹, Œ) and the terms tk can be evaluated recursively: (‹, 0) = 1 ; t0 = 1 ;

(‹, k) =

4‹ 2 − (2k − 1)2 · (‹, k − 1); 4k

4‹ 2 − (2k − 1)2 · tk−1 . tk = 8k z

(39)

The fraction of Bessel functions in Eq. (35b) can be written (if ‹ = odd integer) as: 1 Q(‹, i) cos i − sin i + P(‹, i ) cos i + sin i + . . . J‹ (i )/J‹ (a ) = √ Š Q(‹, a ) cos a − sin a + P(‹, a ) cos a + sin a + . . . (40) . . . + tan(‹/2) Q(‹, i) cos i + sin i − P(‹, i ) cos i − sin i . . . . + tan(‹/2) Q(‹, a ) cos a + sin a − P(‹, a ) cos a − sin a

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Duct Acoustics

Multiply tan(‹/2) by the identity (‹/2)/(‹/2) and use for (‹/2) · tan(‹/2) the well convergent continued fraction: z2 z 2 z 2 . (41) 1− 3− 5 − . . . In total, the terms J‹ (i )/J‹ (a ) in Eq. (35b) can be written as a fraction with polynomials of ‹ in the numerator and denominator. Next one expands the cross product z · tan z =

J‹ (a )Y‹ (i ) − J‹ (i )Y‹ (a ) in Eq. (35b) to

J‹ (a )Y‹ (i ) − J‹ (i )Y‹ (a )

−2 √ P(‹, i ) Q(‹, a ) cos(a − i ) + P(‹, a ) sin(a − i ) a Š + Q(‹, i ) Q(‹, a ) sin(a − i ) − P(‹, a ) cos(a − i ) .

=

(42)

This too can be expanded as a polynomial in ‹. Thus the characteristic equation can be substituted by a polynomial equation in ‹.This transformation here is more complicated than for the Grigoryan method; however, its advantage is an extension of range of a to large values. Transformation with Hankel asymptotics for modes with Hankel functions: Part of the complexity of the previous transformation can be avoided when the bow modes are formulated with Hankel functions: (2) −j‹œ , p‹ (r, œ) = H(1) ‹ (kr r) + H‹ (kr r) e Z0 v‹r (r, œ) = j

kr (1) kr ∂p =j H‹ (kr r) + H (2) (k r) e−j‹œ . r ‹ k0 ∂(kr r) k0

The boundary conditions

(2) kr ri,a H (1) ‹ (kr ri,a ) + H‹ (kr ri,a ) Z0 v‹r (ri,a , œ) ! =j = ∓k0 ri,a Z0 Gi,a (1) (2) p‹ (ri,a , œ) H‹ (kr ri,a ) + H‹ (kr ri,a ) lead to the characteristic equation for mode eigenvalues ‹ in the form: (1) (2) (2) ( ) − H ( ) (‹ + jU )H ( ) − H ( ) (‹ − jUi )H(1) i i i a a a a 1+‹ 1+‹ ‹ ‹ (1) (2) (‹ − jUi )H(2) − (‹ + jUa )H(1) ‹ (a ) − a H1+‹ (a ) ‹ (i ) − i H1+‹ (i ) = 0. Apply the asymptotic expansions to the Hankel functions: 2/(z) · P(‹, z) ± jQ(‹, z) · e±j(z−(‹/2+1/4)·) H(1,2) ‹ (z) =

(43)

(44)

(45)

(46)

with the component series P(‹, z) = P‹ (z); Q(‹, z) = Q‹ (z) from Eq. (38). Because only mixed products with both kinds of Hankel function as factors will appear in Eq. (45), these products produce as common factors e±j(a −i ) and the exponential factors e±j‹/2 will cancel. Thus, Eq. (45) becomes a polynomial in ‹ if the asymptotic series Eq. (38) is truncated at khi . Tests for Re(‹) ≥ 0 and Im(‹) ≤ 0, and for |char.eq.| < lim (with lim ≈

Duct Acoustics

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775

80) should be made with the polynomial solutions before they are applied as Muller starters. The numerical coefficients in the polynomials are of moderate magnitude [in contrast to the modes in the form of Eq. (31) and the previous transformation, where the coefficients may become extremely large].

J.45

Lined Bow Duct Between Lined Straight Ducts


Turning-vane splitters in the corners of wind tunnels may have locally reacting absorber surfaces; such turning-vane splitter silencers may be designed so that they have a broad-band middle and high-frequency attenuation. The elementary ducts between the turning-vane splitters can be considered as a sequence in the direction of sound propagation of a circular bow duct between straight entrance and exit ducts. The three zones (I),(II),(III),with their co-ordinate systems,wall lining admittances,and component sound waves,are depicted in the sketch.The linings and their admittances Gi , Ga may be different on both sides in (II); the admittances GI , GIII are assumed (for simplicity) to be the same on both sides in (I) and (III). The incident wave pi is supposed to be the Œ-th mode of the entrance duct (I) (more complicated excitations can be synthesised with such modes). The exit duct (III) is anechoic (for simplicity also) for the transmitted wave pt , which is a sum of modes of (III), analogously to the reflected wave pr in (I), which is synthesised with modes of (I). The sound field pII in the bow duct is composed as a mode sum of forward and backward (œ direction) running bow modes.

The sound fields in general are unsymmetrical with respect to the central planes of the straight sections and to r = (ri + ra )/2 in the bow duct, even if the excitation is symmetrical in (I) and if the linings are the same on both sides. Therefore the mode sums in (I) and (III) must include both symmetrical and anti-symmetrical modes of

776

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Duct Acoustics

those sections. For ease of writing we attribute even mode indices n = ns = 0, 2, 4, . . . to symmetrical modes, and odd mode indices n = na = 1, 3, 5, . . . to anti-symmetrical modes. The index n = 0 is attributed to the least attenuated mode. Field formulation in (I): Cross profiles of modes: cos(—n y) ; symmetrical mode ; n = 0, 2, 4, . . . . qn (—n y) = sin(—n y) ; anti-symmetrical mode ; n = 1, 3, 5, . . .

(1)

Incident mode: = Pi · qŒ (—Œ y) · e−Œ xI ; Œ2 − —2Œ + k02 = 0, j ∂pi (xI , yI ) −jŒ Z0 vix (xI , yI) = = pi (xI , yI). k0 ∂xI k0

pi (xI , yI)

(2)

Wave reflected at xI = 0 (summation and mode index n ≥ 0): pr (xI , yI) = An qn (—n y) · e+n xI ; n2 − —2n + k02 = 0, n

Z0 vrx (xI , yI ) = j

An

n

n qn (—n y) · e+n xI . k0

(3)

The wave numbers —ns , —na of the symmetrical and anti-symmetrical modes are solutions of the characteristic equations: —ns h · tan(—ns h) = jk0 h · Z0 GI ;

—na h/ tan(—na h) = −jk0 h · Z0 GI .

(4)

See earlier sections in this chapter for the evaluation of sets of modes. The range of the summation and mode index n must include, at least, the order Œ of the incident mode. Field formulation in (III): Cross profiles of modes: cos(†ny) ; symmetrical mode ; n = 0, 2, 4, . . . . qn (†n y) = sin(†n y) ; anti-symmetrical mode ; n = 1, 3, 5, . . .

(5)

Transmitted wave (summation and mode index n ≥ 0): pt (xIII , yIII) = Dn · qn (†n yIII ) · e−‚n xIII ; ‚n2 − †n2 + k02 = 0, n

Z0 vtx (xIII , yIII) = −j

n

Dn

‚n qn (†n yIII ) · e−‚n xIII . k0

(6)

The wave numbers †ns ,†na of the symmetrical and anti-symmetrical modes are solutions of the characteristic equations: †ns h · tan(†ns h) = jk0 h · Z0 GIII ;

†na h/ tan(†na h) = −jk0 h · Z0 GIII .

(7)

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777

Field formulation in (II): pII+ (r, œ) =

B‹ J‹ (k0r) + ‹ · Y‹ (k0 r) e−j‹œ ,

‹

pII− (r, œ) =

C‹ J‹ (k0r) + ‹ · Y‹ (k0r) e+j‹œ .

(8)

‹

If the same manifold of modes is assumed in both directions of propagation (reasonable, but not necessary), the sound field in (II) is: pII (r, œ)

= pII+ (r, œ) + pII− (r, œ) =

J‹ (k0r) + ‹ · Y‹ (k0 r) B‹ e−j‹œ + C‹ e+j‹œ ,

‹

Z0 vIIœ (r, œ) = =

j j ∂pII (r, œ) gradœ pII (r, œ) = k0 k0 r ∂œ

(9)

1 ‹ J‹ (k0 r) + ‹ · Y‹ (k0r) B‹ e−j‹œ − C‹ e+j‹œ . k0 r ‹

Abbreviation: Ui,a : = k0 ri,a Z0 Gi,a .

(10)

The mode eigenvalues ‹ are solutions of the characteristic equation; see > Sect. J.44: k0ri J1+‹ (k0 ri ) − (‹ − jUi )J‹ (k0 ri ) · k0ra Y1+‹ (k0 ra ) − (‹ + jUa )Y‹ (k0ra ) (11) − k0 ra J1+‹ (k0 ra ) − (‹ + jUa )J‹ (k0 ra ) · k0 ri Y1+‹ (k0 ri ) − (‹ − jUi )Y‹ (k0 ri ) = 0. The ratios of the radial component waves are: (‹ − jUi )J‹ (k0 ri ) − k0 ri J1+‹ (k0 ri ) ‹ = − (‹ − jUi )Y‹ (k0 ri ) − k0 ri Y1+‹ (k0 ri ) (‹ + jUa )J‹ (k0ra ) − k0 ra J1+‹ (k0 ra ) . = − (‹ + jUa )Y‹ (k0ra ) − k0 ra Y1+‹ (k0ra )

(12)

∧

Matching of sound pressures at œ = 0 = xI = 0: !

pi (0, yI) + pr (0, yI ) = pII (r, 0), An qn (—n yI ) = Pi · qŒ (—Œ yI ) + J‹ (k0r) + D‹ · Y‹ (k0 r) B‹ + C‹ . n

(13a) (13b)

‹

Make use of the mutual orthogonality of the modes in (I), i. e. perform on both sides the integral 1 2h

h qm (—m y) · . . . dy, −h

(14)

J

778

Duct Acoustics

giving on the left-hand side of Eq. (13b) (with ƒm,n = Kronecker symbol): 1 2h

h −h

0 qm (—m y) · qn (—n y) dy = ƒm,n · Nn

the mode norms Nn : sin 2(—n h) 1 Nn = 1± ; 2 2—n h

;

symm. and anti-symm symm. or anti-symm.

n = ns symm . n = na anti-symm

(15)

(16)

Integral Eq. (14), with a fixed index m from within the range of n, applied on both sides of Eq. (13b) will result in: Pi · ƒm,Œ · NŒ + An ƒm,n · Nn = IJm,‹ + ‹ · IYm,‹ B‹ + C‹ , (17a) n

‹

or, with the left-hand side simplified, one obtains the linear system of equations with running m for the mode amplitudes An , B‹ , C‹ : Pi · ƒm,Œ · NŒ + Am Nm = IJm,‹ + ‹ · IYm,‹ B‹ + C‹ . (17b) ‹

There appear the mode coupling integrals: IJm,‹

1 = 2h

h qm (—m y) · J‹ (k0r) dy −h

−1 −−−−−−−−−−−→ y→(ra +ri )/2−r 2h −1 = 2k0h

IYm,‹

1 = 2h

k0 ra qm ( k0 ri

ra qm (—m (ra + ri ) /2 − —m r) · J‹ (k0r) dr

(18a)

ri

—m —m (k0 r)) · J‹ (k0 r) d(k0r), (k0 ra + k0ri ) /2 − k0 k0

h qm (—m y) · Y‹ (k0r) dy −h

−1 −−−−−−−−−−−→ y→(ra +ri )/2−r 2h −1 = 2k0h

k0 ra qm ( k0 ri

ra qm (—m (ra + ri ) /2 − —m r) · Y‹ (k0 r) dr

(18b)

ri

—m —m (k0r)) · Y‹ (k0r) d(k0 r). (k0 ra + k0ri ) /2 − k0 k0 ∧

Matching of axial particle velocities at œ = 0 = xI = 0: !

Z0 vix (0, yI ) + Z0 vrx (0, yI) = Z0 vIIœ (r, 0),

(19a)

Duct Acoustics

J

n Œ Pi · qŒ (—Œ y) − An qn (—n y) k0 k0 n ‹ J‹ (k0r)/k0r + ‹ · Y‹ (k0r)/k0r B‹ − C‹ . =j

779

(19b)

‹

Perform again integral (14) on both sides taking m ∈ {n}, with the following result: n Œ Pi · ƒm,Œ · NŒ − An ƒm,n · Nn = j ‹ KJm,‹ + ‹ · KYm,‹ · B‹ − C‹ . k0 k0 n ‹

(20a)

With the left-hand side simplified, one gets the linear system of equations for the mode amplitudes An , B‹ , C‹ : m Œ Pi · ƒm,Œ · NŒ − Am Nm = j ‹ KJm,‹ + ‹ · KYm,‹ B‹ − C‹ . k0 k0 ‹

(20b)

The mode-coupling integrals here differ from those in (18a) and (18b) by a division with k0 r in the integrands: KJm,‹

1 = 2h

h qm (—m y) · J‹ (k0r)/k0r dy −h

−1 −−−−−−−−−−−→ y→(ra +ri )/2−r 2h −1 2k0h

=

KYm,‹ =

1 2h

qm (—m (ra + ri ) /2 − —m r) · J‹ (k0r)/k0r dr

(21a)

ri

k0 ra —m —m qm ( (k0ra + k0 ri ) /2 − (k0r)) · J‹ (k0r)/k0r d(k0 r), k0 k0

k0 ri

h qm (—m y) · Y‹ (k0 r)/k0r dy −h

−1 −−−−−−−−−−−→ y→(ra +ri )/2−r 2h =

ra

−1 2k0h

k0 ra qm ( k0 ri

ra qm (—m (ra + ri ) /2 − —m r) · Y‹ (k0r)/k0r dr

(21b)

ri

—m —m (k0 r)) · Y‹ (k0 r)/k0r d(k0 r). (k0ra + k0ri ) /2 − k0 k0 ∧

Matching of sound pressure at œ = Ÿ = xIII = 0: !

pt (0, yIII) = pII (r, Ÿ), n

Dn · qn (†nyIII ) =

(22a)

‹

J‹ (k0 r) + ‹ · Y‹ (k0 r) B‹ e−j‹Ÿ + C‹ e+j‹Ÿ .

(22b)

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780

Duct Acoustics

Use the orthogonality of modes in section (III) and perform the integral on both sides, with m from the applied mode range in (III): 1 2h

h qm (†my) · . . . dy,

(23)

−h

defining on the left-hand side the mode norms Mn : 1 2h

h −h

0 qm (†my) · qn (†n y) dy = ; ƒm,n · Mn

with values Mn =

sin(2†n h) 1 1± ; 2 2†nh

symm. and anti-symm symm. or anti-symm

n = ns symm. , n = na anti-symm

(24)

(25)

and on the right-hand side the mode coupling integrals: RJm,‹

1 = 2h

h qm (†m y) · J‹ (k0r) dy ;

RYm,‹

−h

1 = 2h

h qm (†m y) · Y‹ (k0 r) dy. (26a,b) −h

They differ from the mode-coupling integrals in Eq.(18a,b) by the substitution —m → †m and may therefore be evaluated analogously to those integrals. Thus one gets a linear system of equations for the amplitudes Dm , B‹ , C‹ : RJm,‹ + ‹ · RYm,‹ B‹ e−j‹Ÿ + C‹ e+j‹Ÿ . (27) Dm Mm = ‹ ∧

Matching of axial particle velocities at œ = Ÿ = xIII = 0: !

Z0 vtx (0, yIII ) = Z0 vIIœ (r, Ÿ), −j

n

Dn

(28)

1 ‚n qn (†n yIII ) = ‹ J‹ (k0 r) + ‹ · Y‹ (k0r) · B‹ e−j‹Ÿ − C‹ e+j‹Ÿ . (29) k0 k0 r ‹

With the integral of Eq. (28) applied on both sides one gets the linear system of equations for the amplitudes Dm , B‹ , C‹ : ‚m ‹ SJm,‹ + ‹ · SYm,‹ B‹ e−j‹Ÿ − C‹ e+j‹Ÿ , (30) Dm Mm = j k0 ‹ where the mode-coupling integrals are: SJm,‹

1 = 2h

1 SYm,‹ = 2h

h qm (†my) · J‹ (k0 r)/k0r dy ; −h

(31)

h qm (†my) · Y‹ (k0r)/k0r dy. −h

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In total one has four linear systems of equations (17b), (20b), (27), (30) for four sets of mode amplitudes An , Dn , B‹ , C‹ ; systems (17b) and (20b) are inhomogeneous (the value of Pi can be choosen arbitrarily, e. g. Pi = 1). In both first systems (17b) and (20b) the index m sweeps through the range of mode indices in (I); in (27) and (30) m passes through the mode orders in (III). It is plausible to eliminate the Am by the operation (20b) + (17b) · m /k0, giving: m1 B‹ IJm1 ,‹ + ‹ · IYm1 ,‹ + j‹ KJm1 ,‹ + ‹ · KYm1 ,‹ k0 ‹ (32) Œ m1 IJm1 ,‹ +‹ · IYm1 ,‹ −j‹ KJm1,‹ +‹ · KYm1 ,‹ = 2 Pi ·ƒm1 ,Œ · NŒ , + C‹ k0 k0 and analogously to eliminate the Dm by the operation (30) − ‚m /k0 · (27): ‚m B‹ e−j‹Ÿ j‹ SJm3 ,‹ + ‹ · SYm3 ,‹ − 3 RJm3 ,‹ + ‹ · RYm3 ,‹ k0 ‹ ‚m − C‹ e+j‹Ÿ j‹ SJm3 ,‹ + ‹ · SYm3 ,‹ + 3 RJm3 ,‹ + ‹ · RYm3 ,‹ = 0. k0

(33)

Both systems together form a combined, inhomogeneous, linear system of equations for the amplitude sets B‹ , C‹ . With them, An and Dn follow from (27) and (30). Evidently, the task of sound field evaluation in a sequence of a bent lined duct between straight lined ducts consists of the principal steps: • determination of wall admittances; • evaluation of sets of mode eigenvalues in the ducts; • evaluation of the mode-coupling integrals (here by numerical integration); • solution of the linear systems of equations for the mode amplitudes; • insertion of mode amplitudes and mode wave numbers in the field formulations.

J.46

Zero-Order and First-Order Transmission Loss of Turning-Vane Splitter Silencers


Turning-vane splitter silencers rarely satisfy the requirement of the sketch and the analysis in the previous > Sect. J.45 that both walls of the elementary bent duct section must occupy concentric arcs. The silencers often are formed by juxtaposition of bent splitters with about constant thickness (left-hand sketch). A better approach to the requirement may be achieved with curved splitters of variable thickness (right-hand sketch), and a more broad-band attenuation is possible with such constructs. Regarding these and other visible deviations from the theoretical assumptions, it may be sufficient to evaluate the transmission loss of the silencer in an approximation of either zero order or first order. For this it is assumed that the straight duct sections have the lengths L1 , L3 and the bent duct section is bent with an angle Ÿ (Ÿ = /2 in the sketches). The zero-order transmission loss is the sum of the sound pressure level reductions of the least attenuated modes in the sections over the lengths L1 , L2 and the bend angle Ÿ, respectively. Additional losses by reflections at the section limits are then neglected.

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The first-order transmission loss again uses only the least attenuated modes of the sections (instead of sums of modes as in the previous > Sect. J.45), but the additional reduction by reflections of these modes at the section limits are considered. The symbols of the previous section will be used below. The linear systems of equations (17b), (20b), (27), (30) of the previous > Sect. J.45 for the four amplitudes of the least attenuated modes A0 , D0, B‹ , C‹ simplify to: Pi · N0 + A0 N0 = IJ0,‹ + ‹ · IY0,‹ B‹ + C‹ ,

(1)

0 0 Pi · N0 − A0 N0 = j‹ KJ0,‹ + ‹ · KY0,‹ B‹ − C‹ , k0 k0

(2)

D0 M0 = RJ0,‹ + ‹ · RY0,‹ B‹ e−j‹Ÿ + C‹ e+j‹Ÿ ,

(3)

D0

‚0 M0 = j‹ SJ0,‹ + ‹ · SY0,‹ B‹ e−j‹Ÿ − C‹ e+j‹Ÿ , k0

and after elimination of A0 , D0 as in (32) and (33): 0 B‹ IJ0,‹ + ‹ · IY0,‹ + j‹ KJ0,‹ + ‹ · KY0,‹ k0 0 0 IJ0,‹ + ‹ · IY0,‹ − j‹ KJ0,‹ + ‹ · KY0,‹ = 2 Pi · N0 , + C‹ k0 k0 ‚0 B‹ e j‹ SJ0,‹ + ‹ · SY0,‹ − RJ0,‹ + ‹ · RY0,‹ k0 ‚0 RJ0,‹ + ‹ · RY0,‹ = 0. − C‹ e+j‹Ÿ j‹ SJ0,‹ + ‹ · SY0,‹ + k0

(4)

(5)

−j‹Ÿ

(6)

Combining the mode-coupling integrals like: IJY0,‹ = IJ0,‹ + ‹ · IY0,‹ ; KJY0,‹ = KJ0,‹ + ‹ · KY0,‹ , RJY0,‹ = RJ0,‹ + ‹ · RY0,‹ ; SJY0,‹ = SJ0,‹ + ‹ · SY0,‹ ,

(7)

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one gets explicit solutions for B‹ , C‹ : 0 N0 ‚0 /k0 · RJY0,‹ + j‹ · SJY0,‹ k0 / · j‹ · KJY0,‹ ‚0 /k0 1 + e−2j‹Ÿ RJY0,‹ + j‹ 1 − e−2j‹Ÿ SJY0,‹ 0−1 , + 0 /k0 · IJY0,‹ ‚0 /k0 1 − e−2j‹Ÿ RJY0,‹ + j‹ 1 + e−2j‹Ÿ SJY0,‹

(8)

0 C‹ = −2Pi N0 e−2j‹Ÿ ‚0 /k0 · RJY0,‹ − j‹ · SJY0,‹ / k0 · j‹ · KJY0,‹ ‚0 /k0 1 + e−2j‹Ÿ RJY0,‹ + j‹ 1 − e−2j‹Ÿ SJY0,‹ 0−1 + 0/k0 · IJY0,‹ ‚0 /k0 1 − e−2j‹Ÿ RJY0,‹ + j‹ 1 + e−2j‹Ÿ SJY0,‹ .

(9)

B‹ = 2Pi

With these the amplitudes A0 , D0 are obtained from: A0 = −Pi + IJY0,‹ /N0 · B‹ + C‹ ; D0 = RJY0,‹ /M0 · B‹ e−j‹Ÿ + C‹ e+j‹Ÿ . (10a,b) If the attenuation in the bent duct section is high, i. e. |e−2j‹Ÿ | 1, one may further simplify: B‹ ≈ 2Pi

0 1 N0 , k0 0 /k0 · IJY0,‹ + j‹ · KJY0,‹

‚0 /k0 · RJY0,‹ − j‹ · SJY0,‹ 0 N0 e−2j‹Ÿ k0 ‚0 /k0 · RJY0,‹ + j‹ · SJY0,‹ 1 · . 0/k0 · IJY0,‹ + j‹ · KJY0,‹

C‹ ≈ −2Pi

(11)

(12)

And even the reflected wave in the bow duct with the amplitude C‹ may be neglected if the attenuation in the bow is sufficiently high. The resulting sound pressure ratio (on the duct axis) of the transmitted wave pt (L3 , 0) to the incident wave pi (−L1 , 0) is: pt (L3 , 0)/pi (−L1 , 0) = D0 e−‚0 L3 = e−‚0 L3 RJY0,‹ /M0 · B‹ e−j‹Ÿ + C‹ e+j‹Ÿ . (13) The transmission loss D in dB becomes in the first-order approximation: * * D = −20 · lg *pt (L3 , 0)/pi (−L1 , 0)* = 8.686 · Re(0 )L1 + Re(‚0 )L3 − Im(‹)Ÿ * * * N0 ‚0 /k0 · RJY0,‹ − j‹ · SJY0,‹ * 0 /k0 · RJY0,‹ * * − 20 · lg *2 1− M0 0 /k0 · IJY0,‹ + j‹ · KJY0,‹ ‚0 /k0 · RJY0,‹ + j‹ · SJY0,‹ * = 8.686 · Re(0 )L1 + Re(‚0 )L3 − Im(‹)Ÿ * * * * N0 0 /k0 · RJY0,‹ j‹ · SJY0,‹ * − 20 · lg **4 M0 0 /k0 · IJY0,‹ + j‹ · KJY0,‹ ‚0 /k0 · RJY0,‹ + j‹ · SJY0,‹ * = 8.686 · Re(0 )L1 + Re(‚0 )L3 − Im(‹)Ÿ * * * M0 j‹ · KJY0,‹ ‚0 /k0 · RJY0,‹ * *. * 1+ 1+ + 20 · lg * * 4N0 0 /k0 · RJY0,‹ j‹ · SJY0,‹

(14)

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The first lines of Eq. (14) represent the transmission loss in the zero-order approximation; the second lines represent the additional terms of the first-order transmission loss due to reflections at the section limits.

Sound transmission loss DL of the least attenuated mode in a bow duct with bending angle Ÿ = 90◦ ; the linings consist of glass fibre boards, covered with porous limp foils and perforated metal sheet. The foil on the outer lining has three values of flow resistance ratio Rfa An important contribution to the sum of transmission losses of the sections may come from the bent section,as the above diagram illustrates,showing the transmission loss DL of the least attenuated mode in a single bow duct section with bending angle Ÿ = 90◦ , radii ri = 1.9 [m], ra = 2.7 [m], linings consisting of ti = ta = 0.2 [m]-thick glass fibre boards with flow resistivities ¡i = ¡a = 7000 [Pa · s/m2 ], covered with porous, limp foils having surface mass densities mfi = 0.4 [kg/m2 ], mfa = 1.0 [kg/m2], the inner foil with a fixed flow resistance ratio (relative to Z0 ) Rfi = 2.0, and the outer foil with one of the alternative resistance ratio values Rfa = {0.5, 1.0, 2.0}. The linings are covered (towards the duct) with perforated metal sheet, di,a = 1.5 [mm] thick, with holes of diai,a = 5 [mm] and a porosity of i,a = 0.4. It is conspicuous that, assuming a proper layout of the linings is choosen, the usual high-frequency decrease in the attenuation in straight ducts, as a consequence of ray formation in straight ducts, can be avoided. It is remarkable that such extraordinary frequency curves of attenuation must not be a consequence of multimode compensations but can be obtained also with single-mode fields of the least attenuated mode in the bend.

Duct Acoustics

J.47

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Bent and Straight Ducts with Unsymmetrical Linings


The duct arrangement and the naming of symbols is similar to those in > Sect. J.45. Unlike in that section, the linings in the straight duct sections (I) and/or (III) now are unsymmetrical. As a special case of unsymmetry the straight ducts (I) and/or (III) first are assumed to be one-sided hard, the other side absorbing with local reaction. The origin y = 0 of the Cartesian co-ordinates is placed in the hard wall; the width of the duct is h (in contrast to > Sect. J.45, where it is 2h). Only symmetrical modes relative to y = 0 will appear in the field sums in sections (I) and (III). The mode formulations in these sections are: ; n2 − —2n + k02 = 0 ; qn (—n y) = cos(—n y) , pn (xI , yI ) = qn (—n yI ) · e−n xI Z0 vnx (xI , yI) =

(1)

j ∂pn (xI , yI ) −jn = pn (xI , yI ). k0 ∂xI k0

The principal distinctions with respect to > Sect. J.45 arise by the modification of the mode-coupling integrals and the mode norms ( > Sect. J.45 here): Outer wall hard: y = 0 at r = ra : 1 2h

h qm (—m y) · . . . dy −h

r = ra − y ;

1 ⇒ h

0 ≤ y ≤ h ;

h = ra − ri

h qm (—m y) · . . . dy , 0

h sin(2—n h) sin(2—n h) 1 1 1 1± ⇒ Nn = 1+ . cos2 (—n y) dy = Nn = 2 2—n h h 2 2—n h)

(2)

0

Inner wall hard: y = 0 at r = ri : 1 2h

h qm (—m y) · . . . dy −h

⇒

r = ri − y ; 1 h

− h ≤ y ≤ 0 ;

h = ra − ri

0 qm (—m y) · . . . dy , −h

sin(2—n h) 1 1 Nn = 1± ⇒ Nn = 2 2—n h h

0 −h

sin(2—n h) 1 2 1+ . cos (—n y) dy = 2 2—n h

(3)

Substitute —n → †n if the exit duct section (III) is one-sided hard. The one-sided hard ducts are limit cases G1 = 0, G2 = 0 of straight ducts with unsymmetrical locally reacting linings. Straight dut sections with unsymmetrical locally reacting linings G1 = G2: Only formulas for the case of unsymmetrical linings of duct section (I) will be given; unsymmetrical linings in section (III) would be treated analogously. The co-ordinate origin y = 0 is again in the duct centre; the duct width is 2h. Define the symmetrical Us and anti-symmetrical Ua parts of the lining functions U: U1 = k0h · Z0 G1 ; Us = (U1 + U2) /2 ;

U2 = k0 h · Z0 G2 , Ua = (U1 − U2 ) /2

(4)

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Duct Acoustics

(if Re(Ua ) Sect. J.45. If the duct section (III) also has an unsymmetrical lining, the mode formulations there will be: pn (x, y) = cos(†n y) + c · sin(†n y) · e−‚n x ; ‚n2 − †n2 + k02 = 0, (16) j†n Z0 vyn (x, y) = − sin(†ny) + cn · cos(†n y) · e−‚n x , k0 cn = − cot(†n h)

†n h · tan(†n h) − jk0h · Z0 G2 †n h · cot(†n h) + jk0 h · Z0 G2

†n h · tan(†n h) − jk0 h · Z0 G1 = cot(†nh) . †n h · cot(†n h) + jk0 h · Z0 G1

(17)

The other systems of equations for mode amplitudes will transform analogously.

J.48

Silencer with Rectangular Turning-Vane Splitters

This section describes a task which can be solved with results from earlier sections of this chapter, and shows some results. In some sense, this section at the end of the chapter illustrates the application of earlier sections for the solution of a practical task.

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The task: One needs a silencer with a broad-band transmission loss, also at high frequencies. The duct has a corner or bent (e. g. a horizontal duct, with a width H, enters a vertical stack for an exhaust silencer, with a width B). It is advisable to atrribute the high-frequency end of the insertion loss to a turning-vane splitter silencer and to design the silencer in the exhaust stack for the remaining transmission loss, if any.

Details of the splitters are shown in the sketch below. The splitters are assumed to be locally reacting in the numerical example given below, which is indicated by internal partitions; it is also assumed that the splitters have a hard central sheet.

The splitters are treated as a combination of two lined ducts, one with a length L1 and width H1 , the other with L2 and H2 , plus a lined duct corner with dimensions L3 = H2 and L4 = H1 . (The possible additional transmission loss due to the additional lengths L3 and L4 is neglected below.) Such objects are treated in

> Sect. J.26,“Lined duct corners and junctions”.

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The insertion loss of the turning-vane splitter silencer is the sum of the transmission losses of the two straight ducts plus the insertion loss of the corner which is evaluated as in > Sect. J.26. The design makes use of the possibility of avoiding by lined duct corners the high-frequency decrease of attenuation due to ray formation in the ducts. The dimensions of the ducts in the example shown are H = 10 [m], and B = 17 [m] (in a gas turbine test cell, with a meanflow Mach number in the empty stack of M = 0.0424, and in the silencer ducts of M = 0.106).In the parameter list of the example is H1 = 2·h1; D1 = 2 · d1 ; H2 = 2 · h2 ; D2 = 2 · d2 . The splitter branches consist of layers of glass fibre (made locally reacting) with flow resistivity values ¡1, ¡2, covered with porous steel foils of thickness df1 , df2 with normalised flow resistances Rf1 , Rf2, respectively. It is assumed that a plane wave is incident. The lengths Li ; i = 1, 2; should be Li ≥ 2Hi .

Parameters: IL with flow,turning-vane splitter silencer with locally reacting lined duct corner M(B) = 0.0424 empty, mode index limit mhi = 8. Width of ducts: B [m] = 17., H [m] = 10. Duct (1): L1 [m] = 3 . h1 [m] = 0.2, d1 [m] = 0.3 ¡1 [Pa s/m∧2] = 2000 . rof1[kg/m∧3] = 7800 ., df1 [m] = 0.0005, Rf1 = 3 . Duct (2): L2 [m] = 3 . h2 [m] = 0.34, d2 [m] = 0.51 ¡2 [Pa s/m∧2] = 2000 . rof2 [kg/m∧3] = 7800 ., df2 [m] = 0.0005, Rf2 = 1 .

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Corner (3): h3 [m] = 0.2 , d3 [m] = 0.3 ¡3 [Pa s/m∧2] = 2000 . rof3 [kg/m∧3] = 7800 ., df3 [m] = 0.0005 , Rf3 = 3 . Corner (4): h4 [m] = 0.34 , d4 [m] = 0.51 ¡4 [Pa s/m∧2] = 2000 . rof4 [kg/m∧3] = 7800 ., df4 [m] = 0.0005 , Rf4 = 1 .

References Cremer, L.: Theorie der Luftschalldämpfung mit schluckender Wand und das sich dabei ergebende höchste Dämpfungsmaß. Acustica, Akust. Beihefte 3, 249–263 (1953) Coelho,J.L.B.: Acoustic Characteristics of Perforate Liners in Expansion Chambers. Thesis, Fac. Engineer., Inst. Sound Vibr., Southampton (1983)

Mechel, F.P.: Schallabsorber, Vol. III, Ch. 25.4: Superposition of flow. Hirzel, Stuttgart (1998) Mechel, F.P.: Schallabsorber,Vol. III, Ch. 26: Rectangular duct with local lining.Hirzel,Stuttgart (1998) Mechel, F.P.: Schallabsorber, Vol. III, Ch. 27: Rectangular duct with lateral lining. Hirzel, Stuttgart (1998)

Cummings, A.J.: Sound transmission at sudden area expansions in circular ducts with superimposed mean flow. Sound Vibr. 38, 149–155 (1975)

Mechel, F.P.: Schallabsorber, Vol. III, Ch. 28: Rectangular duct with unsymmetrical lining. Hirzel, Stuttgart (1998)

Félix,S.,Pagneux,V.: Multimodal analysis of acoustic propagation in three-dimensional bends. Wave Motion 36, 157–158 (2002)

Mechel, F.P.: Schallabsorber, Vol. III, Ch. 29: Silencers with circular ducts. Hirzel, Stuttgart (1998)

Félix, S., Pagneux, V.; Sound attenuation in lined bends. J. Acoust. Soc.Am. 116, 1921–1931 (2004) Grigoryan,F.E.: Theory of sound wave propagation in curvilinear waveguides. Soviet Phys.-Acoust. 14, 315–321 (1969) Ko, S.-H., Ho, L.T.: Sound attenuation in acoustically lined curved ducts in the absence of fluid flow. J. Sound Vibr. 53, 189–201 (1977)

Mechel, F.P.: Schallabsorber, Vol. III, Ch. 30: Silencers with annular ducts. Hirzel, Stuttgart (1998) Mechel, F.P.: Schallabsorber, Vol. III, Ch. 32: Ducts with cross-layered lining and pine tree silencers. Hirzel, Stuttgart (1998) Mechel, F.P.: Schallabsorber, Vol. III, Ch. 33: Duct with steps. Hirzel, Stuttgart (1998) Mechel, F.P.: Schallabsorber, Vol. III, Ch. 34: Silencer of finite length and silencer cascade. Hirzel, Stuttgart (1998)

Ko, S.-H.: Three-dimensional acoustic wave propagation in acoustically lined cylindrically curved ducts without fluid flow. J. Sound Vibr. 66, 165–179 (1979)

Mechel, F.P.: Schallabsorber,Vol. III, Ch. 35: Locally concentrated absorbers in a duct lining. Hirzel, Stuttgart (1998)

Mechel, F.P.: Schallabsorber, Vol. II, Ch. 10: Sound in capillaries. Hirzel, Stuttgart (1995)

Mechel, F.P.: Schallabsorber, Vol. III, Ch. 36: Splitter type silencer with local lining. Hirzel, Stuttgart (1998)

Mechel, F.P.: Schallabsorber, Vol. II, Ch. 28: Nonlinearities by amplitude and flow. Hirzel, Stuttgart (1995)

Mechel, F.P.: Schallabsorber, Vol. III, Ch. 37: Splitter type silencer with lateral lining. Hirzel, Stuttgart (1998)

Mechel, F.P.: Mathieu Functions; Formulas, Generation, Use. Hirzel, Stuttgart (1997)

Mechel, F.P.: Schallabsorber, Vol. III, Ch. 38: Lined duct junctions. Hirzel, Stuttgart (1998)

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Mechel, F.P.: Schallabsorber, Vol. III, Ch. 39: Sound radiation of exits of ducts and silencers. Hirzel, Stuttgart (1998)

Mechel,F.P.: The scattering at a corner with absorbing flanks and absorbing cylinder. J. Sound Vibr. 219, 581–601 (1999)

Mechel, F.P.: Schallabsorber, Vol. III, Ch. 40: Application of modes. Hirzel, Stuttgart (1998)

Mechel, F.P.: Modal-Analyse im Bogen- oder RingKanal; Teil I: Bogen-Kanal ; Teil II: Ring-Kanal.Acta acustica 94, 173–206 (2008)

Mechel, F.P.: Schallabsorber,Vol. III, Ch. 41: Cremer admittance and hybrid absorbers. Hirzel, Stuttgart (1998) Mechel, F.P.: Schallabsorber, Vol. III, Ch. 42: Influence of flow and temperature on attenuation. Hirzel, Stuttgart (1998) Mechel, F.P.: Modes in lined wedge-shaped ducts. J. Sound Vibr. 216, 649–671 (1998) Mechel, F.P.: Modal analysis in lined wedge-shaped ducts. J. Sound Vibr. 216, 673–696 (1998)

Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes. Cambridge University Press, New York (1989) Ronneberger, D.: Experimentelle Untersuchungen zum akustischen Reflexionsfaktor von unstetigen Querschnittsänderungen in einem luftdurchströmten Rohr. Acustica 19, 222–235 (1967/68) Young-Chung Cho J.; Reciprocity principle in duct acoustics. Acoust. Soc. Am. 67, 1421–1426 (1980) Zwicker, C., Kosten, C.W.: Soundabsorbing materials. Elsevier, London (1949)

K Muffler Acoustics with M.L. Munjal The performance of acoustic mufflers relies heavily on reflections at duct discontinuities, such as steps of the cross section or of the duct lining, following each other in short distances. Such transitions, being mostly neglected in > Ch. J, about long, homogeneous silencers (often called “industrial silencers”), together with the always necessary consideration of mean flow and often of high temperatures, give muffler acoustics a special character.

Conventions in the Present Chapter Muffler acoustics preferably is formulated with the field quantities pressure p and volume flow velocity u in the duct. Therefore, mostly the volume flow impedance (or flow impedance), defined by the ratio p/u = p/(v · S), is used, where v is the particle velocity (as usual in this book) and S is the duct cross section. The flow impedance is indicated by underlining: Zx =

px px = , ux vx Sx

where px , ux , vx , Sx respectively are the sound pressure, volume flow, particle velocity, and duct cross section at a position x of the duct. An exception to this rule may be the symbol Z0 = 0 c0 /S, if the cross section S is unspecified, and Zr , which stands for a radiation impedance in the dimensions of a flow impedance. It is convenient to formulate expressions for sound fields in a steady flow with convected quantities ( > Sect. K.1). Such convected quantities will be indicated by an index c. Matrix formulations play an important role in muffler acoustics. The conventions for writing a vector are {v} = {p, u}, and [M] for a matrix in the running text. A matrix equation may either be written as {u} = [M] · {v} or with the elements as m11 m12 v1 u1 = · . u2 m21 m22 v2 Most graphs in this chapter will indicate by points • u and • d duct cross sections just above the upsound and just below the downsound cross sections, respectively, between which a transformation matrix will be developed.

K.1 Acoustic Power in a Flow Duct In the intake and exhaust systems of reciprocating internal combustion engines and compressors, and ducts of the heating, ventilation and air-conditioning (HVAC) systems,

794

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Muffler Acoustics

the meanflow velocity V is generally small enough for the flow to be assumed as incompressible. Thus, the meanflow Mach number M is limited to the range M = V/c0 < 0.2. For linear waves in a duct with incompressible meanflow, thermodynamic variables p0 , 0 , T, and hence sound speed c0 may be assumed to be constant. For plane waves in such a flow duct, the acoustic power is given by [Morfey (1971)]: M 2 ¢(M) = pu + p + MZ0 u 2 + M2 pu . (1) Z0 With convected field quantities this can be written as follows: ¢(M) = pc uc , pc = p + MZ0 u, uc = u + Mp/Z0 ,

(2)

Z0 = 0 c0 /S. Substitution of M = 0 yields the correponding relationships for a stationary medium. p = sound pressure; u = volume flow velocity; v = particle velocity; V = meanflow velocity; M = V/c0 = meanflow Mach number; S = area of the duct cross-section; Z0 = 0 c0 /S = characteristic flow impedance; R = reflection factor; • subscript c denotes convected quantities; • . . . denotes time average over a cycle; • underlining denotes flow impedances In terms of the amplitudes p(0), u(0) of the sound pressure p and the volume flow u at a position x = 0, and of the sound pressure amplitudes A, B of the forward wave (which is in the direction of V) and reflected/rearward wave components of a plane standing wave in the duct at the same position x = 0 in the duct: p(0) = A + B ;

u(0)= (A − B)/Z0 ,

M Rc ≡ Bc /Ac = B(1 − M) = R 11 − + M, A(1 + M) Ac = A(1 + M) ; Bc = B(1 − M),

R = B/A;

(3)

pc (0) = A(1 + M) + B(1 − M) = Ac +Bc , uc (0) = A(1 + M) − B(1 − M) = (Ac − Bc )/Z0 . Z0 The acoustic power is: |A|2 |A |2 (1 + M)2 − |R|2 (1 − M)2 = 12 c 1 − |Rc |2 ; ¢(M) = 12 Z0 Z0 |A|2 ¢(0) = 21 Z 1 − |R|2 . 0

(4)

Muffler Acoustics

K

795

For given sound pressure amplitudes the convective effect of meanflow is to strengthen the forward wave and weaken the reflected wave. The overall effect is to increase the net acoustic power, i.e. ¢(M) ≥ ¢(0). The reflection factor R, defined with the nonconvective, classical field quantities, may exceed unity if Rc approaches unity [Mechel/ Schiltz/Dietz (1965)].

K.2 Radiation from the Open End of a Flow Duct The object is an unflanged open end of a duct of radius r0 with a flow Mach number M in the duct. The radiation flow impedance of the orifice is given by [Levine/Schwinger (1948); Davies et al. (1980)]: Zr (M) = Rr (M) + jXr (M), Rr (M) ≈ Rr (0) − 1.1 · M · Z0 ,

(1)

Xr (M) ≈ X r (0), R Zr (0) = Rr (0) + jXr (0) = Z0 11 + − R. The inside reflection factor R at the orifice is: R = |R| ej(−2k0 ) , |R| ≈ 1 + 0.01336 k0r0 − 0.59079(k0r0 )2 + 0.33756(k0 r0 )3 − 0.06432(k0 r0)4 , 0 < k0 r0 < 1.5.

(2)

The relative end correction /r0 is approximately: /r 0 = 0.6133 − 0.1168(k0r0 )2 ; /r 0 = 0.6393 − 0.1104 k 0 r0 ;

k0r0 ≤ 0.5, 0.5 < k0 r0 < 2.

(3)

At sufficiently low frequencies, such that k0r0 < 0.5, the stationary impedance radiation flow resistance can be approximated as: Zr (0) ≈ Z0

k02 r02 + 0.6k0r0 ; 4

k0 r0 < 0.5.

(4)

This approximate expression is generally good enough to cover the entire range of plane wave propagation with the cut-off frequency given by k0rsh = 1.84, where rsh , the radius of the muffler shell, is about three to four times r0 , the radius of the radiating tail pipe. Working with the convective state variables, the convective radiation flow impedance is given by the formula: Zc,r = Z0

Zr (M)/Z0 + M . MZr (M)/Z0 + 1

(5)

Zr (M) differs from the corresponding stationary medium Zr (0) not only because of the convective effect, but also the interaction of the outgoing (radiated) wave with an unstable cylindrical vortex layer of the meanflow jet [Munt (1990)]. In fact, the

796

K

Muffler Acoustics

total acoustic power in the farfield ¢F is less than the acoustic power ¢T transmitted out [Howe (1980)], particularly at low frequencies: ¢ F ≈ ¢T

(k0 r0 )2 .∗) 2M + (k0 r0 )2

(6)

K.3 Transfer Matrix Representation Transfer matrix representation is ideally suited for the analysis of cascaded one-dimensional systems like acoustic filters or mufflers. The performance of a muffler may be obtained readily in terms of the fourpole parameters or transfer matrix of the entire system, which in turn may be computed by means of successive multiplication of the transfer matrices of the constituent elements. Transfer matrices of different elements constituting commercial mufflers are given in the subsequent sections of this chapter. Transformation from classical state variables p and u to the convective ones pc and uc may be obtained as follows.

T

T Let {S} = p, u and {Sc } = pc ,uc . Below, vectors are denoted by braces { } and matrices by brackets [ ].

Subscripts u and d denote the upstream end and downstream end respectively of a muffler element, and {S}u = [T]{S}d , {Sc }u = [Tc ]{Sc }d , where [T], [Tc ] are transfer matrices, then [Tc ] = [C]u [T][C]−1 d . −1 [T ][C] Conversely: [T] = [C] d, u c 1 MZ0 where [C] is the transformation matrix: [C] = . M/Z0 1 Thus, one can work with classical state variables or convective state variables as personal preference and skip from one system to the other at the end as necessary.

(1) (2) (3) (4) per

K.4 Muffler Performance Parameters The performance of a muffler is measured in terms of one of the following parameters: • insertion loss (IL), • transmission loss (TL), • level difference (LD) or noise reduction (NR). ∗)


Muffler Acoustics

K

797

Insertion loss (IL) is defined as the difference between the acoustic power radiated without any muffler and that with the muffler (inserted, as it were, between the source and the radiation load impedance), as shown in the figure.

Writing ¢(M) in the form which defines Re , [Prasad/Crocker (1983)]: ¢(M) =

|u|2

2 |u|2 R r + MZ0 + M Zr /Z0 = R, 2 2 e

(1)

the insertion loss may be expressed in terms of the product transfer matrix parameters as:

Re1 1/2

Zrn T11 + T12 + Zrn Zs T21 + Zs T22

IL = 20 log (2)

Ren Zr1 + Zs with Re1 = Rr1 + M1 Z1 + M1 |Zr1 |2 /Z1 , Ren = Rrn + Mn Zn + Mn |Zrn |2 /Zn ,

(3)

where Zrn = R rn + j · Xrn is the (flow) radiation impedance of the nth element, especially Zr1 that of the exhaust pipe without any muffler and Zrn that of the tailpipe, and Zn is the characteristic flow impedance in the nth element. Further, Zs is the source (flow) impedance defined with respect to classical field variables. T11 , T12 , T21 and T22 are the fourpole parameters of the product transfer matrix of the entire muffler, obtained by successive multiplication of the transfer matrices of exhaust pipe (element number 1, next to the source), elements of the muffler proper, ending with the tailpipe (element n), as shown in the figure. The transmission loss (TL) is independent of the source and presumes (or requires) an anechoic termination at the downstream end. It is defined as the difference between the power incident on the muffler proper and that transmitted downstream into the anechoic termination, as shown in the figure. It is given by [Munjal (1987)]:

Zn 1/2 (1 + M1 )

T11 + T12 /Zn + T21 Z1 + T22 Z1/Zn , TL = 20 log Z1 2(1 + Mn )

(4)

798

K

Muffler Acoustics

where M1 is the meanflow Mach number in the exhaust pipe, Mn is the meanflow Mach number in the tailpipe, and all other symbols carry the same connotations as in the expression for IL above. The level difference (LD) (or noise reduction, NR) is the difference in sound pressure levels at two arbitrarily selected points in the exhaust pipe and tailpipe, as shown in the figure.

It is given by [Munjal (1987)]:

T11 + T12 /Zrn

, LD = 20 log cos(k0n ) + jZn sin(k0n )/Zrn

(5)

where n is the distance of the downstream microphone from the radiation end of the tailpipe and k0 = –/c0 ; all other symbols carry the same connotations as in the expression for IL above.

K.5 Uniform Tube with Flow and Viscous Losses For plane linear waves in a uniform-area tube with meanflow Mach number M and viscothermal and turbulent friction loss, the acoustic pressure and particle velocity distributionare given by [Panicker/Munjal, J.Acoust. Soc. India 9 (1981); Munjal (1987)]:

p(z) = ejMkc z Ae −jkc z + Be jkc z , (1) jMkc z

Ae −jkc z − Bejk c z , u(z) = e Z where k0 + − j( + M) k0 + M 1+ , = −j kc = 1 − M2 1 − M2 k0 k0

(2)

Muffler Acoustics

† ‰ K F d Re

= = = = = = =

K

799

dynqamic vicosity; adiabatic exponent; heat conductivity; F/2d; Froude’s friction factor; 2r0 = hydraulic diameter; Vd0 /† = Reynold’s number

+ M + M , Z = Z0 1 + −j k0 k0 –† 1/2 = r01c0 2 e , 0 1/2 2 1 K 1/2 , †e = † 1 + ‰ − 1/2 †Cp ‰ F = 0.0072 + 0.612 ; Re0.35

(3)

Re < 4 · 105.

On elimination of A and B from the field equations (1) for z = 0 (entrance, indicated by subscript u) and z = l (exit of the duct with length , indicated by subscript d), one gets the desired transfer matrix relation: p jZ sin(kc ) p cos(kc ) = e−jMkc . (4) u u (j/Z) sin(kc ) cos(kc ) u d Incidentally, the same transfer matrix would hold for the convective state variables pc and uc at both ends. For inviscid stationary medium, the transfer matrix for a uniform tube reduces to jZ0 sin(k0) cos(k0) . (j/Z0 ) sin(k0 ) cos(k0)

(5)

K.6 Sudden Area Changes Subscripts u and d indicate points just upstream and just downstream of the sudden area discontinuity. Typically,the meanflow Mach number M in the smaller diameter tube is M < 0.2. For sudden expansion and sudden contraction, the equations of mass continuity, momentum balance, stagnation pressure drop and entropy fluctuations [Alfredson/Davies (1970); Panicker/Munjal, J. Indian Inst. Sc. 63, pp. 1–19 (1981)], for plane waves and incompressible mean flow, yield the following transfer matrix relation [Munjal (1987)]: ⎡ ⎤ Kd M2d K d Md Z d ⎢ 1 − 1 − M2 ⎥ pc,d pc,u ⎢ d ⎥ =⎣ , (1) uc,u (‰ − 1)Kd M3d (‰ − 1)Kd M2d ⎦ uc,d 1− 2 2 (1 − Md )Zd 1 − Md

800

K

Muffler Acoustics

where Kd is the K-factor indicating the drop in stagnation pressure of the incompressible mean flow in terms of the dynamic head 0 V2d /2: Kd = (1 − Sd /Su )/2

for sudden contraction,

Kd = [(Sd /Su ) − 1]2

for sudden expansion.

(2)

If M < 0.2 in the smaller diameter tube, the foregoing transfer matrix may be approximated as [Munjal (1987)]: 1 Kd M d Z d pc,u pc,d = . (3) uc,u 0 1 uc,d This relation, on transformation to the classical state variables and on incorporating the low Mach number simplification, becomes: pu pd 1 (1 + Kd )Md Zd − Mu Zu = . (4) uu 0 1 ud In the simplified, approximate, form, the end-correction effect or evanescent higherorder mode effect can be incorporated readily as an inline lumped inertance [Sahasrabudhe et al. (1995)]: pd pu 1 (1 + Kd )Md Zd − Mu Zu + j–H() · 0.85rp /Sp = , (5) uu 0 1 ud where rp is the radius of the narrower pipe, = 0.85rp is the end correction of the orifice in a baffle wall, H() = 1 − 1.25rp /rch for co-axial tubes, rch is radius of the chamber (or the larger diameter tube). If the junction of the pipe (the smaller diameter tube) and chamber (the larger diameter tube) is not co-axial, then the factor H() is given by the polynomial [Panicker, Munjal (1981)]: H() = 1.442 + 3.516 ƒ − 5.403 − 0.068 kr − 11.067 ƒ2 + 10.462 ƒ2 − 0.099(kr)2 + 2.517 ƒ − 0.197 ƒ · kr + 1.024 · kr + 7.774 ƒ3 − 8.15 3 − 0.05(kr)3 − 0.841 ƒ2 + 0.131 ƒ2 · kr − 3.378 ƒ2 − 1.311 2 · kr + 0.141(kr)2 · ƒ − 0.067(kr)2 − 0.031 ƒ · · kr, where kr = k0 rch ; = rp /rch ; ƒ = offset distance/(rch − rp ).

(6)

Muffler Acoustics

For the case of a stationary medium, the transfer matrix would reduce to 1 j–H()0.85rp /Sp . 0 1

K

801

(7)

The description of sudden area changes would become simple if one neglected the evanescent higher-order mode effects, because then the transfer matrix would reduce to a unity matrix, implying pu = pd and uu = ud . Decomposing the standing wave on the entrance side into the forward moving and reflected progressive waves, for anechoic termination, one gets, [Munjal (1987)]: Reflection factor: Transmission loss:

Zd − Z u Su − Sd ≈ ; Zd + Zu Su + Sd (Sd + Su )2 (Zu + Zd )2 ≈ 10 · log . TL = 10 lg 4Zu Zd 4Sd Su

R=

(8) (9)

These relationships represent the principle of impedance mismatch, which is the underlying principle of reflective (or reactive, or non-dissipative) mufflers. When the characteristic impedance undergoes a sudden change or jump, a significant portion of the incident acoustic power is reflected back to the source. This impedance jump may be obtained in several ingenious ways, one of which is sudden area changes. The symmetry of the expression for TL shows that what matters is a sudden jump or change in characteristic impedance, not whether it increases (as in sudden contraction) or decreases (as in sudden expansion).

K.7 Extended Inlet/Outlet See also: > Sect. J.18 for simple expansions or contractions of lined ducts, > Sect. J.19 for sequences of duct sections without feedback (all those sections without flow).

Four types of extended-tube sudden area discontinuities are shown below. Equations of mass continuity, momentum balance, stagnation pressure drop and entropy fluctuations for plane waves and incompressible meanflow [Alfredson/Davies (1970); Panicker/Munjal, J. Indian Inst. Sc. 63 (A) (1981), pp. 1–19, 21–38] yield a transfer matrix relation that is a little too involved. However, for typical cases where the meanflow Mach number M in the smaller diameter tube is M < 0.2, the transfer matrix relation between convective state variables pc , uc is given by [Munjal (1987)]: ⎤ ⎡ 1 K d Md Z d pc,u pc,d ⎦ ⎣ = . (1) Cr Sr Zr − Md Zd (Cd Sd + Kd Su ) Cr Sr uc,u uc,d Cr Sr Zr + Su Mu Zu Cr Sr Zr + Su Mu Zu

802

K

Muffler Acoustics

Therein • subscripts u and d indicate points just upstream and just downstream of the sudden area discontinuity as shown in the figures; • Kd is the K-factorindicating the drop in stagnation pressure of incompressible mean flow in terms of the dynamic head 1/2 0V2d ; • Sx are cross-section areas at x = u, d, r; • Mx are Mach numbers at these positions; • Zd , Zu , Z2 are characteristic flow impedances; • Zr = −j Z 2 cot (k02 ) is the input flow impedance of the annular cavity (2) resonator. Thus, Kd = (1 − Sd /Su )/2 = (Sd /Su − 1)2 = (Sd /Su )2 = 0.5

for extended outlet, for extended inlet, for reversal expansion, for reversal contraction.

(3)

The constants Cr and Cd are given by the area compatibility equation: Su + Cd Sd + Cr Sr = 0. Thus, Cd = −1 ; Cd = −1 ; Cd = +1 ; Cd = +1 ;

Cr Cr Cr Cr

(4) = −1 = +1 = −1 = −1

for extended outlet, for extended inlet, for reversal expansion, for reversal contraction.

(5)

The foregoing transfer matrix relationship between the convective state variables [Tc ] can be transformed into the one between classical state variables [T] making use of the relationship given in > Sect. K.3: [T] = [Cu ]−1 [Tc ][Cd ], where [Cx ] are the transformation matrices.

(6)

Muffler Acoustics

K

803

K.8 Conical Tube With tube length and tube radii ru , rd , the following relations are used: ru , rd − ru 0 c0 . Zd = rd2 z1 =

z2 = z1 +,

For a stationary medium, the transfer matrix relationship for a conical tube is given by [Munjal (1987)]: ⎡ ⎤ ⎡ ⎤ ⎤⎡ z2 sin(k0) z2 pu pd ⎢ ⎥ ⎢ z1 cos(k0 ) − k0 z1 jZd z1 sin(k0) ⎥ ⎥⎢ ⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎢ ⎥ ⎢ j z1 ⎥ ⎢ ⎥ sin (k 0 ) z1 1 ⎢ uu ⎥ = ⎢ ⎥ ⎢ ⎥ ) + cos(k ) u (1) 1 + sin(k 0 d ⎥ . 0 ⎢ ⎥ ⎢ Z z2 2 ⎢ ⎥ k0 z2 z2 k0 z1 z2 ⎢ ⎥ ⎢ d ⎥ ⎥⎢ ⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎣ ⎦ ⎣ ⎦ ⎦⎣ z1 cos(k0 ) − 1− z2 k0 z2 These expression hold for a convergent tube as well as a divergent tube. For a moving medium, assuming that the flare is small enough to avoid separation of boundary layer, Easwaran and Munjal (1992) have solved the wave equation with variable coefficients analytically to obtain the transfer matrix parameters for a conic tube. Dokumaci has obtained these parameters numerically by means of matrizants [Dokumaci (1998)]. The resultant expressions are rather complicated. Fortunately, however, the convective effect of incompressible meanflow (for M < 0.2) is negligible in the case of conical tubes as well as uniform tubes.

804

K

Muffler Acoustics

K.9 Exponential Horn The medium is supposed to be stationary. The horn (or tube section) is characterised by the relations (with r(z) the radius at position z; m is the flare constant): r(z) = r(0)emz ;

S(z) = S(0)e2mz ;

(1) 0 c0 2 ; ru = r(0); ru ⎡ ⎤⎡ ⎤ ⎤ ⎡ pu pd je−m k0 Zu sin(k ) em cos(k )− m sin(k ) k k ⎢ ⎥⎢ ⎥ ⎥ ⎢ ⎦⎣ ⎦ , (2) ⎣ ⎦=⎣ j m k0 m sin(k ) −m cos(k e sin(k ) e ) + uu ud Zu k k Zu =

where k = (k02 − m2 )1/2 ;

k0 = –/c0 .

(3)

These expressions hold for a convergent horn as well as a divergent horn.

For a moving medium, assuming that the flare is small enough to avoid separation of boundary layer, [Easwaran/Munjal (1992)] have solved the wave equation with variable coefficients analytically to obtain the transfer matrix parameters for an exponential tube. Dokumaci has obtained the same numerically by means of matrizants [Dokumaci (1998)]. The resultant expressions are rather too complicated. Fortunately, however, the convective effect of incompressible meanflow (for M < 0.2) is negligible in the case of exponential tubes as well as conical tubes.

K.10 Hose A hose is a uniform tube of interior radius ri with compliant walls. Incorporating the local wall impedance in the mass continuity equation, and the losses due to visco-thermal friction and turbulent eddies in the momentum equation,and neglecting entropy fluctuations for the linear plane waves, yields the following transfer matrix relationship [Munjal/Thawani (1996)]: + − − + − + ej(k −k ) Z− e−j k + Z+ ejk Z+ Z− (e+j k − e−jk ) pu pd = + . (1) − + + − uu ud Z+ e−jk + Z− ej k ejk − e−jk Z + Z−

Muffler Acoustics

K

Here, the wave numbers are given by: ⎡ 1/2 k M ri + Gi 2 M ri + Gi 2 i ± 2 ⎣ k = + (1 − M ) 1−j 1 − M2 ki r i k i ri ⎤ M ri + G i ⎦ . ∓ M 1−j ki ri

805

(2)

The subscript i refers to the medium inside the hose; superscripts + and − refer to the forward wave and reflected wave, respectively. The factor within braces is an empirical factor representing the effect of curvature on the inertance of the hose. The other terms are as follows: k = –/ci ;

M = V/ci ;

M = + MF 4ri ; Gi =

–† 1/2 = r 1c 2 ; i 0 0

(3)

i ci ; Zw

F = 0.0072 + 0.612 ; Re0.35

Re =

Vdi † ;

d = 2ri .

Gi is the normalised wall admittance, Zw the wall impedance. The axial volume flow impedances are given by: Z± = Z0

ki ∓ Mk± − 2jM ; k±

Z0 =

i ci . ri2

(4)

The hose-wall impedance is given by Zw = Zc + Zm + Zo with the component Zc due to compliance (for thin hoses): 2 r0 + ri2 E Et +‹ ≈ , (5) Zc = 2 2 j–ri r0 − ri j –ri2 and the component Zm due to mass reactance: 0.025 2 Zm = j–w t 1 + . ri

(6)

Here E is the complex elastic modulus, Er (–) 1 + j†(–) ; †(–) is the loss factor and Er (–) is the storage modulus of the hose wall material; ‹ is the Poisson ratio of the hose-wall material; w and t are the density and thickness of the wall. Finally, Z0 is the radiation impedance (subscript o refers to the outer medium): Zo = j–

o H(2) 0 (kro ro ) , kro H(2) 1 (kro ro )

(7)

806

K

Muffler Acoustics

where H(2) i (z) = Ji (z) − jYi (z);

i = 0 or 1

are Hankel functions, and the wave numbers are: 1/2 4ji ci ki 2 2 kro = + k 0 − ki ; k0 = –/c0 . dZw

(8)

For the limiting case of rigid walls, Zw → ∞, the transfer matrix would reduce to that for a uniform tube with rigid walls ( > Sect. K.5) if one neglected 2M with respect to ki2. For the limiting case of a hose with inviscid stationary medium inside (M → 0, M → 0), we get: Gi i ci k ± = ki 1 − j = ki 1 − j . (9) ki ri Zw ki ri At low frequencies, Zw would be dominated by its compliance component Zc ; then, for thin hoses, the wave number in either direction would be given by: ri i c2i . (10) k = ki 1 + Et Writing k = –/ceq , the equivalent sound speed in the fluid inside a hose is given by: ci ci ceq = . (11) = 2 c2 rc 1 + i Eti i 1 + rti · i i2 w cw Thus, the effect of wall compliance is to match the characteristic impedance and then to reduce the effective velocity of wave propagation inside a hose pipe.

K.11 Two-Duct Perforated Elements Concentric-tube resonators as well as concentric cross-flow elements have an inner region 1 and outer region 2 which are coupled with each other on a length via a perforated tube.

The mass continuity and momentum equations in the inner region 1 and outer region 2 are combined with the isentropicity equation. These are solved simultaneously for harmonic time dependence, exp(j–t), to obtain coupled quadratic equations in p1 (z) and

Muffler Acoustics

K

807

p2 (z) [Munjal (1987)]. These are then reduced to four linear differential equations so as to apply the standard eigenvalue program [Peat (1988)]. Thence we obtain the following 4 × 4 transfer matrix relation [Munjal (1987)]: ⎡ ⎤ ⎡ ⎤ ⎤⎡ p1 (0) T11 T12 T13 T14 p1 () ⎢ p2 (0) ⎥ ⎢ T21 T22 T23 T24 ⎥ ⎢ p2 () ⎥ ⎢ ⎥ ⎢ ⎥ ⎥⎢ (1) ⎣ Z1 u1(0) ⎦ = ⎣ T31 T32 T33 T34 ⎦ ⎣ Z1 u1() ⎦ , Z2 u2(0) Z2 u2() T41 T42 T43 T44 where [T] = [A(0)] [A(1)]−1 . Elements of the matrix [A(z)] are given by (i = 1, 2, 3, 4): A1,i = •3,i ei z ; A2i = •4i ei z ; • ei z ei z ; A4i = − 2,i . A3,i = − jk0 + M1 i jk0 + M2 i

(2)

[•] and {} are respectively the eigenmatrix (or modal matrix) and eigenvector of the matrix: ⎡ ⎤ −1 −3 −2 −4 ⎢ −5 −7 −6 −8 ⎥ ⎢ ⎥, ⎣ 1 0 0 0 ⎦ 0 1 0 0 where 2 ka + k02 ka2 jM1 ; 1 = − ; 2 = 2 1 − M1 1 − M21 k0 2 2 ka − k02 ka − k02 jM1 3 = = − ; ; 4 k0 1 − M21 1 − M21

(3a)

2 kb − k02 kb2 − k02 ; 6 = − ; k0 1 − M22 2 kb2 kb + k02 jM2 7 = − ; 8 = 2 ko 1 − M2 1 − M22

(3b)

jM2 5 = 1 − M22

with k0 = –/c0 ; M1 = V1 /c0; M2 = V2 /c0; 4jk0 4jk d ka2 = k02 − ; kb2 = k02 − 2 0 21 . di … d2 − d1 …

(4)

… is the normalised partition impedance of the perforate. For different flow conditions, … is given by the following empirical expressions: Perforates with cross flow [Sullivan (1979)]: d1 M p = 0.514 + j0.95k0(t + 0.75dh ) / , …= 0 c0 v

(5)

K

808

where

Muffler Acoustics

d1 is diameter of the perforated tube, M is the mean-flow Mach number in the tube, is the length of perforate, is porosity, f is frequency, t is the thickness of the perforated tube, dh is the hole diameter, v is the (average) radial particle velocity at the perforate.

For perforates in stationary media [Sullivan/Crocker (1978)]: … = [0.006 + jk0 (t + 0.75dh )]/ .

(6)

Perforates with grazing flow [Rao/Munjal (1986)]: … = [7.337 × 10−3 (1 + 72.23M) + j2.2245 × 10−5 (1 + 51t)(1 + 204dh )f ]/ ,

(7)

where wall thickness t and hole diameter dh are in metres. The desired 2 × 2 transfer matrix for a particular two-duct element may be obtained from the 4 × 4 matrix [T] by making use of the appropriate upstream and downstream variables and two boundary conditions characteristic of the element. The final results [Munjal (1987)] are given below for various two-duct elements shown in figures. (a) Concentric-tube resonator:

p1 (0) Z1 u1 (0)

=

Ta Tc

Tb Td

p1 () Z1u1 ()

∗)

(8)

with Ta = T11 + A1 A2 ; Tc = T31 + A1 B2 ;

Tb = T13 + B1 A2 ; Td = T33 + B1 B2 ;

(9)

A1 = (X1 T21 − T41 )/F1 ; B1 = (X1T23 − T43)/F1 ; A2 = T12 + X2T14 ; B2 = T32 + X2 T34 ;

(10)

F1 = T42 + X2T44 − X1(T22 + X2T24 ) ; X1 = −j tan(k0 a ) ; X2 = +j tan(k0 b ) .

(11)

∗)


Muffler Acoustics

K

809

(b) Cross-flow expansion element:

p1 (0) Z1 u1 (0)

=

Ta Tc

Tb Td

p2 () Z2 u2()

(12)

with Ta = T12 + A1 A2 ; Tc = T32 + A1 B2 ;

Tb = T14 + B1 A2 ; Td = T34 + B1 B2 ;

(13)

A1 = (X1 T22 − T42 )/F1 ; B1 = (X1T24 − T44)/F1 ; A2 = T11 + X2T13 ; B2 = T31 + X2T33 ;

(14)

F1 = T41 + X2T43 − X1(T21 + X2T23 ) ; X1 = −j tan(k0 a ) ; X2 = j tan(k0b ) .

(15)

(c) Cross-flow contraction element:

p2 (0) Z2 u2 (0)

=

Ta Tc

Tb Td

p1 () Z1 u1()

(16)

with Ta = T21 + A1 A2 ; Tc = T41 + A1 B2 ;

Tb = T23 + B1 A2 ; Td = T43 + B1 B2 ;

A1 = (X1 T11 − T31 )/F1 ; B1 = (X1T13 − T33)/F1 ; A2 = T22 + X2T24 ; B2 = T42 + X2 T44 ;

(17) (18)

810

K

Muffler Acoustics

F1 = T32 + X2T34 − X1(T12 + X2T14 ) ; X1 = −j tan(k0 b ) ; X2 = j tan(k0b ) .

(19)

(d) Reverse-flow expansion element:

p1 (0) Z1 u1 (0)

=

Ta Tc

−Tb −Td

p2 (0) Z2 u2(0)

.

(20)

The minus sign with Tb and Td is due to reversal in the direction of u2(0),which is needed for making the foregoing relation adaptable to similar relations for other downstream elements. Therein: A1 T a Tb = Tc Td A3

A2 A4

B1 B3

B2 B4

−1 ;

(21)

A1 = T11 + X2T13 ; A3 = T31 + X2T33 ;

A2 = T12 + X2 T14 ; A4 = T32 + X2 T34 ;

(22)

B1 = T21 + X2 T23 ; B3 = T41 + X2 T43 ;

B2 = T22 + X2 T24 ; B4 = T42 + X2 T44 ;

(23)

X2 = j tan(k0b ) . (e) Reverse-flow contraction element:

(24)

Muffler Acoustics

p2 (0) Z2 u2 (0)

with Ta Tc

Tb Td

=

−Tb Td

−Ta Tc

A2 A4

A1 A3

=

−1

p1 (0) Z1 u1 (0)

B2 B4

B1 B3

K

811

(25)

;

(26)

A1 = T32 + Xa T12 ; A3 = T42 + Xa T22 ;

A2 = T34 + Xa T14 ; A4 = T44 + Xa T24 ;

(27)

B1 = T31 + Xa T11 ; B3 = T41 + Xa T21 ;

B2 = T33 + Xa T21 ; B4 = T43 + Xa T23 ;

(28)

Xa = j tan(k0 a ) .

(29)

(f) Reversal expansion, two-duct, open-end, perforated element:

pu Zu uu

with Ta Tc

Tb Td

=

Ta Tc

=

−Tb −Td

A11 A21

pd Z d ud

A12 A22

B11 B21

(30)

B12 B22

−1 ;

(31)

A11 A12 A21 A22

= T11 F11 + T12 + T13 F21 = T11 F12 + T14 + T13 F22 = T31 F11 + T32 + T33 F21 = T31 F12 + T34 + T33 F22

; ; ; ;

(32)

B11 B12 B21 B22

= T21F11 + T22 + T23 F21 = T21F12 + T24 + T23 F22 = T41F11 + T42 + T43 F21 = T41F12 + T44 + T43 F22

; ; ; ;

(33)

F11 = E11 ; F12 = −E12 /Zu ; F21 = E21 Zd ; F22 = −E22 Zd /Zu .

(34)

812

K

Muffler Acoustics

Matrix [E] is the 2×2 transfer matrix of the reversal expansion element of > Sect. K.7(c). [T] is the 4 × 4 matrix for the perforated section of the two interacting ducts derived earlier [Eq. (1)]. (g) Reversal contraction, two-duct, open-end perforated element:

pu Zu uu

with Ta Tc

=

Ta −Tc

Tb Td

=

C11 C21

Tb −Td

C12 C22

pd Zd ud

B11 B21

(35)

B12 B22

−1 ;

[B] = [P] + [Q][F] ; [C] = [R] + [U][F] ; P11 = A11 ; Q11 = A12 ; R11 = A12 ; U11 = A22 ;

P12 = A13 ; Q12 = A14 ; R12 = A23 ; U12 = A24 ;

(36) (37)

P21 = A31 ; Q21 = A32; R21 = A41 ; U21 = A42 ;

P22 = A33 ; Q22 = A34 ; R22 = A43 ; U22 = A44 ;

[A] = [T]−1 .

(38)

(39)

[T] is the 4 × 4 matrix for the perforated section of the two interacting ducts, derived above. F12 = D12 /Zd ; F11 = D11 ; F21 = D21 Zu ; F22 = −D22 Zu /Zd . [D] is the 2 × 2 transfer matrix of the reversal contraction element of

(40) > Sect. K.7(d).

Muffler Acoustics

K

813

(h) Perforated extended outlet:

pu uu

=

C11 C21 /Zu

C12 Zd C22 Zd /Zu

pd ud

(41)

with [C] = [A]−1 [B] ;

(42)

A11 = 1 − f3Z2u ; A12 = Mu (1 − 2f3 Z2u ) ; A21 = Mu − f4Z2u ; A22 = 1 − 2f4Z2u Mu ;

(43)

B11 B12 B21 B22

= T11 + Md Kp1T31 − f3 T21 + Z21 {T11 + 2Mu T31 } ; = T13 + Md Kp1T33 − f3 T23 + Z21 {T13 + 2Mu T33 } ; = Zu2 Tu1 + Zu1 {T31 + Md T11 } − num1 · f4 ; = Zu2 T43 + Zu1 {T33 + Md T13 } − num2 · f4 ;

f3 = num3/den;

f4 = num4/den;

(44)

(45)

num1 = T21 + Z21 {T11 + 2MuT31 } ; num2 = T23 + Z21 {T13 + 2MuT33 } ; num3 = T12 + Xa T14 + Md Kp1 {T32 + Xa T34 } ; num4 = Zu2 {T42 + Xa T44 } + Zu1 [T32 + Md T12 + Xa {T34 + Md T14 }] ;

(46)

den = T22 + Xa T24 + Z21 [T12 + 2Mu T32 + Xa {T14 + 2Mu T34 }] ;

(47)

Z21 = Z2 /Z1 ; Zu2 = Zu /Z2 ; Zd = 0 c0 /Sd ; Z2 = 0 c0 /Sa ;

(48)

Z2u = Z2 /Zu ; Zu1 = Zu /Z1 ; Zu = 0 c0 /Su ; Z1 = Zd ;

S2 = Su − Sd ; Xa = j tan(ka ) ; 2 1 Kc = 2 1 − rrshi ; Kp1 = Kc +1.

(49)

[T] is the 4 × 4 matrix for the perforated section of the two interacting ducts, derived above.

K

814

Muffler Acoustics

(i) Perforated extended inlet:

pu uu

=

D11 D31 /Zu

D12Zd D32Zd /Zu

pd ud

,

(50)

where [D] is a 4 × 2 matrix given by: [D] = [T][A][B].

(51)

[A] is a 4 × 4 matrix and [B] is a 4 × 2 matrix, elements of which are as follows: A11 A21 A31 A41 A43

= 1 ; A12 = 0 ; A13 = Mu ; A14 = 0 ; = M1 /Zw ; A22 = 0 ; A23 = 1/Zu ; A24 = 1/Zd ; = 1/Zu ; A32 = 1/Zd ; A33 = 2Mu /Zu ; A34 = 0 ; = T41 − Xa T21 ; A42 = T42 − Xa T22 ; = T43 − Xa T23 ; A44 = T44 − Xa T24 ;

(52)

B11 = 1 ; B12 = Md (1 + Ke ) ; B21 = Md/Zd ; B22 = 1/Zd ; B31 = 1/Zd ; B32 = 2Md /Zd ; B41 = 0 ; B42 = 0 ;

(53)

Z u = 0 c0 /Su;

(54)

Zd = 0 c0 /Sd ; Z2 = 0 c0 /(Sd − Su ); 2 Xa = −j tan(ka ); Ke = 1 − (ru /rd )2 .

(55)

[T] is the 4 × 4 matrix for the perforated section of the two interactive ducts, derived above.

K.12 Three-Duct Perforated Elements The two-duct perforated section shown in > Sect. K.11 is common to the two-duct perforated muffler elements, and it was treated in > Sect. K.11. Similarly, the three-duct perforated section shown below is common to the three-duct muffler elements discussed below. This common three-duct perforated section is described by the transfer matrix relation [Munjal (1987)]: ⎡ ⎤⎡ ⎤ ⎡ ⎤ p1 (0) p1 () T11 T12 T13 T14 T15 T16 ⎢ p2 (0) ⎥ ⎢ T21 T22 T23 T24 T25 T26 ⎥ ⎢ p2 () ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ p3 (0) ⎥ ⎢ T31 T32 T33 T34 T35 T36 ⎥ ⎢ p3 () ⎥ ⎢ ⎥⎢ ⎥= ⎢ ⎥ (1) ⎢ Z u1(0) ⎥ ⎢ T41 T42 T43 T44 T45 T46 ⎥ ⎢ Z u1() ⎥ , ⎢ 1 ⎥⎢ 1 ⎥ ⎢ ⎥ ⎣ Z2 u2(0) ⎦ ⎣ T51 T52 T53 T54 T55 T56 ⎦ ⎣ Z2 u2() ⎦ Z3 u3(0) Z3 u3() T61 T62 T63 T64 T65 T66

Muffler Acoustics

K

815

where [T] = [A(0)] [A()]−1 .

(2)

The elements of the matrix [A(z)] are given by (i = 1, 2, . . ., 6): A1,i = •4,i ei,z ; A2,i = •5,i ei,z ; A3,i = •6,i ei,z ; A4,i = −ei,z / jk0 + M1 i ; A5,i = −•2,i ei,z / jk0 + M2i ; A6,i = −•3,i ei,z / jk0 + M3i .

(3)

[•] and {} are, respectively, the eigenmatrix (or modal matrix) and eigenvector of the following matrix [Peat (1988)]: ⎡ ⎤ −1 3 0 −2 −4 0 ⎢ −5 −7 −9 −6 −8 −10 ⎥ ⎢ ⎥ ⎢ 0 −11 −13 0 −12 −14 ⎥ ⎢ ⎥, (4) ⎢ 1 0 0 0 0 0 ⎥ ⎢ ⎥ ⎣ 0 1 0 0 0 0 ⎦ 0 0 1 0 0 0 where

2 ka2 ka − k02 jM1 ; 3 = ; 2 = ; k0 1 − M21 1 − M21 2 k 2 − k02 kb2 − k02 kb − k02 jM2 ; = = ; 4 = a ; 5 6 k0 1 − M21 1 − M22 1 − M22 jM2 kc2 − k02 k 2 + kc2 − k02 −jM2 kb2 − kc2 ; 9 = ; 7 = ; 8 = b 2 2 k0 1 − M2 1 − M2 k0 1 − M22 2 kb − k02 k 2 − k02 k 2 − k02 jM3 ; 12 = d 10 = c ; = ; 11 2 2 k0 1 − M2 1 − M3 1 − M23 kd2 −jM3 kd2 + k02 ; 13 = = , 14 k0 1 − M23 1 − M23 −jM1 1 = 1 − M21

ka2 + k02 k0

(5)

K

Muffler Acoustics

M1 = V1 /c0 ;

M2 = V2/c0 ;

816

and

ka2 = k02 −

4jk0 ; d1 …1

kd2 = k02 −

4jk0 . d3 …2

kb2 = k02 −

M3 = V3/c0 ; 4jk0 d1 ; (d22 − d12 − d32 )…1

kc2 = k02 −

4jk0d3 ; (d22 − d12 − d32 )…2

(6)

The desired 2 × 2 transfer matrix for a particular three-duct perforated element may be obtained from the 6 × 6 transfer matrix [T] above, making use of the appropriate upstream and downstream variables and four boundary conditions characteristic of the element. The final results are given below for different three-duct perforated elements shown in graphs. (a) Cross-flow, three-duct, closed-end element

p1 (0) Z1 u1 (0)

=

Ta Tc

Tb Td

p3 () Z3u3 ()

,

(7)

Tb = TT1,4 + B3 C3 ; Td = TT3,4 + B3 D3 ;

(8)

where Ta = TT1,2 + A3C3 ; Tc = TT3,2 + A3 D3 ;

A3 = (TT2,2X2 − TT4,2)/F2 ; B3 = (TT2,4 X2 − TT4,4)/F2 ; C3 = TT1,1 + X1 TT1,3 ; D3 = TT3,1 + X1 TT3,3 ; F2 = TT4,1 + X1 TT4,3 − X2(TT2,1 + X1 TT2,3 )

(9) (10)

Muffler Acoustics

K

817

with TT1,1 = A1A2 + T1,2 ; TT1,2 = B1 A2 + T1,3 ; TT1,3 = C1 A2 + T1,5 ; TT1,4 = D1 A2 + T1,6 ; TT2,1 TT2,4 TT3,1 TT3,4

= A1B2 + T2,2 ; TT2,2 = B1 B2 + T2,3 ; TT2,3 = C1 B2 + T2,5 ; = D1 B2 + T2,6 ; = A1C2 + T4,2 ; TT3,2 = B1 C2 + T4,3 ; TT3,3 = C1 C2 + T4,5 ; = D1 C2 + T4,6 ;

(11)

TT4,1 = A1D2 + T5,2 ; TT4,2 = B1 D2 + T5,3 ; TT4,3 = C1D2 + T5,5 ; TT4,4 = D1 D2 + T5,6 ; A1 = (T3,2X2 − T6,2)/F1 ; C1 = (T3,5X2 − T6,5)/F1 ; A2 = T1,1 + T1,4 X1 ; C2 = T4,1 + T4,4 X1 ;

B1 = (T3,3 X2 − T6,3 )/F1 ; D1 = (T3,6 X2 − T6,6)/F1 ; B2 = T2,1 + T2,4X1 ; D2 = T5,1 + T5,4X1 ;

F1 = T6,1 + X1 T6,4 − X2 (T3,1 + X1T3,4 ) ; X1 = j tan(k0b ) ; X2 = −jtan(k 0 a ) .

(12)

(13)

(b) Reverse-flow, three-duct, closed-end element

p1 (0) Z1 u1 (0)

=

Ta Tc

Tb Td

p3 () Z3u3 ()

,

(14)

where Ta = B1,1D1,1 + B1,2 D2,1 + B1,3 D3,1 ; Tc = B4,1D1,1 + B4,2 D2,1 + B4,3D3,1 ; Bi1,i2 = Ti1,i2 + X1 Ti1,i2+3 ;

Tb = B1,1 D1,2 + B1,2 D2,2 + B1,3 D3,2 ; Td = B4,1 D1,2 + B4,2 D2,2 + B4,3 D3,2 ;

i1 = 1, 2, . . . , 6, i2 = 1, 2, 3 ;

(15)

(16)

818

K

Muffler Acoustics

D1,1 = C1,1D2,1 + C1,2D3,1 ; D1,2 = C1,1 D2,2 + C1,2D3,2 ; D2,1 = C3,2/F4 ; D2,2 = −C2,2/F4 ; D3,1 = −C3,1 /F4 ; D3,2 = −C2,1 /F4 ; C1,1 = B5,2 − X2 B2,2 /F3 ; C1,2 = B5,3 − X2B2,3 /F3 ; C2,1 = B3,2 − C1,1 B3,1 ; C2,2 = B3,3 − C1,2 B3,1 ; C3,1 = B6,2 − C1,1 B6,1 ; C3,2 = B6,3 − C1,2 B6,1 ; F3 = X2 B2,1 − B5,1 ; F4 = C2,1 C3,2 − C2,2 C3,1 ; X1 = jtan(k 0 b ) ; X2 = −j tan(k0 a ) .

(17)

(18)

(19)

The transfer matrices in > Sects. K.11 and K.12 for perforated elements have been derived in the form: Ta T b pd pu = . (20) Zu uu Tc Td Z d ud This can be rewritten in the standard dimensional form as: Ta pd pu Zd Tb = . uu Tc /Zu Zd Td /Zu ud

(21)

(c) Cross-flow, three-duct open-end element For the open-end elements shown below, the boundary conditions are not available as simple end-impedance expressions, but in the form of reversed-flow elements, the transfer matrices for which were given earlier in > Sect. K.11.

Another characteristic of these elements is that the ends of the perforated inlet and outlet ducts being open,the gases moving through the ducts escape (or enter) partially through the perforations and partially through the end. This is expected to result in a much lower pressure drop compared to the closed-end elements. It has been observed that most of the meanflow moves straight grazing the perforates, and therefore the perforation impedance is given by the grazing flow impedance. The transfer matrix relationship between the upstream point u and downstream point d is given by [Gogate/Munjal (1995)]: pd pu = P , (22) uu ud

Muffler Acoustics

K

819

where [P] = [[TT11][X] + [TT12]] [MAT] + [TT13] ; (23) [MAT] = [[[TT21 ][X] + [TT22 ]] −[Y] [TT31 ][X] + [TT32 ]]]−1 [[Y][TT33] − [TT23 ]] . [X]

is the transfer matrix of the reverse-flow expansion element at the right-hand junction, given in > Sect. K.11; and is the transfer matrix of the reverse-flow contraction element at the left-hand junction, given in > Sect. K.11.

[Y]

TTij =

[TM]

TMij TMi+3,j

TMi,j+3 TMi+3,j+3

;

i, j = 1, 2, 3.

is the 6 × 6 transfer matrix for the common perforated section of the three interacting ducts, with respect to state variables p and v. Thus:

TMi,4 = Ti,4 · Z1 ; TMi,5 = Ti,5 · Z2 ; TMi,6 = Ti,6 · Z1 ; TM4,i = T4,i /Z1 ; TM5,i = T5,i /Z2 ; TM6,i = T6,i /Z1 . Z1

(24)

(25)

is the volume-flow impedance of the upstream/downstream or inlet/outlet ducts Z1 = 0 c0 /Su ;

Z2

is the volume-flow impedance of the annular duct Z2 = 0 c0 /(Sshell − 2Su )

neglecting the duct wall thickness;

[T] is the 6 × 6 transfer matrix of the common perforated section of the three interacting ducts, [Eq. (1)]. (d) Reverse-flow, open-end, three-duct element

This element is a combination of elements (b) and (c) inasmuch as on the left-hand end it is like the closed-end element (b) and on the right-hand end it is like the open-end

K

820

Muffler Acoustics

of element (c). The transfer matrix relationship between the upstream point u and the downstream point d is given by [Gogate/Munjal (1995)] pu −Tb Zi Ta pd = , (26) uu Tc /Zi − Td ud where Ta = B11 C11 + (B12 + B15 Xa )C21 + B13 C31 + B14 C41 + B16 C51 ; Tb = B11 C12 + (B12 + B15 Xa )C22 + B13 C32 + B14 C42 + B16 C52 ; Tc = B41 C11 + (B42 + B45 Xa )C21 + B43 C31 + B44 C41 + B46 C51 ; Td = B41 C12 + (B42 + B45 Xa )C22 + B43 C32 + B44 C42 + B46 C52 ;

(27)

Xa = j tan(ka )

(28)

;

Xb = −j tan(kb ).

Here [C] = [A]−1 , and [A] is a 5 × 5 matrix whose elements are related to the 6 × 6 matrix [T] as follows: A11 = T31 ; A15 = T36 ; A24 = T64 ; A33 A41 A51 A54

= T53 − Xb T23 ; A34 = T54 − Xb T24 ; A35 = T56 − Xb T26 ; = 1 ; A42 = 0 ; A43 = −1 ; A44 = Mi ; A45 = Mi (1 + Kre + Krc ) ; = Mi /Zi ; A52 = Xa /Z6 ; A53 = −Mi /Zi ; = 1/Zi ; A55 = 1/Zi .

Kre , Krc Mi Zi [T]

A12 = T32 + Xa T35 ; A13 = T33 ; A14 = T34 ; A21 = T61 ; A22 = T62 + Xa T65 ; A23 = T63 ; (29) A25 = T66 ; A31 = T51 − Xb T21 ; A32 = T52 − Xb T22 + Xa T55 − Xb T25 ;

(30)

are the pressure-loss factors for reversal-expansion and reversal-contraction; is the meanflow Mach number in the inner pipes of radius ri ; is the volume-flow impedance of the inner pipe, Zi = 0 c0 /Si ; Si = ri2 ; is the 6 × 6 transfer matrix for the common perforated section of the three interacting ducts, given above in Eq. (1).

K.13 Three-Duct Perforated Elements with Extended Perforations Three possible configurations in this class of elements are shown below. The derivation of the transfer matrix between the upstream point u and downstream point d calls for simultaneous solution of equations representing (1) the common three-duct perforated section of length 2 ; (2) two-duct extended perforated sections on either end, of lengths 1 and 3 ; (3) the closed-end cavities of lengths a and b , and l1 and l3 in the closed-end configuration (b). The algebraic equations for item (1) are in the form of a 6 × 6 transfer matrix, discussed earlier in > Sect. K.12. Equations for item (2) are in the form of two 4 × 4 transfer matrices discussed in > Sect. K.11; and equations for the end-cavities of item (3) are in the form of an impedance expression.

Muffler Acoustics

K

821

In the open-end elements (a) and (c) below, as indicated for similar elements earlier, most of the meanflow grazes the perforations; very little flows across the perforations. So,for convenience,the entire flow may be assumed to be of the grazing type for selection of the appropriate expression for the perforated impedance. In the closed-end configuration (b) below, however, the entire flow has to get across the perforations, calling for the cross-flow or through-flow expression for perforate impedance. While combining different sets of equations, care has to be taken to account for change of directions of flow and acoustic particle velocities, and also for the transformation of normalised volume velocity and the standard dimensional volume velocity. (a) Cross-flow, open-end, extended-perforation element

The final transfer matrix relation for the element shown is given by:

pu uu

=

Ta Tc /Zu

Tb Zd Td Zd /Zu

pd ud

(1)

where Ta = R11 · (R12 + R14 Xb ) · NP/DN ; Tb = R13 · (R12 + R14 Xb ) · NV/DN ; Tc = R31 · (R32 + R34 Xb ) · NP/DN ; Td = R33 · (R32 + R34 Xb ) · NV/DN

(2)

with NP = Xa R21 − R41 ; NV = Xa R23 − R43 ; DN = R42 − Xa R22 + Xb (R44 − Xa R24 ) ;

(3)

Xa = −j tan(k0 a ) ;

(4)

[R] = [A][Q][C].

Xb = j tan(k0b ) ;

(5)

822

K

Muffler Acoustics

[A] and [C] are 4 × 4 transfer matrices for the extended perforated pipes of lengths 1 and 3 , respectively (see Figure (c) below). Q11 Q12 Q13 Q14 Q21 Q31 Q32 Q33 Q34 Q41

= T11 DE1 + T12 DE2 + T14 DE6 + T15 DE7 + T13 ; = T11 DF1 + T12 DF2 + T14 DF6 + T15 DF7 ; = T11 DG1 + T12 DG2 + T14DG6 + T15DG7 + T16 ; = T11DH1 + T12 DH2 + T14 DH6 + T15 DH7 ; = DE5 ; Q22 = DF5 ; Q23 = DG5 ; Q24 = DH5 ; = T41 DE1 + T42 DE2 + T44 DE6 + T45 DE7 + T43 ; = T41 DF1 + T42 DF2 + T44 DF6 + T45 DF7 ; = T41 DG1 + T42 DG2 + T44DG6 + T45DG7 + T46 ; = T41DH1 + T42 DH2 + T44 DH6 + T45 DH7 ; = DE10 ; Q42 = DF10 ; Q43 = DG10 ; Q44 = DH10 ;

{DE} = [D]−1 {E} ; {DG} = [D]−1 {G} ;

{DF} = [D]−1 {F} ; {DH} = [D]−1 {H} .

(6)

(7)

{E}, {F}, {G} and {H} are 10 × 1 column matrices with all elements zero, except the following: E4 = T23 ; G4 = T26 ;

E5 = T33 ; E6 = T53 ; E7 = T63 ; F3 = 1 ; G5 = T36 ; G6 = T56 ; G7 = T66 ; H2 = 1 .

(8)

[D] is a 10 × 10 matrix whose non-zero elements are: D11 = 1 ; D12 = −1 ; D16 = Mu ; D17 = −(1 + Kre )M6 ; D21 = MuZ4 /Zu ; D22 = M6 Z4 /Z6 ; D26 = Z4 /Zu ; D27 = Z4 /Z6 ; D31 = Z4 /Zu ; D32 = Z4 /Z6 ; D36 = 2MuZ4 /Zu ; D37 = 2M6Z4 /Z6 ; D41 = −T21 ; D42 = −T22 ; D43 = −1 ; D46 = −T24 ; D47 = −T25 ; D51 = −T31 ; D52 = −T32 ; D54 = 1 ; T56 = −T34 ; D57 = −T35 ; D61 = −T51 ; D62 = −T52 ; D66 = −T54 ; D67 = −T55 ; D68 = 1 ; D71 = −T61 ; D72 = −T62 ; D76 = −T64 ; D77 = −T65 ; D79 = 1 ; D83 = 1 ; D84 = −1 ; D88 = M8 ; D89 = −(1 + krc )Md ; D93 = M6Z10/Z8 ; D94 = Md Z10/Zd ; D98 = Z10 /Z8 ; D99 = Z10 /Zd ; D9,10 = −1 ; D10,3 = Z10/Z8 ; D10,4 = Z10 /Zd ; D10,8 = 2M8Z10/Z8 ; D10,9 = 2Md Z10 /Zd ; D10,5 = −1 ;

(9)

Zu = 0 c0 /Su ; Zd = 0 c0 /Sd ; Z4 = 0 c0 /(Ssh −Sd ) ; Z6 = 0 c0 /(Ssh −Su −Sd ) ; Z8 = Z6 ; Z10 = 0 c0 /(Ssh −Sd ) ;

(10)

M6 = −Mu Su /(Ssh − Su − Sd ) ; M8 = M6 ; krc = 0.5 ; kre = {(Ssh − Su − Sd ) /Su }2 .

(11)

[T] is the 6×6 transfer matrix for the common perforated section of the three interacting ducts, given in Eq. (1).

Muffler Acoustics

K

823

(b) Cross-flow, closed-end, extended peroration element

The overall transfer matrix between the upstream point u and downstream point d of the configuration of the figure is given by: Tb Zd pu Ta pd = , (12) uu Tc /Zu Td Zd /Zu ud where Ta = (E12 + X1bE14 )NP/DN + E11 ; Tb = (E12 + X1b E14 )NV/DN + E13 ; Tc = (E32 + X1b E34 )NP/DN + E31 ; Td = (E32 + X1b E34 )NV/DN + E33 ;

(13)

NP = X1a E21 − E41 ; NV = X1a E23 − E43 ; DN = E42 + X1b E44 − (E22 + X1b E24 ) X1a ;

(14)

X1a = −j tan(k0 a1 ) ; X2a = −j tan(k0 a2 ) ;

(15)

X1b = j tan(k0 b1 ) ; X2b = j tan(k0 b2 ) ;

[E] = [A][D][C].

(16)

[A] and [C] are 4 × 4 transfer matrices for the extended perforated pipes of lengths 1 and 3 , respectively (see figure). D11 D13 D21 D23 D31 D33 D41 D43

= T11G2 + T13 + T14 G2X2b ; = T11G4 + T16 + T14 G4X2b ; = T21G2 + T23 + T24 G2X2b ; = T21G4 + T26 + T24 G4X2b ; = T41G2 + T43 + T44 G2X2b ; = T41G4 + T46 + T44 G4X2b ; = T51G2 + T53 + T54 G2X2b ; = T51G4 + T56 + T54 G4X2b ;

D12 D14 D22 D24 D32 D34 D42 D44

= T11 G1 + T12 + T14 G1X2b ; = T11 G3 + T15 + T14G3 X2b ; = T21 G1 + T22 + T24 G1X2b ; = T21 G3 + T25 + T24G3 X2b ; = T41 G1 + T42 + T44 G1X2b ; = T41 G3 + T45 + T44G3 X2b ; = T51 G1 + T52 + T54 G1X2b ; = T51 G3 + T55 + T54G3 X2b .

(17)

824

K

Muffler Acoustics

[T] is the 6×6 transfer matrix for the common perforated section of the three interacting ducts, given in Eq. (1). G1 = (X2a T32 − T62 ) /h ; G3 = (X2a T35 − T65 ) /h ;

G2 = (X2a T33 − T63 ) /h ; G4 = (X2a T36 − T66 ) /h ;

h = T61 − X2a T31 + X2b (T64 − X2a T34 ) ; Zu = 0 c0 /Su; Zd = 0 c0 /Sd .

(18) (19)

(c) Reverse-flow, open-end, extended-perforated element:

For the configuration shown in the figure, the final transfer matrix relationship is given by [Munjal/Behera/Thawani (1997)]: pu Tb Zd Ta pd = , (20) uu Tc /Zu Td Zd /Zu ud where Ta = B11 W11 + B12 U11 + B13 + B14 W21 + B15 U21 ; Tb = B11 W12 + B12 U12 + B16 + B14 W22 + B15 U22 ; Tc = B41 W11 + B42 U11 + B43 + B44 W21 + B45 U21 ; Td = B41 W12 + B42U12 + B46 + B44 W22 + B45 U22 ; [W] = [G][U];

[U] = [Q][R];

R11 = H11B33 + H12 B63 − B23 ; R21 = H21B33 + H22 B63 − B53 ;

[Q] = [X]−1 ; R12 = H11 B36 + H12B66 − B26 ; R22 = H21 B36 + H22B66 − B56 ;

X11 = B21 G11 + B22 + B24 G21 − H11 (B31 G11 + B32 + B34 G21 ) − H12 (B61 G11 + B62 + B64 G21 ) ; X12 = B21 G12 + B25 + B24 G22 − H11 (B31 G12 + B35 + B34 G22 ) − H12 (B61 G12 + B65 + B64 G22 ) ; X21 = B51 G11 + B52 + B54 G21 − H21 (B31 G11 + B32 + B34 G21 ) − H22 (B61 G11 + B62 + B64 G21 ) ; X22 = B51 G12 + B55 + B54 G22 − H21 (B31 G12 + B35 + B34 G22 ) − H22 (B61 G12 + B65 + B64 G22 ) ;

(21)

(22) (23)

(24)

Muffler Acoustics

[G] = [GA][GI] ;

K

[GI] = [GB]−1 ;

825

(25)

GB11 = C21 F11 + C22 + C23F21 ; GB21 = C41 F11 + C42 + C43F21 ;

GB12 = C21 F12 + C24 + C23 F22 ; GB22 = C41 F12 + C44 + C43 F22 ;

(26)

GA11 = C11 F11 + C12 + C13 F21 ; GA21 = C31 F11 + C32 + C33 F21 ;

GA12 = C11 F12 + C14 + C13 F22 ; GA22 = C31 F12 + C34 + C33 F22 ;

(27)

[H] = [HC][HI]; [HI] = [HB]−1 ; [HC] = [HR] + [UF]; [HB] = [HP] + [QF] ; [UF] = [HU][F]; [QF] = [HQ][F] ; HU11 HU12 AI22 AI24 HR11 HR12 AI21 AI23 = ; = ; HU21 HU22 AI42 AI44 HR21 HR22 AI41 AI43 HQ11 HQ12 AI12 AI14 HP11 HP12 AI11 AI13 = ; = ; HQ21 HQ22 AI32 AI34 HP21 HP22 AI31 AI33

(28)

(29)

[AI] = [A]−1 . [T] [see (Eq. K.12.1)] is the 6 × 6 transfer matrix for the common perforated section of the three interacting ducts, and [A] and [C] are 4 × 4 transfer matrices for the extended perforated pipes of lengths a and 3 , respectively [see Eq. (K.11.1)].

K.14 Three-Pass (or Four-Duct) Perforated Elements The figures below show typical three-pass element mufflers where waves in four ducts interact with each other. These comprise three perforated ducts of radius r1 , r2 and r3 and the annular space of equivalent radius r4 . These mufflers have the advantage of good acoustic performance coupled with low back pressure. Following the same procedure as for the two-duct elements in > Sect. K.11 or threeduct elements in > Sect. K.12, we obtain first an 8 × 8 transfer matrix for the common interaction section. Then, making use of the closed-end boundary conditions for the annular duct, we get a 6 × 6 transfer matrix relationship. The common 8 × 8 matrix relationship is given by: {S(0)} = [TM] {S(p )},

(1)

where {S(z)} is the state vector: T p1 (z), p2 (z),p3 (z), p4 (z), V1 (z), V2(z), V3(z), V4(z)

(2)

and V ≡ 0 c0 v =

0 c0 u = Z0u, S

(3)

so that Z0 is the characteristic impedance with respect to volume velocity u and S is the area of cross section of the appropriate duct. [TM] = [A(0)][A(1 )]−1 ,

(4)

826

K

Muffler Acoustics

where [A(z)] is an 8 × 8 matrix with constituent elements given by, i = 1, 2, . . ., 8: A1,i = •1,i ei z ; A2,i = −•5,i ei z / jk 0 + M1 i ; A3,i = •2,i ei z ; A4,i = −•6,i ei z / jk 0 + M2 i ; (5) A5,i = •3,i ei z ; A6,i = −•7,i ei z / jk0 + M3 i ; A7,i = •4,i ei z ; A8,i = −•8,i ei z / jk 0 . [•] is the modal matrix and {} is the eigenvector of the following coefficient matrix [Munjal, Int. J. Acoust. and Vib. 2, pp. 63–68 (1997)]: ⎤ ⎡ 0 0 0 0 1 0 0 0 ⎢ 0 0 0 0 0 1 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 0 0 0 0 1 0 ⎥ ⎥ ⎢ ⎢ 0 0 0 0 0 0 0 1 ⎥ ⎥, ⎢ (6) ⎢ −2 0 0 −4 −1 0 0 −3 ⎥ ⎥ ⎢ ⎢ 0 −6 0 −8 0 −5 0 −7 ⎥ ⎥ ⎢ ⎣ 0 0 −10 −12 0 0 −9 −11 ⎦ 0 −14 −15 −16 0 0 0 0 where

k02 2jk0 M1 2j 4j 1 = 1− ; 2 = 1− ; k0 d1 …1 k0 d1 …1 1 − M21 1 − M21 j4k0 4M1 ; 4 = ; 3 = d1 …1 (1 − M21 ) d1 …1 (1 − M21 ) 5 = 1 , 9 = 1 ,

6 = 2 , 7 = 3 , 10 = 2 , 11 = 3 , j4k0d1 ; 13 = 2 …1 d4 − (d12 + d22 + d32 ) 14 = 13 15 = 13

with with

d1 /…1 d1 /…1

8 = 4 with subscript 1 replaced by 2 ; 12 = 4 with subscript 1 replaced by 3 ;

replaced by replaced by

(7)

d2 /…2 ; d3 /…3 ;

16 = k02 − (13 + 14 + 15) ; M1 = U1 /c0 ;

M2 = U2 /c0 ;

M3 = U3 /c0 ;

k0 = –/c0 .

(8)

Use of the boundary conditions of the annular duct, and rearranging yields a reduced, and more useful, form of the transfer matrix relationship: ⎡ ⎤ ⎡ ⎤ ⎤⎡ p1 (0) T11 T12 T13 T14 T15 T16 p1 () ⎢ V1 (0) ⎥ ⎢ T21 T22 T23 T24 T25 T26 ⎥ ⎢ V1() ⎥ ⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎢ p2 (0) ⎥ ⎢ T31 T32 T33 T34 T35 T36 ⎥ ⎢ p2 () ⎥ ⎢ ⎥=⎢ ⎥ ⎥⎢ (9) ⎢ V2 (0) ⎥ ⎢ T41 T42 T43 T44 T45 T46 ⎥ ⎢ V2() ⎥ ; ⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎣ p3 (0) ⎦ ⎣ T51 T52 T53 T54 T55 T56 ⎦ ⎣ p3 () ⎦ V3 (0) V3() T61 T62 T63 T64 T65 T66 Tij = TMij +

(TMi7 + Xpb TMi8 )(Xpa TM7j − TM8j ) ; TM87 + Xpb TM88 − Xpa (TM77 + Xpb TM78)

i, j = 1, 2, . . . , 6;

(10)

Muffler Acoustics

Xpa = −j tan(k0 a ) ;

Xpb = −j tan(k0b ).

Partitioning the foregoing matrix equation as: ⎡ ⎤ ⎡ ⎤⎡ ⎤ S1 (0) S1 (p ) D E F ⎣ S2 (0) ⎦ = ⎣ G H K ⎦ ⎣ S2 (p ) ⎦ , S3 (0) S3 (p ) P Q R

K

827

(11)

(12)

where {Si } = [pi Vi ]T and D, E, F, G, H, K, P, Q and R are 2 × 2 submatrices as becomes clear from comparison of the two corresponding matrix equations. (a) Flush-tube three-pass perforated element

Now, the desired 2 × 2 transfer matrix relationship between the upstream point u and the downstream point d in the configuration shown is given by [Munjal, Int. J. Acoust. and Vib. 2, pp. 63–68 (1997)] C12 · Zd C11 p1 (0) p3 (p ) = , (13) u1(0) u3 (p ) C21 /Zu C22 Zd /Zu where [C] = [D][A][W] + [E][W] + [F]; [W] = [[G][A] + [H] − [B][P][A] − [B][Q]]−1 [[B][R] − [K]] .

(14)

[A] is the product of the transfer matrices of (1) a duct of length ƒ1 + 4b + tb , (2) reversal expansion element, (3) sudden contraction element, (4) a duct of length ƒ2 + tb + 4b . [B] is the product of transfer matrices of (1) a duct of length ƒ1 + 4a + ta , (2) reversal expansion element, (3) sudden contraction element, (4) a duct of length ƒ3 + ta + 4a . Here, ƒa and ƒb are end corrections. These as well as other constituent transfer matrices are given in > Sects. K.5 and K.6.

828

K

Muffler Acoustics

(b) Extended-tube three-pass perforated element

The foregoing transfer matrix relationship between the state vectors [p1 (0) u1(0)]T and [p3 (p ) u3 (p )]T would hold for the extended-tube three-pass perforated element shown above, with the difference that end matrices [A] and [B] would now be different. For the extended-tube end chambers, [A] would be the product of the transfer matrices of [Munjal, ICSV-5, Adelaide (1997)]: (1) a duct of length ƒ1 + 4b + tb + 1 , (2) reversal expansion element, (3) extended outlet element, (4) a duct of length ƒ2 + tb + 4b + 2b . [B] would be the product of the transfer matrices of (1) a duct of length ƒ2 + 4a + ta + 2a , (2) reversal expansion element, (3) extended outlet element, (4) a duct of length ƒ3 + 3 + ta + 4a . Transfer matrices of the extended-tube elements are given in > Sect. K.7, while the end-correcting ƒ1, ƒ2, ƒ3 and other transfer matrices are given in > Sects. K.5 and K.6.

K.15 Catalytic Converter Elements Catalytic converters are used often in series with exhaust mufflers for control of air emissions by means of oxidation of unburnt carbon particles and carbon monoxide to carbon dioxide. These converters involve area changes and porous blocks of catalyst pellets, or a bank of capillary tubes as shown below.

Muffler Acoustics

K

829

(a) Pellet block element

While transfer matrices of simple uniform tubes and sudden area changes have been given earlier in > Sects. K.5 and K.6, the transfer matrix of a pellet block element is given by: cos (k) jZ sin (k) , (1) j sin (k) cos (k) Z where Z and k for granular pellets are given by Attenborough’s expression [Altenborough (1983)]: k Z q2 1 k0 1 + (‰ − 1)T(C) 1/2 , (2) =q ; = k0 1 − T(B) Z0 1 − T(B) k where q is a tortuosity factor, ‰ is the ratio of specific heats for the gaseous medium, and T(x) =

2J1 (x) ; xJo (x)

B = (−j)1/2 Šp ;

x = B or C ; 1/2

C = BNpr ;

Šp2 = 80 q2 S–/(n2 E). q is the steady flow factor ; n is the dynamic shape factor, n = 2 − S ; NPr is the Prandtl number (b) Capillary tube monolith

(3)

830

K

Muffler Acoustics

The transfer matrix of a monolith or bank of capillary tubes (coated with catalyst), is given by: cos(km ) j Z sin(km )/¥ , (4) j¥ sin(km ) cos(km ) Z where ¥ is the open area ratio. Z = m cm /S ; km = k0 c0 /cm ; 1/2 ‰−1 c0 = (1 + ¥ EG/D) ‰− cm 1 + ¥ EG /(D Pr) with b /4 D = j–0 ; G = −ab/4 ; G = −a ; 1 − 2b/a 1 − 2b /a a = s(−j)1/2 ; b = J1(a)/J0 (a) ; a = s(−j)1/2 Pr1/2 ;

b = J1 (a )/J0 (a );

(5)

(6)

m = 0 + E¥ G/(j–) ; 8– 1/2 s = E¥ 0 . E is the specific flow resistance for laminar flow, E = 32/(¥ d2 ). For air, ‰ = 1.4, Pr = 0.7, ‹ = 1.81 · 10−5 Pa · s. Typically, E ≈ 500 Pa · s/m2 and = 1.07.

K.16 Helmholtz Resonator See > Sects. H.4–H.16 for a more detailed description of Helmholtz resonators without flow and > Sects. J.38–J.39 for non-linear effects of flow.

The transfer matrix of a Helmholtz resonator as shown is given by: pd 1 0 pu = , uu ud 1/Zr 1

(1)

where Zr is the flow impedance of the resonator at the junction, made up of a resistance term, an inertance term and a compliance term: 2 –2 2 − rn + 0.425 Mc0 + j –eq − c0 Zr = 0 c , ru S Sn –Vc 0 (2) n eq = n + tw + 0.85rn 2 − rrn . u

Muffler Acoustics

K

831

ru is the radius of the upstream (or downstream) duct; rn and n are, respectively, the radius and length of the resonator neck; Vc is the volume of the resonator cavity; Sn = rn2 .

K.17 In-Line Cavity The transfer matrix of an inline cavity as shown is given by:

pu uu

=

1 0 1/Z 1

pd ud

,

(1)

where Z is the compliance-type flow impedance and Vc the volume of the cavity: Z=

0 c20 . j–Vc

(2)

K.18 Bellows A bellow is characteriszed by wall compliance coupled with a gradual area change. The matrizant approach leads to the following transfer matrix [Singhal/Munjal (1999)] for the divergent conical part of a single step of a bellow:

T11 T21

T12 T22

,

(1)

832

K

T11 =

e/2 2ƒ cos (ƒ ) − sin (ƒ ) ; 2ƒ

Muffler Acoustics

e/2 k0 Z() sin(ƒ ); 2ƒ 0 ƒ −0 e/2 2 2 T21 = j + ƒ sin(ƒ ); k0 ƒ Z(0) 4 T12 = j

T22 =

(2)

Z(0) e/2 sin(ƒ ) + 2ƒ cos(ƒ ) 2ƒ Z()

with ƒ = jƒ ;

Z(0) =

0 c0 ; ri2

Z() =

0 c0 ; ro2

1 2 − 4 k02 − jk0 ; = 2 a /B ; = p /2 ; 2 B ln(ro /ri ) ; B = 20 c0 /Zw . a = (ro − ri ) / ; = a

(3)

ƒ=

Zw is the wall impedance given earlier in

> Sect. K.10 on hoses.

By interchanging the inlet and outlet radii, we obtain the transfer matrix for the convergent, conical part of the bellows. Successive multiplication of the transfer matrices of the two halves of the step shown above yields the transfer matrix of the full stop (single bellow). Extension of this multiplication process would yield the transfer matrix of multistep bellows. Evaluation of the transfer matrix of flexible-wall bellows, incorporating the convective effect of mean flow is done by means of the same matrizant approach as given in reference [Singhal/Munjal (1999)].However,the more important effect of flow separation and the consequent losses have been neglected; only the convective effect of the mean flow is considered.

K.19 Pod Silencer A pod silencer is a combination of a simple acoustically lined duct and a parallel baffle muffler, as shown. See > Ch. J,“Duct Acoustics”, for lined ducts and especially silencers. > Sect. J.28 for conical duct transitions.

> Sects. J.23–J.25 for baffle

Often a pod is inserted to increase the absorptive contact surface and thereby increase the attenuation or transmission loss of the silencer. Pod silencers are often available in prefabricated form for ready use in a noisy duct. The transfer matrix for a pod silencer of length p , and radii r1 , r2 and r3 , as shown above, is given by: jZ sin(kz p ) cos(kz p ) ; Z = Z0 k0/kz , (1) j Z sin(kz p ) cos(kz p )

Muffler Acoustics

K

833

where the convective effect of meanflow and the presence of highly perforated plate have been neglected; kz is a root of the transcendental determinant equation:

kr2 J1 (kr2 r1 ) kr2 Y1(kr2 r1 )

−kr1 J1 (kr1 r1 )

0 0

kw Z w k0 Z 0 k0 Z0

−J0 (kr2 r1 ) −Y0 (kr2 r1 ) 0 0 J0 (kr1 r1 )

kr2 J1 (kr2 r1 ) −kr2 Y1 (kr2 r2) kr3 J1 (kr3 r2 ) kr3 Y1(kr3 r2 )

0 −

= 0, (2) k0 Z 0 k0 Z 0 kw Z w kw Z w

0 J (k r ) Y (k r ) −J (k r ) −Y (k r ) 0 r2 1 0 r2 2 0 r3 2 0 r3 2

kr3 J1 (kr3 r3 ) kr3 Y1(kr3 r3 ) 0 0 0

kw Z w kw Z w where 1/2 kr1 = kw2 − kz2 ;

1/2 kr2 = k02 − kz2 ;

1/2 kr3 = kw2 − kz2 = kr1

(3)

and Jn (z), Yn(z) denote Bessel and Neumann functions, respectively. Yw and kw are given by Mechel’s formulae presented in > Sect. J.25. The choice of the starting value of kz for the Newton-Raphson iteration process may be done as for a circular lined duct ( > Sects. J.13 and J.15).

K.20 Quincke Tube A Quincke tube is a passive method for generating wave interference in ducts which is more simple, inexpensive and durable than the corresponding active noise control system. The Quincke tube consists of two pipes. The main pipe is connected with a parallel bypass as shown. For plane waves in an incompressible flow, the transfer matrix relationship between the upstream point u = 1 and downstream point d = 6 is given by: 1 1 −Mu Zu 1 Md Z d TM11 TM12 , (1) 1 TM21 TM22 Md /Zd 1 1 − M2u −Mu /Zu where Mu and Zu are the meanflow Mach number and characteristic flow impedance in the main pipe upstream of the truncation and Md = Mu ; Zd = Zu .

834

K

Muffler Acoustics

TM11 = (A11B12 + B11 A12) / (A12 + B12 ) ; TM12 = A12 B12 / (A12 + B12 ) ; A12 + A12B21 + A21 B12 − B11 B22 + B12 B21 ; TM21 = A11B22 + A22 B11 − A11 A22 + A21A 12 + B12 TM22 = (A22B12 + B22 A12) / (A12 + B12 ) ;

A11 A21

A12 A22

B11 B21

B12 B22

= e−jMa kca a = e−jMa kca a

kca = k0/(1 − M2a ) ;

cos(kca a ) j sin(kca a )/Za

jZa sin(kca a ) cos(kca a )

cos(kcb b ) j sin(kcbb )/Zb

jZb sin(kcbb ) cos(kcbb )

(2)

;

(3)

;

(4)

kcb = k0 /(1 − M2b ) .

(5)

a (La ) and b (Lb ) are lengths of the main duct and bypass duct, respectively, and Za = 0 c0/Sa ;

Zb = 0 c0 /Sb .

(6)

Sa and Sb are areas of cross section of the main duct and bypass duct, respectively; Ma and Mb are the meanflow Mach number in the main duct and bypass duct, respectively. These are given by the following expressions: Ma = Mu − SSb Mb ; Sa = Su ; a 1/2 . f11 = da b + 187.5 a db

Mb = Mu /(f11 + S3 /S2 ) ;

(7)

It may be noted that generally Mb will be much less than Ma because of the transverse connection and the requirement of equal pressure drop across the two parallel arms.

K.21 Annular Airgap Lined Duct Automotive exhaust systems are characterised by hot gas flows, often containing some unburnt carbon particles. In an acoustically lined duct with high velocity grazing flow, there is a strong possibility of some fibres of a fibrous absorptive material like glass wool, ceramic wool or mineral/rock wool being swept away continuously and the perforated protective plate getting progressively clogged with unburnt carbon particles and possibly lubricating oil. One of the alternatives would be to make use of another perforated cylinder between the inner flow pipe and the absorptive layer with an airgap in between, as shown. This element involves acoustic interaction between three ducts,

Muffler Acoustics

K

835

viz. the inner flow duct, the annular airgap duct, and the outer acoustically lined duct. Therefore, the analysis of this element runs on the same lines as those of > Sect. K.12, with the important difference that the medium in the outer lined duct is different; its wave number kw and characteristic impedance Zw are given in terms of the specific flow resistance by Mechel’s expressions [Mechel (1976)]. The final transfer matrix is given by [Munjal/Venkatesham/Iutam (2000)].

pu uu

=

ACV11 Zd ACV61Zd /Zu

ACP11 ACP61 /Zu

pd ud

,

(1)

where [ACP] and [ACV] are 10 × 10 matrices given by: [ACP] = [A]−1 {CP} ;

[ACV] = [A]−1 {CV}

(2)

with Zu = Zd = 0 c0 /(d12 /4) and {CP} = {CV} =

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ [A] = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

T11

T21

T31

T41

T51

T61

0 0 0

T14

T24

T34

T44

T54

T64

0

0 0 0

0 0 0 0 0 1 0 0 1 0

−T15 −T25 −T35 −T45 −T55 −T65 0 1 −X4 0

1 0 0 1 0 0 0 0 0 0 0 0 0 −X1 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0

−T12 −T22 −T32 −T42 −T52 −T62 0 −X2 0 −X4

−T13 −T23 −T33 −T43 −T53 −T63 0 0 0 0

0 0 0 1 0 0 0 0 0 0

X1 = −j tan(k0 a );

X2 = −j tan(k0b );

X3 = −j tan(kw a );

X4 = −j tan (kw b ).

0 0 0 0 1 0 1 0 0 0

0

T T

;

(3)

;

−T16 −T26 −T36 −T46 −T56 −T66 0 0 1 1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥; ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(4)

(5)

836

K

Muffler Acoustics

[T] is the 6 × 6 transfer matrix for the common perforated section where all three duct sections interact. It is evaluated exactly as shown in > Sect. K.12; only the values of s in the coefficient matrix are different. These are as follows: 1 = −(0.5f0M1 + f1 )/den; 5 = 0; 11 = 0;

6 = f4 ; f12 = f7 ;

where: f0 = 4jk0 ;

2 = k02 /den;

7 = 0; 13 = 0;

8 =

−k02

14 =

f1 = 4M1 1/d1 ;

3 = f1/den;

− f 4 − f6 ;

kw2

9 = 0;

4 = f2 /den; 10 = f6;

(6)

− f7 ,

f2 = f01 /d1 .

Yw and kw are given by Mechel’s formulae given in

(7) > Sect. J.25 and den

= 1 − M21.

1 is the reciprocal of the non-dimensional grazing-flow impedance of the perforate given in > Sect. K.11, at the interface 1–2 (diameter d1 ); 2 is the reciprocal of the non-dimensional stationary-flow impedance of the perforate given in > Sect. K.12, at the interface 2–3 (diameter d2 ).

K.22 Micro-Perforated Helmholtz Panel Parallel Baffle Muffler This element, in a way, is a combination of a concentric tube resonator [ > Sect. K.11(a)] and a parallel baffle muffler ( > Sect. J.24) without absorbing material. Following Wu (1997) the transfer matrix of this element across a length p is given by: jZ sin (kz p ) cos(kz p ) , (1) j sin(kz p )/Z cos(kz p ) where Z = Z0 k0/kz ;

Z0 = 0 c0 / 2np hW ;

kz = (k02 − ky2 )1/2.

(2)

ky is a root of the transcendental equation ky h · tan(ky h) − jk0 h 0 c0 /Z = 0.

(3)

Muffler Acoustics

K

837

Z is the grazing-flow impedance of the micro-perforated panel:

! 1/2 32Œ0 · th 2 1 + x /32 + 0.177 · x · d /t h h dh2 1 j– · th · 0 0 c0 dh 1+ + − j + 0.85 th tan(k0d) (9 + x2 /2)1/2

Z = (1 + M)

(4)

1/2 with x = d · h 105 · f ; M is the meanflow Mach number in the flow passage; = panel porosity; d, th , dh , h and W are shown in the figure; Œ = 1.56 · 10−5 Pa · s for air.

K.23 Acoustically Lined Circular Duct Circular and annular ducts are extensively treated in > Ch. J. Whereas that chapter describes steps in such ducts with a multimode analysis, the present > Ch. K uses representations with a fundamental-mode analysis.Thus,in order to get some completeness in the collection of muffler elements in the present chapter, this section will present the transfer matrix method for an element which is also contained in > Ch. J, there with higher mode analysis. A uniform circular duct is often lined on the inside for acoustical absorption or dissipation into heat, as shown. For mechanical protection against the eroding effect of the moving medium, the acoustic layer is covered on the exposed side with a very thin membrane like Mylar or a highly perforated thin metallic plate. This protective cover affects the absorptive properties of the acoustic lining at certain frequencies and therefore needs to be included in the acoustic model.

The protective layer is characterised by a radial partition impedance that would support a pressure difference between the two media. The porous/fibrous acoustic layer is characterised by a complex wave number kw (= −j· a in earlier chapters) and a characteristic impedance Zw (= Za in earlier chapters). These in turn may be expressed as a function of the flow resistivity ¡ or of an “absorber variable” E, given by: E=

0 c0 0 f = , ¡ Š0 ¡

(1)

838

K

Muffler Acoustics

with Š0 the free field wave length (see > Ch. G). For the fibrous materials, the characteristic constants Zw and kw are approximately described by the empirical formulae of [Delany/ Bazley (1970)], as modified and improved by [Mechel (1976)]: ⎧ −0.754 ⎪ −j0.087E−0.73 ; E> 60 ⎨1 + 0.0485E Zw = ; (2) 0.5/(E) + j1.4 E< 60 1/2 ; 0 c0 ⎪ ⎩ −1.466 + j0.212/E 1 − j0.189E−0.6185 + 0.0978E−0.6929 ; E> 60 kw = (3) . ∗) 1/2 k0 − 1.466 − j0.212/E ; E< 60 The transfer matrix for a bulk reacting lined duct shown above is given by [Munjal/Thawani (1997)]: pd pu jZ sin(kz ) cos(kz ) = , (4) uu ud (j/Z) sin(kz ) cos(kz ) where Z = Z0 k0/kz ;

Z0 = 0 c0 /S ;

k0 = –/c0 .

The axial wave number for the fundamental mode is the first (lowest) root of the transcendental equation: j

–w J0 (kr,w ri ) + C · Y0(kr,w ri ) –0 J0 (kr,0 ri ) =j + Zp (–) ; kr,0 J1 (kr,0 ri ) kr,w Jo (kr,w ri ) + C · Y1(kr,– ri )

kr,0 = (k02 − kz2 )1/2 ;

kr,w = (kw2 − kz2 )1/2 ;

C=−

J1 (kw ro ) ; Y1 (kw ro )

(5) ro = ri + t .

(6)

Zp ,the partition impedance of a thin impermeable protective foil,is given by Zp = j–p tp where tp are the thickness and material density of the foil, respectively. For a perforated plate, in the presence of a grazing meanflow, the corresponding expression is [Rao/Munjal (1986)]: Zp = 0 c0 7.337 · 10−3 (1 + 72.23M) + j2.2245 · 10−5 f (1 + 51tp)(1 + 204dh ) / , (7) where dh is the hole diameter in m, M is the meanflow Mach number, and is the porosity of the perforated plate. A locally reacting lining would not support waves inside the lining in the axial direction. Therefore, for a locally reacting lining, kr,w = kw , and therefore the right-hand side of the foregoing transcendental Eq. (5) would be independent of the variable kz . In either case (for either type of lining), the appropriate transcendental equation can be solved for the axial wave number kz by means of a Newton-Raphson scheme, making use of the start approximation: 1/2 2 96 + 36jQ ± 9216 + 2304jQ − 912Q2 0 c0 ; Q = (k0 ro ) , (8) kr,0 ro ≈ 12 + jQ Zw and Zw is the right-hand side of the transcendental equation above for kr,w = kw , i.e. for the locally reacting lining case. ∗)


Muffler Acoustics

K

839

K.24 Parallel Baffle Muffler (Multipass Lined Duct) See the introductory comment of the previous

> Sect. K.23.

The figure shows a typical parallel baffle muffle used in general to increase the contact area and to obtain the required attenuation within a short axial length. For plane-wave propagation, each of the passes will act as a two-dimensional rectangular duct as shown in the separate figure. The transfer matrix for axial wave propagation for the element is the same as for a lined circular duct ( > Sect. K.23). The difference lies in the value of the axial wave number, kz . The eigenequation for a two-dimensional rectangular duct with bulk-reacting lining is: jZ0

k0 cot(ky,0h) = Zp − jZw cot(ky,w d) , ky,0

(1)

where Zw = w cw ;

ky,0 = (k02 − kz2)1/2 ;

kw and Zw are as given in

> Sect. K.23, and Zp

Z0 = 0 c0 ;

ky,w = (kw2 − kz2 )1/2 .

(2)

is the impedance of the porous cover.

For locally reacting linings, the foregoing transcendental equation would hold with the simplification ky,w = kw . In either case, the transcendental equation or eigenequation may be solved for kz by means of a Newton-Raphson iteration scheme, with the starting value being given by: 1/2 2.47 + Q + (2.47 + Q)2 − 1.87Q h 0 c0 2 (ky,0 h/2) ≈ ; Q = jk0 . (3) 0.38 2 Zw Incidentally, the transmission loss or attenuation of a lined duct of length may be obtained from the imaginary component of the axial wave number kz , by means of the equation: TL = −8.68Im{kz } [dB],

(4)

or else from the transfer matrix of the lined duct or parallel baffle muffler, making use of the expression given in > Sect. K.4.

840

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References Alfredson, R.J., Davies, P.O.A.L.: The radiation of sound from the engine exhaust. J. Sound Vibr. 13, 389–408 (1970)

Munjal, M.L.: Analysis of a flush-tube three-pass perforated element muffler by means of transfer matrices. Int. J. Acoust. Vibr. 2, 63–68 (1997)

Attenborough, K.: Acoustic properties of rigid fibrous absorbents and granular materials. J.Acoust. Soc. Am. 73, 785–799 (1983)

Munjal,M.L.: Analysis of extended-tube three-pass perforated element muffler by means of transfer matrices. ICSV-5, Adelaide (1997)

Davies, P.O.A.L., Bento Coelho, J.L., Bhattacharya, M., J.: Reflection coefficient for an unflanged pipe with flow. Sound Vibr. 72, 543–546 (1980)

Munjal, M.L., Behera, B.K., Thawani, P.T.: An analytical model of the reverse flow, open end, extended perforated element muffler. Int. J. Acoust. Vibr. 2, 59–62 (1997)

Delany, M.E., Bazley, B.N.: Acoustical characteristics of fibrous absorbent materials. Appl. Acoust. 3, 106–116 (1970) Dokumaci, E.: On transmission of sound in a nonuniform duct carrying a subsonic compressible flow. J. Sound Vibr. 210, 391–401 (1998) Easwaran, V., Munjal, M.L.: Plane wave analysis of conical and exponential pipes with incompressible flow. J. Sound Vibr. 152, 73–93 (1992) Gogate, G.R., Munjal, M.L.: Analytical and experimental aeroacoustic studies of open-ended three-duct perforated elements used in mufflers. J. Acoust. Soc. Am. 97, 2919–2927 (1995) Levine, M., Schwinger: On the radiation of sound from an unflanged circular pipe. J. Phys. Rev. 73, 383–406 (1948) Mechel, F.P.: Extension to low frequencies of the formulae of Delany and Bazley for absorbing materials. (in German) Acustica 35, 210–213 (1976) Mechel, F.P., Schiltz, W.M., Dietz, J.: Akustische ¨ Impedanz einer luftdurchströmten Offnung.Acustica 15, 199–206 (1965) Morfey, C.L.: Sound transmission and generation in ducts with flow. J. Sound Vibr. 14, 37–55 (1971) Howe, M.S.: Attenuation of sound in a low Mach number nozzle flow. J. Fluid Mech. 91, 209–229 (1980) Munjal, M.L.: Velocity ratio cum transfer matrix method for the evaluation of a muffler with mean flow. J. Sound Vibr. 39, 105–119 (1975) Munjal, M.L.: Acoustics of Ducts and Mufflers. Ch. 2 and 3, Wiley-Interscience, New York (1987)

Munjal, M.L., Behera, B.K., Thawani, P.T.: Transfer matrix model for the reverse-flow,three-duct,open end perforated element muffler. Appl. Acoust. 54, 229–238 (1998) Munjal, M.L., Thawani, P.T.: Acoustic performance of hoses – a parametric study. Noise Control Eng. J. 44 (1996) Munjal, M.L., Thawani, P.T.: Effect of protective layer on the performance of absorptive ducts.Noise Control Eng. J. 45, 14–18 (1997) Munjal, M.L.,Venkatesham, B.: Analysis and design of an annular airgap lined duct for hot exhaust systems.IUTAM International Symposium an Designing for Quietness, I.I.Sc., Bangalore Dec. (2000) Munt, R.M.: Acoustic transmission properties of a jet pipe with subsonic jet flow: I.The cold reflection coefficient. J. Sound Vibr. 142, 413–436 (1990) Panicker, V.B., Munjal, M.L.: Acoustic dissipation in a uniform tube with moving medium. J. Acoust Soc. India 9, 95–101 (1981) Panicker, V.B., Munjal, M.L.: Aeroacoustic analysis of straight-through mufflers with simple and extended-tube expansion chambers. J. Indian Inst. Sci. 63(A), 1–19 (1981) Panicker, V.B., Munjal, M.L.: Aeroacoustic analysis of mufflers with flow reversals. J. Indian Inst. Sc. 63(A), 21–38 (1981) Peat, K.S.: A numerical decoupling analysis of perforated pipe silencer element. J. Sound Vibr. 123, 199–212 (1988) Prasad, M.G., Crocker, M.J.: Studies on acoustical modeliong of a multi-cylinder engine exhaust system. J. Sound Vibr. 90, 491–508 (1983)

Muffler Acoustics

Rao,K.N.,Munjal M.L.: Experimental evaluation of impedance of perforates with grazing flow.J.Sound Vibr. 108, 283–295 (1986) Sahasrabudhe,A.D.,Munjal M.L.,Ramu,S.A.: Analysis of inertance due to the higher order mode effects in a sudden area discontinuity. J. Sound Vibr. 185, 515–529 (1995) Selamet A.,Dickey N.S.,Novak,J.M.: The Herschel– Quincke tube: a theoretical, computational, and experimental investigation. J. Acoust. Soc. Am. 96, 77–99 (1994, Selamet, A., Easwaran, V., Novak, J.M., Kach, R.A.: Wave attenuation in catalytical converters: reactive versus dissipative effects. J. Acoust. Soc. Am. 103, 935–943 (1998)

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Singhal,V., Munjal, M.L.: Prediction of the acoustic performance of flexible bellows incorporating the convective effect of incompressible mean flow. Int. J. Acoust. Vibr. 4, 181–188 (1999) Sullivan, J.W.: A method of modeling perforated tube muffler components: II. Applications. J. Acoust. Soc. Am. 66, 779–788 (1979) Sullivan, J.W., Crocker, M.J.: Analysis of concentric tube resonators having unpartitioned cavities. J. Acoust. Soc. Am. 64, 207–215 (1978) Venkatesham, B.: Aeroacoustic analysis of complex muffler elements. M.E. Dissertation, I.I.Sc., Bangalore, Jan (2001) Wu, M.Q.: Micro-perforated panels for duct silencing. Noise Control Eng. J. 45, 69–77 (1997)

L Capsules and Cabins Usually capsules and cabins are combined in one chapter, like here, although their tasks and the analytical methods are quite different. The task of a capsule is to reduce the sound pressure level in the environment from an inside noise source; the task of cabins is to produce a quiet space in a noisy environment. Suppose we have a noise source with constant sound power output, whatever the sound field around the source may be, and suppose we have a capsule surrounding the source with some transmission loss of its walls, but with no sound absorption, either inside or in the walls. The sound pressure level inside the capsule will rise until all the sound power produced is radiated by the capsule again. Thus sound absorption in a capsule plays an important role.

L.1 The Energetic Approximation for the Efficiency of Capsules


Consider two arrangements of a source Q: (a) The source is placed in free space and radiates the (effective) sound power ¢. (b) The source is surrounded by a capsule; the question is, what sound power ¢t will be radiated?

The capsule efficiency is measured by its insertion loss De = −10 · lg (‘e ) = −10 · lg (¢t /¢) [dB] .

(1)

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A widely used proposal by Goesele evaluates De = R − 10 · lg(1/) ,

(2)

where R = −10 · lg (¢t /¢w ) is the sound transmission loss of the capsule wall and is the sound absorption coefficient of the interior side of the capsule wall (measured with a hard backing of the wall). This proposal produces De → −∞ for → 0. By identical transformations: ‘e =

¢t ¢w ¢ t = · = cw · ‘VS ; ¢ ¢ ¢w

cw =

¢w ; ¢

‘VS =

¢t , ¢w

(3)

where ‘VS is the sound transmission coefficient of the capsule wall (possibly a combination of an interior absorber layer and an outer tight wall). If rVS is the symbol for the interior reflection factor of the capsule wall (including its radiation to the outside), the sound power ¢V loss inside the capsule is ¢V = (1 − |rVS |2 ) · ¢W . The “energetic approximation”, which was proposed by Hennig, assumes that at the equilibrium ¢ = ¢V ; thus: ‘e =

‘VS ‘VS = ; 2 1 − |rVS | VS

VS = 1 − |rVS |2 .

(4)

The difference from Goesele’s proposal is the fact that the absorption coefficient VS now also contains the radiated power. If the sound transmission factor tVS through the wall of the capsule is used, with ‘VS = |tVS |2 , the insertion power ratio becomes: ‘e =

|tVS |2 . 1 − |rVS |2

(5)

Simple example: porous interior layer and outer metal sheet It is assumed that the interior sound field can be described by plane waves pe incident on the capsule wall at a polar angle ˜ . The capsule wall consists of an interior porous layer of thickness da and with characteristic values a , Za of its material plus an outer metal sheet with partition impedance ZT .

Capsules and Cabins

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845

In total, it is assumed that the capsule is large and has plane walls. Field formulations: pe (x, y) = Pe · ejk0 y sin ˜ · e−jk0 x cos ˜ , pr (x, y) = Pr · ejk0 y sin ˜ · e+jk0 x cos ˜ , jk0 da cos ˜

pt (x, y) = Pt · e

pa (x, y) = Pa · ea

·e

y sin ã

par (x, y) = Par · e−a

jk0 y sin ˜

· e−a

da cos ã

·e

x cos ã

· ea

(6) −jk0 x cos ˜

,

,

y sin ã

· e+a

with interior angle in the porous layer:

(7)

x cos ã

sin ã =

jk0 · sin ˜ a

(8)

and partition impedance ZT : ZT = 2Zm F[†F2 · sin4 ã + j(1 − F2 · sin4 ã )] ; Z0

Zm =

fcr dp p ; Z0

F=

f , fcr

(9)

where † is the bending loss factor of the sheet, p the density of its material and fcr the critical frequency. The boundary conditions give the following system of equations: ⎛ 1 ⎜1 ⎜ ⎜ ⎝0

−1 b e−a

−e−a −be−a 1

0

be−a

−b

⎞ ⎞ ⎛ ⎞ ⎛ −Pe 0 Pr ⎟ ⎜P ⎟ ⎜ P ⎟ 0 ⎟ ⎜ a⎟ ⎜ e ⎟ ⎟ ⎟·⎜ ⎟ = ⎜ −(1 + ZTn cos ˜ )⎠ ⎝Par ⎠ ⎝ 0 ⎠ −1

(10)

0

Pt

with the abbreviations a = a da · cos ã = k0da

2 + sin2 ˜ ; an

b=

1 Z0 cos ã = Za cos ˜ an Zan

2 + sin2 ˜ an cos ˜

(11)

(an = a /k0; Zan = Za /Z0). It can be solved with Kramer’s rule:

det = −(1 + b)2 + (1 − b)2 · e−2a − b 1 + b + (1 − b)e−2a ZT /Z0 ,

det 1 = −(1 − b2)(1 − e−2a ) + b 1 − b + (1 + b)e−2a ZT /Z0 , det 2 = −2 (1 + b(1 + ZT /Z0 )) ,

(12)

−a

det 3 = 2 (1 − b(1 + ZT /Z0 )) e , det 4 = −4b · e−a , and the desired amplitude ratios are: det 1 Pr ; = rVS = Pe det

Pt det 4 ; = tVS = Pe det

Pa det 2 ; = Pe det

Par det 3 . = Pe det

(13)

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The insertion coefficient for a single incident plane wave (with angle ˜ ) becomes: ‘VS |tVS |2 | det 4|2 = = 2 2 1 − |rVS | 1 − |rVS | | det |2 − | det 1|2 2

= |4b · e−a |2 · (1 + b)2 − (1 − b)2 ·e−2a + b· 1 + b + (1 − b)·e−2a ·ZT /Z0

‘e (˜ ) =

(14)

2 −1

− (1 − b2 )(1 − e−2a ) + b · 1 − b + (1 + b) · e−2a · ZT /Z . The example shows the sound transmission loss R = −10 · lg(‘VS ) of a capsule wall (thin curves) and the insertion loss of a capsule (thick curves) with these walls, for three flow resistivity values ¡ of the absorber layer (sound incidence under ˜ = 45◦ ).

Sound transmission loss R of capsule walls and insertion loss De of a capsule for three flow resistivity values ¡ of the porous layer. Parameters: ˜ = 45◦ ; da = 0.05 [m]; dp = 0.0015 [m]; fcr dp = 12.3 [Hz · m]; p = 7850 [kg/m3]; † = 0.02 Special cases:

‘e (˜ ) −−−−−→ 4

†→0

no abs.

‘e (˜ ) −−−−−−→ |ZT |→∞

|2 + ZT /Z0 |2 − |ZT /Z0 |2 − →1, −−− 2

|tvs | | det 4| , = 2 1 − |rA | | det |2 1 − |rA |2

rVS −−−−−−→ rA = |ZT |→∞

‘e (˜ ) −−−−→ |a|1

2

| det

(1 + b)e−2a + 1 − b , (1 − b)e−2a + 1 + b | det 4|2 = |4b · e−a |2 2 − | det 1||a|1 4Re{b} · |1 + b · (1 + ZT /Z0 )|2 .

|2|a|1

For three-dimensional diffuse sound incidence: ˜hi 2 ‘e (˜ ) cos ˜ sin ˜ d˜ . ‘3−dif = sin2 ˜hi 0

(15)

(16)

Capsules and Cabins

L

847

For two-dimensional diffuse sound incidence: ‘2−dif

L.2

1 = sin ˜hi

˜hi ‘e (˜ ) cos ˜ d˜ .

(17)

0

Absorbent Sound Source in a Capsule


Sound absorption inside a capsule may be produced not only by an absorber layer on the capsule walls, but also by the source itself. This effect will be illustrated with a model in which the capsule and the source are two-dimensional; the source offers at its surface an impedance Zi to incident waves. Let ZF be the field impedance at the source surface; then the sound pressure and particle velocity at its surface are given by: p(xQu ) =

1 · pQu ; 1 + Zi /ZF

v(xQu ) =

1 · vQu , 1 + ZF /Zi

(1)

where pQu , vQu are generally used to characterise a source and belong to the special cases: p(xQu ) −−−−−→ pQu ;

v(xQu ) −−−−→ vQu .

ZF →∞

(2)

ZF →0

The relation pQu = Zi · vQu (Helmholtz’s source theorem) exists and ZF pQu p(xQu ) 1 + ZF /Zi pQu = · = · . v(xQu ) 1 + Zi /ZF vQu Zi vQu

(3)

Because of the finite interior impedance Zi of the source, the condition of the energetic approximation of the previous > Sect. L.1, that the source power ¢ is constant for whatever exterior sound field, no longer holds. The model consists of a plane sound source Qu which radiates a plane wave towards both sides at an angle ˜ , which is given by the wave number kqu along the source surface with sin ˜ = kqu /k0. The walls of the capsule are equal on both sides (for simplicity) but possibly have different distances t± to the source. They consist of a porous layer and a metal sheet or plate. The source will not be transmissive for incident sound (such as big machines).Thus both sides of the source are independent of each other; sound fields will be written only for one side; the fields on the other side follow by simple substitutions. Because of this independence, the source thickness can be taken to be dQu = 0 (or the co-ordinate x is shifted correspondingly). Field formulations: pe+ (x, y) = Pe+ · ejk0 y sin ˜ · e−jk0 (x−t+ ) cos ˜ , pr+ (x, y) = Pr+ · ejk0 y sin ˜ · e+jk0 (x−t+ ) cos ˜ , pt+ (x, y) = Pt+ · ejk0 da pa+ (x, y) = Pa+ · ea

cos ˜

y sin ã

· ejk0 y sin ˜ · e−jk0 (x−t+ ) cos ˜ , · e−a

−a da cos ã

par+ (x, y) = Par+ · e

(x−t+ ) cos ã a y sin ã

·e

,

· e+a

(x−t+ ) cos ã

(4)

848

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Capsules and Cabins

with interior angle in the porous layer sin ã =

jk0 · sin ˜ a

(5)

and partition impedance ZT of the metal sheet: ZT = 2Zm F[†F2 · sin4 ã + j(1 − F2 · sin4 ã )] ; Z0

fcr dp p ; Z0

Zm =

F=

f , fcr

(6)

where † is the bending loss factor of the sheet, p the density of its material and fcr the critical frequency. The amplitudes Pe+ , Pr+ , Pa+ are “defined” at x = t+ , the amplitudes Par+ , Pt+ at x = t+ + da . The source strength is described by its surface velocity profile: vQu (y) = VQu · ejkq y sin ˜ .

(7)

The condition at x = xQu = 0: p(0, y) + Zi · v(0, y) = Zi · vQu , together with the boundary conditions, leads to the following system of equations: ⎛

1 ⎜1 ⎜ ⎜ ⎜0 ⎜ ⎝0 d

−1 b e−a be−a 0

−e−a −be−a 1 −b 0

0 0 −(1 + ZT /Z0) −1 0

⎞ ⎛ ⎞ ⎛ 1 Pr+ ⎜ ⎟ ⎜ −1⎟ ⎟ ⎜ Pa+ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ · ⎜Par+ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎠ ⎝ Pt+ ⎠ ⎝ c

⎞ 0 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎠ Zi · VQu

Pe+

(8)

with the abbreviations (an = a /k0 ; Zan = Za /Z0 ): a: = a da · cos ã = k0da

2 + sin2 ˜ ; an

c: = (1 + cos ˜ · Zi /Z0) · e+jk0 t+ cos ˜ ;

b: =

1 Z0 cos ã = Za cos ˜ an Zan

2 + sin2 ˜ an cos ˜

,

(9)

d: = (1 − cos ˜ · Zi /Z0 ) · e−jk0 t+ cos ˜ .

The transmission and reflection factors of the capsule walls are: tVS =

Pt± ; Pe±

rVS =

Pr± . Pe±

(10)

Capsules and Cabins

L

849

The sound intensities I+ radiated by the capsule (on one side) and I0 by the source, if it is in the free space (ZF (0) = Z0 /cos˜ ), are: I+ =

cos ˜ |Pt+ |2 , 2Z0

cos ˜ cos ˜ I0 = |p+ (0)|2 = 2Z0 2Z0

Zi · ZF (0) 2 |Zi VQu |2 · |VQu |2 = cos ˜ · . Zi + ZF (0) 2Z0 |Zi /Z0 · cos ˜ + 1|2

Thus the insertion power coefficient for one side becomes: 2 I+ 2 Pt+ = |1 + Zi /Z0 · cos ˜ | · , ‘e+ = I0 Zi VQu

(11)

(12)

and for both sides together:

Pt+ 2 Pt− 2 I+ + I− |1 + Zi /Z0 · cos ˜ |2 + . = · ‘e = 2I0 2 Zi VQu Zi VQu

(13)

The determinants needed in Kramer’s rule for the solution of the system of equations are:

det = −(1 + b)2c − (1 − b2 )d + (1 − b)2 c − (1 − b2)d · e−2a (14) − b · ZT /Z0 · ((1 − b)c + (1 + b)d) · e−2a + (1 + b)c + (1 − b)d ,

det 1 = Zi VQu −(1 − b2 ) − b(1 − b)ZT /Z0 + (1 + b) 1 − b(1 + ZT /Z0 ) · e−2a , det 2 = Zi VQu [−2 · (1 + b(1 + ZT /Z0 ))] , det 3 = Zi VQu [2 · (1 − b(1 + ZT /Z0 )) · e−a ] ,

(15)

−a

det 4 = Zi VQu [−4 · b · e ] , det 5 = Zi VQu −(1 + b)2 − b(1 + b)ZT /Z0 + (1 − b) (1 − b(1 + ZT /Z0)) · e−2a , and the required ratio: det 4 Pt± = Zi VQu det

= [4·b·e−a ]· (1 + b)2c± + (1 − b2 )d± − (1 − b)2c± + (1 − b2 )d± ·e−2a

(16)

−1 + b·ZT /Z0 · (1 + b)c± + (1 − b)d± + ((1 − b)c± + (1 + b)d± )·e−2a . Special case: the source is a pressure source, i. e. Zi → 0: ‘e+ → ‘ep+ = |4b · e−a |2 · (1 + b)2 − (1 − b)2 · e−2a

+ b · ZT /Z0 1 + b + (1 − b) · e−2a · e+jk0 t+ cos ˜

+ (1 − b2 ) · (1 − e−2a ) + b · ZT /Z0 1 − b + (1 + b) · e−2a −2 · e−jk0 t+ cos ˜ .

(17)

850

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Capsules and Cabins

Special case: the source is a velocity source, i.e. Zi → ∞: ‘e+ → ‘ev+ = |4b · e−a |2 · (1 + b)2 − (1 − b)2 · e−2a

+ b · ZT /Z0 1 + b + (1 − b) · e−2a · e+jk0 t+ cos ˜

− (1 − b2 )(1 − e−2a ) + b · ZT /Z0 1 − b + (1 + b) · e−2a

(18)

−2 · e−jk0 t+ cos ˜ . Special case: no absorber layer, i. e. di → 0 ; a → 0 ; b → 1: ‘e+ → ‘e0+ = 4

|1 + Zi /Z0 · cos ˜ |2 |2c + (c + d) · ZTnx |2

|1 + Zi /Z0 · cos ˜ |2 =4 2 . 2 + 1 + (1 − Zi /Z0) · e−2jk0 t+ cos ˜ · ZT /Z0 (1 + Zi /Z0)

(19)

If Zi of the source is large: |1 + Zi /Z0 · cos ˜ |2 ‘e0+ → 4 . 2 + (1 + e−2jk0 t+ cos ˜ ) · ZT /Z0 2

(20)

This quantity oscillates strongly with frequency and/or distance t+ : ‘e0+,max → |1 + Zi /Z0 · cos ˜ |2 ;

|1 + Zi /Z0 · cos ˜ |2 . |1 + ZT /Z0|2

‘e0+,min →

(21)

Special case of a narrow capsule, i. e. t± Š0 : c → (1 + cos ˜ · Zi /Z0 ) · (1 + jk0 d+ cos ˜ ) ,

(22)

d → (1 − cos ˜ · Zi /Z0) · (1 − jk0d+ cos ˜ ) , ‘en+ = |4b(1 + Zi /Z0 · cos ˜ ) · e−a |2 · (1 − Zi /Z0 · cos ˜ ) · (1 − jk0t+ cos ˜ ) (1 − b2 )(1 − e−2a )

+ b · ZT /Z0 1 − b + (1 + b) · e−2a

(23)

+ (1 + Zi /Z0 · cos ˜ ) · (1 + jk0 t+ cos ˜ ) (1 + b)2 − (1 − b)2 e−2a

−2 + b · ZT /Z0 1 + b + (1 − b) · e−2a .

Capsules and Cabins

L

851

Insertion loss De of a capsule, for only normal incidence ˜ = 0, with a rather “highohmic” source Zi /Z0 = 10, for three distances t between source and wall. Parameters: ˜ = 0; Zi /Z0 = 10; da = 0.05 [m]; ¡ = 10 [kPa · s/m2 ]; dp = 1.5 [mm]; fcr dp = 12.3 [Hz · m]; p = 7850 [kg/m3 ]; † = 0.02

As before, but with one distance t = 0.5 [m] and three angles of incidence ˜

As before, but with a “low-ohmic” source Zi /Z0

852

L

Capsules and Cabins

Insertion loss De of a capsule for diffuse sound incidence, with three distances t of source and wall. The energetic approximation is shown for comparison. Parameters: ˜ = diff; Zi /Z0 = 10; da = 0.05 [m]; ¡ = 10 [kPa · s/m2 ]; dp = 1.5 [mm]; fcr dp = 12.3 [Hz · m]; p = 7850 [kg/m3]; † = 0.02

Insertion loss De of a capsule for diffuse sound incidence, with three flow resistivity values ¡ of the porous layer material, for a velocity source. Parameters: ˜ = diff; Zi /Z0 = ∞; t = 0.5 [m]; da = 0.05 [m]; ¡ = var.; dp = 1.5 [mm]; fcr dp = 12.3 [Hz · m]; p = 7850 [kg/m3 ]; † = 0.02

As before, but for a pressure source

Capsules and Cabins

L.3

L

853

Semicylindrical Source and Capsule


A better approach to a capsule than the flat model of the previous section may be a semicylindrical model. The model is now a semicircular capsule and a similar source.In principle the source can have an eccentric position. The radiated wave can be expanded in cylindrical harmonics in the coordinate system (, œ) of the source.These harmonics,in turn,can be expanded, by the addition theorem for Bessel functions, in cylindrical harmonics with the coordinates (r, ˜ ). For simplicity this double expansion is avoided here; the source is concentric with the capsule.

The strength of the source is described by its radial particle velocity profile vQu , which, if necessary, is expanded as: vQu (œ, z) = Vn · cos(nœ) · cos(kznz) . (1) n

The interior source impedance Zi is assumed to be constant in z, œ.The following relation holds: p( = aQ , œ, z) + Zi · vr ( = aQ , œ, z) = Zi · vQu (œ, z) .

(2)

Let the field in the interspace between source and capsule wall be pi = pe + pr , in the absorber layer pa , and in the outer space pt . Let the floor of the capsule be hard. Field formulations: H(2) H(1) n (kr r) n (kr r) + Pr,n · (1) Pe,n · (2) pi (r, ˜ , z) = Hn (kr aQ ) Hn (kr aK ) n≥0 · cos(n˜ ) · cos(kznz) , H(2) H(1) n (ka.r r) n (ka.r r) Pa,n · (2) + Qa,n · (1) pa (r, ˜ , z) = Hn (ka.r aK ) Hn (ka.r (aK + da )) n≥0 · cos(n˜ ) · cos(kznz) , pt (r, ˜ , z) =

n≥0

Pt,n ·

H(2) n (kr r) H(2) n (kr (aK + da ))

· cos(n˜ ) · cos(kznz)

(3)

854

with

L

Capsules and Cabins

2 kr2 + kzn = k02 ;

2 2 ka,r + kzn = ka2 = −a2 .

(4)

This relation defines a modal angle of incidence Ÿn : 2 kr kzn 2 + = 1 = cos2 Ÿn + sin2 Ÿn , k0 k0

(5)

which is zero for conphase excitation along the z axis. The relevant angle ”n for the evaluation of the partition impedance ZT of the outer shell of the capsule is given by: 1 2 + (n/a)2 = kzn sin2 Ÿn + (n/k0(aK + da ))2 . (6) sin ”n = k0 The boundary conditions with a concentric source !

pi (aK ) = pa (aK );

!

vir (aK ) = var (aK ) , !

pa (aK + da ) − pt (aK + da ) = ZT · vtr (aK + da ) ,

(7)

!

var (aK + da ) = vtr (aK + da ) , !

pi (aQ ) + Zi · vi (aQ ) = Zi · vQu hold term-wise and produce the following system of equations: ⎞ ⎛ .. . . . . . . . . . .. ⎛ ⎞ ⎛ ⎞ .⎟ Pe,n 0 ⎜. . . ⎟ ⎜ ⎜ .. .. ⎜ ⎟ Pr,n ⎟ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜.. ..⎟ ⎜ ⎜ ⎜ ⎟ · Pa,n ⎟ = ⎜ 0 ⎟ ⎜. ai,k .⎟ ⎟ ⎟ ⎜ ⎜ ⎟ ⎝Qa,n ⎠ ⎝ 0 ⎠ ⎜ . .. . . . ⎠ ⎝ Zi Vn Pt,n . .. . . . . . . . . . . ..

(8)

with coefficients (a prime indicates the derivative): a11 =

H(2) n (kr aK ) H(2) n (kr aQ )

;

a12 = 1;

a13 = −1;

a14 = −

H(1) n (ka,r aK ) ; (1) Hn (ka,r (aK + da ))

(9)

a15 = 0 , a21 = jkr aK · a23 =

H(2) n (kr aK )

;

H(2) n (kr aQ ) (2) ka,r aK Hn (ka,r aK ) an Zan H(2) n (ka,r aK )

a31 = a32 = 0;

a33 = −

a22 = jkr aK · ;

H(1) n (kr aK ) H(1) n (kr aK )

;

H(1) ka,r aK n (ka,r aK ) a24 = ; (1) an Zan Hn (ka,r (aK + da ))

H(2) n (ka,r (aK + da )) H(2) n (ka,r aK )

kr ZT H(2) n (kr (aK + da )) , a35 = 1 + j k0Z0 H(2) n (kr (aK + da ))

;

(10) a25 = 0 ,

a34 = −1 ; (11)

Capsules and Cabins

a41 = a42 = 0 ;

a43 =

ka,r aK H(2) n (ka,r (aK + da )) ; an Zan H(2) n (ka,r aK )

ka,r aK H(1) n (ka,r (aK + da )) ; a44 = an Zan H(1) n (ka,r (aK + da )) kr Zi H(2) n (kr aQ ) ; k0Z0 H(2) n (kr aQ ) = a55 = 0 .

a51 = 1 + j a53 = a54

a45 = jkr aK ·

a52 =

H(1) n (kr aQ ) (1) Hn (kr aK )

H(2) n (kr (aK + da )) H(2) n (kr (aK + da ))

+j

L

855

(12) ,

kr Zi H(1) n (kr aQ ) ; k0 Z0 H(1) n (kr aK )

(13)

The radiated (effective) power is the sum of the modal powers. The radiated modal power of the free source (radius a = aQ ) without z factors (they cancel in ‘) is: Re Zn (aQ )/Z0 aQ (0) ¢n = · · |Zi Vn |2 . (14) 2ƒn Z0 |Zi /Z0 + Zn (aQ )/Z0|2 The radiated (effective) modal power of the capsule (radius a = aK + da ) is: aK + da Z0 · Re {Zn (aK + da )/Z0} · ¢n = 2

0

|vtr,n (aK + da , ˜ )|2 d˜

(aK + da ) |kr Pt,n |2 = Z0 · Re {Zn (aK + da )/Z0 } · 2ƒn (k0Z0 )2

(2) Hn (kr (aK + da )) 2 . (2) Hn (kr (aK + da ))

(15)

Thus the insertion power coefficient of the capsule becomes: ¢n n≥0

‘K =

¢(0) n

n≥0

2 1 kr H(2) 2 n (kr (aK + da )) Re{Zn (aK + da )/Z0 } · · (2) · Pt,n ƒ k n 0 H (k (a + d )) n r K a aK + da n≥0 = · 1 Re{Zn (aQ )/Z0 } aQ 2 2 · |Zi Vn | ƒ n (Z + Z (a ))/Z i n Q 0 n

(16)

with ƒ0 = 1 ; ƒn>0 = 2 and the modal radiation impedances Zn (a) k0 H(2) n (kr a) . = −j (2) Z0 kr Hn (kr a)

(17)

856

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Capsules and Cabins

Insertion loss De for monomodal excitation, n = 0, by a high-ohmic source, Zi/Z0 = 10, with three source radii aQ . Parameters: Ÿ = 0◦ ; n = 0; Zi /Z0 = 10; aK = 1 [m]; da = 0.05 [m]; ¡ = 10 [kPa · s/m2 ]; dp = 0.0015 [m]; fcr dp = 12.3 [Hz · m]; p = 7850 [kg/m3]; † = 0.02

As before, but for a low-ohmic source Zi /Z0 = 0.1 One needs information about the Zi Vn for further valuation if the source pattern is not mono-modal. Heuristic assumptions about the source mode amplitudes could be Zi Vn = const or Zi Vn ∼ 1/(n + 1). The following diagram shows the influence of such assumptions on De for a multimodal excitation with the mode orders n = 0, . . ., 4 (with a low-ohmic source Zi /Z0 = 0.1).

Capsules and Cabins

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857

Insertion loss De for a multimodal excitation, n = 0, . . ., 4, with two assumptions about the modal strength. Parameters: Ÿ = 0◦ ; n = 0 − 4; Zi /Z0 = 0.1; aK = 1 [m]; aQ = 0.5 [m]; da = 0.05 [m]; ¡ = 10 [kPa · s/m2 ]; dp = 0.0015 [m]; fcr dp = 12.3 [Hz · m]; p = 7850 [kg/m3 ]; † = 0.02

L.4

Hemispherical Source and Capsule


The object is similar to the object of the previous > Sect. L.3, but now the source and the capsule are hemispherical. An eccentric source could again be treated with the addition theorem for Bessel functions, but, for simplicity, a concentric source will mainly be considered below. The source strength is described by a surface radial velocity profile vQu .

The sound field inside the capsule is pi = pe + pr , the sound field in the interior absorber layer is pa , and the radiated sound outside the capsule is represented by pt . Field formulation: vQu (˜ , œ) = Vn · Pm n (cos ˜ ) · cos(mœ) , n≥0

(1)

858

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Capsules and Cabins

(for ease of writing, only one azimuthal mode m ≥ 0 is assumed to exist; in the final equations below a sum of azimuthal modes will be considered) h(2) h(1) n (k0 r) n (k0 r) pi (r, ˜ , œ) = · Pm + Pr,n · (1) Pe,n · (2) n (cos ˜ ) · cos(mœ) , h (k a ) h (k a ) n n 0 Q 0 K n≥0 h(2) h(1) (ka r) n (ka r) pa (r, ˜ , œ) = Pa,n · (2) + Qa,n · (1) n hn (ka aK ) hn (ka (aK + da )) n≥0 (2) m ·Pn (cos ˜ ) · cos(mœ) , pt (r, ˜ , œ) =

Pt,n ·

n≥0

h(2) n (k0 r) (2) hn (k0(aK + da ))

· Pm n (cos ˜ ) · cos(mœ).

() h() n (z); = 1, 2; are spherical Hankel functions hn (z) =

() H (z). 2z n

(3)

Pm n (z) are associate Legendre functions with special values: Pm n (z) ≡ 0;

m > n,

dPm n (z) −−−−−−−−→ 0 . n+m=even n+m=odd dz For m = 0 they go over to the Legendre polynomes Pn (z):

Pm n (0) −−−−−−−→ 0 ;

(4)

Pm −−→ Pn (z) , n (z) − m=0

(5) dm Pn (z) ; z = real, |z| ≤ 1 . dzm The partition impedance ZT of the outer, elastic shell follows from the bending wave equation as: B ˜ ,œ ˜ ,œ − kB4 T(˜ ) · P(œ) ƒp B ˜ ,œ˜ ,œ − kB4 v⊥(˜ , œ) = = ZT : = v⊥ j– v⊥ (˜ , œ) j– T(˜ ) · P(œ) (6)

B 4 4 4 4 2 = k − kB = j–m 1 − (ktrace /kB) = j–m 1 − (f /fcr) sin Ÿ , j– trace

m 2 m/2 Pm n (z) = (−1) · (1 − z )

where B is the bending modulus, kB the free bending-wave number, fcr the critical frequency of the shell (if it were a plane plate), ktrace the wave number of the trace of the exciting wave along the shell, and Ÿ the polar angle of incidence on the shell with ktrace = k0 · sin Ÿ. It is (a = shell radius): n(n2 − 1)(2 + n) ktrace 4 n(n2 − 1)(2 + n) f 2 4 ktrace = ; = , (7) a4 kB (k0 a)4 fcr n(n2 − 1)(2 + n) . (8) (k0a)4 The radiated effective power of the source into free space is the sum of modal powers: aQ2 Nm,n Re Zn (aQ )/Z0 (0) · · |Zi Vn |2 (9) ¢m,n = 4Z0 |Zi /Z0 + Zn (aQ )/Z0 |2

and therefore

sin4 Ÿ =

Capsules and Cabins

L

859

with the mode norms Nm,n : a2 Nm,n = 2

2 0

/2 2a2 1 (n + m)! 2 2 dœ a2 sin ˜ · (Pm (cos ˜ )) · cos (mœ) d˜ = n ƒm 2n + 1 (n − m)!

(10)

0

(ƒ0 = 1 ; ƒn>0 = 2), and with the modal radiation impedance (a prime denotes the derivative): Zn (a) h(2) n (k0 a) . = −j (2) Z0 hn (k0 a) The modal effective power radiated by the capsule is correspondingly: (2) hn (k0(aK + da )) 2 (aK + da )2 Nm,n |Pt,n |2 . ¢m,n = · Re {Zn (aK + da )/Z0 } · (2) 4Z0 hn (k0(aK + da )) The insertion power coefficient for multimodal excitation finally becomes: ¢m,n

(11)

(12)

m,n≥0

‘e =

m,n≥0

¢(0) m,n

(2) h (k0 (aK + da )) 2 · Pt,m,n 2 Nm,n Re{Zn (aK + da )/Z0 } · n(2) hn (k0 (aK + da )) (aK + da )2 m,n≥0 = · . 2 2 Re{Zn (aQ )/Z0} aQ Nm,n 2 · Zi Vm,n (Zi + Zn (aQ ))/Z0 m,n≥0

(13)

The amplitudes Pt,m,n therein follow from the system of equations of the previous () > Sect. L.3 after substitutionH() n (kr r) → hn (k0 r) in the coefficients aik of that system, and Zi Vn → Zi Vm,n on the right-hand side. The examples shown below illustrate the influence of the source interior impedance Zi and of the source radius aQ (with a fixed capsule radius aK = 1 [m]) for monomodal and multimodal excitations.

Insertion loss De for monomodal excitation with two source impedances Zi.Parameters: m = 0; n = 0; Zi /Z0 = var; aK = 1 [m]; aQ = 0.5 [m]; da = 0.05 [m]; ¡ = 10 [kPa·s/m2 ]; dp = 0.0015 [m]; fcr dp = 12.3 [Hz · m]; p = 7850 [kg/m3 ]; † = 0.02

860

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Capsules and Cabins

Insertion loss De for monomodal excitation with three source radii aQ . Parameters: m = 0; n = 0; Zi /Z0 = 10; aK = 1 [m]; aQ = var; da = 0.05 [m]; ¡ = 10 [kPa · s/m2 ]; dp = 0.0015 [m]; fcr dp = 12.3 [Hz · m]; p = 7850 [kg/m3]; † = 0.02

Insertion loss De for multimodal excitation and three source radii aQ . The excitation mode amplitudes decay as Zi Vn = 1/(n + 1). Parameters: m = 0; n = 0 − 5; Zi /Z0 = 10; aK = 1 [m]; aQ = var; da = 0.05 [m]; ¡ = 10 [kPa · s/m2 ]; dp = 0.0015 [m]; fcr dp = 12.3 [Hz · m]; p = 7850 [kg/m3]; † = 0.02

As before, but for a source which is “matched” to the exterior field, ZI/Z0 = 1

Capsules and Cabins

L.5

L

861

Cabins, Semicylindrical Model


Cabins are exposed to an exterior sound field pe .A suitable quantity for the qualification of the efficiency of the cabin is the sound pressure level difference (sound protection measure): L = −10 · lg

|pi |2 VK [dB] ,

|pe |2VK

(1)

where pi is the interior sound field, and . . .VK indicates the spatial average over the volume VK of the cabin. However, because cabins are often relatively small with a low mode density in them, the measurement of the average may be difficult. Alternatively a sound pressure level in some defined point r0 in the cabin could be used: L0 = −10 · lg

|pi (r0)|2 [dB] . |pe (r0 )|2

(2)

This definition reduces the amount of numerical computation considerably. The model to be treated below is a semicylindrical cabin on a hard floor. It consists of an outer elastic shell with partition impedance ZT and an interior layer of porous material with characteristic constants a , Za (or, in normalised form, an = a /k0 , Zan = Za /Z0 ). A plane wave p+e is assumed as incident wave (see below for diffuse sound incidence); a mirror-reflected wave p−e simulates the hard floor.

A variation of the incident wave in the z direction could be cos(kz z), sin(kz z), e±jkz z or a linear combination thereof. It would influence the radial wave number kr by kz2 + kr2 = k02 . The field factor in the z direction can be dropped because it is the same in all field parts.

862

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Capsules and Cabins

The exciting field pe = p+e + p−e is: pe (r, ˜ ) = Pe ƒn(−j)n · Jn (kr r) · [cos (n(˜ + Ÿ0 )) + cos (n(˜ − Ÿ0 ))]

(3)

n≥0

= 2Pe :=

n≥0

ƒn (−j)n · cos(nŸ0 ) · Jn (kr r) · cos(n˜ )

(4)

Pen · Jn (kr r) Jn (kr (aK + da )) · cos(n˜ ) .

n≥0

The last form uses the abbreviation Pen = 2Pe · ƒn (−j)n · cos(nŸ0 ) · Jn (kr (aK + da ))

(5)

with ƒ0 = 1 ; ƒn>0 = 2. The interior field pi , in the absorber layer pa , and the exterior scattered field ps are formulated as: pi (r, ˜ ) = Pin · Jn (kr r) Jn (kr aK ) · cos(n˜ ) , n≥0

H(2) n (kar r)

H(1) n (kar r)

· cos(n˜ ) , + Qan · (1) H(2) Hn (kar (aK + da )) n (kar aK ) (2) ps (r, ˜ ) = Psn · H(2) n (kr r) Hn (kr (aK + da )) · cos(n˜ ) .

pa (r, ˜ ) =

Pan ·

(6)

n≥0

n≥0 2 The radial wave number kar in the absorber layer is given by kar + kz2 = ka2 = −a2 . The unknown amplitudes Pin , Psn , Pan , Qan follow from the boundary conditions which hold term-wise: !

pin (aK ) = pan (aK );

!

virn (aK ) = varn (aK ) , !

(7)

pan (aK + da ) − pen (aK + da ) − psn (aK + da ) = ZTn · varn (aK + da ) , !

varn (aK + da ) = vern (aK + da ) + vsrn (aK + da ). The modal partition impedance ZTn of the shell is evaluated from: ZTn Z0

= 2Zm F[†F2 · sin4 ”n + j(1 − F2 · sin4 ”n )];

sin ”n =

1 2 kz + (n/(aK + da ))2 k0

Zm =

fcr dp p ; Z0

F=

f , fcr

(8)

(dp = shell thickness, p = shell material density, fcr = critical frequency of the shell as a plane plate, † = bending loss factor of the shell). The system of equations of the boundary conditions has the following form, with the abbreviation Cn = kr (aK + da )

Jn (kr (aK + da )) Jn (kr (aK + da ))

(9)

Capsules and Cabins

L

(a prime indicates the derivative): ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 −1 a1,4 Pin 0 ⎜a ⎜ ⎟ ⎜ ⎟ 0 a2,3 a2,4⎟ ⎜ 2,1 ⎟ ⎜ Psn ⎟ ⎜ 0 ⎟ ⎜ ⎟·⎜ ⎟ = ⎜ ⎟. ⎝ 0 −1 a3,3 a3,4⎠ ⎝ Pan ⎠ ⎝ Pen ⎠ 0 a4,2 a4,3 a4,4 Qan Cn · Pen

863

(10)

The matrix coefficients are: a1,4 = −

H(1) n (kr aK ) H(1) n (kr (aK + da ))

a2,1 = kr aK a2,4 = a3,3 =

Jn (kr aK ) ; Jn (kr aK )

a2,3 =

−jkar aK H(2) n (kar aK ) ; an Zan H(2) n (kar aK )

−jkar aK H(1) n (kar aK ) , (1) an Zan Hn (kar (aK + da )) H(2) n (kar (aK + da )) H(2) n (kar aK )

a3,4 = 1 +

+

kar /k0 · ZTn /Z0 H(2) n (kar (aK + da )) ; an Zan H(2) n (kar aK )

(11)

kar /k0 · ZTn /Z0 H(1) n (kar (aK + da )) , an Zan H(1) n (kar (aK + da ))

a4,2 = −kr (aK + da ) a4,4 =

,

H(2) n (kr (aK + da )) H(2) n (kr (aK + da ))

;

a4,3 =

jkar (aK + da ) H(2) n (kar (aK + da )) ; an Zan H(2) n (kar aK )

jkar (aK + da ) H(1) n (kar (aK + da )) . an Zan H(1) n (kar (aK + da ))

Cn and some matrix coefficients have the form: z

Zn+1 (z) Zn (z) = −z +n Zn (z) Zn (z)

(12)

with Zn (z) some cylinder function. The desired quantity Pin is: Pin = −

(a1,4 · a2,3 + a2,4 ) · (a4,2 + Cn ) Pen . (a1,4 · a2,1 − a2,4 ) · (a3,3 · a4,2 + a4,3 ) + (a2,1 + a2,3 ) · (a3,4 · a4,2 + a4,4 )

(13)

With this, the field inside the cabin is known. Factors with a variation in the z direction will cancel in the ratio for L after averaging; therefore the average over the area AK = aK2 is sufficient. The average of the exterior field is (with ky = kr · sin Ÿ0 ): 2

|pe | AK

2 = AK

+aK −aK

y(x) J1 (2ky aK ) dx |pe |2 dy = 2|Pe|2 1 + ; 2ky aK

y(x) =

aK2 − x2 .

(14)

0

The average inside the cabin is: 1 Jn−1 (kr aK ) · Jn+1 (kr aK ) 2 2 . |Pin | 1 −

|pi | AK = ƒ J2n (kr aK ) n≥0 n

(15)

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Capsules and Cabins

The sound protection measure, based on the average squared pressure magnitudes, for a single incident plane wave p+e is, finally: 1 Pin 2 1 − Jn−1 (kr aK ) · Jn+1 (kr aK ) ƒ Pe J2n (kr aK ) n≥0 n L = −10 · lg . (16) J1 (2ky aK ) 2 1+ 2ky aK The sound protection measure, based on the level difference in the cabin centre, is: 1 1 Pi0 2 . (17) L0 = −10 · lg 4 Pe J20 (kr aK ) Because the angle of incidence Ÿ0 is not contained, it also holds for two-dimensional diffuse incidence.

Sound protection measure L0 for sound incidence normal and oblique to the cabin axis. Parameters: ¥ = var.; aK = 2 [m]; da = 0.05 [m]; ¡ = 10 [kPa · s/m2 ]; dp = 0.0015 [m]; fcr dp = 12.3 [Hz · m]; p = 7850 [kg/m3 ]; † = 0.02

Sound protection measure L0 for oblique sound incidence and three cabin radii aK . Parameters: ¥ = 45◦ ; aK = var.; da = 0.1 [m]; ¡ = 10 [kPa · s/m2 ]; dp = 0.0015 [m]; fcr dp = 12.3 [Hz · m]; p = 7850 [kg/m3]; † = 0.02

Capsules and Cabins

L.6

L

865

Cabin with Plane Walls


The model is two-dimensional: two walls consisting of an exterior plate and an interior porous absorber layer. This model is an approximation to reality if the lateral dimension of the cabin in the x direction is large (at least compared to t) and if sound incidence comes mainly from the half space in front of one wall.

Under these conditions the cabin is just a multilayer absorber; the desired interior sound field is the field in one layer. This view of the task permits the application of the equivalent network method (see > Ch. C).

The source pressure is p0 = 2Pe with the amplitude Pe of the incident plane wave. The interior impedance of the source and the load impedance are Z0 /cosŸ0. The sound pressure inside the cabin, i. e. in the layer named “Air” above, is: pi (x, y) = Pi1 · e−jkx (x−da1 ) + Pi2 · e+jkx (x−da1 −t) · e−jky y , kx = k0 cos Ÿ0 ;

ky = k0 sin Ÿ0

with the relation between amplitudes and circuit node pressures: ! pi (x = da1 ) = Pi1 + Pi2 · e−jkx t = 2Pe · p2 , ! pi (x = da1 + t) = Pi1 · e−jkx t + Pi2 = 2Pe · p4

(1)

(2)

or Pi1 =

p2 − p4 · e−jkx t · 2Pe ; 1 − e−2jkx t

Pi2 =

p4 − p2 · e−jkx t · 2Pe . 1 − e−2jkx t

(3)

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Capsules and Cabins

If one includes in the exciting wave the ground-reflected wave, the factor e−jky y simply changes to cos(ky y); it cancels anyway in the averages for L and L0 . The equivalent circuit elements are (i = 1, 2): Z,i cos Ÿ,i cosh(a,i dai · cos Ÿ,i ) − 1 sinh(a,i dai · cos Ÿ,i ) , = Zan,i ; Z0 G,i = Z0 cos Ÿ,i Zan,i sinh(a,i dai · cos Ÿ,i ) (4) 2 an,i + sin2 Ÿ0 2 ; a,i dai · cos Ÿ,i = k0dai an,i + sin2 Ÿ0 , cos Ÿ,i = an,i Z 1 − cos(k0t · cos Ÿ0 ) ; =j Z0 cos Ÿ0 · sin(k0 t · cos Ÿ0 )

Z0 G = j cos Ÿ0 · sin(k0 t · cos Ÿ0 ) ,

ZT,i = 2 · Zm,i · Fi †i F2i · sin4 Ÿ0 + j 1 − F2i · sin4 Ÿ0 Z0 with

Zm,i =

fcr,i di i ; Z0

Fi =

f f · di = fcr,i fcr,i di

(5) (6) (7)

(di = plate thickness, i = plate material density, fcr,i = critical frequency, †i = bending loss factor) With the abbreviations z1 = 1 +

ZT,2 cos Ÿ0 ; Z0

Za,2 · ; z2 = z1 + Z0

= cos Ÿ0 + Z0 G,2 · z1 , (8a) = + Z0 G,2 · z2 ,

Z · ; z4 = z2 + z3; ‚ = + Z0 G · z4 , Z0 Za,1 Z · ‚; ƒ = ‚ + Z0 G,1 · z5; z6 = z5 + ·ƒ, z5 = z4 + Z0 Z0 z3 =

(8b)

one gets z5

p2 =

, 1 ZT,1 + Z0 cos Ÿ0 Za,1 1 ZT,1 ·ƒ+ + p4 = z2 · z5 + Z0 Z0 cos Ÿ0 Za,1 Z · ‚ + Z0 G,1 · z2 + · + Z0 G,2 · z1 · ƒ + Z0 G,1 · z5 + Z0 Z0 Za,2 Z Z + · + · + Z0 G · z2 + · + Z0 G,2 · z1 Z0 Z0 Z0 −1 Za,1 2 · (cos Ÿ0 + Z0 G,2 · z1 ) . + Z0 z6 + (ƒ + Z0 G,1 · z6)

(9)

Capsules and Cabins

L

867

Thus the square of the sound pressure magnitude in the cabin will be (kx , ky real): |pi (x, y)|2 =

4|Pe |2 2 |p2 | · sin2 (kx (x − da1 − t)) + |p4 |2 · sin2 (kx (x − da1 )) sin2 (kx t) − 2Re{p2 p∗4 } · sin (kx (x − da1 − t)) · sin (kx (x − da1 ))

with an average

2

sin(2kx t) 2 2 |p2 | + |p4 |

|pi (x, y)| = 1− 2kx t sin2 (kx t) sin(kx t) − 4Re{p2 p∗4 } cos(kx t) − . kx t |Pe |2

2

(10)

The sound protection measure L is, finally:

2 sin(2kx t) 1 2 | + |p | 1 − 2 |p L = −10 · lg 2 4 2kx t sin2 (kx t) sin(kx t) − 4Re{p2 p∗4 } cos(kx t) − . kx t

(11)

(12)

With the sound pressure in the cabin centre: |pi (x = da1 + t/2, y)|2 =

4|Pe |2 sin2 (kx t/2) 2

sin (kx t)

|p2 |2 + |p4 |2 + 2Re{p2 p∗4 } ,

(13)

the sound protection measure L0 becomes: L0 = −10 · lg

2 (1 − cos(kx t)) 2 |p2 | + |p4 |2 + 2Re{p2 p∗4 } . 2 sin (kx t)

(14)

Until now it was tacitly assumed that the shadow field of the cabin, i. e. the field behind the cabin which is generated there by scattering of pe , was much lower than the sound pressure of the incident wave at the front side of the cabin. A different extreme situation would be a scattered sound field behind the cabin, which would be strong enough to inhibit the radiation of the sound which has traversed the cabin at the back side of the cabin. Then the equivalent network ends in the node with p5 . One gets in this case with the abbreviations = 1 + G,2 · Z,2 ;

g1 = Z0 G,2 · (1 + ) ,

= +

Z · g1 ; Z0

g2 = g1 + Z0 G · ,

‚ = +

Z · g2 ; Z0

g3 = g2 + Z0 G,1 · ‚ ;

(15) ƒ = ‚ +

Z,1 · g3 Z0

the node pressures: −1

ZT1 1 p2 = ‚ · ƒ + g3 + Z0 G,1 ‚ + Z0 G,1 · g3 + , Z0 cos Ÿ0 −1

ZT1 1 + . p4 = · ƒ + g3 + Z0 G,1 · ƒ Z0 cos Ÿ0

(16)

868

L

Capsules and Cabins

The sound protection measures then follow as above. Numerical checks in a number of examples have shown that the results agree with the former results within the precision of graphical representation. A further special case can be easily treated: a coherent sound incidence with equal strength takes place on both sides of the cabin. Then the equivalent circuit ends in the node with p3 after substitution t → t/2. The required node pressures are: −1

ZT1 1 p2 = · + g1 + Z0 G,1 · + , Z0 cos Ÿ0 (17) −1

ZT1 1 p4 = + g1 + Z0 G,1 · + Z0 cos Ÿ0 with the abbreviations = 1 + G · Z ;

g1 = Z0 G + Z0 G,1 · ;

=+

Z,1 · g1 . Z0

(18)

L follows as before,except for an additional factor 1/2 in the argument of the logarithm. With incoherent sound incidence from both sides the contributions of each side to the interior sound pressure magnitude are evaluated separately and added. In the examples shown below,the walls on both sides of the cabin are equal,for simplicity.

The two sound pressure measures L and L0 in a 2-dimensional cabin. Parameters: Ÿ0 = 0◦ ; t = 4 [m]; da = 0.1 [m]; ¡ = 10 [kPa · s/m2 ]; dp = 0.0015 [m]; fcr dp = 12.3 [Hz · m]; p = 7850 [kg/m3]; † = 0.02

Capsules and Cabins

L

869

Influence of the absorber layer thickness da on the sound protection measure L for normal sound incidence on the cabin wall. Parameters: Ÿ0 = 0◦ ; t = 4 [m]; da = var.; ¡ = 10 [kPa·s/m2 ]; dp = 0.0015 [m]; fcr dp = 12.3 [Hz·m]; p = 7850 [kg/m3 ]; † = 0.02

As above, but for diffuse sound incidence

L.7

Cabin with Rectangular Cross Section


The cabin is still two-dimensional, but with a ceiling which (for simplicity) is locally reacting with a surface admittance G. The incident wave pe with pe = p+e + p−e considers also the ground reflection. The field pi inside the cabin is formulated as a mode sum of a locally lined flat duct with the cabin ceiling as lining.The lateral wave numbers —m are solutions of the characteristic equation of the duct. With the usual dimensions of cabins it makes no significant difference whether the back wall radiates a wave pt into the free space or if the back wall has a hard termination. Therefore this possibility of simplification will be used below.

L

870

Capsules and Cabins

Field formulations: pe = Pe · e−jkx x · cos(ky y) ;

kx = k0 · cos Ÿ;

ky = k0 · sin Ÿ ,

pr = Pr · e+jkx x · cos(ky y) , Am · e−‚m x + Bm · e+‚m x · cos(—m y) ; pi = m

pa1 =

m

pa2 =

−‚am x

Pam · e

+‚am x

+ Qam · e

· cos(—m y) ;

‚m =

—2m − k02 ,

‚am =

—2m

−

(1) a2 ,

Ram cosh ‚am (x − t − da ) · cos(—m y) .

m

The exciting and the reflected field at x = 0 are synthesised with duct modes: pe (0, y) = Pe · cos(ky y) =

Pem · cos(—m y) ;

m

pr (0, y) = Pr · cos(ky y) =

(2)

Prm · cos(—m y)

m

with mode amplitudes:

Pem

Pe 1 = · Nm h

h cos(ky y) · cos(—m y) dy = 0

1 using mode norms: Nm = h

h

Sm · Pe ; Nm

cos2 (—m y) dy =

0

1 and mode coupling coefficients: Sm = 2

!

Prm =

Sm · Pr Nm

sin(2—m h) 1 1+ 2 (2—m h)

(3)

(4)

" sin (—m − ky )h sin (—m + ky )h + . (5) (—m − ky )h (—m + ky )h

Capsules and Cabins

L

871

The boundary conditions (which hold term-wise) at the front and back side walls !

pe (−da ) + pr (−da ) − pa1 (−da ) = ZT · va1x (−da ) , !

vex (−da ) + vrx (−da ) = va1x (−da ) , !

pa1 (0) = pi (0); !

(6)

!

va1x (0) = vix (0) , !

pi (t) = pa2 (t);

vix (t) = va2x (t) ,

(ZT is the partition impedance of an outer cover plate of the walls; see previous > Sect. L.6) lead to the following system of equations: ⎛

..

.

···

⎜ ⎜ ⎜. ⎜ .. a i,k ⎜ ⎜ ⎝ . .. · · ·

..

.

.. . ..

.

⎛ ⎞ ⎞ Prm 1 ⎟ ⎜ Am ⎟ ⎜1⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ Bm ⎟ ⎜ ⎟ ⎟·⎜ ⎟ = e+jkx da · Pem · ⎜0⎟ ⎟ ⎜ Pam ⎟ ⎜0⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎠ ⎝Qam ⎠ ⎝0⎠ 0 Ram ⎞ ⎛

(7)

using the matrix coefficients: a1,1 = −e−jkx da ; a1,2 = a1,3 = a1,6 = 0 ; ‚am da ZT /Z0 ‚am da ZT /Z0 a1,4 = e+‚am da 1 + ; a1,5 = e−‚am da 1 − , k0 da · an Zan k0da · an Zan a2,1 = +e−jkx da ; a2,4 = e+‚am da

a2,2 = a2,3 = a2,6 = 0 ;

‚am da ; kx da · an Zan

a2,5 = −e−‚am da

‚am da , kx da · an Zan a3,4 = a3,5 = −1 ,

a3,1 = a3,6 = 0 ;

a3,2 = 1 ;

a3,3 = e−‚m t ;

a4,1 = a4,6 = 0 ;

a4,2 = 1 ;

a4,3 = −e−‚m t ;

j‚am da /‚m t t a4,4 = − ; an Zan da

a4,5

a5,1 = a5,4 = a5,5 = 0 ; a5,2 = e−‚m t ;

−1 +‚am da e + e−‚am da , a5,6 = 2 a6,1 = a6,4 = a6,5 = 0 ; a6,6 =

(8)

j‚am da /‚m t t =+ , an Zan da

a6,2 = e−‚m t ;

a5,3 = 1 ;

a6,3 = −1 ;

−j‚am da /‚m t t +‚am da e − e−‚am da . 2an Zan da

With the abbreviations Xm : = jan Zan · ‚m t + ‚am t ;

Ym : = jan Zan · ‚m t − ‚am t ,

(9)

L

872

Capsules and Cabins

the required amplitudes Am , Bm will be:

Am = 4an Zan · k0 da · kx t · ‚am da Xm · e+‚am da + Ym · e−‚am da · e+jkx da · Pem ·

e−2‚am da + e−2‚m t · Ym + 1 + e−2(‚am da +‚m t) · Xm

· an Zan · k0da · kx da − ‚am da · (k0da + kx da · ZT /Z0 ) · Xm

(10)

−1 − an Zan · k0 da · kx da + ‚am da · (k0da + kx da · ZT /Z0) · e+2‚am da · Ym ,

Bm = 4an Zan · k0da · kx t · ‚am da Xm · e−‚am da + Ym · e+‚am da · e−‚m t · e+jkx da · Pem ·

−2‚ d

e am a + e−2‚m t · Ym + 1 + e−2(‚am da +‚m t) · Xm

· an Zan · k0 da · kx da − ‚am da · (k0da + kx da · ZT /Z0 ) · Xm

−1 − an Zan · k0da · kx da + ‚am da · (k0 da + kx da · ZT /Z0 ) · e+2‚am da · Ym .

References Mechel, F.P.: Schallabsorber, Vol. III, Ch. 20: Capsules and cabins. Hirzel, Stuttgart (1998)

(11)

M Room Acoustics with M. Vorländer Both deterministic and statistic methods of sound field evaluations in rooms will be described. Because of the complexity of room shapes and acoustic qualities of walls, evaluations in general will be approximative.

M.1

Eigenfunctions in Parallelepipeds

Rooms with uniform, rectangular walls belong to the few examples in room acoustics in which a modal analysis can be performed with a reasonable amount of analytical and numerical work. As such they may serve as gauge objects for conceptions and methods.

The corner lengths are x , y , z , respectively. The wall surface admittances are Gx , Gy , Gz , if the walls on opposite sides are equal, otherwise Gx1 , Gx2 , etc. The aim is to find elementary solutions (eigenfunctions or modes) with which sound fields for an arbitrary sound source in the room can be synthesised. They must obey the wave equation, symmetry conditions and boundary conditions. Alternative writing: x, y, z = x1 , x2 , x3 ;

x , y , z = 1 , 2 , 3 ;

Gx , Gy , Gz = G1 , G2, G3 .

Wave equation: ∂ 2p ∂ 2p ∂ 2p + + + k02p = 0 . ∂x2 ∂y 2 ∂z2

(1)

Fundamental solutions separate p(x1 , x2, x3 ) = q1(k1 x1 ) · q2 (k2x2 ) · q3 (k3x3 )

(2)

M

874

with

qi =

Room Acoustics

cos(ki xi );

symmetrical rel. xi = 0

sin(ki xi );

anti-symmetrical rel. xi = 0

;

i = 1, 2, 3 .

(3)

They satisfy the wave equation if (secular equation): !

k02 = k12 + k22 + k32 .

(4)

If the room is symmetrical in the direction of xi (i.e. Gi1 = Gi2 = Gi ) and the field is symmetrical (depending on the directivity and position of the source): qi = cos(ki xi ); with room symmetry in the direction of xi and anti-symmetrical field: qi = sin(ki xi ). Else: qi = ai cos(ki xi ) + bi sin(ki xi ). The boundary conditions at the walls lead to the characteristic equations for ki . Define Ui = (k0i /2) · Z0 Gi Ui1 = (k0i /2) · Z0 Gi1 ;

for symmetrical walls , Ui2 = (k0i /2) · Z0 Gi2

for anti-symmetrical walls .

General case (unsymmetrical room): With:

Usi =

1 (Ui1 + Ui2 ) ; 2

Uai =

1 (Ui1 − Ui2 ) , 2

the characteristic equation is written as: ! (ki i /2) · tan(ki i /2) − jUsi · (ki i /2) · cot(ki i /2) + jUsi = U2ai

(5)

with the amplitude ratio of the anti-symmetrical to the symmetrical part of the mode: bi (kii /2) · tan(ki i /2) − jUi2 = − cot(ki i /2) . ai (ki i /2) · tan(ki i /2) + jUi2

(6)

Special case: symmetrical room (Gi1 = Gi2 = Gi ), symmetrical mode; characteristic equation: (ki i /2) · tan(ki i /2) = jUi .

(7)

Special case: symmetrical room (Gi1 = Gi2 = Gi ), anti-symmetrical mode; characteristic equation: (ki i /2) · cot(ki i /2) = −jUi .

(8)

Special case: both walls normal to xi are hard (Gi1 = Gi2 = Gi = 0), symmetrical mode: ki i /2 = mi ;

mi = 0, 1, 2, . . . .

(9)

Special case: both walls normal to xi are hard (Gi1 = Gi2 = Gi = 0), anti-symmetrical mode: ki i /2 = (mi + 1/2);

mi = 0, 1, 2, . . . .

(10)

In these equations Gi = 0 may represent either a locally reacting or a bulk reacting wall. In the first case Gi is independent of ki ; in the second case Gi = Gi (k1, k2, k3). In both

Room Acoustics

cases the modes are orthogonal (see field synthesis.

>

M

875

Sects. B.6 and B.7) and as such are suited for

The three characteristic equations (i = 1, 2, 3) and the secular Eq. M.1.(4) in general cannot be solved simultaneously for all frequencies. In the special case of only hard walls and symmetrical modes, one finds (with l = m1 , m = m2 , n = m3 ) eigenfrequencies flmn: k02 = (2flmn /c0)2 = ki2 = (2mi /i )2 , i i (11) 2 2 flmn = c0 (l/1) + (m/2 ) + (n/3)2 . If the mode is anti-symmetrical in some direction xi , substitute mi → mi + 1/2. In cases with Gi = 0 the secular Eq. M.1.(4) with the solutions ki of the characteristic equations must be solved numerically for eigenfrequencies. Under some restrictive conditions one can derive approximations. This will be shown for symmetrical, locally reacting walls and symmetrical modes, i.e. for Eq. M.1.(7). Write that equation as zi · tan zi = j Ui = j k0i · Z0 Gi ;

zi = ki i /2 .

(12)

With the continued-fraction expansion of tan(zi ) = tan(zi − m); mi = 0, 1, 2. . . writing a1 / (a2 − a3 / (a4 − a5 / (a6 − . . .))) = one gets (see

a1 a3 a5 ... , a2 − a4 − a6 −

> Sect. J.7),

(zi − mi )2 (zi − mi )2 zi2 = mi · zi + j k0 i · Z0 Gi · 1 − ... 3− 5−

(13a)

−−−−−−→ (mi )2 + j k0 i · Z0 Gi −−−−−−−−−−−−−−−−−→ (mi )2 . zi →mi

k0 i → 0 and/or Gi → 0

Thus: ki2 = zi2 (2/i )2 ≈ 4

(mi )2 + j k0 i · Z0 Gi . 2i

(13b)

This inserted into the secular equation gives an approximate equation for the eigenfrequencies (represented by k0): k02 − 4j k0 (Z0 G1 /1 + Z0 G2/2 + Z0 G3 /3)

! − 42 (m1 /1)2 + (m2 /2 )2 + (m3 /3 )2 = 0 .

(14)

Similar procedures may be applied for other cases of symmetry. It should be noticed that the condition used zi → mi implies small admittance values |Gi |. But even this equation cannot be discussed further without knowledge of the functions Gi (k0 ). In a formal manner one can write for the solutions (with klmn = 2flmn/c0 ; flmn from Eq. M.1.(11)):

k0 → klmn = –lmn + jƒlmn /c0 , (15) where ƒlmn represents a modal damping constant (which itself generally is complex).

M

876

Room Acoustics

Half-widths of modes (resonance curve): (f )lmn =

ƒlmn .

(16)

The transfer function between two points in a room is calculated by superposition of damped modes (resonance curves): p (x1 , x2, x3 , –) =

l,m,n

–2

−

Almn(–)

2 –lmn − 2jƒlmn–lmn

with ƒlmn –lmn and Almn (–) depending on the positions (x1 , x2 , x3 ): Almn (–) = p (x1 , x2 , x3 )p (xS1 , xS2 , xS3 ) lmn

lmn

(17) source (xS1 , xS2 , xS3 ) and receiver 2 plmn (x1 , x2 , x3 ) dx1 dx2 dx3 . (18)

V

M.2

Density of Eigenfrequencies in Rooms

Let N be the number of eigenfrequencies below the frequency f ; let n = dN/df be the number of eigenfrequencies in an interval of 1 Hz. Volume with smallest corner length >Š0 /2 (V = volume, S = room surface, L = sum of corner lengths, f = frequency): N=

4V · f 3 S · f 2 L · f + + , 2c0 3c30 4c20

4V · f 2 S · f L n= + + . 2c0 c30 2c20

(1)

Flat volume with smallest corner a < Š0 /2, other corners b, c > Š0 /2: n=

2 · b · c · f . c20

(2)

Tube of length with hard walls: n = 2/c0 .

(3)

The mode overlap m is defined as the ratio of the half-value bandwidth of a room resonance and the average frequency separation between neighbouring resonances around a frequency f . The mode overlap in the diffuse reverberant field is (T = reverberation time): m = 0.69

V (f /1000)2 . T

(4)

The mode overlap must exceed some lower limit value for the application of statistical methods in room acoustics. This defines a lower limit frequency fs for such methods. (5) Limit frequency for m ≥ 10: fs > 4000 T/V .

M

Room Acoustics

Limit frequency for m ≤ 10: fs > 2000 T/V .

877

(6) The modulus of the room transfer function, p(–), can be estimated by (see also Eq. M.1.(17)): p(–) ≈

|Almn | (–2 − –2lmn )2 + 4–2 ƒ2lmn

.

(7)

The probability density of the transfer function modulus z = |p(w)| (Rayleigh distribution) is: P(z) dz =

−z 2 /4 z dz , e 2

(8)

and the probability density of transfer function phase œ = arg p(–) is:

1 P œ dœ = dœ . 2

M.3

(9)

Geometrical Room Acoustics in Parallelepipeds

Assumptions: • The room is a parallelepiped with corner lengths x , y , z . • A small isotropic source is placed in the centre of the room. Room volume:

V = x · y · z .

Walls:

Sx = y · z ; Sy = z · x ;

S = 2 Sx + Sy + Sz .

Interior room surface:

(1) Sz = x · y .

(2) (3)

Number of mirror sources up to the order n = 1, 2, . . .:

sn = 4n2 + 2.

(4)

Positions of the mirror sources:

{±nx , ±ny , ±nz }.

(5)

Number of reflections including order n:

2 £n = n 2n2 + 3n + 4 . (6) 3

Estimation of temporal density of reflections [Cremer (1948)]:

4c30 2 n(t) ≈ t t V

(7)

and of the total number of reflections between times 0 and t:

n(t) ≈

4c30 3 t . 3V

(8)

Mean free path length of sound: energetic average: 2me

4n2 + 2 = lim n→∞ n2

2 4n +2

i=12

(1/2i,n )

3 ≈ 4

1 1 1 + 2 + 2 2 2 2 x + y y + z z + 2x

−1 ,

(9)

M

878

Room Acoustics

geometrical average: 4n +2 1 i,n 1 2 2 + 2 + 2 + 2 + 2 . ≈ + x y y z z x n→∞ n 4n2 + 2 6 2

mg = lim

(10)

i=12

Value, often used in room acoustics [Kosten (1960)]:

ma ≈ 4V/S.

(11)

(Effective) intensity of the direct sound field: (¢q = effective power of source, d = distance source to receiver) ID =

¢q . 4d

(12)

Intensity of reverberant field: IR =

∞

IRn = ¢q

n=1

∞ (1 − ) ¯ n (4n2 + 2) n=1

4n2 2me

−−−−→ ¢q 0.1: 1 − ¯ 2 2 (1 − )(6 ¯ + 5) ¯ . IR ≈ ¢q 2 + ≈ ¢q me ¯ 6 3 ¯ 2me

(16)

Level steps in reverberation plot:

3 ¯ 2me L1 = 10 · lg 1 + 2 2d (1 − )(6 ¯ + 5) ¯

[dB] ,

(17)

if an order of reflection is missing (independent of n!): L2 = −10 · lg(1 − ) ¯ [dB] .

(18) c0 . me

Decay rate (slope of level decay):

m = −L

Reverberation time:

T = −60/m.

(19) (20)

Room Acoustics

M.4

M

879

Statistical Room Acoustics

The diffuse sound field is a scientific artefact; it is a model for sound fields in large rooms. Z0 = p0 c0 r Q w A V S a, b, c ¯ Si i d

= = = = = = = = = = = = =

free field wave impedance; distance source to field point; source directivity; energy density; total absorption area; room volume; room interior surface area; corner lengths of a cubic room; average absorption coefficient; absorber surface areas; absorption coefficient of Si ; propagation attenuation of power; distance of limit between direct and reverberant field

A sound field is said to be diffuse if on average over some time interval the effective sound intensity (as a vector) in any field point is omnidirectional with constant magnitude for all directions. An immediate consequence is a zero effective power through a (small) reference volume around a field point.Without the permission of a finite time averaging the necessary consequence would be a zero sound field. Energy balance in steady state conditions [Kuttruff (2000)]: V

dw = ¢q − V · n¯ · ¯ · w dt

(1)

with n¯ denoting the average reflection rate, i.e. the expected number of reflections per time unit. It is calculated from the mean free path (Eq. M.3.(11)) by: n¯ =

c0 . ma

(2)

The energy components in parallelepiped rooms can be related to reflections which depend on room shape, the reflection rate and wall absorption. The expectation value of the magnitude of the intensity of a reflection of order i is: I(t) =

¢q (1 − ) ¯ i, 4(c0 t)2

(3)

where ¯ denotes the average absorption coefficient. With the mean reflection rate n¯ it follows: I(t) =

¢q ¯ (1 − ) ¯ nt . 4(c0 t)2

(4)

In a diffuse field the intensity vectors are independent of the direction. Therefore the total sound field is obtained by superposition of incoherent contributions of intensity

880

M

Room Acoustics

moduli. With the number of reflections per time interval dt as given in above (Eq. M.3.(7)), the total time-differential intensity is: c0 · ¢q dI(t) ¯ . = (1 − ) ¯ nt dt V

>

Sect. M.3

(5)

The expectation value of the total time-differential energy density, dw(t), is accordingly (differential energy density impulse response or: energy time curve): ¢q dw(t) ¯ . (6) = (1 − ) ¯ nt dt V The energy density is thus:

∞ w = t0

¢q ¯ (1 − ) ¯ nt dt V

(7)

with t0 = 0, and n¯ according to Eq. M.3.(11) yields: w =

4¢q , c0 S¯

(8)

2 which can also be expressed in terms of the mean sound pressure p = Z0 c0 w

2 p = Z0 c0 w = 4Z0 ¢q = 4Z0 ¢q . S¯ A

(9)

With t0 = 1/n¯ (see Eq. M.4.(2)) this yields: 2 4Z0 ¢q 4Z0 ¢q p = (1 − ) ¯ = (1 − ) ¯ . S¯ A

(10)

More generally, including air attenuation as well as wall absorption: dw(t) =

¢q · dt ¯ −0 c0 t e , (1 − ) ¯ nt V

(11)

¯ and again, with t0 = 1/n: 2 4Z0 ¢q −A/S 4Z0 ¢q −A/S p = e e = . S¯ A

(12)

The relation between the sound pressure p(r) at a field point with distance r to the source of a diffuse sound field and the effective power ¢q of a (small) source is: p(r)2 = Z0 c0 ¢q Q + 4 e−A/S (13) 4r2 A with Q being the directivity of the source. A = S + 4V

is the equivalent absorption area

(14)

Room Acoustics

S =

Si i .

M

881

(15)

i

Expectation of level decay: ¯ · ln (1 − ) ¯ [dB] . L(t) = L0 + 4.34 nt

(16)

Decay rate (slope of the level decay): m = 4.34 n¯ · ln (1 − ) ¯ .

(17)

Reverberation times (for level decay over 60 dB): T=−

60 , 4.34 n¯ ln (1 − ) ¯

(18)

according to Eyring TEy = − TEy =

24 ln(10) V , c0 S ln(1 − ) ¯

V 60V = −0.161 ; −1.086 c0 S · ln(1 − ) ¯ S · ln(1 − ) ¯

(19) (20)

according to Sabine: TSab =

60V V ; = 0.161 1.086 c0 i Si S · ¯ Sab + 4V

(21)

according to Millington–Sette: TMS =

V 60V = −0.161 ; −1.086 c0 Si · ln(1 − i ) Si · ln(1 − i )

according to Pujolle (for rectangular rooms): 2 + 2 + 2y + 2z + 2z + 2x x y 6mg TPu = − ≈− ; c0 log(1 − ) ¯ c0 log(1 − ) ¯

(22)

(23)

according to Pujolle (for rectangular rooms) including absorption of air: TPu =

13.8mg 6mg = . c0 mg − log(1 − ) ¯ c0 0.43mg − log(1 − ) ¯

(24)

The distance, d, of the limit between the direct and the reverberant field is determined by the equilibrium of direct and reverberant sound (Eq. M.4.(13)): QA A/S QV A/S e = 0.1 e (25) d= 16 T or, in approximation of low absorption (A S): QA QV = 0.1 . d= 16 T

(26)

882

M

Room Acoustics

M.5 The Mirror Source Model


The task is the evaluation of the sound pressure at a receiver point P inside a room for a simple source placed at Q. Analytically the sound field must satisfy the source condition (e.g. agreement of volume flow of the field at source position with volume flow of the source) and the boundary conditions (matching of field admittance with wall admittance) at all room walls. Analytical solutions, however, are possible only for very simple room geometries (see > Sect. M.3). The classical tool for sound field evaluations in room acoustics is the mirror source model. It will be displayed here in some detail, going farther than the usual textbook example of parallelepipedic rooms. The mirror source model is often unjustly said to be inapplicable in practical tasks due to the supposedly enormous number of mirror sources which it is said to need. One can find in the literature the number nw (nw − 1)(o−1) of mirror sources “needed” for the reflection order o in a room with nw walls (e.g. 27 000 mirror sources for the low numbers of nw = 6 walls up to order o = 4). It will be shown that the number of sources actually needed is much smaller. The mirror sources here will be described as a sequence of algorithms and programming rules for their evaluation, most of them being very elementary. The word “mirror source” will be abbreviated as MS because of its frequent occurrence. The word“source” and the symbol q may denote both the original source Q and a mirror source (mostly symbolised by S).Sometimes we speak of a“mother source”which creates at a wall (or “reflecting wall” if necessary) a “daughter source”.

M.5.1

Foundation of Mirror Source Approximation

The MS method is exact only for a single, infinite, plane wall with ideal reflection (either hard or soft). Then the superposition of the fields of source Q and MS S satisfy the boundary condition at the wall. (See > Sects. D.15 through D.20).

The analysis of that elementary task returns the result for the reflected field pr : (2) H0 (k0 r ); line source pr (r , ‡s) −−−− −→ R(‡s ) · (2) k0 r 1 point source h0 (k0 r ); with

H(2) 0 (z)

zero-order cylindrical Hankel function of second kind ;

h(2) 0 (z)

zero-order spherical Hankel function of second kind .

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R(‡s ) is the reflection factor of a plane wave incident on the wall under the angle which the connection of the mirror point S with the field point P includes with the normal to the wall. The forms (Eq. M.5.(1)) do not satisfy the wave equation if R(‡s ) = const(‡s ) because an azimuthal factor strictly cannot be associated with a Hankel function of zero order. This should also be kept in mind if the original source has a directivity factor D(˜ ) (in 2D) or D(˜ , œ) (in 3D). This is the MS approximation. One should keep in mind the following: • The MS solution is only approximate if R(‡s ) = const(‡s ), i.e. for G = 0 or |G| = ∞. • Then it violates the wave equation. • It determines precisely the meaning of R(‡s )! • With that definition (and only with that definition) it satisfies the wall boundary condition. • It supposes k0r 1, i.e. large distances dist(S, P), or more precisely: a great sum of the heights of Q and P over the wall. • It supposes that P is not under an angle ‡s with a strong angular variation of R(‡s ). • For grazing incidence, i.e. Q and P on the wall, the influence of higher terms R(n) (‡s ) in Eqs. (D14.10) or (D20.11) is important! • The derivation further supposes that the wall does not guide a surface wave [but this is only rarely the case in a restricted frequency range below a high-quality resonance; but even then Eq. M.5.(1) is an approximation to the field in points not too close to the wall]. The mentioned facts have important consequences: • On the one hand, it makes no sense to try to compute with a higher precision than the precision of the fundamental process of the MS method. • On the other hand, the approximate character of this process does not give a justification to fantastic modifications of the MS method.

M.5.2 General Criteria for Mirror Sources Mirror sources are created at a wall by a source which is on the interior side of the wall by the steps: • Mirror-reflect the source position to behind the wall. • Multiply its source factor by R(‡s ). • If the source has a directivity factor D(˜ ), reflect that directivity, i.e. rotate it. The continued multiplication, for increasing order o of reflection, creates a product of reflection factors R(‡s ), which will be called the “source factor” and symbolised by ¢R. The form of the MS and the co-ordinates used should, if possible, be such that these steps can be performed easily in the computations. The right of a MS to exist is the satisfaction, together with its mother source, of the boundary condition at the wall at which it was created by its mother source – nothing else! The first criterion for the generation of a daughter MS at a wall is that the mother source irradiates the interior surface of that wall. As a consequence, if a source is outside a wall, it does not create a daughter source at that wall. We call this rule the “inside criterion”. The chain of MS production is terminated if this criterion is violated; otherwise the daughter source would be “illegal” (see below for other criteria of interruption of MS

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generation). In particular, a MS will never be mirror-reflected back to the position of its mother source.

M.5.3 Field Angle of a Mirror Source The field angle of a MS gives a further important criterion for the interrupt of MS production. The field angle is explained for the case of reflection of a source Q at a plane wall which is subdivided into two sections with different reflection factors Ri in them. Although the source Q has only one position S for a MS, there are indeed positioned two MSs with different angular ranges ¥1 , ¥2 of their fields because there are two different source factors R1 , R2 in both ranges.

The field is unsteady at the common flank of the field angles. This is a consequence of the character of the MS method as an approximate solution which must be tolerated. In 3D the field angle ¥ is given by a polygonal pyramid subtended by a wall W (or wall section) and with the source S in the apex. The “field angle criterion” states two things: • A MS creates in P a field contribution only if P is in its field angle (more precisely: . . . and on the interior side of the creating wall); otherwise we say “the MS is ineffective”. • A MS generates a daughter MS at a wall W only if that wall is inside the range of its field angle (again, on the interior side of the creating wall); otherwise no boundary condition must be satisfied at W for MS, and therefore the daughter MS would be “illegal”. The additions “on the interior side” in parentheses will be important for convex corners (see below). One can describe the effect of the field angle with the word “visibility”. A source q sees an object only if that object is inside its field angle. If q does not see P, then q is ineffective; if q does not see a wall W, then q does not produce a daughter source with W. The chain of MS generation for increasing order of reflection is continued for an ineffective MS (because a daughter MS may become effective), but the chain is terminated at an illegal MS. Some problems are caused by walls W which are only partially inside ¥ . In a strict procedure one would have to subdivide the wall at the intersection with the flank of ¥ , but such a “dynamical” definition of walls would produce much computational work. It is sufficient, within the framework of precision of room acoustical computations, to check

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whether the wall section inside ¥ exceeds some size limit (e.g. Š0 ); if not, that wall is neglected for that MS. It is a good compromise between precision and computation load to check whether the centre C of W is inside ¥ .This check is done by a repeated test whether C is inside the walls of the polygonal pyramid with the MS at its apex and subtended by W. The repetition can be interrupted if C is outside one of the pyramid walls. In general, MSs with increasing order are displaced farther and farther away from the interior of the considered room. Thus their field angles become smaller and smaller; so fewer walls have to be considered for the production of further MSs with increasing order of reflection. The mentioned additional condition that either P or another wall must be in the field angle on the interior side of the generating wall is important, as can bee seen from the next figure.

Here the source Q creates at W1 a MS, S1 , which in the depicted case is outside both walls W1 , W2 . The inside criterion would interrupt a further production of MSs anyway.If, however, Q is displaced farther away from the wall W1 , then S1 may fall on the interior side of W2 . Nevertheless S1 will not produce a MS at W2 because W2 would be in ¥ but not on the interior side of the generating wall W1 The additional condition (W2 inside W1 ) is relevant only for convex corners.

M.5.4 Multiple Covering of MS Positions In the first sketch of > Sect. M.5.3 two MSs occupy the same position; both are legal; they are different from each other. If two walls form a space wedge, and if the wedge angle Ÿ of two walls is a rational multiple of , the MS beginning with some higher order will fall upon positions of MSs of lower orders, so they again occupy same positions.

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The outer sides of the (possibly extended) walls are hatched. The chain of the MS production ends with S2 , S4 because both MSs are outside both walls.The source factors of both MS S2 and S4 are equal. Such coincident MSs with the same source factors are illegal because they violate the source condition of the boundary value problem

In this example the MS production ends with S5 , S6 because both MSs are outside both walls. Depending on the wedge angle Ÿ and on the position of Q, different numbers of MSs can be constructed (higher for small Ÿ, infinitely high for parallel walls, i.e. for Ÿ = 0).

M.5.5

Convex Corners

Convex corners, i.e. with wedge angles Ÿ > , cannot be treated with the “traditional” MS method. Only one MS can be constructed at this convex rectangular corner. That is evidently not enough to represent the sound field at such a corner. Consequently, the traditional MS method fails at convex corners!

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M.5.6 Interrupt Criteria in the MS Method The MS method is often blamed for the apparent exorbitantly high numbers of MSs involved.In such statements inherent interruption criteria of the MS method are ignored. Conditions for the interrupt of the chain of MS production are as follows: 1) The source q is outside the mirror wall W: there exists no boundary condition for q at W; the MS chain must be interrupted. 2) A wall W is outside the field angle ¥ of q: q does not produce a daughter MS at W. For walls with a common corner this is equivalent to 1). 3) A MS would fall within the interior room space: then there would be, except for Q, a new pole position of the field; this violates the condition of regularity of the field outside Q. This case is encountered in the case of convex corners only. 4) If the new MS falls on Q: from then on the MS positions would be repeated; this violates the source condition which demands that the volume flow through a small enclosure around Q must be the same as that of the original source. 5) The product ¢|R|, which is the source factor of q, would become < limit, which is a preset limit. This would make the field contribution negligible, also for all daughter sources of q. 6) If the sound field is the target quantity, the distance dist(q, P) of a source q from P may be restricted to being < dmax. For point sources the amplitude ratio of q at P, relative to the amplitude of Q at P, is dist(Q, P)/dist(q, P). This ratio generally becomes even smaller for daughter sources of q. 7) A limitation ¢|R| · dist(Q, P)/dist(q, P) < limit · dmax would be more significant. 8) With some arbitrariness one sets an upper limit of the orders o = 1, 2, . . ., omax of the MS. In interrupt checks using ¢|R| it may be sufficient to use approximate values for the reflection factors R (or their magnitudes |R|),for example the reflection factor for normal sound incidence, or the reflection coefficient |R|2 from the absorption coefficient for diffuse sound incidence. Then the construction of the MS becomes independent of the position of P.The reflection factors are the only quantities which introduce the frequency into the construction of the MS. If one takes for the interruption check a lower limit or an average value of |R| over the frequency interval considered, the construction of the MS is also independent of the frequency. One should apply tests with the true |¢R| while evaluating the field contributions of the sources, when the ¢R are available. In the phase of field evaluation these tests must be applied on a smaller number of MSs than in the phase of MS construction.

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Computational Parts of the MS Method

The traditional MS method consists of three computational parts: • Find the positions of the MS (considering the inside and field angle criteria). • Determine the source factors of the MS,i.e.of the reflection factors R(‡s) (depending on the acoustical quality of the mirror wall and of the relative positions of the MS and P). • Evaluate the contributions of the MS to the field at P. Most programming is needed to find the MS positions, although the single steps are elementary geometrical tasks. Less intensive in programming is the evaluation of the reflection factors of absorbent walls. This task may be delegated to subroutines for the wall surface admittance G. Most simple is the evaluation of the field contributions; only a number of Hankel functions of zero order must be evaluated; this subtask is fast computing for spherical Hankel functions (which are given by cos(x), sin(x)) and is fast computing also for cylindrical Hankel functions when using the known polynomial approximations for Bessel and Neumann functions of zero order. The computational MS method proceeds with the order o of mirror reflections. • At the order o = 1 the MSs S(1) are determined in turn for all walls (consider the inside criterion for convex corners!). • At the order o = 2 all S(1) are potential mother sources for the generation of the MSs of the second order S(2) at all walls (except for the wall at which S(1) was produced) unless the inside and field angle criteria exclude S(2). • Continue until a final interrupt criterion is met.

M.5.8 Inside Checks Checks for interruption and efficiency form the main part of the computational work in the computational MS method. They are fundamental tasks of computational geometry. But because they are repeated very often, they should compute fast. We break down all geometrical tests into “inside checks”. An inside check examines whether a point q is on the interior side of a wall plane, in the wall plane, or on the exterior side of the wall plane (the sides are defined by the rotational sense of the edges Ek of a wall W = {E1 , E2 , E3 , . . .}). One could do the inside check with the help of direction cosines of the connecting lines between q and the Ek . The evaluation of angles, however, is slow. An inside check in 3D uses the vector triple product (scalar product of a vector and a vector product) if the wall W is given by three of its edges. If the parameters a, b, c, d of the reduced normal form of the wall equation a·x+b·y +c·z+d = 0 are known, the inside check pendix) with q = {x, y, z}: ⎧ ⎪ ⎨> 0; a · x + b · y + c · z + d = 0; ⎪ ⎩ < 0;

(2) needs three multiplications and three additions (see apq inside the W plane q on the W plane , q outside the W plane

if the edges of W and q form a right-handed system; otherwise the signs change.

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Another, often used, test examines if a point P is inside the polygonal pyramid which has the point q as apex and is subtended by a wall W. This test is done by a repetition of inside checks for P and the triangles forming the sides of the pyramid. The loop over the triangles can be aborted with a negative answer for the test if one of the inside checks fails (distinguish whether W and q are a right-handed or left-handed system). The shading of a point P or a wall W by a convex corner and the visibility of P or W from a point q through an aperture (formed by convex corners with a free interspace between them) are also tested with inside checks.

M.5.9 What Is Needed in the Traditional MS Method? One needs as input the following: • the list of walls {W1 , W2 , W3 , . . .}, which themselves are lists of edges Ww = {Ew1 , Ew2 , Ew3 , . . .}, Ei = {xi , yi , zi }; • the source point Q = {xQ , yQ , zQ }; • the field point P = {x, y, z}; • the list of wall admittances G = {G1, G2 , G3, . . .}; • the limits omax, limit, dmax for the order o, the source factors |¢R|, and dist(q, P), respectively. It is supposed that the dist(Q, P), the wall centres Cw , and the parameters a, b, c, d of the reduced normal forms of the wall equations a · x + b · y + c · z + d = 0 are evaluated (see appendix).One needs,for the evaluation of the reflection factors and of the contributions in P the following: 1) position q of a source (either Q or a MS); 2) counting index w of the wall W at which q was generated; 3) distance dist(q, P); 4) amplitude factor ¢R(‡s ); 5) a flag which signals with flag = 0 that q is an effective source (i.e. with a field contribution in P) and with flag = 1 that q is ineffective (no field contribution). These data are collected in “source lists” {q, w, dist(q, P), ¢R(‡s ), flag}, and the source lists for a given order o = 0, 1, 2, . . ., omax are collected in tables tab(o) = {. . ., {q, w, dist(q, P), ¢R(‡s ), flag}, . . .}. Let the counting index of a source list within tab(o) be s. The source table for the order o = 0, i.e. for the original source q = Q, has the form tab(0) = {{Q, 0, dist(Q, P), 1, 0}} for rooms with concave corners (see below for rooms with convex corners). One can delegate the task of mirror reflection of a mother source qm , represented by its source list {qm , wm , dist(qm , P), ¢mR(‡s ), flagm }, at a wall Ww, given by its index w, including all tests of interrupt and effectivity, to a subroutine, which should be carefully checked and economised with respect to computing time. That subroutine returns • the source list {q, w, dist(q, P), ¢R(‡s ), flag} of the daughter source if no interrupt criterion is met; • the value 0 if an interrupt criterion is met. Such a subroutine for 3D rooms with concave corners is a program of about 25 program lines in the Mathematica language (the geometrical subtasks inside the subroutine are delegated to subroutines).

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The traditional MS method works in three nested loops: 1) The outer loop over the order o = 1, 2, . . ., omax: it produces the source table tab(o). 2) The middle loop over the counting index s = 1, 2, . . . of the sources in tab(o − 1). 3) The innermost loop over the counting index w = 1, 2, . . . of walls; it calls the above-mentioned subroutine; if that subroutine does not return 0, the new source list is appended to tab(o). This traditional MS method is attractive for its computational simplicity. A frame program for the evaluation of the tab(o) in Mathematica typically is a program of about 12 lines, if the frame program calls a subroutine for the MS evaluation with all checks inside the subroutine. It may be of some advantage to perform the checks of legitimacy of a new MS (mother source inside the mirror wall; mirror wall in the field cone of the mother source) in the frame program (this is true especially when the MS method is applied to rooms with convex corners). The returned tables tab(o) also contain ineffective sources (flag = 1). One can select the effective sources with flag = 0 and collect them in tables tabeff(o). So one has available all data which are needed to evaluate and sum up the field contributions in P of the effective sources.

M.5.10 The Object The geometrical object is a room formed by plane walls Ww . What is inside and outside of the room is clearly defined. The (original) source Q and the field point P are always inside. A right-handed Cartesian system of co-ordinates x, y, z is laid over the room. The walls Ww are plane. They are described by lists of edges, Ww = {Ew1 , Ew2 , Ew3 , . . .}, which are ordered such that the sense of rotation in that order and the direction pointing to the inside of the room make a right-handed system. Because the first three edges are used for the determination of the unit normal vector of the wall, these edges should not be collinear and should agree with the general sense of rotation of wall edges. Edges may be cyclically interchanged in a wall list. The counting order w = 1, 2, 3, . . . of the walls is arbitrary. Wall couples form a room wedge; they either have a (straight) real corner if the walls succeed each other, or they have a virtual corner if other walls are placed between the couple walls. The walls of a couple include a wedge angle Ÿ (measured inside the room). Real corners are “concave” when 0 < Ÿ ≤ and “convex” for < Ÿ ≤ 2. The acoustic qualification of a wall will use its surface admittance Gw . The MS method applies to the acoustic qualification of a wall its reflection factor Rw (‡), which is the reflection factor for a plane wave with incidence under the polar angle ‡ formed by the normal to the wall and the connection line of the MS with P. Rw (‡) depends on that angle, and thereby on the position of P, as: Rw (‡) =

cos ‡ − Z0 Gw . cos ‡ + Z0 Gw

If the wall is bulk reacting, one further has Gw = Gw (‡).

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It is strongly recommended not to use too small parts of walls.Not only does the subdivision of the room envelope into too small wall sections produce analytical and numerical nonsense, but the computational work is increased immensely by the subsequent generation of MSs at such small faces. If one likes to consider the acoustic effect of e.g. pillars and/or handrails, it would be easier and faster to solve the task of scattering at suitable scatterers (e.g. cylinders or spheres). As a general rule for the dimensions of walls to be considered one can neglect walls with dimensions smaller than about Š0 (if the room itself is much larger than Š0 ).

M.5.11 A Concave Model Room, as an Example We consider a 3D model room which could function as a simple concert hall (see figures below). It has w = 1, 2, . . ., 19 walls, two of them coplanar, and two couples have parallel walls on opposite sides of the room. The floors of the stage and of the seat area are inclined. Balconies cannot be modelled with concave rooms. The preceeding 3D plots are computed from the input data (such plots are parts of the checks of input data). The first figure shows the room as a 3D wire plot, together with the source Q and the field point P; the second figure shows an outside view of the room. The co-ordinate units are arbitrary.

The enumeration of the walls (w = 1, 2, . . ., 19), the plan and side elevations of the “concert hall”, the positions of source Q and receiver point P, are shown below in scaled co-ordinates:

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The normalised wall admittances Z0 Gw for the example shown would produce a list of absorption coefficients dif for diffuse sound incidence as given below for the list of the walls in the above enumeration. They are not exceptional in any sense. dif ≈ {0.10, 0.10, 0.40, 0.71, 0.20, 0.60, 0.20, 0.40, 0.20, 0.40, 0.40, 0.20, 0.20, 0.20, 0.20, 0.20, 0.20, 0.20, 0.50}.

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The following diagrams show MSs (as points) for two orders o = 1, 3 if only backreflection (into the position of the mother source) is avoided. Such diagrams would correspond to the mentioned numbers of MSs being claimed in the literature as “needed”.

MSs of order o = 1, with only back-reflection criterion. Number of MSs: 19

MSs of order o = 3, with only back-reflection criterion. Number of MSs: 6156

The next diagrams show the effective MSs with all interrupt criteria applied. In the test of exclusion of a wall as a mirror wall, it was checked whether the wall centre was inside the field angle cone of the mother source. It should be noticed that already for the order o = 1 the number of MSs is reduced from 19 to 10 (mainly by the efficiency check).

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Effective mirror sources of order o = 1, with all interrupt criteria. Number of MSs: 10

Effective mirror sources of order o = 3, with all interrupt criteria. Number of MSs: 42

Effective MSs of order o = 6, with all interrupt criteria. Number of MSs: 141

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The published estimates nw · (nw − 1)(o−1) for the MSs of order o needed in a room with nw walls would give for the order o = 6 (with nw = 19 for our room) a number 35 901 792(!!). The computing time for the effective MSs up to o = 6 with all criteria of interrupt and effective source selection applied was 23 s (on an 800-MHz laptop computer with non-compiled Mathematica programs). The numbers of MSs in orders o for different applied criteria of interrupt are collected in the following Table 1. Table 1 Numbers of MSs in several orders o for different interrupt criteria applied o

Back-reflection

& q and wall inside

& effective

1

19

19

10

2

342

97

25

3

6256

261

42

4

110 808

478

69

5

1 994 544

755

99

6

35 901 792

1059

141

The next Table 2 collects, for each order o, minimum and maximum reductions in the level of the sound pressure contribution in P due to |¢R|, to dist(Q, P)/dist(q, P), and to their product. Table 2 Minimum and maximum reductions in dB of level of field contributions in the order o by the source factor ¢R, the distance ratio dist(Q; P)=dist(q; P), and their product j¢Rj

o min

max

dist(Q,P)/dist(q,P) min

max

j¢Rj dist(Q; P)=dist(q; P) min

max

1

7.34

0.566

4.69

0.122

7.46

0.948

2

21.05

1.32

10.23

0.331

25.57

4.36

3

22.61

1.69

14.2

4.36

32.29

6.93

4

22.96

3.06

15.91

4.86

36.02

8.89

5

36.21

3.58

18.86

7.34

49.29

12.86

6

37.72

4.19

19.01

8.98

52.91

14.53

The limits in the examples were set to |¢R| < 0.01 ∼ −40 dB; dist(q, P) > 10 · dist(Q, P) (corresponding to −20 dB of the distance ratio); so neither of the two limitations restricted the number of effective MSs. The table also shows that a limitation of the orders to o ≤ omax = 6 is reasonable because the highest contribution of a MS for o = 6 is −14.53 dB below the contribution of the original source Q. As an example of application, we plot the profiles of the sound pressure level in places X = (x, y, zP ) around P = (xP , yP, zP ) as 3D plots of 20 · lg|p(X)/p(P)| over k0 x, k0 y

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for a fixed frequency. It is supposed that the distances dist(X, P) are small enough to neglect the influence of the variation of X on R(‡s ). (Such patterns cannot be computed with modified MS methods using |ps (P)| or rays or sound particles as field descriptors because they lack phase.)

Profile of sound pressure level around P, for a higher frequency

M.5.12 The MS Method in Rooms with Convex Corners As shown above, the traditional MS method fails for the description of the sound field around convex corners (we shall solve this problem below). But one can apply the traditional form of the MS method in such rooms if one supposes that the scattered sound field in the shadow zone behind a convex corner can be neglected in comparison with the contributions of MSs which radiate into the shadow zone without scattering at the convex corner. The mentioned supposition introduces the concept of “shading” into the MS method (see below). It is an important statement that a convex corner can be treated in the computations like a concave corner if the source q “sees” both flanks of the corner (from inside) because then no shadow is created. This condition is easily checked by “q inside both flanks”. The next figure shows a possible situation at a strongly convex corner in which the daughter source S1 of Q lies inside the room; this makes S1 illegal. Shading plays a role with a possible contribution of a source at P (P can be shaded or not), as well as with a possible continuation of MS production, if a wall is shaded. Q in the example below is ineffective if the field point is at P ; P is in the shade angle §Q which Q subtends with the wall W1 . If we consider the MS q, it does not produce a daughter source at W2 , although it is inside that wall. Now ¥q is the field angle of q as defined and used in previous sections.

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The requirement that q must not produce a daughter source at W2 is covered when we expand the condition for walls W at which a source q can produce a daughter source (W is inside ¥q ) by the additional requirement that W be inside the reflecting wall (here W1 ) as well With that expanded rule, q can legally produce a daughter source at W3 . The convex corner in the figure is treated like a concave corner if the source is in the range indicated with“no shadow”.This“no shadow”range gets larger for“mildly convex” corners. Thus the sound field evaluated in a room having only mildly convex corners will not be much different from the field in a similar room with only concave corners, as would be expected. The situation is more complicated if two (or more) convex corners form an “aperture” which subdivides the room.

The picture shows, as an example, a 3D view from inside the room on the stage and into an orchestra pit (this model will be used later for application of the MS method).

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The head of the stage and the balustrade of the orchestra pit form an aperture A. One must not only distinguish if the original source Q and the receiver P are on different sides of the aperture (for efficiency), but also a wall (for further MS production) must be seen by a source through the aperture, i.e. either P or a wall must be inside the aperture angle ¥A which is the cone subtended by the aperture A and with the source q in its apex.

Stage and orchestra pit from above (reversed side of view), with wall numbers and aperture angle ¥A subtended by the aperture A, and the source Q The “inside check” for P inside the cone(q, A) is simple (see above). In concave rooms one can similarly test if a mirror wall is inside the field angle ¥ of a source by checking whether the wall centre is inside ¥ . This kind of check for the visibility of a wall through an aperture would be too inaccurate,however; important sound paths from one subspace (e.g. the orchestra pit) to the other subspace (e.g. the stage or the auditorium) could be missed with that kind of test. The source q and the aperture A produce a “bright patch” F in the plane of an opposite wall W (F is the polygon formed in the plane of W by the intersection points Xi of the side corners of the cone(q, A) with the plane of W; they can be evaluated). Three cases of visibility should be distinguished; they are shown in the figure below.

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One could describe the condition of visibility by the requirement that at least one edge of F is within W or at least one edge of W is within F. The implementation of this test would need the evaluation of the intersection points Xi and the test of whether a“point is within a polygon” (which should not be confused with “a point is inside the polygon plane”). One can avoid these (computer-intensive) subtasks by using the cone(q, W). Then the visibility check reduces to the tests “at least one edge of A inside the cone(q, W)” or “at least one edge of W inside the cone(q, A)”. The MS method in rooms with convex corners (in the supposed approximation which neglects corner scattering) has the same aim as in concave rooms, namely to find source lists {q, w, dist(q, P), ¢R, flag} for legal sources q. As previously the frame program operates in three nested loops over the order o, the counting index s of the sources in the order o − 1, the counting index w of the walls. Because of the many decisions which must be made by the frame program anyway, it is advisable to write a subroutine for the MS evaluation, which is applicable for flanks of concave and convex corners, i.e. which internally only makes interrupt checks related to |¢R| and to dist(q, P) and efficiency tests for the new MS, whereas the frame program performs all interrupt tests. One can summarise the modifications of the MS method in rooms with convex corners as follows: 1) Find the lists of wall couples forming convex corners and of exclusive couples. 2) Determine the aperture A (if any). 3) An efficiency check must be performed already for the order o = 0, i.e. for q = Q. (Q is ineffective if P is on the other side of A, but not in ¥A of Q; or, for a single convex corner when A is not defined, if P is in the shade angle §Q (a pyramid with Q at the apex and subtended by a wall W which is a flank of the convex corner)). Therefore determine tabo(0) in the frame program. 4) The source lists of the order o = 1 (i.e. with Q as mother source) must also be determined separately in the frame program (because Q has no reflecting wall). 5) For orders o > 1 the frame program in its innermost loop over the wall indices w has to check the interrupt conditions: • w = wm , the index of the reflecting wall; • the mother source qm outside the wall with W(w); • for walls w, wm on the same side of A if the centre of the wall w is outside the cone(qm , W(wm )); • for walls w, wm on different sides of A, if W(w) is not visible for qm through A. 6) If none of the tests in 5) is positive, the frame program calls a subroutine for the evaluation of the source list of a new MS. The subroutine causes an interruption (skip of w) if |¢R| < limit or dist(q, P) > dmax. It also performs the efficiency tests. These efficiency checks are explained above.

M.5.13

A Model Room with Convex Corners

The model room of this section widely corresponds to the concave model room in the previous paragraph, but an orchestra pit is added below the stage (see the 3D view of the stage and the orchestra pit in the paragraph above; it shows a part of the present model room). A 3D wire plot of the walls and a 3D view from outside is shown in the next picture. The next page contains the plan and elevation views. The original source Q is placed in the orchestra pit; the receiving point P has the same position as in the previous

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paragraph. The number of walls of the room is nw = 29. There are three important convex corners: the upper corner of the balustrade at the orchestra pit and the upper and lower corners of the head of the stage floor.

Plan and elevation views of the “concert hall” with an orchestra pit, showing the enumeration of the walls, w = 1, 2, . . ., 29, and the positions of the source Q and of the receiver point P (scaling of the the co-ordinates different from the other graphs)

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3D wire plot of a room, showing the positions of the source Q and of the field point P. Table 3 gives the numbers of the legal and of the effective mirror sources in the orders o = 1, 2, . . ., omax = 6. Table 3 Number of legal and effective MSs of orders o = 1; 2; : : :; 6 Type

o=1

o=2

o=3

o=4

o=5

o=6

Legal

24

109

286

637

1306

2467

0

19

44

96

186

272

Effective

All MSs of the order o = 1 are ineffective, like the original source Q. The total number of effective sources is 617 up to o = 6; the conventional prediction would require (with 6 nw = 29) as the number of MSs: 1 + nw (nw − 1)(o−1) = 517 585 882. o=1

The following diagrams show legal and effective MSs in 3D plots for some orders o (legal first, then effective, except for o = 1, where no effective MS exists).

Legal MSs with order o = 1

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Legal and effective mirror sources MS with order o = 2 (pictures above) and o = 5 (pictures below).

Table 4 presents the lowest values for the source factor |¢R| (in dB) and the lowest and highest values of |¢R| · dist(Q, P)/dist(q/P) (in dB) within the listed orders. The table shows that, when using the product |¢R| · dist(Q, P)/dist(q/P) as interrupt criterion, some MSs in the orders o = 5, 6 could be dropped.

Table 4 Level changes of contributions in the order o Level Change by

o=3

o=4

o=5

o=6

−4.38

−19.02

−23.81

−31.05

−34.59

j¢Rj dist(Q; P)=dist(q=P), min

−12.37

−25.17

−30.88

−37.89

−47.13

j¢Rj dist(Q; P)=dist(q=P), max

−3.37

−4.22

−3.38

−4.25

−6.98

j¢Rj, min

o=2

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Again, a final plot shows the sound pressure level profile around P (see previous paragraph for the assumptions made) and refers the level to the sound pressure which the free source Q would produce at P. The figure is for a higher frequency.

The contributions of the higher-order MSs lift the sound pressure level in the room significantly over the level in free space

M.5.14 Other Grouping of Mirror Sources This section is based on the easily proven fact that the original source Q and all MSs which are created by it and its daughter sources at a couple of walls F1, F2 are arranged on a circle (“MS circle”) which contains Q and has its centre in the foot point Z of Q (normal projection) on the intersection line of the walls. So we are dealing with flanks of a corner; the corner is “real” if the flanks are subsequent walls of the room or “virtual” if other walls are arranged between the flanks. The corners (either real or virtual) may be concave or convex, but for the moment we consider only concave corners. A special case form antiparallel flanks: their (necessarily virtual) corner line is at infinity; the MS circle then becomes a straight line through Q normal to the flanks. Because the plane containing the MS circle is normal to the flank corner, we are dealing with a two-dimensional problem (which is a further advantage of grouping the mirror sources in groups of MSs at wall couples). This suggests that the problem should be discussed within the context of a cylindrical co-ordinate system r, ˜ centred at Z and with the reference for ˜ preferably (but not necessarily) in one of the flanks. The radius of the MS circle will be symbolised by rq and the angle for Q by ˜Q , whereas the angle for a MS will be called ˜q . The co-ordinates of the field point P are r, ˜ , …, where … is the co-ordinate along the corner line, with … = 0 for Z, Q, q. The transformation between the Cartesian co-ordinates x, y, z of the room and the cylindrical co-ordinates r, ˜ , … of a flank couple is described in the appendix.

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The next figure shows a couple of flanks F1 , F2 , the original source Q and its mirror sources on the MS circle.

The figure shows all legal MSs (the number is rather high because the wedge angle Ÿ is small, by intention). The source at 6Ÿ − ˜Q formally could be mirror-reflected once again, but then the daughter source would coincide with the MS at 6Ÿ + ˜Q with the same source factor, so by the coincidence criterion this daughter source would be illegal. The indicated source factors as products of R1 , R2 should not be interpreted too literally; the factors in the powers of these reflection factors may contain different angles ‡s . The MSs on both circular arcs ˜ < 0 and ˜ > 0 have source angles ˜q within the limits: !

!

Ÿ − < ˜q = −2s · Ÿ ± ˜Q < −;

s = 0, ±1, ±2, . . . .

(5)

The mirror sources lastly were generated at F1 for s < 0; they would produce a daughter source on the lower circular arc with F2 if that is not excluded by the inside criterion, and vice versa for s > 0. The range (Eq. M.5.(5))∗) leads to limits for the counter s: 1 ± ˜Q on the upper arc ˜ < 0: 0 ≥ s > −1 ; 2 Ÿ ∓ ˜Q . on the lower arc ˜ > 0: 0<s< 2Ÿ 1 The sum of the counters is restricted to s + |s | < − . Ÿ 2 ∗)


(6a) (6b) (7)

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Thus the number of MSs decreases with increasing Ÿ. If Ÿ = , then only the MS with n = 0 is legal. Evidently, anti-parallel wall couples with Ÿ = 0 are a special case, which will be discussed separately below. One needs, for the implementation of the MS method, the distance between q and P and the angle ‡s formed by the connection line (q, P) with the normal of the flank. The distance between q and P is: dist(q, P) = …2 + r2 + rq2 − 2r rq cos(˜ − ˜q ) . (8) The cosine cos(‡s ) is easily evaluated in the Cartesian co-ordinates, but this would need a co-ordinate transformation for all q. It is better to use the cylindrical co-ordinates of P and to evaluate cos(‡s ) in that system. This task will be described in the appendix. The advantages of the described grouping are evident: one need not find the MSs by trial and error; they are evaluated in a straightforward method if the flanks have a real corner (where exclusion by interrupt and efficiency checks play no role). Further, one is sure to have covered all MSs for a wall couple. The question is whether one possibly introduces too many MSs if the flanks have a virtual corner. See the next figure for that question.

This graph illustrates the case that both flanks F1 , F2 are on different sides of the MS circle. The last MSs generated at the outer flank F1 never “see” the other flank F2 , and the last MSs generated at F2 may or may not have F1 in their field angles ¥ . Conclusion: for flanks with a virtual corner the tests “Fi in cone(q, Fj )” must be made as interrupt checks, except in situations which can be described and programmed easily in which the number of these tests can be reduced. Similarly, not only are the efficiency checks, which ask whether the projection P of P into the plane of the MS circle is in the angle ¥ , simplified as compared to the traditional MS construction because this check now is a 2D check, but also conditions which make such checks unnecessary can be formulated. The tests “Fi in cone(q, Fj )” remain three-dimensional if the flanking walls have a virtual corner and no intersection with the MS circle. The set of mirror sources constructed in this way (we call it “corner set”) is “complete”; the sum of their contributions to p(P) is a precise field description (with the principal limitation of precision of the MS method) if only the flanking walls of the couple exist.We

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call their sound field the “corner field”. If P lies in between the flanks, it is plausible that this field represents the most important contributions of wall reflections at P. Further contributions come from other wall couples and their corner sets if P is in the field angle ¦ of those couples (see below) and by reflections of the corner set of a wall couple at other walls within ¦ . The combination of corner sets to the ensemble of mirror sources of a room (“room set”), or the completion of corner fields to the room field, will be discussed in the next paragraph.

M.5.15

Combination of Corner Fields to Obtain the Room Field

The room may have nw walls W. Find solutions for corners with couples Wi , Wj ; i, j = 1, 2, . . ., nw ; i = j; that is nw · (nw − 1)/2 combinations (the combination Wi , Wj is equivalent to the combination Wj , Wi ). For the preparation of the next idea we take the simplest examples of two-dimensional triangular and rectangular rooms. If the room has a 3D tetrahedral shape, for example, the main difference as compared to a two-dimensional room would be an inclination of the MS circles relative to each other (it is simpler to compute the situation in a 3D room than to present it in a graph). Below, MSs from the traditional MS method are drawn with interrupts according to the inside criterion. This will be sufficient for the ensuing argumentation. The MSs created at a wall couple are collected on circles.In the next figure for a triangular room, three of the MSs appear twice (they are marked with a grey fill); they are the MSs of first order. In the subsequent figure for a rectangular room the vertical and horizontal lines through Q represent the limit cases of MS circles for the two antiparallel wall couples. In that figure the four MSs of the first order appear three times. Long arrows in the figures indicate where the MSs come from. Short arrows indicate at which wall the MSs shown should be mirror-reflected in the next step of MS production. These arrows point to opposite walls.

The original source Q appears three times in the three corner sets; the MSs of first order appear twice.

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The original source Q appears four times in the four corner sets; the MSs of the first order appear three times. The facts that the MSs of a wall couple are placed on a circle around the wall corner with Q on the circle, and that the continuation of MS production would imply opposite walls, suggest that one should collect the MSs of a wall couple into one equivalent source for the corner field. The following rules are evident from the above discussion: • exclude the original source from the field of that source; • exclude the MSs of the first order from that source; • define the field angle ¦ of the source. We shall see in the next section that the new source is placed in the foot point Z of Q on the corner line. The field angle ¦ therefore is the angle defined by the two flanks and the corner line as apex line.

M.5.16 Collection of the MSs of a Wall Couple in a Corner Source Up to now the graphs in the previous paragraph indicate nothing more than an involved, but legitimate, procedure of traditional MS production; only the collection of the MSs is special. Now we take advantage of the fact that the MSs of a wall couple are arranged on a circle around the intersection line (normal to that line) which also contains Q (and thereby defines the radius rq of the circle). The intersection line between the walls must not really exist (see the rectangular room above). But first we exclude the special case of parallel walls (it will be treated below as a special case). Let the radius of the MS circle be designated as rq . The field of a MS at P is described by:

¢R pq (P) = ¢R · h(2) · sin k0 dqP + j cos k0dqP k0dqP = 0 k0 dqP e−jk0 dqP = j ¢R k0dqP

(9)

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with dqP = dist(q, P). The addition theorem for spherical Bessel functions, when applied to the above expression, leads to: ⎧ ⎨jn (k0 rp) · h(2) n (k0 rq ); rp < rq r h(2) (k d ) = (2n + 1) · P cos(˜ − ˜ ) · , (10) 0 qP n q 0 ⎩ rp n≥0 jn (k0 rq ) · h(2) (k r ); r > r 0 p p q n where jn (x) are spherical Bessel functions of order n, h(2) n (x) are spherical Hankel functions of the second kind,Pn (x) are Legendre polynomials,and the geometrical quantities are taken from the next figure.

The circle is the MS circle; q is a source on it with the cylindrical co-ordinates rq , ˜q , … = 0; P is the field point with the cylindrical co-ordinates r, ˜ , …; P is the projection of P on the plane of the MS circle. The following relations exist among the geometrical quantities: dq2 P = rq2 + rp2 − 2rq rp cos ” ,

(11)

rp2 = r2 + …2 , dq2 P = d2 + …2 ,

(12)

d2 = rq2 + r2 − 2rq r cos(˜q − ˜ ) , from which it follows that: r cos ” = cos(˜q − ˜ ) . rp

(13)

The first line in Eq. M.5.(11) was used for the addition theorem. Summation over the sources on the MS circle gives for the field contribution at P of that sources: p(P) = j ¢s R(‡s ) (2n + 1) s

n≥0

· Pn

⎧ ⎨jn (k0 rp )·h(2) n (k0 rq ); r cos(˜qs − ˜ ) · ⎩ rp jn (k0 rq )·h(2) n (k0 rp );

r p < rq r p > rq

(14) .

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The summation index s can be taken from Eq. M.5.(5) if all sources are added. As explained above, it is recommended that one avoid the summation over Q and the MSs of the first order (their field contributions will be added to the field in their traditional forms); and illegal or inefficient sources will also be left out of the summation over s. Eq. M.5.(14) represents an important group of mirror sources in an explicit formula. The terms represent radially standing waves in rp if rp < rq and outward propagating waves if rp > rq . The sound field is continuous at rp = rq . The sum satisfies the boundary conditions at the flanks, Sommerfeld’s far field condition, the source condition and the edge condition (which requires that the volume flow through a small cylinder around a corner or a sphere around an edge does not exceed the volume flow through a similar cylinder or sphere, respectively, around the original source Q; the edge condition mostly is used for the selection of permitted radial functions, like Sommerfeld’s far field condition). But Eq. M.5.(14) in general does not satisfy the wave equation because the factors ¢s (‡s ) in general are neither constant nor do they have the form required by the wave equation for angular factors to Bessel functions of order n. However, satisfaction of the wave equation could not be expected with the MS method as a basis for Eq. M.5.(14). This representation, however, has a numerical problem: the convergence and the precision are critical for rp = rq , i.e., if the field point lies on the sphere which has the MS circle as equator circle. Physically the numerical problem comes from the fact that the sphere surface contains the poles of the sources q. The evaluation of Eq. M.5.(14) at rp = rq needs a careful check of the summation limit for n (a detailed discussion of the convergence check can be found in Mechel (2002)). This problem in general does not appear after a mirror-reflection of Eq. M.5.(14) at an opposite wall (see below). One could avoid it by using the traditional MS method at or near rp = rq , but this would require the programming of both methods. An easier method in the case rp = rq is the evaluation of Eq. M.5.(14) on both sides of that limit, at some distance, and then to take the mean value. The fact that a set of spherical Bessel and Hankel functions with integer orders must be evaluated causes no problem; the set can be obtained from two start values of the order by the known recursions of such functions, and also the Legendre polynomials are easily computed. The radial arguments are constant for all sources on the MS circle and for a fixed immission point P. Important conclusions can be drawn from Eq. M.5.(14). The components of the sum over n are spherical wave terms, which are centred at the centre Z of the MS circle on the corner line. Therefore we say that Eq. M.5.(14) represents the field of a “corner source”. It can be introduced into a continued MS generation like any directional source. The advantages of the corner source are evident. Its position need not be found in a complicated search algorithm; it is explicitly defined by the room geometry and the position of Q. Also, its field angle ¦ is immediately given; it is the angle between the flanking walls. The difference with the field angle ¥ of a traditional MS is remarkable (¥ is the angle of the cone subtended by the reflecting wall and the MS in the apex). Efficiency checks (P in ¦ ?) and interrupt checks (an opposite wall in ¦ ?) are much easier to perform. It is not proven, but plausible, that one can stop the field evaluation after the corner sources of all wall couples have been evaluated and mirror-reflected once at their op-

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posite (visible) walls. All really important field contributions will be obtained with that procedure. The mirror reflection of the corner source is done by a simple modification of Eq. M.5.(14) if one applies the reciprocity of the next paragraph. One must not evaluate the positions of the mirror-reflected sources, but one mirror-reflects P, multiplies the ¢R(‡s ) in the sum over s by the new reflection factor, and evaluates Eq. M.5.(14) with the geometrical parameters of the new position of P (which can also be determined directly from the room input data and the position of P). This procedure avoids the reflection of the directivity of the corner source.

M.5.17 A Kind of Reciprocity in the MS Method Evidently, a source q generated by a mother source qm at a mirror wall W will produce at a receiver point P the same contribution p(P) that the mother source qm would produce at the point P , which is the mirror-reflected point to P with respect to W, after multiplication by R(‡s ). So one could set up a “mirror-receiver method” instead of a “mirror-source method” by recasting all interrupt and efficiency rules. Numerically there would be no advantage as compared with the MS method for isotropic sources Q. If, however, the original source Q has a directivity D(˜ , œ), the mirror-receiver method avoids the mirror reflection of the directivity.

M.5.18 Limit Case of Parallel Walls Because only antiparallel walls will be considered here,we use the abbreviation“parallel” walls. In principle, the number of MSs needed for the sound field in the space between parallel walls is infinitely high (on both sides of the walls). Up to now we have had only an interrupt criterion if the walls are absorbent due to the reduction of the source factor ¢R with increasing order. In the limit case of parallel walls in Eq. M.5.(14) the limits r, rp , rq → ∞ and ˜ → 0 are assumed; the MS circle becomes a straight line normal to the walls. First we derive with the next figure a plausible interrupt criterion for the MS production with parallel walls by consideration of allowable errors; then we sum up the MSs to a single equivalent source. The arrows indicate which couples of sources satisfy the boundary condition at which wall. After an interrupt one has a couple without a “partner” for the boundary condition (this is the lowest couple in the graph). An interruption makes an error in the boundary condition at a wall. The absolute error is in the order of magnitude of the field contribution of the uncompensated couple. It decreases with increasing order of reflection • because of the increasing distance of the couple to the wall (geometrical reduction), • because of the decrease of ¢R with absorbent walls (acoustical reduction).

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The relative error, which is important, further decreases with increasing order because the reference quantity is the sum of contributions of “complete” couples. From experiences of field evaluations in flat ducts (which, indeed, is the object at hand) one can conclude that a relative boundary value error rel ≈ 1/10 to 1/20 is tolerable. Neglecting geometrical and acoustical reductions, this leads to a permitted interrupt at about shi = 10 to 20 MSs. This will be sufficient if geometrical reduction with increasing order is taken into account and if one considers that real walls never are ideally reflecting. A reflection coefficient |R| = 0.9 produces for an order o = 15 a source factor of about 0.915 = 0.206. One can perform an analysis as in > Sect. M.5.15 for parallel walls as well. Its principal result is a recommendation to write the sound field of the MS for parallel walls as the field of the original source multiplied by a directional factor. But, knowing this goal, it is easier to derive that form of the MS field directly. The problem is characterised geometrically by the straight line, normal to the flanks F1 , F2, which contains the original source Q and the MS q, and the field point P.We therefore take a right-handed Cartesian co-ordinate system x , y , z with the y axis on the source line and the x axis in one of the flanks so that P is in the x , y plane (it follows from the system x, y, z of the room by a rotation and shift); see next figure.

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The source positions are given by x = 0, z = 0 and xq = (2s + 1) · H ± h;

s = 0, ±1, ±2, . . . ,

(15a)

if the co-ordinate origin is chosen on F1 as in the figure, otherwise by xq = 2s · H ± h;

s = 0, ±1, ±2, . . . .

(15b)

The field contributionpq (P) at P of a source q can be written in terms of the contribution pQ (P) of Q as: pq (P) = ¢q R(‡s ) · with dqP =

k0dQP −jk0 (dqP −dQP ) e · pQ (P) k0dqP

(xP − xq )2 + (yP − yq )2 ;

dQP =

(xP − xQ )2 + (yP − yQ )2 .

(16)

(17)

The angles ‡s for the reflection factors are easily obtained in the x , y co-ordinates. The equivalent source and its contribution representing all mirror sources are given by: p(P) = pQ (P)

s

¢q R(‡s ) ·

k0 dQP −jk0 (dqP −dQP ) e . k0 dqP

(18)

For the combination with corner sources of non-parallel wall couples it is again recommended to leave the original source and the MSs of the first order out of the summation over s. For the mirror reflection of this source at an opposite wall use the reciprocity, i.e. determine the position of the reflected field point P in the co-ordinates x , y , z . If the parallel flanks F1 , F2 are not directly opposite to each other, but with a parallel offset, interrupt checks for mirror sources are preferably performed in the co-ordinates x , y , z (skip s for illegal sources in Eq. M.5.(18)), as well as efficiency checks (skip s for inefficient sources). The total source Eq. M.5.(18) is ineffective if P is not in the space between the planes of the flanks F1 ,F2,and the mirror-reflected combined source is

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ineffective if P is not in that space.So indeed one has to form two sums as in Eq.M.5.(18), one for the “legal” equivalent source, which is used in its mirror reflection, and one for the final evaluation of the field contribution.But the decision about whether Eq.M.5.(18) must be evaluated at all can be made before any computation. One still has the problem with convex corners and their scattered field. This problem can be solved rather easily, within the frame of a MS method, by combination of the MS method with the “second principle of superposition” (PSP). See > Sect. B.10 for that principle. It will be briefly repeated below because it will take special forms in combination with the MS method.

M.5.19 The Second Principle of Superposition (PSP) It should be stated in advance: • The PSP, when applied to single concave corners, does not result in significant computational savings as compared to the traditional MS method; • unless it is globally applied to symmetrical rooms (see below); • but its application is necessary with convex corners; • because it is applicable to both convex and concave corners, and some of its features are more easily explained with concave corners, these are treated here also. The objects of the second PSP are an arbitrary (also multimodal) source Q and a scattering object which has a plane of symmetry M.When the PSP is applied to room edges, the flanking walls are assumed to extend from their line of intersection to infinity. If they are absorbent, the absorption should be the same at both flanks (symmetrical flanks; see below for unsymmetrical flanks). It is not necessary that the flanking walls of a room have a real line of intersection; they may be couples of walls, with other walls between them on the apex side.

We assume equal wall admittance values Gi at both flanks Fi . We further assume a coordinate system with a co-ordinate z (which preferably is an azimuthal co-ordinate) normal to the median plane M. This assumption is not necessary; it just simplifies the description. The median plane M subdivides the wedge into two halves: • (I) on the source side of M, • (II) on the back side of M (as seen from Q).

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With the above choice of z, mirror-reflected points in both halves are distinguished just by ±z. The PSP composes the solution of the original scattering task by the solutions of two subtasks = h, w of scattering in zone (I): • in the first subtask, = h, the plane of symmetry M is assumed to be hard; • in the second subtask, = w, the plane of symmetry M is assumed to be soft. The field of each subtask is composed of the source field pQ and a scattered field p() s : p() = pQ + p() s . The fields of the original task in both zones (I) and (II) then are:

1 (h) ps (x, z) + p(w) s (x, z) , 2

1 (h) ps (x, −z) − p(w) pII (x, z) = s (x, −z) 2 pI (x, z) = pQ (x, z) +

(19a)

(x represents the co-ordinates other than z). One can also decompose the field as p() = pQ + pr + p() s , where pr is the same in both subtasks; then we have:

1 (h) ps (x, z) + p(w) s (x, z) , 2

1 (h) (w) pII (x, z) = ps (x, −z) − ps (x, −z) . 2

pI (x, z) = pQ (x, z) + pr (x, z) +

(19b)

The scattered fields here are generally different from those of Eq. M.5.(19a), but in our problem pr is just a member of the scattered field terms (see below), so that these remain unchanged. In summary: the PSP solves the scattering task on the source side (I) for the subtasks = h, w and computes with them the field on the back side (II) (by simple mirror reflection at M). One should keep in mind this “detour” of the field evaluation in (II) via zone (I). The derivation of the PSP uses, in addition to the source Q(x, z), sources Q(x, −z), which are mirror-reflected at M; in the first subtask the MS has the same amplitude as Q; in the second subtask it is multiplied by −1. Equations M.5.(19a) just describe the superposition of both subtasks with Q and ±MS as sources. The following features should be observed: • The simplicity of the derivation shows the general validity of the PSP (under the mentioned condition of object symmetry). • The PSP is an exact description if the scattered fields of the subtasks can be determined exactly. • The PSP is suitable for combination with the MS method! • If, in the course of applying the PSP, mirror sources are created at M (to satisfy the boundary conditions there), one should remember that mirror sources at ideally reflecting planes give an exact description of the field. With absorbing flanks, the errors of the MS method remain. • The fields of the mirror sources Si form the “scattered fields”. If Si is created by mirror reflection at F1 on the source side, the sign of the MS is the same in both subtasks; this will be indicated in the sketches below with (+). If Si is created by

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mirror reflection at M, the sign of Si is different in both subtasks; this will be marked in the sketches with (±), i.e. (+) for = h, (−) for = w. If a MS created at M (i.e. marked with (±)) afterwards is reflected at F1 , the double sign (±) remains. The resulting fields of the PSP are: (w) 1 (h) Si (x, z) + Si (x, z) , pI (x, z) = Q + 2 i i (w) 1 (h) Si (x, −z) − Si (x, −z) . pII (x, z) = 2 i

(20a)

i

The change h → w of the median plane M does not influence the position and number of the MSs; therefore the sums have the same counting and summation index i. If the first MS is created at F1 (on the source side), it has in both subtasks the same sign (it will be indicated by S0 ). In pI (x, z) it can be pulled to outside the parentheses and needs no superscript (h) or (w). The remaining MSs under the sums (with superscripts) have undergone at least one mirror reflection at M. Thus one can write: (w) 1 (h) Si (x, z) + Si (x, z) , pI (x, z) = Q + S0 (x, z) + 2 i i (w) 1 (h) Si (x, −z) − Si (x, −z) . pII (x, z) = 2 i

(20b)

i

With the production of MSs in the PSP one must distinguish the following: 1.) “Mirror reflected at. . . ” (a) the wall F1 ; then the sign is the same as for the mother source; if F1 is absorbent, a reflection factor R arises; (b) at M, then the sign in the subtask h is that of the mother source; in the subtask w the sign is changed, i.e. the new factor in the source factor ¢R is R = ±1. 2.) Two subtasks (a) = h: no sign change at mirror reflections; (b) = w: sign change for mirror reflections at M (but not at F1). 3.) Two “paths” of MS production: (a) first path: begins with mirror reflection at F1; MSs on this path will be designated with even indices i = 0, 2, 4, . . .; (b) second path: begins with mirror reflection at M; MSs on this path will have odd indices i = 1, 3, 5, . . .. Whereas the subtasks h, w do not change the position of a MS of some order, the positions of the MSs of both paths are generally different from each other. As a detailed example we take the concave rectangular corner of the next figure.

M

916

Room Acoustics

Here the MS production is continued (irrespective of the inside criterion) until the MSs begin to fall on positions of previously created MSs. The paths from then on would begin to be followed backwards (with multiple covering of the positions). The following schemes represent the chains with multiple reflections at F1 and M for both paths. Mirror reflection at F1 is marked by a simple arrow → and reflection at M by a double arrow ⇒. The cases = h are arranged in the upper line and = w in the lower line. MSs in vertical columns of the scheme have equal spatial positions. The flanking wall F1 is initially assumed to be rigid. Under the symbols of the MSs in the schemes below are written the sums appearing in the PSP (for one path; recall that MSs in a column of the scheme have equal spatial positions): (h) Si + S(w) in zone (I) , S= i S =

i≥0 (w) S(h) i − Si

in zone (II).

i≥0

First path: h

S=

i≥0

S =

i≥0

h +S2 → +S4 ⇒ +S6 → +S8 ⇒ +S10 → +S12 Q→ − S0 w w −S2 → −S4 ⇒ +S6 → +S8 ⇒ −S10 → −S12 (w) S(h) i + Si

= 2S0

(w) S(h) i − Si

=0

0

0

2S6

2S8

0

0

2S2

2S4

0

0

2S10

2S12

(21a)

Second path: h

S=

i≥0

S =

i≥0

+S1 → +S3 ⇒ +S5 → +S7 ⇒ +S9 → +S11 Q −S1 → −S3 ⇒ +S5 → +S7 ⇒ −S9 → −S11 w (w) S(h) = i + Si

0

0

2S5

2S7

0

0

(w) S(h) = i − Si

2S1

2S3

0

0

2S9

2S11

(21b)

Room Acoustics

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917

If one sums up the MSs of both paths, the PSP gives: pI (x, z) = Q(x, z) + S0 (x, z) + pII (x, z) =

4

n=0,2,4... k=1

4

n=1,3,5... k=1

S4n+k (x, z) , (22)

S4n+k (x, −z) .

Now we complete the scheme for the case of (symmetrical) absorption at the walls. The reflection factors Ri in an order of MS are not changed by h → w, except for the sign. First path: h

+R0 S2 → +R0 R4 S4 ⇒ +R0 R4 S6 → +R0 R4 R8 S8 ⇒ . . . Q→ − R0 S0 w w −R0 S2 → −R0 R4 S4 ⇒ +R0 R4 S6 → +R0 R4 R8 S8 ⇒ . . . h

S=

i≥0

S =

i≥0

(w) S(h) i + Si = 2R0 S0 (w) S(h) =0 i − Si

0

0

2R0 R4 S6

2R0 R4 R8 S8

2R0S2

2R0 R4 S4

0

0

(23a)

Second path: h

S=

i≥0

S =

i≥0

+S1 → +R3 S3 ⇒ +R3 S5 → +R3 R7 S7 ⇒ +R3 R7 S9 → . . . Q w −S1 → −R3 S3 ⇒ +R3 S5 → +R3 R7 S7 ⇒ −R3 R7 S9 → . . . (w) S(h) = i + Si

0

(w) S(h) = 2S1 i − Si

0

2R3 S5

2R3R7 S7

0

2R3 S3

0

0

2R3 R7 S9

(23b)

The most important advantage of the combination MS and PSP is the fact that with it convex corners become tractable; the traditional MS method fails there completely. Since the wedge angle of a couple of walls always is Ÿ ≤ 2, the angle between F and M is always ≤ ; the zone (I) in which the scattered field has to be determined is concave; the MS method can be applied there. In concave corners the application of the PSP possibly produces a higher precision because the MSs have new positions. But this should be checked. The following sketches mainly assume hard flanks (for simplicity reasons); absorbing flanks will be specially dealt with.

918

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Room Acoustics

This is the simple case of a source above a plane wall. The MSs are shown until interruption by the inside criterion. There is coincidence of S2 , S3 (with the same source factor R for absorbent F1 , F2). It is an important question whether such coinciding MS have to be counted a multiple of times or if the MS production is interrupted when coincidence begins. S1 , S2 , S3 compensate each other in the sum £S of p(I) . The field in (I) is built up by Q and S0 , as expected. In the difference S of p(II) the contributions of S1 , S2, S3 would sum up if the MSs were used as drawn. As a consequence, the field would be unsteady at M, which is in contradiction to reality. Consequently, coincidence of MS with the same source factor ¢R must be avoided! Taking this interrupt criterion into account, the field is correctly given by the MS and PSP method. This method thus has a further interrupt criterion, as compared with the traditional MS method. This example also illustrates well the procedure in the MS and PSP method for the evaluation in zone (II): one first evaluates with the significant MSs the scattered field in zone (I) and then mirror-reflects that field at M into zone (II). Another instructive example is the concave, rectangular edge.

The MSs are drawn as far as is permitted by the inside and coincidence criteria (a further reflection of S5 at F1 , which would be permitted by the inside criterion, would produce coincidence with S6 ). According to Eq. M.5.(22), the field pI (x, z) is created by the superposition of the fields of Q, S0 , S5 , S6 and the field pII (x, −z) by the mirror sources S1 , S2 , S3 , S4 . The boundary

Room Acoustics

M

919

conditions at the flanks F1 , F2 are evidently satisfied. Because both groups of MS can be transformed into each other by a rotation by /4 and a mirror reflection, the field is also steady at M; thus it is a solution of the task. A special case which can easily be understood is obtained if the source Q approaches the flank F1. With a hard flank F1 again the case of a source above a hard wall F2 is achieved.

The effective source is a double source Q + S0 . As expected, the field is symmetrical relative to F1 and F2 . It is also steady at M. The field is completely and precisely represented. In a further limiting case Q approaches the median plane M; the field again is correctly represented, as can be seen from the sketch below.

The MSs are again drawn according to the inside and coincidence criteria. The source Q and the MSs marked with (+) determine the field in zone (I); the MSs marked with (±) contribute (first in zone (I)) to the field in zone (II). The boundary conditions at F1 , F2 and the condition of steadiness at M are satisfied. The examples presented again illustrate the importance of the determination of pII via the detour over zone (I). The examples contain MSs with (±) in (II). Without the detour, this would mean poles of the field in zone (II). That must be excluded according to the condition of regularity of the scattered field. But with the detour trough (I) no problems exist.

920

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Room Acoustics

M.5.20 The PSP for Unsymmetrical Absorption The condition of symmetrical absorption for the application of the PSP to sound fields between couples of walls is a sensible restriction in applications. It will be attempted, therefore, to resolve that restriction, if necessary by an approximation, which, however, should not introduce errors exceeding the errors of the traditional MS method with absorbing walls. If the flanks are symmetrical, the field in the subtask = h is symmetrical with respect to M (i.e. M is hard); in the subtask = w the field is anti-symmetrical (i.e. M is soft). Symmetry and anti-symmetry of the fields are solely determined by the source Q and the auxiliary sources of the PSP. If the walls are different, a new anti-symmetrical part of the field will arise due to the asymmetry of the walls. It is always possible to compose an unsymmetrical field p(x, z) from two geometrically equal halves (I) and (II) of a wedge with symmetrical and anti-symmetrical field components: p(x, z) = p(sy) (x, z) + p(as) (x, z) , p(sy) (x, −z) = p(sy) (x, z) ;

p(as) (x, −z) = −p(as) (x, z) .

From this it follows that:

1 p(x, z) + p(x, −z) , p(sy) (x, z) = 2

1 p(x, z) − p(x, −z) . p(as) (x, z) = 2

(24)

(25)

M by definition is hard for the part p(sy) and soft for the part p(as) . Therefore these parts (w) have a common characteristic with p(h) s , ps of the PSP with symmetrical flanks. Further, the field p(x, z) shall satisfy the boundary conditions at the flanks F1, F2 : !

p(F1 ) = p(sy) (F1 ) + p(as) (F1 ) = v⊥ (F1)/G1 , !

(26)

p(F2 ) = p(sy) (F1 ) − p(as) (F1 ) = v⊥ (F2)/G2 . In these boundary conditions the parts p(sy) , p(as) appear in the same combination as (w) p(h) s , ps in the PSP. Thus, from formal considerations one arrives at an approximation for the PSP when applied to wall couples with different absorption of the flanks.

Room Acoustics

M

921

Complete the PSP for acoustically different walls as follows: • Apply in the PSP for the evaluation of the field p(I) in zone (I) the sum (w) 1 (h) p(I) (x, z) = Q + ps,i (x, z) + ps,i (x, z) 2 i

i

using for the reflection factors the admittance of the flank F1. Apply in PSP for the evaluation of the field p(II) in zone (II) the difference (w) 1 (h) p(II) (x, z) = ps,i (x, −z) − ps,i (x, −z) 2 •

i

i

using for the reflection factors the admittance of the flank F2. This rule creates a field which • satisfies the wave equation in the approximation of the MS method for absorbent walls, • satisfies the source condition, • satisfies the boundary conditions at F1, F2 , • but is not steady at M for unsymmetrical absorption. It is possible to derive a better approximation which is steady also at M, but it is more complicated to handle and therefore will not be presented here. The errors of the above approximation are of the same order as the errors of the conventional MS method.

M.5.21 A Global Application of the PSP Most auditoriums have a constructional plane of symmetry M, so the room as a whole satisfies the condition for the application of the PSP with symmetrical couples of walls, also with respect to absorption. That means: one solves the task of field evaluation twice in the half of the room containing source Q; once when M is assumed to be hard and once when M is assumed to be soft. The computational advantage can be easily quantified in 2D (similar relations hold in 3D). The room is supposed to have nw walls. In the subtasks of the PSP will appear the following numbers of walls (M included): • nw /2 + 2 walls, if M ends on both sides on walls; • (nw − 1)/2 + 2 walls, if M ends on one side with a wall, and on the other side in a corner; • nw /2 walls, if M ends on both sides in corners. With this remark ends that part which is concerned with the evaluation of the stationary sound field in rooms. It should be noted that the methods described yield complex sound pressures in field point P. The computations will be somewhat simplified if one is satisfied with the magnitudes |pq | of the contributions of the effective sources (MS and corner sources). This is mostly done in room acoustical papers, although it is impossible to conclude from |pq | to |p(P)| (the magnitude of a sum mostly is different from the sum

922

M

Room Acoustics

of magnitudes. . . ). We shall be confronted with that difference in the next section which deals with the determination of the room reverberation using the results of the field evaluation. Although reverberation is a non-stationary process, it should be possible to evaluate the most important room acoustical qualifier, the reverberation time, from the results of a computational field model. In doing that, one will be confronted with some lack of definition of reverberation in common textbook descriptions.

M.5.22 Reverberation Time with Results of the MS Method The method described above delivers the stationary,monochromatic field of a stationary, harmonic source in a room. It returns the complex sound pressure p(P) at a point P, i.e. with magnitude and phase. The most important room acoustical qualifier is the reverberation time; it is described in the literature (more or less) by: “The reverberation time is the elapsed time for a decay of the sound pressure (or sound pressure level) by 60 dB after termination of a stationary sound excitation.” It is tacitly understood that“sound pressure”means the magnitude of the sound pressure, and in most experimental determinations of the reverberation time band noise is used with the sound pressure magnitude of an average over the bandwidth.It is not mentioned (but it is important, as we will see below) that the rectification of the received signal (for the magnitude and band average) implies averaging over time intervals. The reverberation process is non-stationary; our field evaluation is for stationary fields. Therefore one needs a “switch-off model” for the evaluations. It should be recalled in that context that the sound field in the room is created in the MS method (as well as in the modified MS and PSP methods) by equivalent sources, which means that, after the equivalent sources have been installed in the right places and with the right amplitudes, the walls of the room can be taken away. The equivalent sources radiate into the free space! One can imagine the distributed sources as a network of loudspeakers driven by the same signal generator. The network lines contain attenuators which model the source factors ¢R. If the signal generator is switched off, all loudspeakers are switched off instantly, but the sound waves radiated before switch-off still propagate. This model has the advantage that the boundary conditions at the walls are satisfied every time because as long as a sound wave from a source hits a wall, the sound wave from its daughter source with that wall will be present also. The end of the contribution of a source q arrives at P at a time t after switch-off: dist(q, P) k0 · dist(q, P) ; = c0 – k0 · dist(q, P) t . = Tp 2

t=

– = 2/Tp ,

(27)

Here Tp is the time period of the harmonic sound wave. It is reasonable to measure t in units of periods Tp.

Room Acoustics

M

923

Imagine all evaluated MSs (and their field contributions ps (P)) sorted with increasing k0 ·dist(q, P) and indexed in this order with a number s (s = 0 may represent the original source Q).After some elapsed time t/Tp , those contributionsthat are still travelling from their source to P will be summed at P. The decay curve L(P) at P expressed as sound pressure level is therefore: 2 L(t/Tp ) = 10 · lg ps (P) . (28) k0 ·dist(q(s),P)≥2·t/Tp With increasing t/Tp the summation is performed over smaller and smaller remainders of the set of effective MSs. This evaluation will therefore produce a steeper slope of L(t/TP ) at the end of a time interval of observation when this end approaches fewer and fewer remaining MSs. This increase in slope must not be confused with the slope produced by the decreasing amplitudes of MS with increasing distance (due to geometrical and/or acoustical reduction) but is a consequence of the finite size of the set of MSs. Equation M.5.(28) is a direct transcription of the reverberation process defined above verbally. Formation of the magnitude and square is applied to the sum of contributions. Instead of proceeding on the t/Tp axis in unit steps of s, one can proceed in steps t/Tp . Contributions within the interval t/Tp are summed up (linearly!). Below we shall see that the decay curves thus evaluated have only a slight similarity to an expected reverberation curve.This is a consequence of an improper definition.A clearer definition describes the reverberation as the “decay of the average energy density”. The (effective) energy density implies the square of the sound pressure; averaging is performed over directions of the sound intensity and time intervals which are short compared with the reverberation time. Because tacitly the contributions of different sources are also assumed to be incoherent, the contributions of sources to the energy density can be added and they are proportional to the magnitude squared of their contributions pq (P) to the sound pressure p(P): Eq (P) =

1 |pq (P)|2 . 20 c20

(29)

This definition leads to a reverberation curve ⎡ 1 ⎢ 1 L(t/Tp ) = 10 · lg ⎣ 2 s 20 c0 t/Tp n> t/Tp

⎤ ⎥ |p(s)|2 ⎦ .

(30)

k0 ·dist(q(s),P)⊂2·t/Tp

The outer sum indicates summation in steps of time intervals t/Tp in which s sources are found and that this outer summation includes a decreasing number n of such steps. The inner sum forms a squared average (with the factor 1/s) of the contributions in a time interval (this summation is skipped if s = 0).

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Room Acoustics

M.5.23 A Room with Concave Edges as an Example We take the concave room of the previous > Sect. M.5.10 as a model room, with the same positions of the source Q and the receiver P as there. The evaluation of the mirror sources is performed for orders up to omax = 8. The lower limit of |¢R| was set to limit = 0.01; the upper limit for the distances was with dmax = 100 · dist(Q, P) set so high that it did not exclude a legal source. This produces 837 effective sources. The next diagram shows over k0r, with r = dist(q, P), the sound pressure levels pq (P) of the contributions at P, relative to the contribution pQ (P) of the original source Q. The cloud of points has a typical triangular shape: the upper border has an approximately constant slope after a somewhat steeper slope for smaller k0r. The lower border lines are not so well defined, because there the points are disperse. The upwards trending lower border line towards the right is predominantly determined by the termination with omax . One can expect that the upper border line has some similarity with the reverberation curve.

Levels of sound pressure contributions at P of the original and mirror sources up to order omax = 8 The evaluation of Eq. M.5.(28) returns the reverberation curves of the next figures which differ from each other only in the value used for the time interval t/Tp . They do not resemble reverberation curves with which one is acquainted.

Room Acoustics

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925

Summation of complex contributions, with time interval t/Tp = 1

Summation of complex contributions, with time interval t/Tp = 4

The next diagram combines results of the evaluation of Eq. M.5.(30) with squared averaging in time intervals, again for different time intervals t/Tp (within which now an averaging of squared magnitudes takes place); the values of the curves from high to low are t/Tp = 1, 2, 4. The points represent centres of the time intervals. The constant factor is omitted. Except for the steep ends of the curves, which come from the termination of the MS evaluation with omax , as explained above, the curves now represent common reverberation curves. They have a steeper “early reverberation” and not so steep “late reverberation”. The choice of t/Tp may influence to some degree the value of the reverberation time obtained from such curves, for example as the coefficient of a linear regression through the points within a given interval of observation for t/Tp .

926

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Reverberation curves with Eq. M.5.(30), for different values t/Tp = 1, 2, 4 (from high to low) The last diagram combines the points of the reverberation curve for t/Tp = 1 with the linear regression within the interval 20 ≤ t/Tp ≤ 50. The reverberation time Tr in units of Tp is Tr /Tp = 67.46.

Appendix 1: Geometrical Subtasks A three-dimensional, right-handed Cartesian co-ordinate system x, y, z is assumed. Points, lines and planes will be considered in 3D. Corresponding relations in 2D are obtained either by setting one co-ordinate to a zero value, identically, or by easy direct derivations. Walls are defined by a list of subsequent edge points W = {E1 , E2 , . . .}; the sequence of the edges Ee in the list is such that they define a rotation which with the direction towards the interior of the room forms a right-handed system. Below, mostly not the polygon of a wall is considered, but the plane which contains the wall.

Room Acoustics

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927

The direction angles , , ‚ of an oriented line L are the angles between the axes and the line. (1) Distance d between two points P1 (x1 , y1, z1 ), P2 (x2 , y2, z2 ): dPP = (x2 − x1 )2 + (y2 − y1)2 + (z2 − z1 )2 .

(A.1)

(2) Direction cosines of the line from P1 (x1 , y1 , z1) to P2 (x2 , y2 , z2): cos =

x2 − x1 ; dPP

cos =

y2 − y 1 ; dPP

cos ‚ =

z2 − z1 . dPP

(A.2)

(3) Cosine of the angle œ between two lines: The directions of the lines given by their direction angles i , i , ‚i . cos œ = cos 1 cos 2 + cos 1 cos 2 + cos ‚1 cos ‚2 .

(A.3)

(4) Normal form A · x + B · y + C · z + D = 0 of a plane: The plane is given by three points P1 , P2 , P3 on it. A possible form of the plane equation (coming from a zero value of the vector triple product of the vectors (P, P1 ), (P2 , P1), (P3 , P)) is: x − x1 y − y1 z − z1 x2 − x1 y2 − y1 z2 − z1 = 0 , (A.4) x3 − x1 y3 − y1 z3 − z1 whence follow the parameters A, B, C, D: A = y1(z2 − z3 ) + y2(z3 − z1 ) + y3(z1 − z2 ) , B = −x1 (z2 − z3 ) − x2 (z3 − z1) − x3 (z1 − z2 ) , C = x1 (y2 − y3 ) + x2 (y3 − y1 ) + x3 (y1 − y2 ) , D = x1 (y3 z2 − y2 z3 ) + y1 (x2 z3 − x3 z2) + z1 (x3 y2 − x2 y3 ).

(A.5)

(5) Reduced normal form a · x + b · y + c · z + d = 0 of a plane: a= √ c= √

A A2

+ B2

+ C2

C A2 + B2 + C2

; ;

B b= √ ; 2 A + B2 + C2 D d= √ . 2 A + B2 + C2

(A.6)

M

928

Room Acoustics

This reduced normal form should not be confused with Hesse’s normal form: a · x + b · y + c · z − p = 0 , A B ; b = √ ; √ 2 2 2 2 ± A +B +C ± A + B2 + C2 D ≥ 0, p= √ ± A2 + B2 + C2 a =

c =

C ; √ 2 ± A + B2 + C2

(A.7)

where the sign of the root is taken so that p is positive. This additional convention in Hesse’s normal form makes it unsuited for inside checks. The parameters a, b, c of the reduced normal form are the direction cosines of the normal vector on the plane (pointing to the interior side). (6) Foot point P = (x, y, z) of a point P1 = (x1 , y1, z1 ) on a plane: The “foot point” P is the orthogonal projection of P1 on a plane. Let the plane be given by the parameters of its reduced normal form: x = (b2 + c2 )x1 − a(d + by1 + cz1) , y = (a2 + c2 )y1 − b(d + ax1 + cz1 ) , 2

(A.8)

2

z = (a + b )z1 − c(d + ax1 + by1 ) . (7) Mirror point P = (x, y, z) of a point P1 = (x1 , y1, z1 ) on a plane: Let PF be the foot point of P1 on the plane. Then P = 2 · PF − P1 .

(A.9)

(8) Direction cosines of intersection line of two planes: The planes Wi be given by the parameters ai , bi , ci , di of their reduced normal form: 2 3 1 ; cos = ; cos ‚ = ; c1 a1 a b1 c1 ; 2 = ; 3 = 1 b 2 c2 c2 a2 a2

cos = 1 =

b1 ; = 21 + 22 + 22 . b2

(A.10)

The rotation W1 → W2 and the direction of the intersection line form a right-handed system. (9) Point of intersection X = (x, y, z) of a line through two points Pi = (xi , yi , zi ) with a plane: Let the plane W be given by the parameters a, b, c, d of its reduced normal form: x = −d(x1 − x2 ) + b(x2y1 − x1 y2 ) + c(x2 z1 − x1 z2)/xx , y = −d(y1 − y2) + a(x1 y2 − x2 y1 ) + c(y2 z1 − y1z2 )/xx , z = −d(z1 − z2 ) + a(x1 z2 − x2 z1 ) + b(y1 z2 − y2 z1 )/xx , xx = a(x1 − x2 ) + b(y1 − y2 ) + c(z1 − z2 ) .

(A.11)

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929

(10) Foot point P = (x, y, z) of a point P1 = (x1 , y1, z1 ) on the intersection line of two planes W1 , W2 : Let the planes Wi be given by the parameters ai , bi , ci , di of their reduced normal form:

x = 3 (b1 d2 − b2 d1 ) + 2 (c2 d1 − c1 d2 ) + 1 · (1 x1 + 2 y1 + 3 z1 ) /2 ,

(A.12) y = 3 (a2 d1 − a1 d2 ) + 1 (c1 d2 − c2 d1 ) + 2 · (1 x1 + 2 y1 + 3 z1 ) /2 ,

2 z = 2 (a1 d2 − a2 d1 ) + 1 (b2 d1 − b1 d2 ) + 3 · (1 x1 + 2 y1 + 3 z1) / with and i from Eq. (A.10). (11) Bisectrice plane between two intersecting planes W1 , W2 : Let the planes Wi be given by the parameters ai , bi , ci , di of their reduced normal form. The parameters of the bisectrice plane (containing the intersection line) are: a = (a1 + Ša2 ); b = (b1 + Šb2 ) ; c = (c1 + Šc2 ); d = (d1 + Šd2 ) ; Š = ±1 .

(A.13)

(12) Two planes parallel or anti-parallel: Parallel: the three parameters a, b, c of the reduced normal form are pairwise equal; anti-parallel: two of the parameters are pairwise equal, the other differs in sign. (13) Distance between two antiparallel planes: Let the planes Wi be given by the parameters ai , bi , ci , di of their reduced normal form. Distance ƒ: ƒ = |d1 − d2 | .

(A.14)

(14) Inside check of a point P = (x, y, z) relative to a plane given by three points Pi = (xi , yi , zi ): The check is performed with and returns: ⎧ x − x1 y − y1 z − z1 ⎪+1 if P is inside W ⎨ if P is on W (A.15) sign x − x2 y − y2 z − z2 = 0 ⎪ x − x3 y − y3 z − z3 ⎩−1 if P is outside W , where |. . .| indicates a determinant and sign(x) checks the sign of x. (15) Inside check of a point P = (x, y, z) relative to a plane given by its reduced normal form parameters: The check is performed with and returns ⎧ ⎪ ⎪+1 if P is inside W

⎨ sign a · x + b · y + c · z + d = 0 if P is on W (A.16) ⎪ ⎪ ⎩ −1 if P is outside W . (16) Co-ordinate transformation: The system x, y, z is rotated and shifted as shown in the sketch. The new axis z = … in the applications is the intersection line of two planes F1 , F2; the new origin Z is the foot point of the original source Q on the intersection line.

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x , y , z is a right-handed system, like x, y, z. The rotation F1 → F2 forms with z a right-handed system. The transformation x, y, z → x , y , z is done by: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x cos x cos x cos ‚x x − xZ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ y ⎠ = ⎝ cos y cos y cos ‚y ⎠ · ⎝ y − yZ ⎠ . z cos z cos z cos ‚z z − zZ

(A.17)

The inverse transformation x , y , z → x, y, z is done (with the transposed matrix) by: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x cos x cos y cos z x xZ ⎜ ⎟ ⎝ cos x cos y cos z ⎠ · ⎝ y ⎠ + ⎝ yZ ⎠ . (A.18) ⎝ y ⎠= cos ‚x cos ‚y cos ‚z z zZ z

Appendix 2: Algorithm of Mirror Source Scouting in Concave Rooms “Scouting” means finding the positions of mirror sources. In “concave rooms” with only concave corners the traditional mirror source (MS) scouting is applied. This method can also be applied in moderately convex rooms in which the wedge angles of convex ∼ corners are close to >. Then a wall, possibly shaded by a convex corner, will mostly be excluded from the chain of MS production by the inside criterion. The sound field evaluation with the traditional MS method consists of five main parts: • input and evaluation of geometrical room data (e.g. wall co-ordinates, wall centres, normal form parameters of wall planes); • input and evaluation of acoustical wall parameters (e.g. wall admittance values, normal incidence reflection factors); • scouting for legal mirror sources. Selection of effective mirror sources from list of legal mirror sources; • evaluation of sound field contributions of MSs in field point P.

Room Acoustics

M

931

Symbols: list = {item1 , item2 ,. . . ,itemn,. . . } list of items with item counter n; list[[m]] list member number m; tab[o] a table (or list) with index o; ((n)) see entry number n in Appendix 1. entry to a loop with loop counter n, ending at

branching after test ? for positive or negative test result. Room input: {Ww } = {W1 , W2, . . ., Ww , . . .} Ww = {E1 , E2 , . . ., Ee , . . .} Ee = {xe , ye , ze} Derived room parameters: {Cw } = {C1 , C2, . . ., Cw , . . .} {. . ., {aw , bw , cw , dw }, .. . .} Acoustical wall parameters: {Gw } = {G1, G2 , . . ., Gw , . . .} {|R0w |} = {. . ., |R0w |, . . .}

list of walls Ww , index w; defines the room; the sequence of walls is arbitrary; list of edges Ee , index e; defines a wall; the sequence of edges forms a right-handed system with wall normal towards interior room side; list of Cartesian co-ordinates of edge Ee . list of wall centres Cw ; coefficients of reduced normal form of walls Ww ; see Appendix 1, no. ((5)). list of (normalised) wall admittance values; list of reflexion factor magintudes of walls Ww ; |R0w | = |1 − Gw |/|1 + Gw | for normal sound incidence; may be used for scout interrupt checks.

Input of source and immission point co-ordinates: Q = {xQ , yQ , zQ } P = {xP , yP, zP }

co-ordinates of original source Q; co-ordinates of field point P.

Input for scouting interrupt: omax = upper limit of MS order o = 0, 1, . . ., omax ; o = 0 belongs to original source Q; dmax = upper limit of distance dist(q, P) between a MS and P; limit = lower limit of magnitude of source amplitude factor |¢R|. Goal of scouting: Output of tables tab[o] for MS orders o = 0, 1, 2, . . ., ohi consisting of MS lists qlist which contain all needed parameters for the evaluation of the mirror sources of the next order o + 1 (if any) and for the evaluation of the field contribution of effective mirror sources in the field point P:

932

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Room Acoustics

qlist = {q, w, dqP = dist(q, P), pR = ¢R(Ÿ), flag} q = co-ordinates of the source (original or mirror); w = counting index of the “mother wall” if q is a MS; w = 0 for original source; dqP = dist(q, P) = distance from q to field point P; pR = ¢R(Ÿ) = “source factor” = product of reflection factors R(Ÿ) of increasing orders for sound incidence on Ww under the angle Ÿ from q; flag = control flag for source efficiency: flag = 0: q is an efficient source; flag = 1: q is inefficient because P is not in the field angle ∠(q, Ww ). Looping: The algorithm works with nested loops. The outer loop runs through the orders o = 1, 2, . . ., omax of MS reflection; the middle loop runs over the sources qs , s = 1, 2, . . ., in the table tab[o − 1]; the innermost loop runs over the wall indices w of the room list {Ww }. The source list of the mother source (index s in tab[o − 1]) is: qlist(s) = {qs, ws, dqsP = dist(qs, P), pRs = ¢R(Ÿs ), flags}. The wanted source list of a daughter source is: qlist = {q, w, dqP = dist(q, P), pR = ¢R(Ÿ), flag}. Break-off criteria for scouting: • w = ws back-reflection of daughter source on mother source: • qs is not inside Ww ; • centre Cw of mirror wall Ww is not in field angle ∠(qs , Wws ); • |R(Ÿ) · ¢R(Ÿs )| < limit, i.e. expected field contribution too small by absorption; • dist(q, P) > dmax, i.e. field contribution too small by distance; • o > omax mirror source order would become too high; • tab[o − 1] is empty, i.e. no legal MS of order o − 1 exists.

Room Acoustics

M

933

Room geometry data; admittance of walls; position source Q and field point P; limits for order o, distances dist(q, P), MS amplitude in P. Invariable parameters.

Source list of original source Q. Upward looping over order o. Looping through sources q in tab[o − 1]. Call source list of source no. s in tab[o − 1].

934

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Room Acoustics

Loop over walls of room. Jump w-loop if back-reflection. Jump w-loop if qs is not inside wall. Jump w-loop if centre Cw of wall Ww is not in field angle of qs . Evaluate position q; distance dist(q, P); direction cosine of angle of sound incidence from q on Ww ; reflection factor R(Ÿ), amplitude ¢R. Jump w-loop if |¢R| is too small. Jump w-loop if q is too far from P. q is inneffective if P is not in its field angle; such sources are marked with flag = 1. Compose source list of newly found MS. Scouting stops if tab[o] remains empty. Scouting is finished.

Evaluation of the sound field in P by the summed contributions of the MSs in the lists tab[o] is easy. The sound field contributions are phased.

Room Acoustics

M.6

M

935

Ray-Tracing Models

Ray-tracing models are based on geometrical acoustics. Accordingly the wavelengths are considered small compared with characteristic dimensions of the room. Sound is regarded as an energetic process. Calculations of quantities proportional to the sound pressure are estimations with the assumption of: a) broadband stationary or transient signals, b) superposition of energy or other quadratic field quantities. These assumptions are both related to a consideration of sound energy in terms of a particle rather than a wave. Accordingly, ray tracing cannot be used for simulation of interference effects like standing waves, modes or diffraction. ƒ: e: N: rd : t: W: z, z1, z2 :

random-incidence scattering coefficient; ray energy; number of rays; detector radius; sampling interval; total sound energy in a volume V; random numbers

Scattering, however, is often accounted for by a statistical approach by means of modelling the statistical case. The statistical case of scattering is described by Lambert’s law. The direction of scattered sound is independent of the direction of incidence. Furthermore, the directions of scattering are distributed according to a cosine law, thus resulting in a constant emission of energy into all spatial angles. (In the equivalent phenomenon in optics, a surface with sound scattering according to Lambert’s law would be observed with constant light intensity from all directions.) The crucial parameter for surface scattering is the size and the shape of the surface corrugation.Sound wavelengths are distributed from being “large” to “small”, compared with the surface corrugations. This fact can be related to assumption of two ideal cases: (a) geometrical reflection and (b) random scattering. Therefore low frequencies are best treated with specular reflections, intermediate frequencies with random scattering and high frequencies with small-scale geometrical reflections.

Reflection from rough surfaces, at low (a), mid (b) and high (c) frequencies Since reflections in room acoustics and particularly the reverberation process are built up by numerous reflections, the average behaviour is much more important than each individual scattering characteristic. It is thus sufficient to consider a mixed model of

936

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Room Acoustics

geometrical and diffuse reflections with a “switching” parameter and a random process with appropriate probability distribution. The parameter which decides which model is used is the random-incidence scattering coefficient ƒ: ƒ=1−

Espec . Etotal

(A.18)

Monte Carlo Method: N rays are radiated from a source point. Typically N is larger than 10 000. Each ray is carrying a portion of sound energy e0 . In this method the ray detection is provided by counting the rays hitting a detector (for instance, a sphere with radius rd) and sampling the counts in time intervals t.

In a diffuse sound field the expectation value of the energy decay (energy time curve) is: rd2 c0 t (A.18) V with room volume V and n¯ the mean reflection rate and ¯ the average absorption coefficient of the room (see > Sect. M.4). ¯ ¯ nt e(t) = e0 N(1 − )

Wall absorption can be modelled by energy reduction according to multiplication of the ray energy by (1 − ), being the random-incidence absorption coefficient of the wall. Alternatively, a random number z ∈ [0; 1] can be chosen and the ray can be absorbed if z < . The probability density that the ray is absorbed at the next wall reflection is: ¯ . ¯ w(nt) ¯ = (1 − ) ¯ nt−1

(A.18)

Room Acoustics

M

937

Whether Lambert scattering or specular reflection is used is decided by a random number z ∈ [0; 1]. The ray is scattered if z < ƒ. Uncertainty of the Monte Carlo Method Relative standard deviation of the statistical counts in the energy decay curve: ) e V = (energy absorption by multiplication), or e

Nrd2 c0 t e = e

)

V ¯ r 2 c t N(1 − ) ¯ nt d 0

(energy absorption by random absorption).

(A.18)

(A.18)

Relative standard deviation of the total energy W in the energy decay curve (energy integral): ) |p|2 A W = * + = (energy absorption by multiplication), (A.18) 2 W

8Nr 2 |p| d which is related to a sound level variation of ) A L = 4.34 8Nrd2 and |p|2 W = = 2 W

p

)

A 4Nrd2

(energy absorption by random absorption),

which is related to a sound level variation of ) A . L = 4.34 4Nrd2

(A.18)

(A.18)

(A.18)

The Cone, Beam or Pyramid Approach: In contrast to the Monte Carlo approach of ray tracing, deterministic models of ray tracing are known. Here, the energy time curves are not calculated by counting rays but by determination of ray paths and corresponding geometrical energy reductions. There are several ways of finding physically correct paths by associating the rays with cones or beams with constant solid angle and, thus, increasing spatial spread. In a diffuse sound field the expectation value of each ray energy is calculated by: e(t) = e0

¯ (1 − ) ¯ nt r2

(A.18)

with r denoting the distance between the ray source and the receiving point. Another important difference with the Monte Carlo approach is that the ray energy can be recorded in an arbitrarily high time resolution. In the case of a time resolution

938

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Room Acoustics

that is sufficient for audio processing, impulse responses can be composed from a set of reflections (or image sources). Each contribution contains a frequency function Hj which is based on the Fourier transform of the reflection pulse (a Dirac pulse) and multiplied by various frequency functions corresponding to the filter effects on the ray path: nj

, e−jk0 rj H S H R Ha Ri Hj = rj

(A.18)

i=1

with rj denoting the distance between image source and receiver, HS the (directional) spectrum of the source, HR the directional head-related transfer function (HRTF, right or left ear) of the receiver person (in the case of binaural processing), Ha the spectrum of air attenuation, and Ri the reflection factors of the walls involved in the ray path (or the mirror source). The total binaural impulse response (r, l = right, left ear) is then obtained by inverse Fourier transformation: ⎧ ⎫ N ⎨ ⎬ pr,l (t) = F−1 Hj,r,l . (A.18) ⎩ ⎭ j=1

Ray Sources (Deterministic, Random): Consider an omnidirectional source, with angles of ray direction in relation to sourcerelated spherical co-ordinates œ, ˜ . The angles are: œ = 2z1 ,

(A.18)

˜ = arccos z2

(A.18)

with z1 the random number of the interval [0; 1] and z2 the random number of the interval [0; 1] independent of z1 . Algorithm of reflections (scouting of reflection points): Last point of ray history P (vector p ), calculation of next wall hit at point S (vector s):

n i : normal vector of wall i (all wall normal vectors direction towards the interior of the room), r: vector of actual flight direction.

Room Acoustics

Step 1:

M

939

Ray must hit the wall plane facing the room (inside) n i · r < 0 .

Step 2:

(A.18)

All walls passing step 1, ray must hit the next wall in positive flight direction: Calculation of (p − s) · r < 0 .

(A.18)

¯ i (i = 3, 5, 4, 2 in the figure above), starting with After sorting of distances PS the nearest wall intersection point, test of “intersection point within polygon” (see > Sect. M.5). In the case of failure (wall 3 in the figure above), the next point Si is to be checked. If a special wall type is defined, “non-shadowing wall”, this test can be skipped for each wall that is non-shadowing. Shadowing walls can block the free line of sight between two arbitrary observer points in the room; non-shadowing walls never block any two arbitrary lines within the room (in the figure above, i = 3, 4, 7, 8, 9 are shadowing walls, i = 1, 2, 5, 6 are non-shadowing walls). It is worthwhile to divide the wall polygons into the two categories. If a non-shadowing wall is reached or if Si lies within the wall polygon, the ray is reflected (or absorbed) at wall i.

Step 3:

Step 4:

Wall scattering (Lambert’s law). Independent of the angle of incidence, the reflection angle §(œ, ˜ ) is randomly chosen according to: w(˜ ) d§ =

1 cos ˜ d§

(A.18)

with w(˜ ) the probability density of the reflection polar angle ˜ . The azimuth angles are equally distributed in [0, 2]. This is obtained by two independent random numbers z1,2 ∈ [0; 1] in: √ ˜ = arccos z1 ; œ = 2z2 . (A.18)

M.7

Room Impulse Responses, Decay Curves and Reverberation Times

The results of mirror source or ray-tracing algorithms are discrete energy density impulse responses: w(t)t ≈ w(t) dt ∝ p2 (t) dt.

(A.18)

r2(t): decay curve; T30, Tx: reverberation time; EDT: Early Decay Time Decay curve (expectation value for interrupted noise decay): 2

∞

r (t) =

2

t

p (‘) d‘ − 0

2

∞

p (‘) d‘ = 0

2

t

p (‘) d‘ = t

∞

p2 (‘) d(−‘).

(A.18)

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M

Room Acoustics

Reverberation Time T30: Linear regression from −5 dB to −35 dB of 10 log[r2 (t)] for determination of t−5 and t−35 , extrapolated to −60 dB by T30 = 2(t−35 − t−5 ).

General Reverberation Time Tx: Linear regression from −5 dB to −(x + 5) dB of 10 log[r 2 (t)] for determination of t−5 and t−(x+5) , extrapolated to −60 dB by Tx = 60/x(t−(5+x) − t−5 ). Early Decay Time EDT (characterising the subjectively perceived reverberance of a room): Linear regression from 0.1 dB to −10.1 dB of 10 log[r2(t)] for determination of t−0.1 and t−10.1 , extrapolated to −60 dB by EDT = 6(t−10.1 − t−0.1 ).

M.8

Other Room Acoustical Parameters

Impulse responses can be evaluated for calculation of quantities with specific correlation to subjective impressions. All quantities are based on integrals of the squared impulse responses. p(t): p10 (t):

sound pressure impulse response; sound pressure impulse response in a reference source-to-receiver distance of 10 m; pr (t): sound pressure impulse response obtained for the right ear of a test subject or a dummy head (see Eq. M.6.(A.18)); pl (t): sound pressure impulse response obtained for a test subject or a dummy head (see Eq. M.6.(A.18)); pL (t): lateral sound pressure impulse response (figure-of-eight directionality) with its directional null pointed towards the source See > Sect. M.4 for definitions of other symbols.

Room Acoustics

M

941

The sound strength G is the relative level between the sound field in the room and the level in a free sound field at 10 m distance, with the power output of the source and the direction of the axis of source and receiver remaining the same: ⎛∞ ⎞ ∞

G = 10 log ⎝ |p(t)|2 dt |p10 (t)|2 dt⎠ . (A.18) 0

0

Definition D or early-to-late energy ratio (characterising the speech intelligibility): 50

ms

∞

2

|p(t)| dt

D= 0

|p(t)|2 dt .

(A.18)

0

Clarity Cx , early-to-late energy ratio for music (x = 80 ms) and for speech (x = 50 ms) (characterising the subjective transparency or speech intelligibility, respectively): 80

ms

C80 = 10 log

∞

2

|p(t)| dt 0

|p(t)|2 dt [dB] ,

(A.18)

|p(t)|2 dt [dB] .

(A.18)

80 ms

50

ms

C50 = 10 log

∞

2

|p(t)| dt 0

50 ms

Relation between D and C50 : D . C50 = 10 log 1−D

(A.18)

Centre time (first moment of the impulse response, characterising the reverberance and speech intelligibility):

∞ TS =

∞

2

t|p(t)| dt 0

|p(t)|2 dt.

(A.18)

0

Lateral energy fraction: Early lateral sound ratio (characterising the subjective spatial impression “apparent source width”): 80

ms |pL (t)| dt |p(t)|2 dt

80

ms

LF =

2

5 ms

(A.18)

0

with pL (t) denoting the sound pressure weighted with a figure-eight directionality, with its directional null pointed towards the source. Interaural Cross-Correlation Function: 0 1

t2

t2 1 t2 1 2 IACFt1 ,t2 (‘) = pl (t) · pr (t + ‘) dt 2 pl (t) dt p2r (t) dt. t1

t1

t1

(A.18)

942

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Room Acoustics

Interaural Cross-Correlation Coefficient (characterising the subjective spatial impression): IACC = max IACF−1 ms,1 ms (‘) . (A.18) Late Lateral Sound Level: (characterising the subjective spatial impression “listener envelopment”) [Bradley/Soulodre (1995)]: LG∞ 80

∞ =

2

∞

|pL (t)| dt 80 ms

|p10 (t)|2 dt .

(A.18)

0

Estimates of the room acoustical parameters using just the reverberation time can be achieved by consideration of a purely exponential decay [Barron/Lee (1988)]: Energy integral from time t to infinity: it = (31200T/V) · e−13.82t/T

(A.18)

with r denoting the source-to-receiver distance in m, T the reverberation time in s, and V the room volume in m3 . Contribution of direct, early and late (limit 80 ms) sound energy at a source-to-receiver distance r: ed = 100/d2 ,

(A.18)

ee = (31200T/V) · e−0.04d/T(1 − e−1.11/T ) ,

(A.18)

el = (31200T/V) · e−0.04d/T · e−1.11/T ,

(A.18)

and accordingly: G = 10 · log(ed + ee + el ) = 10 · log(100/d2 + 31200T/V)

(A.18)

and C80 = 10 · log[(ed + ee )/el ].

(A.18)

In another formulation of the sound strength G and in agreement with Eq. M.8.(15) and M.8.(16) we find: T 1 50 1 + 4.34A/S + 20 = 10 log 2 + + 4.34A/S + 20, (A.18) G = 10 log 2 + 310 r V r A or, in approximation for large distances r, √ G = 37 − 10 log A + 4.34A/S; r A/7

(A.18)

with A the total absorption area and S the room interior surface area. Furthermore, the early and late energy densities ee , el can be calculated as follows, similar to Barron’s revised theory [Barron/Lee (1988)], with the n¯ the average reflection rate (see Eq. M.4.(2)) and ¯ the average absorption coefficient (see Eq. M.4.(14)): (1/ n)+0.05 ¯ 276 +A/S − 1 1 1 T ee = e−A/S − e T + e−13.8t/T dt + = , (A.18) 2 V 4c0 r 13.8V 4c0 r2 1/n¯

Room Acoustics

1 el = V

∞ e 0.05+1/n¯

−13.8t/T

T − dt = e 13.8V

276 +A/S T .

M

943

(A.18)

Accordingly, definition D and clarity C80 result in: 276 276 − +A/S T 1 −A/S 1 T − e −e T + T 1 − e + 2 13.8V 4c0 r 13.8V 4c0 r2 D= ≈ , (A.18) 1 1 T −A/S T + e + 13.8V 4c0 r2 13.8V 4c0 r2 13.8 0.08+A/S − 1 T −A/S T e + −e 13.8V 4c0 r2 C80 = 10 log 13.8 T − T 0.08+A/S (A.18) e 13.8V 13.8V 13.8T/V −1 for A S or ¯ 1 , ≈ 10 log e 1+ 4c0 r2T the latter with the influence of the direct field neglected.

References Barron, M., Lee, L.-J.: Energy relations in concert auditoriums. J. Acoust. Soc. Am. 84, 618 (1988)

Mechel, F.P.: Schallabsorber, Vol. II, Ch. 10: Sound in capillaries. Hirzel, Stuttgart (1995)

Bradley, J. Soulodre, G.A.: Objective measures of listener envelopment. J. Acoust. Soc. Am. 98, 2590 (1995)

Mechel, F.P.: Improved mirror source method in room acoustics. J. Sound Vibr. 256, 873–940 (2002)

Cremer, L.: Die wissenschaftlichen Grundlagen der Raumakustik, Band I: Geometrische Raumakustik. Hirzel, Stuttgart (1948) Kosten, C.W.: The mean free path in room acoustics. Acustica 10, 245 (1960) Kuttruff,H.: Room Acoustics, 4th edn.E&FN SPON, London (2000)

Pujolle: Nouvelle Théorie der la Réverberation des Salles.Rev.Techn.Radio Télevis.25,254–259 (1972) Pujolle: Nouveau point de vue sur l’acoustique des salles. Revue d’Acoustique 18, 21–25 (1972) Pujolle: Détermination des dimensions optimales d’une salle. Revue d’Acoustique 28, 13–18 (1974)

N Flow Acoustics with P. Költzsch Sound propagation in a flowing medium is treated also in > Ch. J, “Duct Acoustics”, and K,“Acoustic Mufflers", but there mostly with simplifying assumptions. This chapter uses throughout the “double subscript summation rule”, i.e. terms in an expression in which a subscript (for example i) appears twice represent a sum over that term with the multiple subscript as summation index. Terms containing xi2 , for example, are also cases of the summation rule. The general convention to symbolise the density of air with 0 and sound velocity with c0 must be suspended in this chapter because these quantities may be used in other than standard conditions. These conditions will always be defined in the context.

N.1

Concepts and Notations in Fluid Mechanics, in Connection with the Field of Aeroacoustics

See also: Morfey (2001); Lauchle (1996); Douglas (1986); Roger (1996)

N.1.1 Types of Fluids Ideal fluids: Newtonian fluid: Non-Newtonian fluid:

N.1.2

‹ = 0, Š = 0

‹: dynamic viscosity Š: thermal conductivity

‹ = constant ‹ = constant The relationship between shear stress ‘ and velocity gradient ∂v ∂n is non-linear.

Properties of Fluids

Density Pressure

Viscosity

kg , mass per volume, = 3 m p, normal force pushing against a plane area divided by the N area, p = 2 = Pa m N· s = Pa · s dynamic viscosity ‹, ‹ = m2 kinematic viscosity

Gas constant

R, [R] =

J kg · K

Œ, [Œ] =

m2 s

946

N

Flow Acoustics

Specific heats

∂s ∂u = at constant volume cv = T ∂T ∂T ∂s ∂h = at constant pressure cp = T ∂T p ∂T p

(1) (2)

cp , cv =

J kg · K with: s specific entropy u specific internal energy h specific enthalpy Specific heat ratio ‰ = cp cv , ratio of the specific heat at constant pressure (3) to that at constant volume m Speed of sound c, [c] = s Bulk modulus K, expresses the compressibility of a fluid, [K] = Pa, ∂p (4) adiabatic or isentropic bulk modulus: Ks = ∂ s ∂p isothermal bulk modulus: KT = (5) ∂ T The reciprocal 1 Ks or 1 KT is the adiabatic or isothermal compressibility. 1 1 ∂ Coefficient of expansion = − = (6) ∂T p K Thermal conductivity Shear stress

N.1.3

W m ·K ‘, [‘] = Pa

Š, [Š] =

Models of Fluid Flows

Real flow: Ideal flow:

flow without any assumptions flow without viscosity and thermal conductivity

Inviscid flow: Viscous flow:

flow without viscosity ‹ = 0

Incompressible flow: Compressible flow:

= constant = constant

Adiabatic flow:

flow without heat transfer Ds = 0, the specific entropy of each fluid particle along its path Dt is constant, but may vary from one particle to another (Roger); inviscid and non-heat-conducting gas flow, also frictionless adiabatic flow

Isentropic flow:

N

Flow Acoustics

Homentropic flow:

s = constant throughout the flow, uniform specific entropy

Isothermal flow:

T = constant

Steady flow:

no time dependence for v, p, , T, . . .;

Stationary flow:

∂... =0 ∂t T ∂ A¯ 1 ¯ . . . . and A ¯ = v¯ , p¯ , ¯ , T, ¯ = lim = 0, with A Adt T→∞ T ∂t

947

thus

(7)

0

Unsteady flow:

¯ ∂A ∂A = 0, possibly also = 0 ∂t ∂t

Uniform flow:

∂ v¯ =0 ∂s

Non-uniform flow:

∂ v¯ = 0 ∂s

Rotational flow:

– = rotv = curlv = ∇ × v = with:

ex ∂ ∂x u

ey ∂ ∂y v

ez ∂ ∂z w

= 0

(8)

vx = u, vy = v, vz = w

Vorticity:

– = rotv = ∇ × v is a measure of local fluid rotation.

Irrotational flow:

– = rotv = ∇ × v = 0

(9)

Comment: From Crocco’s form of the momentum equations it follows that (stationary flow with constant stagnation enthalpy) – × v = T · grad s. Consequences: a rotational flow cannot exist with uniform entropy; a homentropic flow must be irrotational (except when vorticity field and velocity field are parallel). Laminar flow:

viscous or streamline flow, without turbulence; the particles of the fluid moving in an orderly manner and retaining the same relative positions in successive flow cross sections.

Turbulent flow:

a random, non-deterministic motion of eddying fluid flow;

Turbulence: characterised by (Morfey):

• three-dimensional velocity fluctuations field; • unsteady flow;

948

N

Flow Acoustics

• • • •

viscous flow; rotational flow; flow with viscous dissipation of energy; viscous dissipation takes place at the smallest length scales of eddies, far removed from the larger scales eddies contain most of the kinetic energy; the smallest scales molecular scales; fluctuations cover a wide frequency range and a wide • range of eddy sizes or length scales; • occurring at high Reynolds numbers. Turbulence level:

based on the averaging of the specific kinetic energy:

1 1 2 1 1 vi = v¯ i + vi v¯ i + vi = v¯ i2 + vi2 2 2 2 2

(10)

three-dimensional:

1 1 2 1 2 u¯ + v¯ 2 + w¯ 2 + vi = u2 + v 2 + w 2 2 2 2

(11)

turbulence level:

1 2 + v 2 + w 2 u 3 Tu = (u¯ 2 + v¯ 2 + w¯ 2 )

(12)

in the special case of isotropic turbulence and unidirectional flow v¯ i = {¯vx = U; 0; 0}: Tu =

u 2 urms = U U

Transition:

the fluid flow change from laminar to turbulent flow

Boundary layer flow:

•

(13)

in the mean flow sense (Morfey): flow next to a solid surfaces within which the mean flow u¯ (y) varies with distance y from the wall, from zero at the wall (at y = 0) to 99% of its free-stream value at y = ƒ, ƒ is the boundary layer thickness;

Flow Acoustics

N

949

•

Reynolds stress:

N.2

in the acoustic sense (Morfey): a thin region produced by a sound field next to a solid boundary, within which the oscillatory velocity parallel to the wall drops to zero as the wall is approached, as a result of viscosity. The acoustic boundary layer thickness is 2Œ Š (14) ƒ= – • vi vj in unsteady fluid flow; vi , vj are fluid velocity components in any of the three orthogonal Cartesian coordinate directions; vi vj represents the transfer rate of j-component fluid momentum per unit area; the double divergence of vi vj represents a source term in Lighthill’s inhomogeneous wave equation (acoustic analogy for aerodynamic sound generation); • in turbulent flows: the time-average Reynolds stress vi vj is a term in the timeaveraged momentum equation, as the negative of an effective stress; vi vj represents the mean momentum flux due to turbulent eddies; the Reynolds stress tensor is ‘ij = vi vj with normal stress if i = j and shear stress if i = j (Morfey)

Some Tools in Fluid Mechanics and Aeroacoustics

See also: Telionis (1981); Johnson (1998); Schlichting (1997); Lauchle (1996); Liu (1988); Hussain (1970); Reynolds (1972)

N.2.1 Averaging General quantity:

f (x , t)

Spatial average:

1 f¯spatial = V

Time average:

Root mean square:

f (x , t) dV = f¯ s (t)

(1)

V

1 f¯time = lim T→∞ T

T

f (x , t) dt = f¯ t (x) = f¯

(2)

0

with: abbreviation: f¯

∞ 1 2 f 2 (x , t) dt = f¯ rms (x) frms = f = lim T→∞ T 0

the square root of the mean square value

(3)

950

N

Flow Acoustics

Ensemble average:

Phase average:

over N repeated experiments N 1 () f¯ensemble = f (x , t) = lim (4) f (x, t) = f¯ en (x , t) N→∞ N =1 for periodic flows N 1 f¯phase = f (x , t) = lim (5) f (x , t + n‘) = f¯ ph (x, t) N→∞ N n=0 with: ‘ period of an externally imposed fluctuation N

1 f¯phase = f (x, t) = lim f x , œ0 + n–‘ = f¯ ph x , œ0 (6) N→∞ N n=0 with: 0 < œ0 < 2 ; œ = œ0 + n–‘ phase of the periodic flow

Reynolds averaging: decomposition of a general quantity in the flow in the following form: f = f¯ + f 1 with: f¯ = f0 = lim T→∞ T f = lim

T→∞

1 T

T

T f dt

mean quantity

(7)

fluctuating quantity

(8)

0

f dt = 0

0

Mass-weighted or Favre averaging: decomposition of a general quantity in the flow in the following form: f = ˜f f + f f f with: ˜f f = ¯ f f f = 0. f

N.2.2

filtered part of f ; unresolved or subgrid part of f :

Decomposition (in General)

Decomposition of a general flow quantity in three (or four) parts: f (x, t) = f¯ (x) + ˜f (x , t) + f (x , t) with: f¯ (x)

time-averaged or mean component, obtained by Reynolds averaging: f˜¯ = 0; f

˜f (x, t)

(9)

1 f¯ (x ) = lim T→∞ T

T f (x , t) dt 0

= 0;

organised fluctuation: periodicities are in time, periodic mean component of flow can be split into odd modes ˜f odd and even modes ˜feven (see Liu):

Flow Acoustics

˜f (x , t) = ˜f odd (x, t) + ˜f even (x , t); f (x, t)

N

951

(10)

random fluctuations, incoherent fluctuating flow quantities, e.g. small-scale stochastic fluctuations of fine-grained turbulence; N

1 f (x , t + n‘) = f¯ + ˜f f¯phase = f (x , t) = lim N→∞ N n=0 with: f (x , t) = 0 ˜f = f (x , t) − f¯ .

(11)

odd

= 0. Phase averaging with period 2‘ is denoted by

so that f˜ odd Therefore the even modes are obtained from f˜ + ˜f even = f˜even, and the odd modes from subtracting fõdd = ˜f − ˜f odd + ˜f even .

(12)

(13) (14)

N.2.3 Decomposition of the Physical Quantities in the Basic Equations = ¯ + p = p¯ + p vi = v¯ i + vi Continuity equation:

∂ ∂ vi + =0 ∂t ∂xi Decomposition:

(15)

∂ ∂ + ¯ v¯ i + ¯ vi + v¯ i + vi = 0 ∂t ∂xi with assumptions:

(16)

, vi ¯ , v¯ i and ¯ = f (x)

Continuity equation in the case of mean flow:

∂ v¯ i =0 ∂xi

(17)

∂v ∂ ∂ + ¯ i + v¯ i = 0 (18) ∂t ∂xi ∂xi p and with the equation of state = 2 : c0 ∂p ∂p 2 ∂vi + v¯ i + ¯ c0 =0 (19) ∂t ∂xi ∂xi and in the case of fluctuating flow:

¯ ∂v Dp + ¯ c20 i = 0 Dt ∂xi with:

¯ ∂ ∂ D = + v¯ i Dt ∂t ∂xi

(20) (21)

952

N

Flow Acoustics

Momentum equation (without viscosity)

∂

∂ vi + vi vj + pƒij = 0 ∂t ∂xj

(22) ∂ v¯ i ∂ p¯ + =0 ∂xj ∂xi

(23)

Mean flow:

¯ v¯ j

Fluctuating flow (with linearisation):

¯

∂v ∂vi ∂ v¯ i ∂ v¯ i ∂p + ¯ vj + = 0 (24) +v j i + v¯ j ∂t ∂xj ∂xj ∂xj ∂xi

¯

∂vi ∂v ∂p + vj i + =0 ∂t ∂xj ∂xi

(25)

∂v ∂p ∂p + v¯ i + ¯ c20 i = 0 ∂t ∂xi ∂xi

(26)

With constant mean flow (assumption: v¯ i is uniform): Wave equation Following from the continuity equation

∂vi ∂v ∂p + vj i + =0 ∂t ∂xj ∂xi ∂ 2 1 ∂ ∂ 2 p + v ¯ − p =0 Result: convective wave equation: i ∂xi ∂xi2 c20 ∂t 2 ∂ 2 ∂2 ∂2 ∂ ∂ + v¯ i + v¯ i v¯ j = + 2v¯ i with: ∂t ∂xi ∂t2 ∂xi ∂t ∂xi ∂xj and the momentum equations

¯

(27) (28) (29)

Navier–Stokes equation: Double decomposition of quantities and time averaging: Reynolds averaging Assumptions: for the mean flow:

with stress tensor of fluctuating flow:

incompressible flow v¯ j

2

∂ vi vj

∂ v¯i 1 ∂ p¯ ∂ v¯ i =− +Œ 2 − ∂xj ∂xi ∂xj ⎛ ⎛ ⎞ x ‘xy ‘xz ⎝ ‘xy y ‘yz ⎠ = − ⎝ ‘xz ‘yz z

∂xj u2 u v u w

Reynolds equation u v v 2 v w

Triple decomposition of quantities and time averaging (Telionis): f (x, t) = f¯ (x) + ˜f (x , t) + f (x , t)

⎞ u w v w ⎠ w 2

(30) (31)

Flow Acoustics

2

1 ∂ p¯ ∂ v¯ i ∂ v¯ i v¯ j =− +Œ 2 − ∂xj ∂xi ∂xj

for the mean flow:

∂ vi vj ∂xj

∂ v˜ i v˜ j − ∂xj

N

953

(32)

with two Reynolds stress terms on the right-hand side of the equation: non-linear contributions due to the random fluctuations and due to organised fluctuations; for the organised fluctuations:

∂ vi vj ∂ vi vj ∂ v˜i ∂ v˜i ∂ v¯ i ∂ v˜i 1 ∂ p˜ ∂ 2 v˜i ∂ v˜ i v˜ j + v¯ j + v˜ j + v˜j =− +Œ 2 + + − (33) ∂t ∂xj ∂xj ∂xj ∂xi ∂xj ∂xj ∂xj ∂xj

N.2.4 Correlations 1 T→∞ T

T

Rp (‘) = lim

p(t)p(t + ‘)dt

autocorrelation function

0

Rp (0) = p2 (t) 1 Rp () = L

(35)

L p(x)p(x + )dx

1 Rp (0)

x =

correlation function, fluctuations of velocity

(37)

Scales

1 ‘c = Rp (0) =

(36)

0

v v R= 1 2 v12 v22

N.2.5

(34)

∞ Rp (‘)d‘

integral time scale

(38)

Rp ()d

integral length scale

(39)

0

∞ 0

E11 (k1 )k1 →0 2 4vrms

integral length scale, limiting value of the power spectrum as k1 approaches zero

(40)

954

N

v 2 E(k) = rms ke

Flow Acoustics

4 2 k ke −2 (k/kŒ ) e power spectral density for isotropic 17/6 2 1+ k ke turbulence (von Kármán spectrum) (41) with (Longatte): ≈ 1.453 ke ≈ 0.747/ = (v 2 )3/2 — — = dissipation rate of turbulent kinetic energy kŒ = (— Œ 3 )1/4

p2 (t) l‘2 = ∂p 2 ∂t l‘2 = −

differential time scale

Rp (0) ∂ 2 Rp (0) ∂‘2

(42)

(43)

N.3 The Basic Equations of Fluid Motion

See also: Bangalore/Morris (1996); Bailly/Lafon/Candel (1996)

N.3.1 Continuity Equation, Momentum Equation, Energy Equation Continuity equation:

∂ ∂ vi + =0 ∂t ∂xi

respectively

∂ + div v = 0 ∂t

∂ ∂ ∂vi + + vj =0 ∂t ∂xi ∂xj ∂vi D + =0 Dt ∂xi

(1) (2)

with

∂ ∂ D = + vj Dt ∂t ∂xj

(3)

Momentum equation: Euler equations (viscous terms are neglected): 1 ∂ v ∂vi 1 ∂p ∂vi + (v · ∇) v + gradp = 0 + vj + = 0 respectively ∂t ∂t ∂xj ∂xi

(4)

Dvi 1 ∂p + =0 Dt ∂xi

(5)

Reformulation with help of continuity equation:

∂ vi vj ∂ vi ∂p + + =0 ∂t ∂xj ∂xi

(6)

Flow Acoustics

Energy equation:

∂e ∂ vi e + p = 0 + ∂t ∂xi 1 with: e = u + |vi |2 2 p u= (‰ − 1) with:

N

955

∂e + div v e + p = 0 ∂t

(7)

fluid energy density (per unit volume)

(8)

specific internal energy

(9)

respectively

u = cv T = c v

1 p p p = cv = R cv (‰ − 1) (‰ − 1)

p 1 + |vi |2 ‰−1 2 Reformulation with help of continuity and momentum equation: ∂vi ∂vi Dp Du +p + ‰p = 0 respectively =0 Dt ∂xi Dt ∂xi e=

(10) (11)

(12)

N.3.2 Thermodynamic Relationships The law of energy conservation with: dq supplied heat du internal energy pdv mechanical work dr friction loss

dq = du + pdv − dr

(13)

Internal energy: u specific internal energy per unit volume 1 du = Tds − pd = cv dT u=

p (‰ − 1)

(14) (15)

Fluid energy: e fluid energy density per unit volume 1 p 1 + |v |2 e = u + |v |2 = 2 ‰−1 2

(16)

Enthalpy: 1 B = h + v 2 (= hs ) total enthalpy, stagnation enthalpy, per unit volume 2 p specific enthalpy per unit volume h=u+ dh = Tds +

dp = cp dT

(17) (18) (19)

N

956

Flow Acoustics

Entropy: s specific entropy per unit volume ds =

dq + dr T

(20)

Further relationships: p = RT

equation of state of an ideal gas

R = cp − cv

gas constant, equal to the difference in the specific heats (22)

cp ‰= cv

ratio of the specific heats

(23)

isentropic speed of sound

(24)

c2 = ‰

p0 0

(21)

Pressure-density relation: ∂T ∂ dp dp T dp ∂ ∂ d = dp + ds = 2 + ds = 2 − ds = 2 − ds (25) ∂p s ∂s p c ∂T p ∂s p c cp c cp with: 1 ∂ = 2 ∂p s c ∂s cp = T ∂T p 1 1 ∂ = =− ∂T p T

isentropic sound speed c

(26)

specific heat at constant pressure

(27)

coefficient of expansion

(28)

N.3.3 Non-linear Perturbation Equations, non-linear Euler Equations General form of the three-dimensional fluid flow equations: ∂E ∂F ∂G ∂H + + + =0 ∂t ∂x ∂y ∂z

(29)

The general flow quantity A is split into a mean quantity A0 and a perturbation quantity A , so that the non-linear disturbance equations follow: ∂E ∂F ∂G ∂H ∂Fn ∂Gn ∂Hn ∂F0 ∂G0 ∂H0 + + + + + + =Q=− + + (30) ∂t ∂x ∂y ∂z ∂x ∂y ∂z ∂x ∂y ∂z with: F , G , H linear perturbation terms Fn , Gn , Hn non-linear perturbation terms F0 , G0, H0 Q: sum of the divergence of mean convective fluxes. The abbreviations are calculated from the following three-dimensional equations

Flow Acoustics

Continuity equation:

∂ v ∂ w ∂ ∂ u + + + =0 ∂t ∂x ∂y ∂z Momentum equations:

∂ u ∂ uu ∂ uv ∂ uw ∂p + + + + =0 ∂t ∂x ∂y ∂z ∂x

∂ uv ∂ vv ∂ wv ∂ v ∂p + + + + =0 ∂t ∂x ∂y ∂z ∂y

∂ uw ∂ vw ∂ ww ∂ w ∂p + + + + =0 ∂t ∂x ∂y ∂z ∂z Energy equation:

∂ v e+p ∂ w e+p ∂e ∂ u e + p + + + =0 ∂t ∂x ∂y ∂z

N

957

(31)

(32)

(33)

p 1 + u2 + v 2 + w2 (34) ‰−1 2 The abbreviations in the general form of the non-linear disturbance equations mean in the sequence of the equations below: • Continuity equation • Euler equation in x direction • Euler equation in y direction • Euler equation in z direction • Energy equation ⎧ ⎧ ⎪ ⎪ ⎪ ⎪ 02u + u0 ⎪ ⎪ ⎪ ⎪ ⎨ 0 u + u0 + u ⎨ u0 + 20 u0u + p (35,36) 0 v + v0 + v F 0 u0v + 0 v0 u + u0v0 E ⎪ ⎪ ⎪ ⎪ w + w + w u w + w u + u w ⎪ ⎪ 0 0 0 0 0 0 0 0 ⎪ ⎪

⎩ ⎩ e u e0 + p0 + u0 elin + p ⎧ ⎧ 0 v + v0 0 w + w0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 0 u0 u + 0 u0v + u0v0 ⎨ 0 w0 u + 0 u0 w + u0 w0 2 v + 2 v v + p 0 w0 v + 0 v0w + v0 w0 G H (37,38) 0 0 0 ⎪ ⎪ 2 ⎪ ⎪ v w + w v + v w w + 2 w w + p ⎪ 0 0 ⎪ 0 0 0 0 0 0 0 ⎪ ⎪

⎩ ⎩ v e0 + p0 + v0 elin + p w e0 + p0 + w0 elin + p ⎧ u ⎪ ⎪ ⎪ ⎪ ⎨ 2u0 u + 0 u 2 + u2 0 u v + u0 v + v0 u + u v (39) Fn ⎪ ⎪ u w + u w + w u + u w ⎪ 0 0 ⎪

0 ⎩ u e + p + u0 enonlin

with:

e=

N

958

Flow Acoustics

⎧ v ⎪ ⎪ ⎪ ⎪ ⎨ 0 u v + u0 v + v0 u + u v Gn 2v0 v + 0 v 2 + v 2 ⎪ ⎪ 0 v w + v 0 w + w0 v + v w ⎪ ⎪ ⎩ v e + p + v0enonlin ⎧ w ⎪ ⎪ ⎪ ⎪ ⎨ 0 u w + u0 w + w0 u + u w 0 v w + v0 w + w0 v + v w Hn ⎪ ⎪ ⎪ 2w 0 w + 0 w 2 + w 2 ⎪ ⎩ w e + p + w0 enonlin ⎧ ⎧ 0 u0 0 v0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 0 u02 + p0 ⎨ 0 u 0 v 0 G0 0 u0 v0 0 v02 + p0 F0 ⎪ ⎪ ⎪ ⎪ 0 u 0 w0

0 v 0 w0

⎪ ⎪ ⎪ ⎪ ⎩ ⎩ u0 e0 + p0 v0 e0 + p0

(40)

(41)

⎧ 0 w0 ⎪ ⎪ ⎪ ⎪ ⎨ 0 u0w0 H0 0 v0w0 ⎪ ⎪ 0 w 02 + p0

⎪ ⎪ ⎩ w0 e0 + p0

(42,43,44)

Fluctuating part of energy density: e =

1

1 p + u02 + v02 + w02 + 0 + u 2 + v 2 + w 2 ‰−1 2 2

+ 0 + (u u0 + v v0 + w w0 )

subdivided in two terms: • linear term

1 p + u02 + v02 + w02 + 0 u u0 + v v0 + w w0 elin = ‰−1 2

(45)

(46)

• non-linear term enonlin

1 0 + u 2 + v 2 + w 2 = 2

(47)

Stationary part of energy density: e0 =

1 p0 + 0 u02 + v02 + w02 ‰−1 2

N.3.4

(48)

Formulation of Euler Equations to Use in Computational Aeroacoustics (CAA)

Basic equations: D ∂vi =0 + Dt ∂xi

(49)

Dvi 1 ∂p + =0 Dt ∂xi

(50)

Decomposition:

v (x , t) = v0 (x , t) + v (x, t)

(51)

Flow Acoustics

with:

v0 (x , t) =

1 T

N

t+T v (x , t) dt

(52)

t

T 1 T T2 ≈ T1 time scale of turbulent fluctuations

(53)

time scale of large variations in mean flow

T2 1 T

959

t+T v (x , t) dt = v = 0

(54)

t

The short forms v = v0 + v ; • in the continuity equation:

p = p0 + p ;

= 0 + are introduced:

(55)

∂v ∂v ∂0 ∂ ∂ ∂0 ∂v0i ∂0 ∂ ∂v0i + v0j − v0j + vj + 0 i + =− − vj − 0 − i (56) ∂t ∂xj ∂xj ∂xi ∂xi ∂t ∂xj ∂xj ∂xi ∂xi with condition of the incompressibility:

0 = const. and

∂v0i = 0: ∂xi

∂v ∂v ∂ ∂ ∂ + 0 i = −vj − i + v0j ∂t ∂xj ∂xi ∂xj ∂xi

(57)

• in the momentum equation: ∂vi ∂v ∂v0i ∂v0i ∂p0 1 ∂p ∂v0i ∂vi ∂p 1 ∂p0 − 2 + =− −vj + − + v0j i +vj − v0j ∂t ∂xj ∂xj 0 ∂xi 0 ∂xi ∂t ∂xj ∂xj 02 ∂xi 0 ∂xi

(58)

1 1 1 0 ≈ − 1 − = 0 0 0 02 1+ 0

(59)

with:

1 1 = = 0 + ( 0 )

Averaging over time T1 : ∂v ∂p 1 ∂p0 ∂v0i ∂v0i + v0j + = −vj i + 2 ∂t ∂xj 0 ∂xi ∂xj 0 ∂xi

(60)

∂vi ∂v ∂v ∂p ∂v ∂p 1 ∂p ∂v0i ∂p0 − 2 + = −vj i + 2 + vj i − 2 + v0j i + vj ∂t ∂xj ∂xj ∂xj 0 ∂xi ∂xj 0 ∂xi 0 ∂xi 0 ∂xi

(61)

Introducing decomposition in turbulent and acoustic fluctuating quantities: = t + a ;

v = vt + va ;

p = pt + pa

(62)

furthermore introducing the approximate condition of incompressibility of turbulent flow: t ≈ 0 and ∂vti ∂xi = 0 results:

N

960

Flow Acoustics

• for the continuity equation: ∂a ∂a ∂vai cont.eq. cont.eq. + v0j + 0 = Qat + Qaa ∂t ∂xj ∂xi with:

cont.eq.

Qat

= −vtj

cont.eq.

Qaa

∂a ∗) ∂xj

= −vaj

∂a ∂vai − a ∂xj ∂xi

(63) (64) (65)

• for the momentum equation: ∂vai ∂vai 1 ∂pa ∂v0i a ∂p0 mom.eq. mom.eq. mom.eq. mom.eq. +v0j + +vaj − = Qot +Qtt +Qat +Qaa (66) ∂t ∂xj 0 ∂xi ∂xj 02 ∂xi with:

mom.eq.

Q0t

=−

∂vti 1 ∂pt ∂vti ∂v0i − v0j − vtj − ∂t ∂xj ∂xj 0 ∂xi

(67)

sound source due to the interaction between the mean flow and turbulent flow: partly shear noise; ∂vti ∂vti mom.eq. Qtt = −vtj + vtj (68) ∂xj ∂xj sound generated by turbulent interaction: self-noise; ∂vai ∂vti a ∂pt ∂vai ∂vti a ∂pt mom.eq. Qat = −vtj − vaj + 2 + vtj + vaj − ∂xj ∂xj 0 ∂xi ∂xj ∂xj 02 ∂xi sound generated by interaction between turbulence and sound; ∂vai a ∂pa ∂vai a ∂pa mom.eq. = −vaj + 2 + vaj − 2 Qaa ∂xj ∂xj 0 ∂xi 0 ∂xi

(69)

(70)

sound generated by sound, e.g. scattering of sound.

N.4 The Equations of Linear Acoustics The basic equations of fluid mechanics to use in the acoustics are as follows: • equation of continuity, law of conservation of mass: ∂ ∂(vi ) ˙ + =M ∂t ∂xi

(in tensor notation, double suffix summation convention)

˙ with: M

mass flux per unit volume

(1)

• equation of motion, law of conservation of momentum, Newton’s law of motion, momentum equation, Navier–Stokes equations (for a viscous fluid):

∂ ∂vi ∂vi + vj pƒij − ‘ij = Fi − ∂t ∂xj ∂xj ∗)


(2)

Flow Acoustics

N

961

and in the form due to Reynolds:

∂

∂ ˙ i vi + vi vj + pij = Fi + Mv ∂t ∂xj with:

pij = pƒij − ‘ij

(3) (4)

• equation of state (the pressure-density): d =

dp − ds c2 cp

(5)

and with the isentropic condition ds = 0: d =

dp c2

(6)

Premises of linear acoustics: Assumptions, applied to the basic equations of fluid mechanics: ˙ = 0; • without mass sources: M • without external forces: Fi = 0; • inviscid flow: pij = p (‘ij = 0); • decomposition of all physical quantities in mean values and fluctuating components: p = p¯ + p =¯+ vi = v¯ i + vi with the following assumptions and definitions: p¯ , ¯ constant in time and space; p p¯ (with: p sound pressure); ¯ (with: acoustic fluctuation of mass density); v¯ i = 0 without mean flow; vi acoustic part of fluid velocity, particle velocity; • linearisation of equations of fluid mechanics; • irrotational flow: – = rotv = ∇ × v = 0. The basic equations of linear acoustics: ∂v ∂ + ¯ i = 0 linearised continuity equation; ∂t ∂xi ¯

∂vi ∂p + = 0 linearised Euler equation (momentum equation); ∂t ∂xi

p = c2 with: p , , vi

linearised equation of state sound pressure, acoustic density fluctuation, particle velocity.

(7) (8) (9)

N

962

Flow Acoustics

The homogeneous wave equation of linear acoustics; there are the following homogeneous wave equations: ∂ 2p 1 ∂ 2p • pressure fluctuations (sound pressure): − 2 2 =0 2 ∂xi c0 ∂t 1 ∂2 ∂ 2 − 2 2 =0 2 ∂xi c0 ∂t

• density fluctuations: • velocity fluctuations: with: •

=

∂ 2 vi 1 ∂ 2 vi − 2 2 =0 2 ∂xj c0 ∂t

∂2 ∂2 ∂2 ∂2 = + + ∂x2 ∂y 2 ∂z2 ∂xi2

velocity potential:

∂2¥ 1 ∂ 2¥ − =0 ∂xi2 c20 ∂t2

Wave equation for uniform flow: 1 ∂ U∞ ∂ 2 ¥ − + ¥ =0 c0 ∂t c0 ∂x ¥ −

1 D2 ¥ =0 c20 Dt2

with:

D2 = Dt2

resp.

v −

(11) 1 ∂ 2 v =0 c20 ∂t2

Laplace operator resp.

¥ −

(10)

(12) (13)

1 ∂ 2¥ =0 c20 ∂t2

convective wave equation

(14)

(15) (16)

∂ ∂ + U∞ ∂t ∂x

2 =

2 ∂2 ∂2 2 ∂ + U + 2U ∞ ∞ ∂t2 ∂x∂t ∂x2

U∞ uniform time-averaged velocity in the x direction; 1 ∂2¥ U∞ ∂ 2 ¥ U2 ∂ 2 ¥ ∂ 2 ¥ ∂ 2 ¥ − 2 2 =0 + + 2 −2 2 1− ∞ 2 2 2 c0 ∂x ∂y ∂z c0 ∂x∂t c0 ∂t $ % 1 ∂ 2 ∂ + − M ¥ =0 ∂x c0 ∂t

(17)

with:

M=

U∞ c0

¥ k = –/c

(19)

Mach number.

With harmonic components, separating the time factor ej–t : & ' ∂2 ∂ − M2 2 − 2jkM + k 2 ¥ = 0 ∂x ∂x with:

(18)

(20)

complex amplitude of the velocity potential, wave number.

Cylindrical co-ordinate system r, x, Ÿ: (with Laplace operator in cylindrical co-ordinates) 1 D2 ¥ = 0 convective wave equation of velocity potential ¥ − 2 c0 Dt2

(21)

Flow Acoustics

N

becomes 2 ∂ 1 ∂2 ∂2 1 ∂ 1 D2 ¥ = 0. + + + − ∂x2 ∂r2 r ∂r r2 ∂Ÿ 2 c20 Dt2

(22)

Spherical co-ordinate system r, ˜ , œ (with Laplace operator in spherical co-ordinates): & ' 1 ∂ 1 ∂ 1 D2 1 ∂ ∂2 2 ∂ r + sin ˜ − + ¥ = 0. r2 ∂r ∂r ∂˜ c20 Dt2 r2 sin2 ˜ ∂œ2 r2 sin ˜ ∂˜

N.5

963

(23)

Inhomogeneous Wave Equation, Lighthill’s Acoustic Analogy

See also: Lighthill (1952); Howe (1998); Crighton et al. (1992); Goldstein (1976); Curle (1955)

N.5.1 Lighthill’s Inhomogeneous Wave Equation Equation of continuity, the law of conservation of mass: ∂ ∂(vi ) + =0 ∂t ∂xi

(1)

Equation of motion, in fact in the form due to Reynolds:

∂ ∂

vi vj + pij = 0 vi + ∂t ∂xj with:

pij = pƒij − ‘ij ∂vi ∂vj 2 ∂vk ‘ij = ‹ + ƒij − ‹ ∂xj ∂xi 3 ∂xk

(2) compressive stress tensor

(3)

viscous stresses

(4)

Lighthill’s equation is obtained by taking the time derivative of the continuity equation and subtracting the divergence of the momentum equation:

∂ ∂ momentum equation continuity equation − ∂t ∂xi eliminating the term vi , that is the mass density flux in the continuity equation but the momentum density in the momentum equation:

∂2 ∂ 2 vi vj + pij = 2 ∂t ∂xi ∂xj

(5)

∂ 2 (with c0 characteristic speed of sound in the medium ∂xi2 surrounding the flow region), gives

addition of the term −c20

2

∂2 ∂ 2 2∂ − c = vi vj + pij − c20 ƒij = q 0 2 ∂t2 ∂xi ∂xj ∂xi

Lighthill’s inhomogeneous wave equation

(6)

N

964

with:

Flow Acoustics

Tij = vi vj + pij − c20 ƒij = vi vj + p − c20 − ‘ij Tij q

(7)

Lighthill’s stress tensor source term.

The fluid mechanical problem of calculating the aerodynamic sound is formally equivalent to solving this equation for radiation into a stationary ideal fluid produced by a distributionof quadrupole sources whose strength per unit volume is the Lighthill stress tensor Tij (Howe). Incompressible approximation: Tij ≈ 0 vi vj (8) with assumptions: • low Mach number, velocity fluctuations are of order 0 Ma2 , • isentropic flow, • high Reynolds number, viscous effects are much smaller than inertial effects, the viscous stress tensor is neglected compared with the Reynolds stresses vi vj , • furthermore: viscous terms in Tij are ∂ 2 Tij ∂vi ∂ 3 vi , so that =‹ , (9) ‘ij = ‹ ∂xj ∂xi ∂xj ∂xj ∂xi ∂xj corresponding to a octupole source (a very ineffective sound radiator). Tij ≈ 0 vi vj can be used as a source term, generating the acoustic field. Lighthill’s development can be expanded as an inhomogeneous wave equation in general form, starting from: • equation of continuity with external sources: ∂ ∂(vi ) ˙ =M + ∂t ∂xi with:

˙ M

(10)

external source flux of mass (per unit volume);

• equation of motion with external forces:

∂vi ∂ ∂vi = Fi − pƒij − ‘ij + vj ∂t ∂xj ∂xj

with:

Fi

(11)

external forces (per unit volume),

or in the form due to Reynolds:

∂

∂ ˙ i. vi + vi vj + pij = Fi + Mv ∂t ∂xj

(12)

Inhomogeneous wave equation in general form: ˙

∂ ∂2 ∂2 ∂M ∂ 2 ˙ i + − c20 2 = − Fi + Mv vi vj + pij − c20 ƒij = q 2 ∂t ∂t ∂xi ∂xi ∂xj ∂xi

(13)

Source terms: q=

˙

∂ ∂2 ∂M ˙ i + Fi + − Tij Mv ∂t ∂xi ∂xi ∂xj

(14)

Flow Acoustics

with:

˙ ∂M ∂t

∂ ˙ i − Fi + Mv ∂xi ∂ 2 Tij ∂xi ∂xj

N

965

monopole source

(15)

dipole source

(16)

quadrupole source

(17)

N.5.2 Solutions of Inhomogeneous Wave Equation Using the generalisation of Kirchhoff ’s equation: 1 ∂p q 1 1 1 ∂r ∂p 1 ∂r + 2 p+ dS + dV, p(xi , t) = 4 c0 r ∂n ∂t r ∂n r ∂n ‘ 4 r ‘ S

(18)

V

a formal solution of the inhomogeneous wave equation follows: ˙ ∂M ∂(vi ) 1 1 p (xi , t) = dV − ni dS 4r ∂t ‘ 4r ∂t ‘ V S

∂ ∂ 1 1 ˙ i dV + Fi + Mv vi vj + pij ‘ nj dS − ‘ ∂xi 4r ∂xi 4r V S ∂2 1

+ Tij ‘ dV ∂xi ∂xj 4r

(19)

V

with:

(....)‘ r ‘=t− c0 r = xi − y i xi yi

retarded source strength, retardation time, distance between source point and observer point, vector from origin of co-ordinates to the observer point, vector from origin of co-ordinates to the source point.

Free-space solution: ˙

1 1 1

∂ ∂2 ∂M ˙ i dV+ Fi + Mv Tij ‘ dV (20) dV− p (xi , t) = ‘ 4r ∂t ‘ ∂xi 4r ∂xi ∂xj 4r V

V

V

Decomposition into the near-field and the far-field solution: ˙ ∂M 1 p (xi , t) = dV 4r ∂t ‘ V

(xi − yi )

(xi − yi ) 1 1∂ 1 ˙ ˙ i dV+ Fi + Mv dV Fi + Mvi ‘ ‘ 2 r r 4c0 r ∂t r V V 1 1 3 Tij ‘ 2 (xi − yi )(xj − yj ) − ƒij dV + 4 r2 r V

1 ∂ 2 Tij ‘ 1 1 (x − y )(x − y ) dV + i i j j r ∂t2 r2 4c20 1 + 4

V

(21)

966

N

Flow Acoustics

with the far-field solution: ˙

∂M 1 1∂ 1 ˙ i (xi − yi ) dV Fi + Mv p (xi , t) = dV + ‘ 4r ∂t ‘ 4c0 r ∂t r V V

2 1 1 ∂ Tij ‘ 1 + (xi − yi )(xj − yj) dV 2 r ∂t2 r2 4c0

(22)

V

Solution with solid boundaries in flow (Curle): ˙ ∂M ∂(vi ) 1 1 p (xi , t) = dV − ni dS 4r ∂t ‘ 4r ∂t ‘ V S

1 1 ∂ ∂ ˙ Fi + Mvi ‘ dV + vi vj + pij ‘ nj dS − ∂xi 4r ∂xi 4r V S 2

1 ∂ Tij ‘ dV + ∂xi ∂xj 4r

(23)

V

with:

surface S is stationary, the body can have mass injection or suction: vi ni = vn = 0, ∂vn = 0. body vibrations: ∂t Solution when the boundaries S are rigid and impermeable: ˙ ∂M 1 p (xi , t) = dV 4r ∂t ‘ V

1 1

∂ ∂ ˙ i dV + Fi + Mv pij ‘ nj dS − ‘ ∂xi 4r ∂xi 4r V S 1

∂2 Tij ‘ dV + ∂xi ∂xj 4r

(24)

V

Sound from free turbulence: 1 ∂ 2 Tij 1 dV p (xi , t) = 4 r ∂yi ∂yj ‘

(25)

V

p (xi , t) =

1 ∂2 4 ∂xi ∂xj

1 p (xi , t) = 4c20

V

1 r

V

(

in the far field xi yi : p (xi , t) =

1 4r

1

Tij ‘ dV r

(26)

) ∂ 2 Tij 1 (xi − yi )(xj − yj ) dV ∂t2 r2

(27)

‘

V

∂ 2 Tij ∂yi ∂yj

dV ‘

(28)

Flow Acoustics

1 ∂2 p (xi , t) = 4r ∂xi ∂xj 1 xi xj p (xi , t) = 4c20 r r2 p (xi , t) =

1 4c20 r

V

$ V

V

N

Tij ‘ dV

967

(29)

% ∂ 2 Tij dV ∂t2

(30)

‘

∂ 2 (Trr ) ∂t2

dV

(31)

‘

xi xj with: Trr = 2 Tij r Solutions using a Fourier transformation: –2 e−j–x/c0 p (xi , –) = − r 4c20

T 0

ejk·y e−j–t Trr yi , t dVdt

(32)

V

Solutions neglecting viscous stresses: Tij = vi vj :

2 ∂ vi vj ‘ xi xj dV p (xi , t) = ∂t2 4c20 r3

(33)

V

N.6

Acoustic Analogy with Source Terms Using Pressure

See also: Ribner (1959); Meecham (1958, 1981)

N.6.1 Lighthill’s Representation of the Source Term with Use of Pressure Reformulation of the source strength using equations of continuity and momentum to introduce the pressure as a source (see Lighthill):

∂vj ∂vi ∂ ∂ vi vj = pik + pjk − vi vj vk + pik vj + pjk vi , ∂t ∂xk ∂xk ∂xk neglecting the viscous stresses and the octupole source:

∂vi ∂vj ∂ vi vj ≈ p , + ∂t ∂xj ∂xi

(1)

(2)

calculating the source term:

∂2 ∂2 Tij ≈ 2 vi vj 2 ∂t ∂t $ * +% ∂vi ∂vj ∂p ∂ v¯ i ∂ v¯ j ∂ ∂ ∂vi ∂vj p p = + ≈ + + + ∂t ∂xj ∂xi ∂t ∂xj ∂xi ∂t ∂xj ∂xi first term: second term:

shear noise, self-noise.

(3)

968

N

Flow Acoustics

N.6.2 Pressure-Source theory (Ribner) Decomposition of pressure fluctuations inside flow p − p0 = p + pa with: p − p0 pressure fluctuations, p0 constant pressure, p pseudo-sound: pressure fluctuations in a nearly incompressible flow, that is, inside the flow pressure fluctuations are dominated by inertial effects rather than compressibility, acoustic pressure. pa Pseudo-sound pressure is a solution of Poisson’s equation: application of divergence operator to the momentum equation

∂ ∂2 ∂

vi + vi vj + pij = 0 ∂xi ∂t ∂xi ∂xj Incompressible flow

(4)

∂vi = 0 (without viscosity) leads to the Poisson equations: ∂xi

∂ 2 vi vj ∂ 2p =− ∂xi ∂xj ∂xi2

∂ 2 vi vj ∂ 2 p =− ∂xi ∂xj ∂xi2

∂ 2 v i vj ∂ 2 p ∂vi ∂vj = −0 = −0 2 ∂xi ∂xj ∂xj ∂xi ∂xi

(5)

(6)

(7)

Inhomogeneous wave equation: 1 ∂ 2 p ∂ 2 p ∂ 2p − = ∂xi2 c20 ∂t2 ∂xi2

(8)

and with p − p0 = p + pa : 1 ∂ 2 pa 1 ∂ 2 p ∂ 2 pa − = Ribner’s inhomogeneous wave equation ∂xi2 c20 ∂t2 c20 ∂t2 The far-field solution: 1 ∂ 2 p 1 dV p (xi , t) = − 4c20 r ∂t2 ‘

(9)

(10)

V

in comparison with Lighthill: 1 1 ∂ 2 Tij ‘ 1 p (xi , t) = (xi − yi )(xj − yj ) dV r ∂t2 r2 4c20 V

(In far field both equations are identical!)

(11)

Flow Acoustics

N.6.3

N

969

Pressure-Source Theory (Meecham)

Expanding the field about an incompressible flow, low Mach number fluctuating flow p = p0 + pa

(12)

= 0 + 1 + a

(13)

v = v 0 + va

(14)

with:

subscript 0 pa , a , va

for incompressible flow quantities acoustic field

Definition of 1 : The change in density 1 is caused by the nearly incompressible pressure change p0 : ¯ 0 ¯ 1 D 1 Dp = 2 Dt c Dt with:

(15)

¯ D ∂ = + v0 · ∇ Dt ∂t

substantial derivative, following the incompressible flow v0 .

Wave equation: ∂ 2 1 ∂ v0 1 ∂ 2 pa 2 − ∇ pa = − 2 + ∇ · 1 ∂t ∂t c20 ∂t2

(16)

The second source term on the right side may be neglected. ∂p0 ∂1 = c20 ∂t ∂t

Reformulation with

(17)

leads to: 1 ∂ 2 pa 1 ∂ 2 p0 2 − ∇ p = − a c20 ∂t2 c20 ∂t2 Solution: 1 p (xi , t) = − 4c20

V

1 r

∂ 2 p0 ∂t2

Meecham’s inhomogeneous wave equation.

(18)

dV

(19)

‘

(see Ribner’s solution mentioned above). The incompressible pressure fluctuations are the solution to Poisson’s differential equation: ∂ 2 vi vj ∂ 2 p0 = −0 . 2 ∂xi ∂xj ∂xi

(20)

970

N.7

N

Flow Acoustics

Acoustic Analogy with Mean Flow Effects, in the Form of Convective Inhomogeneous Wave Equation

See also: Phillips (1960); Pao (1972); Lilley (1958, 1993, 1999);Legendre (1981); Morfey (2000); Goldstein/Howes (1973); Ribner (1996); Albring (1981); Detsch (1976); Dittmar (1983)

N.7.1 Phillips’s Convective Inhomogeneous Wave Equation

& ' ∂ ‰ ∂‘ij ∂vi ∂vj ∂ D ‰ Ds D2 ¢ 2 ∂¢ − c =‰ − + Dt2 ∂xi ∂xi ∂xj ∂xi Dt cp Dt ∂xi ∂xj & ' ∂ ‰ D ‰ ˙ − Fi M + Dt ∂xi p with: ¢ = ln definition of a dimensionless logarithmic pressure ratio p0 constant reference pressure p0 ∂vi ∂vj 2 ∂vk + ƒij viscous stresses − ‹ ‘ij = ‹ ∂xj ∂xi 3 ∂xk ∂ ∂ D = + vi Dt ∂t ∂xi Left-hand side: Right-hand side:

(1)

(2)

substantial derivative.

corresponding to a wave equation in a moving medium with variable speed of sound (Phillips); contains propagation terms (in the first member) and source terms, generation of pressure fluctuations by velocity fluctuations in the fluid, by effects of entropy fluctuations, of fluid viscosity and by external mass and force sources (Goldstein).

Neglecting the effects of viscosity and thermal conductivity, furthermore the external mass and force sources: ∂ ∂vi ∂vj D2 ¢ 2 ∂¢ − (3) c =‰ 2 Dt ∂xi ∂xi ∂xj ∂xi in comparison with Lighthill’s equation:

2 ∂ 2 vi vj ∂ 2 2∂ − c = 0 ∂t2 ∂xi ∂xj ∂xi2

(4)

(Both equations are identical with assumptions: low Mach number and constant speed of sound.) , v¯ i = v¯ x (y); 0; 0 Example: shear flow vi = v¯ i + vi ∂vj ∂vi ¯ 2¢ ∂ v¯ x ∂vy D ∂ ∂¢ 2 = 2‰ + ‰ − c (5) Dt2 ∂xi ∂xi ∂y ∂x ∂xi ∂xj ¯ ∂ ∂ D = + v¯ i with: Dt ∂t ∂xi c2

mean value the square of the speed of sound

Flow Acoustics

right-hand side:

first term: second term:

N

shear noise self noise

In comparison with the equation of sound propagation in non-uniform flow: $ % ∂vj ∂vi ∂vj ∂vi ¯ 2 ¢ ∂ ∂ v¯ x ∂vy D 2 ∂¢ +‰ − − c = 2‰ Dt2 ∂xi 0 ∂xi ∂y ∂x ∂xi ∂xj ∂xi ∂xj with:

¢ = ln

p ; p0

971

(6)

p = pa

N.7.2 Lilley’s Convective Inhomogeneous Wave Equation Continuity equation: with:

‰ Ds D¢ + ‰div v = Dt cp Dt p , ¢ = ln p0

(7)

following from continuity equation in the form 1 D + div v = 0 Dt with:

d 1 dp ds = − ‰ p cp

Momentum equation:

‰

∂¢ ‰ ∂‘ij Dvi = −c2 + Dt ∂xi ∂xj

Lilley’s wave equation: ∂vj ∂ ∂vj ∂vk ∂vi ∂ D D2 ¢ 2 ∂¢ 2 ∂¢ c +2 c = −2‰ − 2 Dt Dt ∂xi ∂xi ∂xi ∂xj ∂xi ∂xi ∂xj ∂xk & ' ∂vj ∂ ‰ ∂‘ij ∂ ‰ ∂‘ij D D2 ‰ Ds +2 − + 2 ∂xi ∂xj ∂xj D‘ ∂xi ∂xj Dt cp Dt Interpretation:

(8)

(9)

third-order equation, left-hand side contains all propagation effects, right-hand side includes all source terms.

Neglecting the effects of viscosity and thermal conductivity, furthermore introducing the mean values in the propagation terms on the left-hand sides, replacing vi by v¯ i and c2 by c2 : 2 ¯ D ¯ ¢ ∂ v¯ j ∂ ∂vj ∂vk ∂vi ∂ D ∂¢ ∂¢ 2 2 − c c (10) +2 = −2‰ Dt Dt2 ∂xi ∂xi ∂xi ∂xj ∂xi ∂xi ∂xj ∂xk Example: shear flow: ¯ D Dt

vi = v¯ i + vi

, v¯ i = v¯ x (y); 0; 0

¯ 2¢ ∂ v¯ x ∂ ∂ D 2 ∂¢ 2 ∂¢ c =q c +2 − Dt2 ∂xi ∂xi ∂y ∂x ∂y

(11)

N

972

with: q

Flow Acoustics

different source terms but no terms which are linear in fluctuating velocities.

In comparison with equation applicable to the sound propagation in a shear flow: ¯ ¯ 2 ¢ ∂ 2 ¢ ∂ v¯ x ∂ 2 ¢ D 1 D +2 − =0 (12) 2 2 2 Dt c0 Dt ∂y ∂x∂y ∂xi ¯ D Dt

¯ 2 pa ∂ 2 pa 1 D − c20 Dt2 ∂xi2

+2

∂ v¯ x ∂ 2 pa =0 ∂y ∂x∂y

(13)

(This equation follows as a result of Lilley’s equation with assumptions: p p0 + pa pa q=0 ¢ = ln = ln ≈ = ¢ with pa p0 c2 = c20 ) p0 p0 p0

N.7.3 Lilley’s Wave Equation with a New Lighthill Stress Tensor Definition of a new Lighthill stress tensor:

Tij = vivj − ‘ij + p − c2∞ ƒij

(14)

v

velocity fluctuations; that is: the Lighthill tensor involves only quadratic fluctuations of the velocity field. , Assuming a uniform mean flow: v¯ i = v¯ x (y); 0; 0 ; with:

decomposition:

vi = v¯ i + vi;

convective operator for mean flow

¯ ∂ D ∂ = + v¯ x (y) . Dt ∂t ∂x

This leads to the generalised linear convective wave equation, a third-order equation: ¯ D Dt

2 ¯2 ¯ ∂ 2 Tij dv¯ x ∂ Tyi D D ¯x ∂ 2 2 2 dv − c + 2c − 2 , = 0 0 Dt2 dy ∂x∂y Dt ∂xi ∂xj dy ∂x∂xj

left-side hand: right-hand side:

(15)

all linear fluctuating terms; generation terms, all quadratic in the fluctuations.

N.7.4 Convected Wave Equation for the Dilatation (Legendre) D Dt

with:

1 DŸ c2 Dt

1 ∂vi ∂vj ∇ ln h Dvi − c2 ∂xj ∂xi c2 Dt ∂vj ∂ vi Dvi 1 Dvi −2 − 2 c Dt ∂xi ∂xj c2 Dt

− Ÿ = −

D Dt

D 1 D =− ln Ÿ = ∇ · v = − Dt Dt 0 h specific enthalpy

dilatation

(16)

Flow Acoustics

N

973

N.7.5 Goldstein’s Third-Order Inhomogeneous Wave Equation Assumptions: parallel shear flow in the x direction, mean velocity U, density ¯ , sound speed c¯ : all independent of x and t: ¯ 1 D ¯ 2p

1 ∂ D − ¯ ∇ · ∇p +2 ∇U · ∇p 2 2 Dt c¯ Dt ¯ ∂x (17) & 2 ' ¯ ¯ q ∂ D D ∇ ·f +2 f · ∇U − = ¯ Dt2 Dt ∂x with:

¯ ∂ ∂ D = +U Dt ∂t ∂x

material derivative

f

force per unit mass

q

mass flux per unit mass

∇ · v = q −

1 D Dt

source term in the continuity equation

N.7.6 Goldstein-Howes Inhomogeneous Wave Equation Assumptions: high Reynolds number, no entropy fluctuations; wave operator is linearised by assuming that only the first-rder interaction between the mean flow and the fluctuating field is retained; the mean flow is decomposed into a mean part and a fluctuating part:

vi = U y ƒxi + vi . Introducing the convective derivative:

∂ ∂ D = +U , Dt ∂t ∂x

use of Phillips wave equation:

∂vj ∂vi D2 ¢ ∂U ∂vy 2 2 = ‰ − c ∇ ¢ − 2‰ , 0 Dt2 ∂y ∂x ∂xi ∂xj

convective derivative and introduction the momentum equation (no viscosity) leads to: + + * * & ' 2 ∂vy ∂U ∂ D D2 ¢ D ∂vi ∂vj 2 2 2 dU ∂ ¢ − 2‰ . (18) =‰ vi − c0 ∇ ¢ + 2c0 Dt Dt2 dy ∂x∂y Dt ∂xj ∂xi ∂y ∂x ∂xi This is an acoustic analogy, like Lighthill’s equation.

974

N

Flow Acoustics

Further assumptions: acoustic fluctuations are negligible compared to the turbulent fluctuations in the source volume;turbulent velocity is incompressible, these lead to another formulation of the wave equation: & ' 2 D D2 ¢ 2 2 2 dU ∂ ¢ − c ∇ ¢ + 2c 0 0 Dt Dt2 dy ∂x∂y * * + + (19) Dvi vj ∂vy ∂2 ∂U ∂ ∂ d2 U 2 v . =‰ − 4‰ vi −‰ ∂xi ∂xj Dt ∂y ∂x ∂xi ∂x dy 2 y New form of inhomogeneous convective wave equation: * + Dvi vj ∂ ∂U ∂ ∂2 c20 D2

2 2 ƒ − 4‰ − c ∇

= ‰ v + v yi 0 i y Dt2 ∂xi ∂xj Dt ∂x ∂y ∂xi ‰ 2 c20 ∂ dU 2 + v −‰ y ∂x dy 2 ‰ with new variable:

=

Alternatively: ∂2 D2

2 2 − c ∇

= ‰ 0 Dt2 ∂xi ∂xj

(20)

D¢ ∂¢ ∂¢ = +U Dt ∂t ∂x

*

Dvivj Dt

+

∂ + 4‰ ∂x

$

(21)

∂U Dvy ∂y Dt

% −‰

∂ ∂x

Simplification of source term:

d dy follows the source term q (right-hand side of wave equation): $ % * + Dvi vj ∂2 ∂ ∂U Dvy q=‰ + 4‰ . ∂xi ∂xj Dt ∂x ∂y Dt

In mixing layer of a jet: gradient of mean velocity is constant:

c20 d2 U 2 + (22) v y dy 2 ‰

dU dy

= 0 from which

(23)

N.7.7 Ribner’s Recent Reformulation of Lighthill’s Source Term Lighthill’s wave equation: ∂ 2 vi vj 1 ∂ 2p 1 ∂2p ∂2 − p = + − 2 ∂xi ∂xj ∂t c20 ∂t2 c20 ∂t2

(24)

Assumptions: • instantaneous local velocity:

vi = v¯ i + vi

• unidirectional, transversely sheared, mean flow:

v¯ i = U y , 0, 0

∂ 2 vivj ¯ 2 1 ∂ 2p 1 ∂ 2p D ∂U ∂vy + − p = + 2 − 2 2 ∂xi ∂xj ∂y ∂x Dt2 c0 ∂t2 c0 ∂t2

with:

¯ ∂ ∂ D = +U convective derivative following mean flow. Dt ∂t ∂x

(25)

Flow Acoustics

With approximation:

N

975

¯ 2p ¯ 2 1 D D = (¯c = c¯ (x) local time-averaged sound speed): Dt2 c¯ 2 Dt2

¯ 2p ∂ 2 vi vj ∂U ∂vy 1 D − p = + 2 c¯ 2 Dt2 ∂xi ∂xj ∂y ∂x

(26)

In the case of an exact wave equation with the following notations: vi = v¯ i + vi , vi av = v¯ i (x ), = ¯ (x ) + , av = ¯ (x), there follows the wave equation: ¯ 2 ∂ 2 vi vj 1 ∂ 2p ∂ v¯ i ∂ ∂ v¯ i ∂vj 1 ∂2p D − p = 2 + − + + 2vj 2 ∂t 2 2 2 2 ∂xj ∂xi c0 ∂t Dt ∂xi ∂xj ∂xj ∂xi c0 2 ∂vi vj ∂ ∂ ∂ ∂ v¯ i +2 + vi vj +2 vj ∂xj ∂xi ∂xi ∂xj ∂xj ∂xi ∂ v¯ j ∂ v¯i ∂ v¯ i ∂ v¯ j + + ∂xi ∂xj ∂xi ∂xj

¯2 D = Dt2

with:

N.7.8

∂ ∂ + v¯ i ∂t ∂xi

2 =

∂2 ∂2 ∂2 + v ¯ + 2 v ¯ v ¯ i i j ∂t2 ∂xi ∂t ∂xi ∂xj

(27)

(28)

Inhomogeneous Wave Equation Including Stream Function (Albring/Detsch)

Inhomogeneous wave equation in approximated form of Lighthill:

∂ 2 vi vj 1 ∂ 2p ∂ 2p − = 0 =q ∂xi ∂xj c20 ∂t2 ∂xi2

(29)

Introducing the stream function in a two-dimensional flow vx =

∂¦ ; ∂y

leads to: $ q = 20

vy = −

∂ 2¦ ∂x∂y

2

∂¦ ∂x

∂ 2¦ ∂ 2¦ − ∂x2 ∂y 2

% (30)

Decomposition ¦ = ¦¯ + ¦ gives for the source term: ( ) 2 2 ∂ 2 ¦¯ ∂ 2 ¦ ∂ 2 ¦¯ ∂ 2 ¦ ∂ 2 ¦¯ ∂ 2 ¦ ∂ 2 ¦ ∂ 2 ¦ ∂ ¦ + − − − q = 20 2 ∂x∂y ∂x∂y ∂x∂y ∂x2 ∂y 2 ∂y 2 ∂x2 ∂x2 ∂y 2 with:

term number 1, 3, 4: term number 2, 5:

shear noise self-noise.

(31)

976

N.8

N

Flow Acoustics

Acoustic Analogy in Terms of Vorticity, Wave Operators for Enthalpy

See also: Powell (1963, 1964); Howe (1975, 1998); Crightonet al. (1992); M¨ ohring(1978, 1999); Möhring/Obermeier (1980); Doak (1995, 1998)

N.8.1 Powell’s Theory of Vortex Sound Assumption: fluid motion is isentropic. Use of vector identities: 1 2 v = (v · ∇) v − (∇ × v ) × v = (v · ∇) v − – × v ∇ 2 with: – = ∇ × v vorticity vector, or: ∂vj ∂vi ∂vk ∂vi ∂vi 1 2 v = vj − − − vk − vj ∇ 2 ∂xj ∂xi ∂xk ∂xi ∂xj

(1)

(2)

of the continuity equation: ∂ + (v · ∇) + ∇ · v = 0, ∂t and of the momentum equation (neglecting the viscous stress tensor ‘ij ): 1 2 ∂ v + (– × v ) + ∇ |v | = −∇p ∂t 2

(3)

(4)

leads to an inhomogeneous wave equation: & 2 '

v2 ∂ v 1 ∂ 2 pa 2 − ∇ − v + ∇ p − c0 . − pa = ∇ (– × v ) + ∇ c20 ∂t2 2 2 ∂t With the assumption: |v | c0 Powell’s equation follows in the theory of vortex sound: & 2 ' v 1 ∂ 2 pa (5) − pa = ∇ (– × v ) + ∇ 2 c20 ∂t2 Solution in free space: ∂ 1 1 1 ∂2 p (x, t) = (– × v ) ‘ dV+ 2 |v |2 dV ∂xi 4r 4r 2 ∂xi ‘ V

0 xi ∂ p (x, t) = 4c0 r2 ∂t

V

V

0 ∂ 2 × v )i ]‘ dV+ [(– 4c20 r ∂t2

V

1 2 |v | dV 2 ‘

(6)

(7)

Source terms: first term: dipole source, vorticity distribution, incompressible approximation of fluctuating flow, rate of change of vortex stretching by fluid flow → principal source of sound at low Mach number second term: quadrupole source, isotropic quadrupole (three longitudinal quadrupoles with undirected radiation characteristic), rate of change of kinetic energy of source flow

Flow Acoustics

N.8.2

N

977

Howe’s Formulation of Acoustic Analogy Equation for Total Enthalpy

Momentum equation (Crocco’s formulation, neglecting effects of external forces): ∂vi ∂B ∂s 1 ∂‘ij = − (– × v ) + T + + ∂t ∂xi ∂xi ∂xj with:

(8)

thermodynamic relations: 1 B = h + v 2 total enthalpy, stagnation enthalpy, per unit volume 2 p specific enthalpy (per unit volume) h=u+ u

specific internal energy (per unit volume)

vector relations: – = rotv = curlv = ∇ × v vorticity Dv ∂ v ∂ v = + (v · ∇) v = +– × v + ∇ Dt ∂t ∂t L = (– × v )

1 2 v vector identity 2

Lamb vector, unsteady vortical lifting force (per mass unit)

1 v=– L = (v · ∇) × v + grad v 2 2

or

(9)

Continuity equation: 1 D + div v = 0 Dt with:

or

1 Dp T Ds + div v = c2 Dt cp Dt

(10)

D by means of relations: Dt dp dp dp T ∂ ∂ ∂T d = 2 + ds = 2 + ds = 2 − ds c ∂s p c ∂T p ∂s p c cp

elimination of

(11)

Subtracting the divergence of the momentum equation from the time derivative of the continuity equation gives: ∂ ∂t

with:

1 Dp c2 Dt

∂ − B = div (– × v − T∇s − ) + ∂t

1 ∂‘ij , ∂xj ‘ij = viscous stress tensor.

i =

T Ds cp Dt

(12)

978

N

Flow Acoustics

Howe’s inhomogeneous wave equation in terms of total enthalpy: & ' D 1 D ∇p · ∇ ∇p − − B = div + · (– × v − T∇s − ) Dt c2 Dt c2 c2 D 1 ∂ T Ds Ds + + v · + T ∂t cp Dt Dt c2 Dt Special cases: • Absence of viscous dissipation and heat transfer, with momentum equation Dv = −∇p: Dt & ' D 1 D 1 Dv 1 Dv + · ∇ − B = div − × v − T∇s) (– Dt c2 Dt c2 Dt c2 Dt • High Reynolds number, homentropic flow (dissipation and heat transfer are neglected, s = const.): & '

D 1 D 1 1 − ∇ · ∇ B = div – × v 2 Dt c Dt

(13)

(14)

(15)

• Low Mach number, = 0 and c = c0 , neglecting non-linear effects of propagation and scattering of sound by vorticity: 1 ∂2 − B = div (– × v ) (16) c20 ∂t2 • Non-homentropic source flow, fluid is temporarily incompressible, dissipation is ignored, mean flow is irrotational, mean velocity v (x), density (x), sound velocity c(x ): &

∂ + v · ∇ ∂t

1 c2

∂ + v · ∇ ∂t

'

1 − ∇ · ∇ B = div (– × v − T∇s) +

• At very low Mach number: ' & ∂ T Ds 1 ∂2 − B = div (– × v − T∇s) + ∂t cp Dt c20 ∂t2

∂ ∂t

T Ds cp Dt

(17)

(18)

right-hand side: sources of noise are vorticity, entropy gradients and unsteady heating of the fluid; the T Ds last source is equivalent to a volume monopole of strength q (xi , t) = . cp Dt

Flow Acoustics

N

979

• At low Mach number, mean density, entropy and sound velocity are constant, isentropic flow: ( ) 2 1 ∂ + v · ∇ − B = div (– × v ) (19) c20 ∂t The acoustic pressure p can be calculated from the fluctuations in total enthalpy by: p = B. 0

(20)

This is a linearised relation of acoustic pressure p in the far field, to the first order in Mach number, furthermore considering a very low mean flow Mach number. Formulation of the wave equation in terms of the pressure: ' & 1 ∂2 − p = 0 div (– × v ) c20 ∂t2

(21)

The right-hand side represents an aerodynamic source for incompressible flow; the generated sound is dipole in nature; – × v is a force distribution. Lamb vector: L = (– × v ), quantity L vortex force (per unit volume); v can be calculated directly from –: $ % –( y , t) dV v (x, t) = curl (Biot-Savart law). (22) 4 x − y Therefore the source term depends only on the vorticity. See also the Helmholtz vorticity equation: ∂– + curl (– × v ) = 0. ∂t

(23)

It is a direct relationship between the non-linear term – × v and the linear term in the vorticity. Solution:

1 ∂ r × v )i yi , t − dV (– r ∂yi c0 ∂ 1 r dV p (xi , t) = −0 × v )i yi , t − (– ∂xi r c0

p (xi , t) = −0

(24)

or:

(25)

In the far field (with the approximations of low Mach number flow, compact turbulent eddies): 0 xi ∂ r p (xi , t) = − × v )i yi , t − dV (26) (– 4c0 x2 ∂t c0

0 xi ∂ 2 r p (xi , t) = − dV (27) x · y x · – × v y , t − ( ) i c0 4c20 x3 ∂t2

980

N

Flow Acoustics

N.8.3 Mohring’s ¨ Equation with Source Term Linearly Dependent on Vorticity Field = grad G (with: G a scalar Green’s funcDefinition of a vector Green’s function: curl G tion). Solution of wave equation: 1 ∂2 − B = div (– × v ) (28) c20 ∂t2

Introduce a Green’s function G y , ‘ |x, t which satisfies & '

∂2 1 ∂2 − 2 G = ƒ x − y , t − ‘ . (29) 2 2 c ∂‘ ∂yi Use the combination

div (– × v ) G dVd‘

B (x, t) = V

in the far field

pa (x, t) = −0

× v )i (– V

with the vector Green’s function

∂G dVd‘ ∂yi

pa (x , t) = −0

with the Helmholtz vorticity equation to obtain

pa (x, t) = −0

pa (x , t) = 0

∂ ∂G dVd‘ = 0 – ∂‘ ∂t

(30)

V

dVd‘ × v ) curlG (–

(31) (32)

V

∂– GdVd‘ ∂‘

– GdVd‘.

(33) (34)

This equation does not contain the flow velocity. It depends linearly on the vorticity field, that is, the contributions from several vortices add linearly. Vorticity sound in a low Mach number flow in unbounded space (see Dowling): 0 ∂3 p(x , t) = (x · y )y · [– × x ] x dV 12c20 r3 ∂t3 t− c0

(35)

right-hand side: only components of vorticity perpendicular to the observer’s position vector x contribute to the sound far field.

N.8.4

Convected Wave Operators for Total Enthalpy in Comparison

Möhring: ∂ DB ∂ ∂ ∂s ∂ ∂s v DB − = −divL + +div v LMöhring B = ∇ · ∇B− 2 2 c Dt ∂t c Dt ∂t ∂s ∂t ∂s ∂t with: L = – × v − T∇s

(36)

Flow Acoustics

Möhring/Obermeier:

D LMöhring/Obermeier B = ∇ · ∇B − Dt

1 DB c2 Dt

= −divL −

N

∂ v ∂ ∇s ∂t ∂s

981

(37)

with: L = – × v = curlv × v Howe: 1 Dv D LHowe B = B − 2 · ∇B − c Dt Dt

1 DB c2 Dt

(38)

Doak:

1 ∂2B ∂ LDoak B = B − 2 × v + T∇s − 2∇h ∇B + v v · ∇ · ∇B + 2v + – c ∂t2 ∂t

(39)

Those terms of these three convected wave operators which contain second derivatives of B agree. This means that they agree in the high frequency limit of geometrical acoustics (Möhring/Obermeier).

N.8.5 Doak’s Theory of Aerodynamic Sound Including the Fluctuating Total Enthalpy as a Basic Generalised Acoustic Field for a Fluid Continuity equation:

∂ ∂ vi ˙ + =M ∂t ∂xi with:

˙ M

(40)

rate of mass creation per unit volume

Momentum equation:

∂ vi ∂ vi vj ∂pij ˙ i + I˙i + Fi + + = Mv ∂t ∂xj ∂xj with:

I˙ i Fi

(41)

rate of production of momentum per unit volume by internal processes such as chemical reactions, rate of production of momentum per unit volume due to external forces such as gravitational and electromagnetic fields.

Energy equation: 1 ∂T 1 ∂ uvj + vi2vj + pij vi − Š ∂ u + vi2 2 ∂xj 2 ˙ 2 + I˙i vi + = Mv i ∂t ∂xj + Fi vi + q + Q with:

u Š Q q Q

internal energy per unit volume coefficient of thermal conductivity rate of external heat addition per unit mass rate of energy production per unit volume due to internal processes rate of energy production per unit volume due to external processes

(42)

982

N

Flow Acoustics

Inhomogeneous convected scalar wave equation for fluctuating total enthalpy: ' & 2 ∂ 2 B ∂ ∂ 2 B ∂h ∂B 1 ∂ B − 2 + 2vi − (v × –) + vi v j i + Vi − 2 c ∂t2 ∂t ∂xi ∂xi ∂xi ∂xj ∂xi2 & . / ∂ ∂ ∂h 1 i + Vi − 2 = + vi v j j + V j − 2 − (v × –) (v × –) ∂xi c ∂xi ∂xj %) $ ∂vi · (v × –) j + V j + ∂t $ % ˙ 1 Dh ∂ ∂ 1 Ds M 1 ∂V i + + − − 2 vi ∂t c2 Dt ∂t R Dt c ∂t with:

(43)

fluctuating part of the quantity is denoted by the prime,all quantities without primes are the sum of the respective mean and fluctuating parts; ∂ D ∂ = + vi material derivative; Dt ∂t ∂xi B fluctuating total enthalpy, basic generalised acoustic field for a fluid (Doak); ∂s 1 ∂‘ij I˙i Fi Vi = T + + + sum of accelerations (“forces” per unit mass) ∂xi ∂xi with: first term vanishes if the motion is homentropic; second term vanishes if the fluid is inviscid; third term vanishes if there is no production of momentum by internal processes such as chemical processes; fourth term vanishes if there are no external forces.

Rewriting in a more compact form: & 2 ' ∂h 1 ∂ ∂ ∂ 2 B ∂ ∂2 − v × –) − + 2v + V − 2 + v v B ( i i i j i c2 ∂t2 ∂t ∂xi ∂xi ∂xi ∂xj ∂xi2 & 1 ∂h ∂ ∂ = − + vi vj − (v × –) i + Vi − 2 ∂xi c2 ∂xi ∂xi * +) . / ∂v i · j + V j + (v × –) ∂t $ % ˙ ∂ ∂ 1 Ds M 1 ∂V i 1 Dh + − − 2 vi + ∂t c2 Dt ∂t R Dt c ∂t

(44)

Reformulation: The idealised case of the homentropic flow of the lossless fluid, under no external force, mass creation or heat addition: & 2 ' ∂h 1 ∂ ∂ 2 B ∂ ∂ ∂2 − v × –) − + 2v − 2 + v v B ( i i j i c2 ∂t2 ∂t ∂xi ∂xi ∂xi ∂xj ∂xi2 & 1 ∂h ∂ ∂ − (v × –) i−2 = − + vi vj (45) ∂xi c2 ∂xi ∂xi * +) ∂vi 1 Dh ∂ + · (v × –) j+ ∂t ∂t c2 Dt

Flow Acoustics

N

If the flow is also irrotational: & 2 ' ∂ ∂2 ∂ 2 B ∂h 1 ∂ ∂ − + 2vi − 2 + vi vj B ∂t ∂xi ∂x* ∂x ∂x ∂xi2 ( c2 ∂t2 i i j +) ∂vi 1 ∂ ∂ ∂h 1 ∂ Dh = − 2 −2 + v i vj + ∂xi c ∂xi ∂xi ∂t ∂t c2 Dt If the flow is also time-stationary: & 2 ' ∂h 1 Dh 1 ∂ ∂ ∂ ∂ 2 B ∂ ∂2 − 2 − + 2v + v v = B i i j ∂xi2 c2 ∂t2 ∂t ∂xi ∂xi ∂xi ∂xj ∂t c2 Dt For flows in which temperature variations are rather negligible: & 2 ' ∂ 2 B 1 ∂ ∂ ∂ ∂2 i − + 2vi − (v × –) + vi v j B ∂t ∂x* ∂xi ∂xj ∂xi2 ( c2 ∂t2 i +) ∂vi ∂ 1 ∂ = − 2 − (v × –) i + v i vj j+ (v × –) ∂xi c ∂xi ∂t

983

(46)

(47)

(48)

In the field of aeroacoustics: • External forces and heat addition are of relatively less significance, • like the effects of temperature and entropy fluctuations, mean entropy gradients and viscous diffusion on propagation. • Also the dependence of the inhomogeneous source terms on these quantities is of less significance. • Therefore one can suggest neglecting them and linearise the equation mentioned above in the fluctuations: & 2 ∂ ∂2 2 ∂ c¯ 2 1 ∂ B 1 ∂2 ∂ ∂ ¯ − − 2 + 2v¯ i − v × – + v¯ i v¯ j − 2 B= 2 2 i ∂t ‰ − 1 ∂xi ∂xi ∂xi ∂xj ∂xi c¯ ∂xi c¯ ∂t . ' / 2 ∂ 2 ∂ c¯ · − v¯ × – ¯ − + v¯ i v¯ j ¯ + v¯ × – v × – (49) i i i ‰ − 1 ∂xi ∂xj % $ ∂ ∂v 2 ∂c2 1 i ¯ ¯ − v × – − + vi v¯ j + v¯ i vj − 2 − v × – i i c¯ ‰ − 1 ∂xi ∂xj ∂t * + 2 ¯ c ¯ ‰R T with the approximation h¯ = = . (50) ‰−1 ‰−1 If the flow is time-stationary, then the last line of the last equation is zero.

984

N

N.9

Acoustic Analogy with Effects of Solid Boundaries

Flow Acoustics

See also: Ffowcs Williams/Hawkings, Sound generation by turbulence (1969); Howe (1998); Crighton et al. (1992); Farassat (1986); Prieur/Rahier (1998); Long/Watts (1987); Pilon/Lyrintzis (1998); Farassat/Myers (1988); Brentner/Farassat (1998); Brentner (1997); Farassat (2001)

N.9.1 Ffowcs Williams–Hawkings (FW-H) Inhomogeneous Wave Equation, FW-H Equation in Differential and Integral Form The aeroacoustic analogy in the representation of Ffowcs Williams–Hawkings is a very general Lighthill acoustic analogy. Primarily it is used for problems of sound generation by flow with moving boundaries and by moving sources interacting with such boundaries. Notations in connection with moving surface S: f = f (xi , t) = 0, ui (xi , t)

velocity of the surface,

n = ∇f

subscript n indicates projection of a vector quantity in surface normal direction,

u n = u i ni , |∇f | = 1

vn = vi ni , on surface,

∂f Df ∂f = + ui = 0. Dt ∂t ∂xi Heaviside function H(f ): H(f ) = 1 for f (xi , t) > 0, H(f ) = 0 for f (xi , t) < 0,

that is in the fluid region, exterior to S that is in the volume enclosed by the surface S interior of S

∂f ∂H(f ) ∂H(f ) ∂f = = ƒ(f ) ∂t ∂f ∂t ∂t ∂H(f ) ∂H(f ) ∂f ∂f = = ƒ(f ) = ƒ(f )ni (1) ∂xi ∂f ∂xi ∂xi

In what follows the function − 0 H(f ) is defined in the continuity and momentum

equations. That is, the derived equations are valid for all space. The term − 0 H(f ) is determined only in the region of space that is of interest, i.e. space occupied by fluid. Continuity equation:

∂ ∂

− 0 + vi = 0 ∂t ∂xi

∂ ∂f ∂ − 0 H(f ) + vi H(f ) = 0 ui + (vi − ui ) ƒ(f ) ∂t ∂xi ∂xi ∂H(f ) = 0 ui + (vi − ui ) = Qƒ(f ) ∂xi

(2)

(3)

Flow Acoustics

N

∂f Q = 0 ui + (vi − ui ) = 0 un + (vn − un ) ∂xi un + vn = 0 Un = 0 1 − 0 0 with:

Q 0 un

985

(4)

mass flux per unit volume density of undisturbed medium local normal velocity of the surface

Momentum equation:

∂

∂ vi + vi vj + pij = 0 ∂t ∂xj with:

pij = pƒij − ‘ij

∂ ∂ vi H(f ) + vi vj + pij H(f ) = Fi ƒ(f ) ∂t ∂xj with:

(5)

∂f = pij nj + vi (vn − un ) Fi = pij + vi vj − uj ∂xj Fi force per unit area acting on medium

Eliminate vi H(f ) by cross differentiation:

∂ 2 − 0 H(f ) ∂ ∂ ∂ 2 2 = − 0 H(f ) − c0 [Qƒ(f )] − [Fi ƒ(f )] 2 2 ∂t ∂t ∂xi

∂xi 2

− 0 H(f ) ∂ 2 2∂ + vi vj + pij H(f ) − c0 ∂xi ∂xj ∂xi2

and with Lighthill tensor Tij = vi vj + pij − c20 − 0 ƒij

(6) (7)

(8)

(9)

follows: Ffowcs Williams–Hawkings equation in differential form & 2 ' 2

∂ ∂ ∂ ∂2 2 ∂ )] − − H(f ) = Tij H(f ) . − c ƒ(f )] + [Qƒ(f [F 0 i 0 2 2 ∂t ∂t ∂xi ∂xi ∂xj ∂xi

(10)

This equation is valid throughout the whole of space. Inside the fluid, H = 1, are Lighthill’s quadrupole sources. In addition to these volume sources there exist surface sources in the form of dipoles and monopoles with: dipole strength density: monopole strength density:

surface stress rate at which mass is transmitted across unit area of surface.

Source contributions caused by surface permeability: ∂ (vn − un ) ƒ(f ) • monopole source: ∂t ∂ • dipole source: − vi (vn − un ) ƒ(f ) ∂xi

N

986

Flow Acoustics

Ffowcs Williams–Hawkings equation in integral form:

∂ ∂2 dV 2 − Hc0 − 0 = Tij ‘ vi vj − uj + pij ‘ ∂xi ∂xj ∂xi 4 xi − yi V(‘)

dSj (yi ) ∂ + · ∂t 4 xi − yi xi − y i

S(‘)

vj − uj + 0 uj ‘

S(‘)

dSj (yi ) 4 xi − yi

(11)

x − y with: ‘ = t − = t− retarded time, surface integrals over the retarded c0 c0 surface S(‘) defined by f (yi , t) = 0, surface element dSi directed into the region V(‘) where f > 0. The control surface is a non-porous surface (impenetrable): (vi − ui ) = 0 Monopole term:

Q = 0 u i

Dipole term:

Fi = pij

with:

∂f = 0 u n ∂xi

(12)

∂f = pij nj ∂xj

(13)

pij = pƒij − ‘ij ∂f Fi = p = pni , if viscous stresses are neglected; ∂xi p local surface pressure, ni local unit outward normal to surface.

Sources can be interpreted • as a volume distribution of quadrupoles due to the turbulent flow,

∂ 2 Tij in the outer region of the surfaces, ∂xi ∂xj

∂ [Fi ƒ(f )] due to the interaction of the flow ∂xi with moving bodies, especially due to surface pressure and stress fluctuations on the bodies in the flow, ∂ • as a surface distribution of monopoles [Qƒ(f )], due to the kinematics of the ∂t bodies, especially from normal accelerations of the body surfaces. • as a surface distribution of dipoles −

Detailed representation of the Ffowcs Williams–Hawkings equation in differential form: 2 [“H (f )] : =

1 ∂2 ∂2 − 2 [“H (f )] 2 ∂t 2 [“H (f )] c0 ∂xi

∂ , ∂ ∂2 Tij H(f ) − p − p0 ƒij − ‘ij nj ƒ(f ) − vi (vn − un) ƒ(f ) (14) ∂xi ∂xj ∂xi ∂xi ∂ ∂ + 0 un ƒ(f ) + (vn − un ) ƒ(f ) ∂t ∂t

wave variable in linear acoustics corresponding to sound with: “ = c20 − 0 pressure. =

Flow Acoustics

Reformulation (Pilon/Lyrintzis): ∂ 1 ∂ ∂“ Mn ∂“ 2 [“H (f )] = − ƒ(f ) − + [Mn “ƒ(f )] − [“ni ƒ(f )] ∂n c0 ∂tx c0 ∂t ∂xi ∂ 2 Tij + H (f ) ∂xi ∂xj with:

N

987

(15)

subscript x in the time derivative denoting differentiation with respect to time, holding the observer co-ordinates fixed.

In comparison to the generalised wave equation, which is the governing equation for the Kirchhoff formulation: • the domain is considered in terms of wave propagation, & p f >0 • the generalised pressure perturbation: p = . 0 f 1: 2 2 3 DU 0 † ∼ 0 P ∼

Power spectral density of far-field noise: (statistical source models in jet noise of Ribner 1969)

(24) (25)

N

1004

Flow Acoustics

Notations: Density autocorrelation function: 1 (x , t + ‘) − 0 (x, t) − 0 3 0 c0 2 0 xi xj xk xl ∂ ∂ 2 v v y , t v v y , t dV dV C (x, ‘) = 5 162c0 x6 ∂t2 i j ∂t2 k l C (x, ‘) =

(26) (27)

V

with:

y ,

y tworunning points in domain source x − y x − y t =t− , t =t+‘− c0 c0

C (x, ‘) =

with:

0 xi xj xk xl 162 c50 x6

V

∂4 † · x dV dV† R , † , ‘ + y ijkl ∂‘4 c0x

(28)

, ‘ = vi vj y , t vkvl y , t + ‘ Rijkl y , † two-point time-delayed fourth-order correlation tensor

Directional acoustical intensity spectrum, which is the temporal Fourier transform of the density autocorrelation C (x, ‘): 1 I– (x ) = 2

+∞ C (x, ‘) ej–‘ d‘

(29)

−∞

Acoustical power spectrum (emitted from a unit volume located at y ):

P– y = 2r2

I– x , Ÿ y sin ŸdŸ

(30)

0

Total acoustic power: +∞

P= P– y dVd–

(31)

V −∞

Far-field noise radiated: acoustic intensity from an elementary volume of jet: self 2 0 u 2 L3 –4t noise dI (x) ∼ c50 r2 C5

(32)

Flow Acoustics

shear 0 u 2L5 –4t ∂U 2 dI noise (x ) ∼ DŸ c50 r2C5 ∂y2

N

1005

(33)

respectively the intensity spectrum: self noise

dI–

x , Ÿ y =

shear noise

dI–

with:

x , Ÿ y =

DŸ =

1 2

2 0 u2 L3 –4 –2 C2 exp − 1285/2 c50 r2 –t 8–2t 0 u2L5 247/2 c50 r2

cos2 Ÿ + cos4 Ÿ

C –t = 2 –t L≈

— k

∂U ∂y2

2

–4 –2 C2 DŸ exp − –t 4–2t

(34)

(35)

directivity of shear-noise component (isotropic directivity of self noise is a necessary consequence of isotropy of turbulence) convection factor (at Goldstein/Howes: C−3 instead of C−5 ) local characteristic frequency of turbulence, related to eddy lifetime

k3/2 —

L u

U y2

integral length scale of turbulence turbulent velocity mean velocity in direction of y1, dependent on co-ordinate y2

Other developments (Goldstein/Rosenbaum): Acoustic intensity per unit source volume: self 2 0 u 2 L1 L22 –4t noise dI D1 (x) ∼ c50 r2 C5

(36)

shear 0 u12 L1 L42 –4t ∂U1 2 D2 dI noise (x ) ∼ c50 r2C5 ∂y2

(37)

respectively: self noise

dI–

x , Ÿ y =

2 –2 C2 0 u12 L1 L22 –4 D exp − √ 1 8–2t 40 23/2 c50 r2 –t

(38)

1006

N

shear noise

dI–

Flow Acoustics

0 u 2 L1 L4 –4 x , Ÿ y = 3/21 5 22 c0 r –t

∂U1 ∂y2

2

–2 C2 D2 exp − 4–2t

(39)

with:

M D1 = 1 + 2 − N cos2 Ÿ sin2 Ÿ 9 3N 3 1 M2 2 +M− 3 − 3N + 2 − sin4 Ÿ + 3 7 2 2 2 1 1 2 2 2 − 2N sin Ÿ D2 = cos Ÿ cos Ÿ + 2 2 with:

anisotropic structure of turbulence L2 = L1 3 1 2 M= − 2

(40) (41)

(42)

N=1−

u22

(43)

u12

u12 , u22

axial and transversal turbulent kinetic energy

U1

axial mean flow velocity

L1 , L2

integral length scale in direction of flow and in transverse direction

L2 ≈

1 L1 3

(2k/3)3/2 — — –t = 2 k

(44) (45)

L1 ≈

angular frequency of turbulence

Flow Acoustics

N

1007

for isotropic turbulence:

2 u12 = k 3

(46)

for anisotropic turbulence:

∂ U¯ 1 2 u12 = k − Œt 3 ∂x1

(47)

∂ U¯ 2 2 u22 = k − 2Œt 3 ∂x2

(48)

k2 — kinematic turbulent viscosity

with: Œt = 0, 09

(49)

N.12.2 Rotor Noise Computation of rotor noise, based on Ffowcs Williams–Hawkings equation: 2 pa (x, t) =

∂ ∂ ∂2 Tij H(f ) [Qƒ(f )] − [Li ƒ(f )] + ∂t ∂xi ∂xi ∂xj

(50)

Assumptions: • (at first) neglecting quadrupole sources, • moving surface is non-porous. Blade thickness noise: 2 pT =

∂ 0 un ƒ(f ) ∂t

(51)

Solution: pT (xi , t) =

∂ ∂t

S

0 un ∂ 0 un 1 dS = dS 4r (1 − Mr ) ‘ 1 − Mr ∂‘ 4r (1 − Mr ) ‘

(52)

S

for open, rotating blades with a subsonic tip Mach number: pT (x , t) = f =0

) ( ˙ r + c0 M r − M 2 0 un rM 0 (u˙ n + un˙ ) dS+ dS 4r (1 − Mr )2 ‘ 4r2 (1 − Mr )3 ‘ f =0

(53)

1008

with:

N

Flow Acoustics

un

local velocity of blade surface in directional normal to f =0

u˙ n = u˙ i ni un˙ = ui n˙ i = ui n

∂ni ∂‘

M = M M

unit outward normal vector to surface, with components ni

local Mach number vector of source, with components Mi Mr component of velocity in radiation direction normalised by c0 f =0 function describing rotor blade surface c0 sound speed in quiescent medium The dot over a symbol implies source-time differentiation of that symbol, e.g. ∂Mi ˙r= ri . M ∂‘ Loading noise: 2 pL = −

∂ [Fi ƒ(f )] ∂xi

(54)

Solution:

Fi dS 4r (1 − Mr ) ‘ f =0 1∂ Fr Fr dS + dS = c ∂t 4r (1 − Mr ) ‘ 4r 2 (1 − Mr ) ‘

pL (xi , t) = −

∂ ∂xi

f =0

(55)

f =0

for open, rotating blades with a subsonic tip Mach number: 1 pL (x, t) = c0

$ f =0

•

Lr 4r (1 − Mr )2

%

dS ‘

Lr − LM dS 4r2 (1 − Mr )2 ‘ f =0

) ( ˙ Lr rMr + c0 Mr − M2 1 + dS (56) c0 4r2 (1 − Mr )3

+

f =0

‘

Flow Acoustics

with:

N

1009

Li

components of local force that acts on fluid (Li is identical with Li used above) Li = pij nj + vi (vn − un ) Lr = Li ri Lr

component of local force that acts on fluid (due to body) in radiation direction

LM = Li Mi Mi

velocity of surface f = 0 normalised to ambient sound speed

Mr

component of velocity in radiation direction normalised to c0

r

distance from source point on surface to observer

The dot over a symbol implies source-time differentiation of that symbol, e.g. • ∂Li ri . Lr = ∂‘ Other formulation: Thickness and loading noise together Integral representation of solution (of FW–H equation): $

pa (x, t) =

% • • Q + Lr c0

Lr − LM dS 4r (1 − Mr )2 4r 2 (1 − Mr )2 ‘ ‘ f =0 f =0

) ( ˙ Q + Lr c0 rMr + c0 Mr − M2 dS + 4r2 (1 − Mr )3 ‘ dS +

(57)

f =0

Quadrupole noise:

2 pa (x, t) =

∂2 Tij H(f ) ∂xi ∂xj

(58)

Solution: ∂ 0 un 0 un 1 dS = dS 4r (1 − Mr ) ‘ 1 − Mr ∂‘ 4r (1 − Mr ) ‘ S S Tij ∂2 pQ (x , t) = dV ∂xi ∂xj 4r |1 − Mr | ‘

pT (xi , t) =

∂ ∂t

V

(59) (60)

1010

N

Flow Acoustics

respectively: pQ (x, t) =

1 ∂ Trr 3Trr − Tii 1 ∂2 dV + dV 4r |1 − Mr | ‘ c0 ∂t 4r2 |1 − Mr | ‘ c20 ∂t2 V V 3Trr − Tii + dV 4r3 |1 − Mr | ‘

(61)

V

with:

Mr

projection of rotational Mach number in source-observer direction

Trr = Tij rî rˆj rˆ

Lighthill stress tensor in radiation direction unit vector in radiation direction, with components rî

Rotor noise in practice: sound far field in terms of source strength spectrum (rotor and stator noise, propeller noise, helicopter rotor noise, etc.) Monopole: +∞ 2–R e−2j– r/c0 −jn(/2−“) p (xi , –) = m (“, – − n–0 ) e Jn − sin Ÿ 4r n=−∞ c0 with:

m (“, – − n–0 )

point monopole, spectrum of the source strength

“

angular position of the point source at time t = 0

–0

rotational angular velocity

R

position vector of the point source

(62)

Dipole: p (xi , –) = − with:

+∞ 2–R e−2j– r/c0 2j– ri fi (“, – − n–0 ) e−jn(/2−“) Jn − sin Ÿ (63) 4r c0 c0 n=−∞

fi (“, – − n–0 ) r ri = |r|

point dipole, spectrum of the source strength component of unit vector in direction i

Quadrupole: p (xi , –) = −

– 2 e−2j– r/c0 2 4r c0 +∞ 2–R ri rj tij (“, – − n–0 ) e−jn(/2−“) Jn − sin Ÿ c0 n=−∞

(64)

Flow Acoustics

with:

tij (“, – − n–0 )

point quadrupole, spectrum of source strength

components of unit vector in direction i, j

ri , rj

N

1011

Rotor monopole sound: ∞ ˙ 2 –2

M P= B 0 (mB)2 J2mB mBMq sin Ÿ sin Ÿ dŸ 80 c0 m=1

(65)

0

with:

B ˙ B = BM ˙ M

number of rotor blades

–0

rotor angular frequency

m

integer

JmB

Bessel function (first kind) of order mB

Mq = –0 R/c0

Mach number

R

radius of rotor, equivalent radius from hub point on rotor blade, in which source strength is concentrated

Ÿ

angle between rotor axis and vector from rotor centre to observer point

overall fluid mass displaced per unit time by all rotor blades

Rotor dipole sound due to stationary rotating forces:

mB–0 mB+1 −jmB –0 r0 /c0 FD FT cos Ÿ − JmB mBMq sin Ÿ −j pm = e 2c0 r0 Mq

(66)

∞

FD 2 –20 FT cos Ÿ − P= (mB)2 J2mB mBMq sin Ÿ sin Ÿ dŸ 3 Mq 40 c0 m=1

(67)

0

with:

pm

m-th sound pressure harmonic

F T , FD

thrust and drag, respectively, which act on air and in site direction, effect of all rotor blades together

Rotor dipole sound due to rotating periodic time-variable forces: • sound radiation of rotor: pm‹ =

Pm‹

jmB–0 mB−‹ mB − ‹ FD‹ FT‹ cos Ÿ − JmB−‹ mBMq sin Ÿ −j 2c0 r0 mB Mq

–20 = 40 c30

(68)

mB − ‹ FD‹ 2 FT‹ cos Ÿ − (mB)2 J2mB−‹ mBMq sin Ÿ sin Ÿ dŸ (69) mB Mq 0

N

1012

with:

Flow Acoustics

pm‹ /Pm‹

mth harmonic of sound pressure / sound power radiated by ‹th harmonic of fluctuating forces (blade loading harmonic) of rotor blades

FT‹ , FD‹

Fourier components of periodic time-variable blade forces (thrust and drag)

in the case of B rotor blades and V stator vanes: ‹ = kV • sound radiation of stator: pmk

jmBV–0 mB−kV mB − kV FDm =− j FTm cos Ÿ − JmB−kV mBMq sin Ÿ 2c0 r0 mB Mq

with:

(70)

m-th harmonic of sound pressure radiated by kth harmonic of fluctuating forces on stator vanes owing to rotor-stator interaction

pmk

Rotor dipole sound by rotating random blade forces: +∞ – −j– r0 /c0 n–0 FD§ n+1 – p– (xi , –) = − e Mq sin Ÿ (71) FT§ cos Ÿ − j Jn 2c0 r0 – Mq –0 n=−∞ zR –2 P– = 40 c30 with:

+∞ 0

n=−∞

FT§ , FD§

n–0 FD§ FT§ cos Ÿ − – Mq

2

J2n

– Mq sin Ÿ sin Ÿ dŸ –0

(72)

Fourier transforms of stochastic time-variable blade forces (thrust and drag)

– = § + n–0 §

N.13

frequency variable of force spectrum

Power Law of the Aerodynamic Sound Sources

See also: Ffowcs Williams,Annual Review of Fluid Mechanics 1 (1969); K¨ oltzsch (1974, 1998)

The power law of aerodynamic sound sources is presented based on developments with the help of dimensional analysis, generalised for multipoles of arbitrary order and of variable number of space dimensions, for compact and non-compact aerodynamic multipoles. Order of multipoles N: Monopole source Dipole source Quadrupole source

→ N = 0; → N = 1; → N = 2.

Flow Acoustics

N

1013

Power law of compact aerodynamic multipoles:

n−1 n+1 L 2 N+ − 0 (x ) ∼ ¯ q Mt 2 r

with:

(1)

− 0

acoustic density fluctuation in far field

0

density of ambient fluid

¯ q

mean density inside flow

L

scale of coherent regions in flow, characteristic length scale

Mt = v /c0

Mach number, ratio of characteristic turbulence velocity v to speed of sound in uniform environment, measure of compactness

N

order of multipoles

n

number of space dimensions in which wave field spreads Mt > kL ;

Measure of compactness →

L 1 < Mt Š 2

L 1; Š acoustic wavelength is much greater than the length scale characteristic of the turbulent source flow;

in general:

Mt is much less than 2, that is

it follows: such sources are acoustically compact. The following overview presents (in the three-dimensional case): • power laws for acoustic density fluctuations, • sound power P, • acoustic-aerodynamic efficiency † (ratio of sound power to flow power): Assumptions:

Mt ∼ M = †ac =

with:

U c0

Pac Pmech

Pac

with: U a characteristic flow velocity; acoustic-aerodynamic efficiency; sound power;

flow mechanical power: Pmech ∼ ¯ q L2 U3 .

L L − 0 (x ) ∼ ¯ q M2+N ∼ ¯ q U 2 MN r r Pmech

General:

P∼

¯ q ¯ q 2 4+2N L U ∼ L2 U3 M1+2N 0 0

†ac ∼

¯ q 1+2N M 0

(2) (3) (4)

1014

N

Monopole:

Flow Acoustics

L L 2 M ∼ ¯ q U2 − 0 (x ) ∼ ¯ q r r

P∼

¯ q ¯ q 2 4 L U ∼ L2 U3 M 0 0

†M ∼ Dipole:

¯ q M 0

− 0 (x) ∼ ¯ q P∼

L L 3 M ∼ ¯ q U2 M r r

¯ q 3 M 0

P∼

L L 4 M ∼ ¯ q U 2 M2 r r

¯ q ¯ q 2 8 L U ∼ L2 U3M5 0 0

†ac ∼

(8) (9) (10)

− 0 (x) ∼ ¯ q

Quadrupole:

(6) (7)

¯ q ¯ q 2 6 L U ∼ L2 U3M3 0 0

†ac ∼

(5)

¯ q 5 M 0

(11) (12) (13)

Power law of non-compact aerodynamic multipoles: 1 L > Mt ; acoustic wavelength Š is smaller Š 2 than 2 L/Mt ; → such sources are acoustically non-compact.

Measure of compactness → Mt < kL;

Power law:

n−1 L 2 − 0 (x ) ∼ ¯ q Mt r

(14)

Power law of moving aerodynamic sources:

M2+N L − 0 (x ) ∼ ¯ q r 1 − Mqr 1+N

with:

xi − y i M Mqr = xi − yi qi

(15)

projection of Mach number vector Mqi in direction of sound radiation r = x − y

Flow Acoustics

N

1015

References Albring, W.: Elementarvorgänge fluider Wirbelbewegungen. Akademie-Verlag, Berlin (1981) Bailly,C.,Lafon,P., Candel, S.: Computation of noise generation and propagation for free and confined turbulent flows. AIAA 96-1732, 2nd AIAA/CEAS Aeroacoustics Conference, State College, PA (1996) Bangalore, A., Morris, P.J., Long, L.N.: A parallel three-dimensional computational aeroacoustics method using non-linear disturbance equations. AIAA 96-1728,2nd AIAA/CEASAeroacoustics Conference State College, PA (1996) Béchara, W., Lafon, P., Bailly, C.: Application of a k − —−turbulence model to the prediction of noise for simple and coaxial free jets. J. Acoust. Soc. Am. 97 6, 3518–3531 (1995) Boineau, Ph., Gervais, Y., Toquard, M.: Application of combustion noise calculation model to several burners. AIAA-Paper 98-2271, 4th Aeroacoustic Conference, Toulouse, France (1998) Brentner, K.S.: A superior Kirchhoff method for aeroacoustic noise prediction: The Ffowcs Williams-Hawkings equation. 134th Meeting of the ASA, San Diego, CA (1997) Brentner, K.S., Farassat, F.: Analytical comparison of the acoustic analogy and Kirchhoff formulation for moving surfaces.AIAA Journal 36 8, 1379–1386 (1998) Crighton, D.G., Ffowcs Williams, J.E.: Sound generation by turbulent two-phase flow. J. Fluid Mech. 36 3, 585–603 (1969) Crighton, D.G.; Dowling,A.P., Ffowcs Williams, J.E., Heckl, M., Leppington, F.G.: Modern methods in analytical acoustics. Springer-Verlag, Berlin etc., 1992 Curle, N.: The influence of solid boundaries upon aerodynamic sound. Proc. Roy. Soc., London (A), 231 505–514 (1955)

Doak, P.E.: Fluctuating total enthalpy as the basic generalized acoustic field. Theoret. Comput. Fluid Dynamics 10, 115–133 (1998) Douglas, J. F. et al. Fluid Mechanics Longman Scientific & Technical, Harlow, Essex, England, 1986 Farassat, F.: The Ffowcs Williams-Hawkings equation – Fifteen years of research. IUTAM Symposium Lyon 1985.Springer-Verlag,Berlin etc.,(1986) Farassat, F., Myers, M.K.: Extension of Kirchhoff’s formula to radiation from moving surfaces. J. Sound Vibr. 123 3, 451–460 (1988) Farassat, F., Brentner, K.S.: The acoustic analogy and the prediction of the noise of rotating blades. Theoret. Comput. Fluid Dynamics 10, 155–170 (1998) Farassat,F.:Acoustic radiation from rotating blades – the Kirchhoff method in aeroacoustics. J. Sound Vibr. 239 4, 785–800 (2001) Ffowcs Williams,J.E.: Hydrodynamic noise.Annual Review of Fluid Mechanics 1, 197–222 (1969) Ffowcs Williams, J.E., Hawkings, D.L.: Theory relating to the noise of rotating machinery. J. Sound Vibr. 10 1, 10–21 (1969) Ffowcs Williams,J.E.,Hawkings, D.L.: Sound generation by turbulence and surfaces in arbitrary motion. Phil. Trans. of the Roy. Soc. London 264, 321– 342 (1969) Fuchs, H.V., Michalke, A.: Introduction to aerodynamic noise theory. In: Progress in Aerospace Sciences, Vol. 14, 227–297 (1973) Goldstein, M.E., Howes, W.L.: New aspects of subsonic aerodynamic noise theory. NASA TN D-7158 (1973)

¨ Detsch, F., Detsch, F.E.: Uber die Schallerzeugung in Wirbelfeldern. Dissertation TU Dresden (1976)

Goldstein, M.E., Rosenbaum, B.: Effect of anisotropic turbulence on aerodynamic noise. J. Acoust. Soc. Amer. 54 3, 630–645 (1973)

Dittmar, R.: Zum Zusammenhang zwischen Turbulenz- und Schallspektrum. Dissertation TU Dresden (1983)

Goldstein, M.E.: Aeroacoustics. McGraw-Hill International Book Company, New York etc. (1976)

Doak, P.E.: Fluctuating total enthalpy as a generalized acoustic field.Acoustical Physics 41 5, 677–685 (1995)

Heckl, M.: Strömungsgeräusche. Fortschr.-Ber. VDI-Z. Reihe 7, Nr. 20. VDI-Verlag D¨ usseldorf, 1969

1016

N

Flow Acoustics

Howe, M.S.: Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute. J. Fluid Mech. 71, 625– 673 (1975) Howe, M.S.: Acoustics of fluid-structure interactions. University Press, Cambridge (1998) Hussain, A.K., Reynolds, M.F., Reynolds, W.C.: The mechanics of an organized wave in turbulent shear flow. J. Fluid Mech. 41 2, 241–258 (1970) Ianniello, S.: Quadrupole noise predictions through the Ffowcs Williams-Hawkings equation. AIAA Journal 37 9, 1048–1054 (1999) Johnson, R.W. (Ed.): The Handbook of Fluid Dynamics CRC Press,Boca Raton,FL,Springer-Verlag, Heidelberg (1998) Költzsch, P.: Strömungsmechanisch erzeugter Lärm. Dissertation B (Habilitationsschrift), Technische Universität Dresden (1974) Költzsch, P.: Berechnung der Schallleistung von axialen Strömungsmaschinen. (Calculation of sound power of axial flow machines.) Freiberger Forschungshefte A 721, Deutscher Verlag für Grundstoffindustrie, Leipzig (1986) Költzsch, P.: Beitrag zur Berechnung des Wirbellärms von Axialventilatoren. In: Ventilatoren (Herausgeber: L. Bommes, J. Fricke, K. Klaes), VulkanVerlag, Essen, S. 434–453 (1994) ¨ Költzsch,P.: Wozu werden Ahnlichkeitskennzahlen in der Akustik verwendet? Preprint ET-ITA-011998, Technische Universität Dresden, Dresden (1998) Lauchle, G.C.: Fundamentals of flow-induced noise Graduate program in acoustics, Penn State University (1996) Legendre,R.: Bruits e´mis par la turbulence.ONERA Publ. 1981–3, 1981 Lighthill, M.J.: On sound generated aerodynamically. Proc. Roy. Soc., London (A), Part I: 211 564– 587 (1952); Part II: 222 1–31 (1954) Lilley,G.M.: On the noise from air jets.Aeronautical Research Council ARC 20376, U.K. 1958 Lilley, G.M.: On the noise radiated from a turbulent high speed jet. In: Hardin, J.C. and M.Y. Hussaini (Editors): Computational Aeroacoustics. SpringerVerlag, New York etc., 85–115 (1993)

Lilley, G.M.: On the refraction of aerodynamic noise. 6th Internat. Congr. on Sound and Vibration, Copenhagen , S. 3581–3588 (1999) Liu, J.T.C.: Contributions to the understanding of large-scale coherent structures in developing free turbulent shear flows. Advances in Applied Mechanics (ed. by J.W. Hutchinson and T.Y. Wu) Vol. 26, Academic Press, Inc., Boston etc., pp. 183–309 (1988) Long, L.N., Watts, G.A.: Arbitrary motion aerodynamics using an aeroacoustic approach. AIAA Journal 25 11, 1442–1448 (1987) Lowson, M.V.: The sound field for singularities in motion. Proc. Roy. Soc., London (A) 2860, 559–572 (1965) Lowson, M.V.: Theoretical analysis of compressor noise. J. Acoust. Soc. Amer. 47 1(2), 371–385 (1970) Lyrintzis, A.S.: Modelling of turbulent mixing noise. Application to subsonic and supersonic jet noise. Lecture Series 1997-07, von Kármán Institute for Fluid Dynamics, Belgium (1997) Meecham, W.C., Ford, G.W.: Acoustic radiation from isotropic turbulence. J. Acoust. Soc. Am. 30, 318–322 (1958) Meecham, W.C.: Discussion of the pressure-source aerosonic theory and of Doak’s criticism. J.Acoust. Soc. Am. 69 3, 643–646 (1981) Möhring, W., Obermeier, F.: Vorticity–the voice of flows. Proceed. 6th Internat. Congr. on Sound and Vibration, Copenhagen (1980), 3617–3626 Möhring, W.: A well posed acoustic analogy based on a moving acoustic medium. Proceedings 1s t Aeroacoustic Workshop (in connection with the German research project SWING), Dresden (1999) Möhring, W.: On vortex sound at low Mach number. J. Fluid Mech. 85, 685–691 (1978) Morfey, C.L., Tanna, H.K.: Sound radiation from a point force in circular motion. J. Sound Vibr. 15 3, 325–351 (1971) Morfey, C.L.: Amplification of aerodynamic noise by convected flow inhomogeneities. J. Sound Vibr. 31 4, 391–397 (1973) Morfey, C.L.: Fundamental problems in aeroacoustics. 7th Internat. Congr. on Sound and Vibration, Garmisch, 59–74 (2000) Morfey, C.L.: Dictionary of acoustics Academic Press, San Diego etc. (2001)

Flow Acoustics

Pao, S.P.: Developments of a generalized theory of jet noise. AIAA Journal 10 5, S. 596–602 (1972) Perrey-Debain, E., Boineau, P., Gervais, Y.: A numerical study of refraction effects in combustiongenerated noise. Proceed. 6th Internat. Congr. on Sound and Vibration,Copenhagen, 3361–3368 1999 Phillips, O.M.: On the generation of sound by supersonic turbulent shear layers. J. Fluid. Mech. 9, 1–28 (1960)

N

1017

Ribner, H.S.: Effects of jet flow on jet noise via an extension to Lighthill model. J. Fluid Mech. 321, 1– 24 (1996) Roger, M.: Applied aero-acoustics: prediction methods Lecture Series 1996-04, von Kármán Institute for Fluid Dynamics, Belgium (1996) Schlichting, H., Gersten, K.: Grenzschicht-Theorie Springer-Verlag, Berlin etc. (1997)

Pilon, A.R., Lyrintzis, A.S.: Development of an improved Kirchhoff method for jet aeroacoustics. AIAA Journal 36 5, 783–790 (1998)

Singer, B.A., Brentner, K.S., Lockard, D.P., Lilley, G.M.: Simulation of acoustic scattering from a trailing edge. AIAA 99-0231, 37th Aerospace Sciences Meeting & Exhibit Reno (1999)

Powell,A.: Mechanisms of aerodynamic sound production. AGARD-Report No. 466 (1963)

Strahle, W.C.: On combustion generated noise. J. Fluid Mech. 49 2, 399–414 (1971)

Powell, A.: Theory of vortex sound. J. Acoust. Soc. Am. 361, 177–195 (1964)

Strahle, W.C.: Some results in combustion generated noise. J. Sound Vibr. 23 1, 113–125 (1972)

Prieur, J., Rahier, G.: Comparison of Ffowcs Williams-Hawkings and Kirchhoff rotor noise calculations. AIAA 98-2376, 984–994 (1998)

Strahle, W.C.: Convergence of theory and experiment in direct combustion-generated noise.AIAAPaper 75-522,2nd Aeroacoustic Conference,Hampton, Va. (1975)

Reynolds, W.C., Hussain, A.K.M.F.: The mechanics of an organized wave in turbulent shear flow. J. Fluid Mech. 54 2, 263–288 (1972) Ribner,H.S.: Aerodynamic sound from fluid dilatations. J.Acoust. Soc. Am. 31, 245–246 (1959), UTIAReport No. 86, Toronto, Canada, 1958 Ribner, H.S.: Quadrupole correlations governing the pattern of jet noise. J. Fluid Mech. 38 1, 1–24 (1969)

Tanna, H.K.: Sound radiation from point acoustic stresses in circular motion. J. Sound Vibr. 16 3, 349– 363 (1971) Tanna, H.K., Morfey, C.L.: Sound radiation from point sources in circular motion. J. Sound Vibr. 16 3, 337–348 (1971) Telionis, D.P.: Unsteady Viscous Flows SpringerVerlag, New York etc. (1981)

O Analytical and Numerical Methods in Acoustics with M. Ochmann Numerous analytical and numerical methods are displayed in this book together with the solutions for special tasks. This chapter contains analytical and numerical methods to be applied in acoustics, going beyond the scope of single examples. The description of a method unavoidably needs more textual explanations than the representation of just the resulting formulas. > Sect. O.1 describes a procedure for optimisation of the parameters of a sound absorber; > Sect. O.2 outlines a method for the evaluation of many concatenated transfer matrices. > Section O.3 will present five standard problems of numerical acoustics which frequently occur in practical applications. In > Sects. O.4–O.6 three important methods for the numerical solution of these problems will be described. The source simulation technique and the boundary element method are mainly used for exterior problems such as the radiation or the scattering problem (see > Sects. O.4 and O.5). The finite element method is especially suited for computing sound fields in interior spaces (see > Sect. O.6). The fluid–structure interaction problem can be treated by a combined finite element and boundary element approach, for example with the method of > Sect. O.6. The transmission problem can be formulated in terms of boundary integral equations (see > Sect. O.5).Analytical field solutions for benchmark models are given in > Sects. O.7, O.8.

O.1

Computational Optimisation of Sound Absorbers

See also:

>

Sect. J.34 for a similar task with duct lining absorbers.

The situation: There exist precise and fast computing algorithms for the evaluation of a variety of absorbers, even those with complicated structures (e. g. multi-layer absorbers with foils and/or resonator neck plates in front of and/or between the layers of the absorber; see > Chs. D, G, H). If one intends to design an absorber with a good performance in some aspect (e. g. sound absorption), one has to optimise by trial and error. This is an optimisation in the space of the absorber parameters which often becomes a 10- to 20-dimensional space, if all parameters are to be optimised in one run. The task: Write a computer program which performs this optimisation of the absorber parameters. What does “optimisation” mean ? First, one has to fix the absorber quantity which should be improved by variation, e. g. the sound absorption coefficient .

1020

O

Analytical and Numerical Methods in Acoustics

Second,optimisation is generally understood with respect to a frequency response curve of that quantity.This introduces a frequency interval (flo,fhi ) for optimisation,and a stepping through the interval, with either linear or logarithmic steps. Logarithmic stepping may be preferable, because it accentuates lower frequencies, which is often wanted. Third, optimisation cannot be understood as a general optimisation of all parameters, because then the result can be found in most cases without any computation. For example, the general optimisation with respect to sound absorption of a multi-layer absorber would produce a porous layer with a huge thickness and a tiny flow resistivity of the material, with no surface cover. Therefore a reasonable optimisation supposes some given structure of the absorber (e. g. multi-layer absorber with locally reacting layers and possibly layer covers, or multiple-layer absorbers with bulk reacting layers, or locally reacting and bulk reacting layers in a mixed sequence, or resonators of different structures, etc.). The following description of the method uses as a special example (just for illustration) a multi-layer absorber with locally reacting layers (in fact the layers must not be partitioned if the flow resistivity values of porous materials come out with sufficiently high values). Next, a “parallel” absorber will be optimised. Fundamentals: The acoustical evaluation begins with the computation of the normalised input admittance Z0 G of the arrangement. This evaluation needs as input most of the absorber parameters. The next step is the evaluation of the reflection coefficient |r|2 from which the absorption coefficient follows as = 1 − |r|2. (Ÿ) = 1 − |r(Ÿ)|2;

r(Ÿ) =

cosŸ − Z0 G . cosŸ + Z0 G

(1)

The absorption coefficient dif for diffuse sound incidence on locally reacting absorbers is obtained with Z0 G = g + j · g by: 1 + 2g g g2 − g 2 arctan ln 1 + − g , dif = 8g 1 + g g + g2 + g2 g 2 + g2 (2) 1 + 2g g − g ln 1 + . −−−−→ 8g 1 + g →0 1 + g g 2 If the absorber is bulk reacting, or has mixed bulk and locally reacting layers, dif is obtained from (Ÿ) by evaluation for a set of incidence angles Ÿ and numerical list integration of the intermediate results. We further write and |r|2 commonly for the different possible kinds of incidence. It is important that one knows an “ideal value” for , i. e. = 1 (this value can be used as the goal, even knowing that for locally reacting absorbers the highest possible absorption coefficient for diffuse incidence is ≈ 0.95). Then in the relation = 1 −|r|2 the reflection coefficient |r|2 can be interpreted as a squared “error” of the actual value compared with the ideal value = 1. The algorithm for optimisation minimises the averaged square error


|r|2 = q (a1 , a2 , . . . ; b1 , b2, . . .) ! 2 wn = Min , = wn · |r(fn ; a1 , a2 , . . . ; b1, b2 , . . .)| n

O

1021

(3)

n

where fn are sampling frequencies over (flo,fhi ); w(f ) is a weight function; wn = w(fn ); the {ai } are the variable absorber parameters; the {bk } are fixed parameters. The minimisation is performed by variation of the {ai }. The use of a weight function w(f ) introduces the possibility to perform either a broadbanded optimisation over (flo,fhi ), e. g. with w(f ) = 1, or a centred optimisation, if w(f ) has a central maximum in (flo ,fhi) (this form of optimisation can be used, for example, for the down-tuning of resonators, together with an improvement). There exist algorithms in the literature for finding a minimum of a real function in the multi-dimensional space (e. g. W.H. Press et al. “Numerical Recipes”, Cambridge University Press). A principal distinction must be made in such algorithms whether the partial derivatives ∂q/∂ai can be evaluated or not. This generally is not possible in the present task. Some start values {ai }start must be given for the start of the minimum search. Most algorithms described for minimisation extend the search over the whole (real) space of the {ai }, i. e. also to negative values. In our case, however, the parameters ai represent geometrical lengths or material data which should be positive; some of them, like the porosity , have to respect lower and higher limits (0 < < 1). Thus range limits for the ai must be transmitted to the minimisation algorithm. The procedure for optimisation of the absorption coefficient (and similarly for other target quantities) works as follows: 1. Make the decision for the target quantity (e. g. the sound absorption coefficient for diffuse incidence). 2. Make the decision for a structure of the absorber (e. g. a multi-layer absorber with locally reacting layers with (possibly) porous cover foils and/or perforated plates on the front sides of the layers). This decision determines the theoretical description of the absorber. 3. Fix the frequency interval (flo, fhi ), the kind of frequency steps (linear f or logarithmic lg(f ), which often is preferable), the step width (it should not be too large, otherwise |r|2 = q(a1 , a2 . . . ; b1, b2 . . .) may not be steady enough for finding a minimum); fix the type of the weight function w(f ) (see above). 4. Conceive a “start configuration” of the absorber, i. e. select values for all required absorber parameters (they define what will be called here the “original absorber”). It is supposed that at this stage the data input is structured as follows (just for giving an example for a two-layer absorber; entries (*text *) indicate comment text): (*Input frequency*) flo= 50.; fhi= 2000. ; (*Input layers*) gn= 2 ; matlist= {1, 1} ; ¡ list= {10000., 10000.}; tlist= {0.05, 0.05 }; (*Input foils*)

flist=

{2700.,

2700.};

lgfstep= 0.1 ; (*number N of layers*) (*types of porous material*) (*flow resistivities ¡ *) (*layer thickness t in meter*) (*foil material density

f in kg/m3 *)

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dflist= {0.0002, 0.0002}; {1000., 2. }; Rflist= (*Input perforates*) shapelist= {1, 1}; list= {0., 0. }; {0., 0. }; dialist= dplist= {0., 0.};

(*foil thickness df in meter *) (*normalised foil flow resistance

Rf *)

(*perforation shape; 1= hole; 2= slit *) (*porosity *) (*diameter of perforation in meter*) (*plate thickness dp in meter *)

(Values df = 0 or dp = 0 indicate that there is no foil or plate in that position; the other foil or plate parameter entries in that position then are neglected). 5. Determine which absorber parameters should be varied for optimisation. Some parameters evidently cannot be varied, such as the number N of layers, the layer material type (in matlist), the shape of the perforations (in shapelist); others should be kept constant, because a variation could lead to values which cannot be realised (such as the foil material density f in flist ). The variable parameters could be signalled by lists of flags like: (*Input layers*) ¡ flag= {1 , 1 }; (*flow resistivities ¡ *) tflag= {0 , 0 }; (*layer thickness t in meter*) (*Input foils*) dfflag= {0 , 0 }; (*foil thickness df in meter *) (*normalised foil flow resistance Rf *) Rfflag= {1 , 1 }; (*Input perforates*) flag= {0 , 0 }; (*porosity *) diaflag= {0 , 0 }; (*diameter of perforation in meter*) (*plate thickness dp in meter *) dpflag= {0 , 0 };

A value 1 indicates that the corresponding parameter belongs to the {ai }, a value 0 signals that the parameter belongs to the {bk }. The number of unit values determines the dimension of the space of variables for the minimisation. 6. Some algorithms for minimisation need more than one set of start parameters, for example to indicate the direction of the search, or to fix the search range limits. They can, for example, be entered by lists like (where xx stands for suitable values at the list positions of the variable parameters ai ; entries 0 signal constant parameters): (*Input layers*) ¡ lo= {xx , xx }; {0 , 0 }; tlo= (*Input foils*) dflo= {0 , 0 }; Rflo= {xx, xx }; (*Input perforates*) glo= {0 , 0 }; dialo= {0 , 0 }; dplo= {0 , 0 };

(*flow resistivities ¡ *) (*layer thickness t in meter*) (*foil thickness df in meter *) (*normalised foil flow resistance

Rf *)

(*porosity *) (*diameter of perforation in meter*) (*plate thickness dp in meter *)

and (*Input layers*) ¡ hi= {xx , xx }; thi= {0 , 0 }; (*Input foils*) dfhi= {0 , 0 }; Rfhi= {xx, xx };

(*flow resistivities ¡ *) (*layer thickness t in meter*) (*foil thickness df in meter *) (*normalised foil flow resistance

Rf *)


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(*Input perforates*) ghi= {0 , 0 }; (*porosity *) diahi= {0 , 0 }; (*diameter of perforation in meter*) (*plate thickness dp in meter *) dphi= {0 , 0 };

7. If one writes a general algorithm for minimisation of a function q({ai }, {bk }), which should be usable for a changing number and composition of the {ai },{bk }, then one should write a subroutine which transmits the variable absorber parameters to the variables {ai }. 8. One needs correspondingly a subroutine which transmits the improved {ai } back to the right positions of the variable absorber parameters (the information comes from the lists of flags). This subroutine, at the same time, takes care of the range limits for the parameters. A lower limit zero, for example, will be respected, if the subroutine transmits |ai |.A lower or a higher non-zero limit can be introduced by the replacement of ai by the limit values . . . lo or . . . hi whenever ai exceeds a limit; this procedure replaces the variable function q(ai ; bk ) outside the limits by a constant value with respect to ai (the minimisation program consequently will avoid ranges outside the limits). Examples of 2-layer absorbers: Some examples will illustrate the procedure. The absorber is a two-layer absorber with locally reacting layers,possibly with foils and/or perforated plates (resonator neck plates) at the front sides of the layers. Diffuse sound incidence is applied. First example: This example uses the data input for the “original absorber” as given in the above lists; the variable parameters are ¡ list[[1]], ¡ list[[2]], Rflist[[1]], Rflist[[2]] (list[[n]] indicates the list element at position n of the list). The diagram below shows dif for both the original absorber (dashed curve) and the optimised absorber (full line); the print-out of the input data indicates the optimised parameters with bold printing.

Input & optimised parameters (*Input frequency*) flo=50., fhi=2000.,

lg(f)=0.1

(*Flags*) Reflection=diffuse, Weight=no (*Number of layers & dimensions*)

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Nlayer=2, dim=4 (*Layer parameters*) mat={1,1}

¡ ={13468.5, 4435.31} t={0.05,0.05} (*Foil parameters*)

f={2700.,2700.} df={0.0002,0.0002} Rf={1.28767, 1.67335} (*Perforate parameters*) shapes={1,1}

={0.,0.} dia={0.,0.} d={0.,0.} (*Minimum value of

< |r|^2>*)

< |r|^2>min=0.2783 (*Varied parameters*){¡ list[[1]],¡ list[[2]],Rflist[[1]],Rflist[[2]]}

The loss of absorption in the resonance maximum could be reduced by choosing a smaller frequency interval (flo ,fhi ) and a weight function w(f) with a central weighting (e. g. in the form of a cosine arc). Second example: The second example uses a resonator neck plate between the layers; its porosity is constant with = 0.15. The second layer makes up the resonator volumes; it consists of air (mat = 0). The Helmholtz resonance is marked by a rather low maximum of of the original absorber at higher frequencies.Varied parameters are ¡ list[[1]], tlist[[2]], Rflist[[1]], dialist[[2]], dplist[[2]].

Input & optimised parameters (in bold) (*Input frequency*) flo=50., fhi=2000.,

lg(f)=0.1

(*Flags*) Reflection=diffuse, Weight=no (*Number of layers & dimensions*) Nlayer=2, dim=5


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1025

(*Layer parameters*) mat={1,0}

¡ ={32183.4, 0.} t={0.02, 0.185033} (*Foil parameters*)

f={2700.,0.} df={0.0002,0.} Rf={0.903345, 0.} (*Perforate parameters*) shapes={0,1}

={0.,0.15} dia={0., 0.000174437} d={0., 0.00628449} (*Minimum value of < |r|^2>*) < |r|^2>min=0.233324 (*Varied parameters*){¡ list[[1]],tlist[[2]],Rflist[[1]],dialist[[2]],dlist[[2]]}

The third example is more of a theoretical than a practical interest. Thickness and flow resistance of the cover foil and of the first layer are variable.The absorption coefficients for diffuse incidence of the original absorber displays a resonance (curve) and the optimised absorber shows the upper nearly horizontal line.

Input & optimised parameters (in bold) (*Input frequency*) flo=50., fhi=2000.,

lg(f)=0.1

(*Flags*) Reflection=diffuse, Weight=no (*Number of layers & dimensions*) Nlayer=2, dim=4 (*Layer parameters*) mat={1,1}

¡ ={328.465, 10000.} t={3.26549, 0.05} (*Foil parameters*)

f={2700.,2700.}

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df={0.000328666, 0.0002} Rf={0.513109, 2.} (*Perforate parameters*) shapes={1,1}

={0.,0.} dia={0.,0.} d={0.,0.} < |r|^2>*) min=0.0492101 (*Varied parameters*){¡ list[[1]],tlist[[1]],dflist[[1]],Rflist[[1]]} (*Minimum value of

As one could have anticipated the result with this choice of the set of variable parameters. The front layer has an enormous thickness and a very small flow resistivity,and the cover foil is thin with a low flow resistance. It is of some interest that the average reflection coefficient < |r|2 > obtained by optimisation is very precisely the theoretical minimum for diffuse sound incidence on locally reacting absorbers. Example of a “parallel” absorber: The examples of multi-layer absorbers, shown above, can be named “series” absorbers. A “parallel” absorber, in contrast, is composed of different elementary absorbers which are placed side by side. As long as the lateral dimensions of the component absorbers and the composition are small compared to the wavelength, only the average admittance G counts.

Boxes of width L − a and depth d contain a porous absorber material (e. g. glass fibres) with flow resistivity ¡. Between the boxes are gaps of width a which form the necks of Helmholtz resonators. The boxes and/or one or both neck orifices may be covered with foils having a surface mass density mf = f · df and a normalised flow resistance Rf . With Gp for the input admittance of the boxes, and Ga for the input admittance of the resonator necks, the average admittance is: G =

G p + · Ga ; 1+

=

Fa a/L . = Fp 1 − a/L

Possibly varied parameters are: pars=

{L,

a/L, d/L, t/L,

¡ , dfp , Rfp , df 1 , Rf 1 , df 2, Rf 2 }

i. e. up to 11 variables in the search for a minimum.

(4)


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The dashed curve of the “original” absorber in the following diagram for the sound absorption coefficient with diffuse sound incidence was obtained after “manual” optimisation. The full curve was obtained after application of the optimisation algorithm on the printed input data.

Input & optimised parameters (*Input frequency*) flo=100. [Hz], fhi=2000. [Hz],

lg(f)=0.1

(*Flags*) Reflection=diffuse, Weight=no (*Dimensions*) L=0.2 [m], a/L=0.1, d/L=0.3, t/L=0.5 (*Layer parameters*) Mat=1

¡ =10000. [Pa·s/m^2] (*Foil parameters*)

fp=2700. [kg/m^3] dfp=0.0005 [m] Rfp=1. Foil Position=1

f1=2700. [kg/m^3] df1=0.0002 [m] Rf1=0.2

f2=0. df2=0. Rf2=0.

< |r|^2>*) < |r|^2>min=0.0886165 (*Minimum value of

(*Varied parameters*)

{a/L,d/L,df1,Rf1}={0.0862694,

0.554809, 0.0000810588, 0.35581}

Post-processing: One should notice that the algorithms for finding a minimum generally search for local minima, a number of which may exist in the range of the variable parameters, especially if the dimension of the parameter space is large.

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9.

You may try to find other minima either by starting with a different “original absorber” and/or by using other parameter ranges. 10. You may modify the optimum found, for example by an exchange of a large resonator volume depth with a smaller neck plate porosity. Then put the other optimised parameter values among the {bk } and append a run of optimisation with the parameters between which a “trading” shall be tried. 11. It is a good practice to evaluate the target quantity with some “manual” modifications of the optimised parameter values, in order to see whether the absorber is sensitive with respect to small changes of the parameter values. In a similar procedure it can be tested, whether parameter values which are evaluated by optimisation, but which are not available in reality, can be replaced by nearby values of available absorber components. 12. Do not forget that the optimum found may depend also on the frequency range used {flo ,fhi}. A final remark should be made. The program for finding the minimum may make wide excursions in the parameter space. Therefore the evaluations for the input admittance Z0 G and the reflection coefficient |r|2 should apply formulas with analytical foundations, in order to avoid false or even nonsense results. This implies that the evaluation of characteristic data of porous materials should not use formulas which stem of regressions through experimental data (like the Delany–Bazley approximation), because the range of the flow resistivity, for example, in which the data were measured may be exceeded. Evaluations are recommended which are based either on analytical models of porous materials or on analytical models which are fitted to experimental data. Further, the diameter of resonator necks may become very small within the search for a minimum. It is recommended to use the propagation constant and wave impedance in capillaries, which include viscous and thermal losses and which go to j · k0 , Z0 for wider necks.

O.2

Computing with Mixed Numeric-Symbolic Expressions, Illustrated with Silencer Cascades

See also: > Sects. J.19, J.20 dealing with silencer cascades, where the problem of this Section is avoided by neglecting acoustical feedback in the ducts, and > Sect. J.29 with a more simple application, due to the monotonic variation of cross-sections.

There are many tasks in acoustics which lead to iterative linear systems of equations. Such tasks are, for example, duct cascades, mufflers, conical ducts, wedges, a medium with spatial variation, etc. In principle one could consider the system of systems of linear equations as a large system for the combined vector of variables of those systems. However, this procedure mostly fails in its numerical realisation. On the other hand, there exist mathematical programs which support both numerical and symbolic computations.This feature can be used to design straightforward solutions for the mentioned tasks, avoiding large systems of equations and many inversions of matrices. The method will be explained and illustrated with the example of a cascade of sections of lined ducts with different cross-sections and/or linings.


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A sequence i = 1, 2, . . ., I of two-dimensional, flat duct sections follow each other, with half heights hi , lengths i , and lined with locally reacting absorbers with input admittances Gi .The fields in the sections are formulated as mode sums of forward propagating modes with amplitudes Ai,n and backward propagating modes with amplitudes Bi,n . The lateral wave numbers —i,n follow from the characteristic equations of the sections, the axial propagating constants ‚i,n from the secular equation. Some kinds of excitation at the entrance x = x0 will be considered, and some kinds of terminations at x = xI . The “heads” of duct steps are assumed to be hard. Field formulations:

pi (x, y) = Ai,n · e−‚i,n (x−xi−1 ) + Bi,n · e+‚i,n (x−xi ) · qi,n (y), n

Z0 vi,x (x, y) = −j

‚i,n k0

n

with the mode profiles:

Ai,n · e−‚i,n (x−xi−1 ) − Bi,n · e+‚i,n (x−xi ) · qi,n (y)

(1)

qi,n (y) = cos(—i,n y)

(2)

and the lateral wave numbers —i,n being solutions of the characteristic equations: (—i,n hi ) · tan(—i,n hi ) = jk0 hi · Z0 Gi

(3)

and the axial propagation constants from the secular equations: ‚i,n hi = (—i,n hi )2 − (k0hi )2 ; Re{‚i,n hi } ≥ 0.

(4)

Mode norms Ni,m : 1 hi

hi qi,m (y) · qi,n (y) dy = ƒm,n · Ni,m ; 0

Ni,m

1 sin(2—i,m hi ) = 1+ . 2 2—i,m hi

(5)

Mode coupling coefficients C(i, m; k, n): 1 C(i, m; k, n) = hi C(i, m; k, n) =

1 2

hi qi,m (y) · qk,n (y) dy; 0

i−1 k= , i+1

sin (—i,m − —k,n )hi sin (—i,m + —k,n )hi + (—i,m − —k,n )hi (—i,m + —k,n )hi

(6)

.

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Boundary conditions at x = xi for sound pressure and axial particle velocity: (hi+1 ≥ hi )

expanding duct: !

pi (xi , y) = pi+1 (xi , y) in y = (0, hi ) ,

! Z v (x , y) in y = (0, hi ) Z0 vi+1,x (xi , y) = 0 i,x i , 0 in y = (hi , hi+1 )

(7)

(hi+1 < hi )

contracting duct: !

pi (xi , y) = pi+1 (xi , y) in y = (0, hi+1 ) ,

! (x , y) in y = (0, hi+1 ) Zv Z0 vi,x (xi , y) = 0 i+1,x i . 0 in y = (hi+1 , hi )

(8)

Notations (for ease of writing): ‚i,n hi+1 = ‚hi,n ;

‚i,n i = ‚i,n ;

Ai,n · Ni,n = A¯ i,n ;

Bi,n · Ni,n = B¯ i,n .

(9)

By use of the orthogonality, with m = any mode order: expanding duct: (hi+1 ≥ hi )

¯ i,m · e−‚i,m + B¯ i,m = A A¯ i+1,n + B¯ i+1,n · e−‚i+1,n n

C(i, m; i + 1, n) , Ni+1,n

¯ i,n · e−‚i,n − B¯ i,n ‚hi+1,m · A¯ i+1,m − B¯ i+1,m · e−‚i+1,m = ‚hi,n · A ·

(10)

n

·

C(i, n; i + 1, m) , Ni,n

contracting duct: (hi+1 < hi )

¯ i+1,m + B¯ i+1,m · e−‚i+1,m = ¯ i,n · e−‚i,n + B¯ i,n A A n

‚hi,m · A¯ i,m · e−‚i,m

C(i + 1, m; i, n) , · Ni,n

¯ i+1,n − B¯ i+1,n · e−‚i+1,n − B¯ i,m = ‚hi+1,n · A

(11)

n

·

C(i + 1, n; i, m) . Ni+1,n

The ratios of mode coupling coefficients and mode norms on the right-hand sides form matrices {matrix}; we symbolise with {matrix}−1 the inverse of a matrix. Then the above systems (of couples of systems) of linear equations for the mode amplitudes lead to the iterative systems:


O

(hi+1 ≥ hi ) −1 C(i, n; i + 1, m) C(i, n; i + 1, m) 1 ‚h i,n A¯ i+1,m = + A¯ i,n · e−‚i,n 2 n Ni+1,m ‚hi+1,m Ni,n ‚hi,n C(i, n; i + 1, m) C(i, n; i + 1, m) −1 + Bi,n − , Ni+1,m ‚hi+1,m Ni,n

1031

expanding duct:

B¯ i+1,m · e−‚i+1,m =

1 A¯ i,n · e−‚i,n 2 n ‚hi,n C(i, n; i + 1, m) C(i, n; i + 1, m) −1 · − Ni+1,m ‚hi+1,m Ni,n ‚hi,n C(i, n; i + 1, m) C(i, n; i + 1, m) −1 , + + B¯ i,n Ni+1,m ‚hi+1,m Ni,n

(hi+1 < hi )

−1 C(i + 1, m; i, n) 1 ‚h C(i + 1, m; i, n) i,n A¯ i+1,m = + A¯ i,n · e−‚i,n 2 n Ni,n ‚hi+1,m Ni+1,m

−1 ‚hi,n C(i + 1, m; i, n) C(i + 1, m; i, n) + B¯ i,n − , Ni,n ‚hi+1,m Ni+1,m

(12)

(13)

contracting duct:

B¯ i+1,m · e−‚i+1,m =

1 A¯ i,n · e−‚i,n 2 n

C(i + 1, m; i, n) −1 ‚hi,n C(i + 1, m; i, n) · − Ni,n ‚hi+1,m Ni+1,m

C(i + 1, m; i, n) −1 ‚hi,n C(i + 1, m; i, n) + Bi,n . + Ni,n ‚hi+1,m Ni+1,m

(14)

(15)

There are more unknown mode amplitudes Ai,n , Bi,n than equations at the duct section limits. One needs source conditions and termination conditions. Alternative source conditions: Sound pressure source: At the entrance x = x0 a sound pressure profile P(y) is given (over the full height h1 , else the task would be ill posed). Expand the pressure profile in modes of the section i = 1: P(y) =

an · cos(—1,n y);

n

1 a¯ n = N1,n · an = h1

h1 P(y) · cos(—1,n y) dy.

(16)

0

The boundary condition for the sound pressure at x = x0 gives: A¯ 1,n + B¯ 1,n · e−‚1,n = a¯ n .

(17)

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One of the amplitudes A¯ 1,n, B¯ 1,n of the first section can be expressed by the other amplitude and a known number a¯ n . Particle velocity source: At the entrance x = x0 a particle velocity profile V(y) is given; the source may cover a height h0 ≤ h1 , assuming the other part (h0 , h1 ) of the entrance plane is hard. Expand in modes of the section i = 1:

Z V(y) in y = (0, h0 ) Z0 v1,x (x0 , y) = 0 an · cos(—1,n y), = 0 in y = (h0 , h1 ) n (18) h1 1 Z0 V(y) a¯ n = N1,n · an = · cos(—1,n y) dy. 0 h1 0

The boundary condition at x = x0 for the particle velocity gives: A¯ 1,n − B¯ 1,n · e−‚1,n =

jk0 h1 a¯ n . ‚h1,n

(19)

Again one of the amplitudes A¯ 1,n , B¯ 1,n of the first section can be expressed by the other amplitude and a known number containing a¯ n . Incident wave from the entrance duct i = 0: A sound wave with given mode amplitudes A0,n is incident from the entrance duct i = 0 which is supposed to be anechoic or infinite for x → −∞. Setting formally 0 = 0, i. e. defining the amplitudes A0,n in the plane x = x0 , the above equations can also be used for that cross-section. The right-hand sides contain only the amplitudes B0,n as unknown quantities. Depending on the selected source condition, the above systems of equations will have the general forms (imagine the iteration to be performed up to i): for pressure or velocity source: 1 B¯ 1,n · i,n + i,n , 2 n 1 B¯ 1,n · i,n + i,n = 2 n

¯ i+1,m = A B¯ i+1,m

(20)

with numerical values i,n , i,n , i,n , i,n and symbolic B¯ 1,n , for incident wave: 1 B¯ 0,n · i,n + i,n , 2 n 1 = B¯ 0,n · i,n + i,n 2 n

A¯ i+1,m = B¯ i+1,m

with numerical values i,n , i,n , i,n , i,n and symbolic B¯ 0,n .

(21)


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Alternative termination conditions: Section i + 1 = I anechoic: i. e. B¯ I,m = 0. Then the last of the above equations (12)–(15) is a system of linear inhomogeneous equations for either B¯ 1,n or B¯ 0,n . After its solution, all other mode amplitudes follow by insertion in the former systems of equations. Section i + 1 = I terminated with an admittance Gt : ¯ I,n , B¯ I,n : This termination condition gives the relations between A

¯ I,n · e−‚I,n − B¯ I,n = jk0 hI · Z0 Gt · A¯ I,n · e−‚I,n + B¯ I,n ‚hI,n A

(22)

¯ I,n , B¯ I,n can be expressed by the other (e. g. A¯ I,n by B¯ I,n ). Thus so one of the amplitudes A the last of the above iterative systems of equations will give two coupled systems of equations for the amplitudes B¯ I,n , B¯ 0,n or B¯ I,n , B¯ 1,n (depending on the source condition). After the solution, all other amplitudes follow by insertion. General termination: With other terminations, e. g. a radiating duct end of the section i = I, one always gets a relation of the form (the termination with an admittance Gt is a special case thereof with a diagonal matrix): {A¯ I,m } = {{matrix}} • {B¯ I,n } ,

(23)

so one will end again in the two coupled systems of equations mentioned above. The general problem with cascades of ducts and layers comes from the fact that one gets enough systems of equations only after the source condition has been concatenated with the termination condition. And one gets solvable systems of equations after the ¯ i,n , B¯ i,n for the intermediate sections are eliminated. This elimination can amplitudes A be done in an analytical manner only for a low number I of sections. A solution which is suited for numerical evaluation is obtained if the systems of equations from the boundary conditions are transformed to iterative systems, and a mixed numerical-symbolic evaluation (“hybrid” evaluation) is applied to that system. The hybrid evaluation makes use of the ability of mathematical programs (like Mathematica or Maple) to handle hybrid expressions. They automatically simplify numerical terms in the expressions and give them canonical forms, so that the numerical coefficients or the symbolic factors can be extracted. Thus the final system(s) of equations for the numerical evaluation can easily be obtained. And if the right-hand sides of the intermediate equations are saved, one gets the numerical values of the left-hand sides (after solution of the final system(s)) just by calling them,because the key solutions (B¯ I,n , B¯ 0,n , B¯ 1,n ) then are automatically inserted.

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O.3

Five Standard Problems of Numerical Acoustics


Five standard problems of numerical acoustics are presented below which frequently occur in practical applications.

O.3.1 The Radiation Problem A vibrating structure radiates sound into the surrounding space. The radiated sound field is characterized by the sound pressure p, particle velocity v , and derived quantities such as the sound intensity I, the radiated sound power ¢, the radiation efficiency etc., which will be calculated by numerical methods. As shown in the figure, the bounded volume of the radiating structure in three-dimensional space is denoted by B (like Body). The interior of B is called Bi and the exterior Be . The surface normal n should be directed into the exterior Be .

The complex sound pressure p, radiated into the free, three-dimensional space has to satisfy the Helmholtz equation p + k02 p = 0

in

Be ,

(1)

where k0 = –/c0 is the wave number, – is the angular frequency, c0 the speed of sound, and is the operator. All time-varying quantities should obey the time √ Laplace dependence exp j–t with j = −1. The fluid in the outer space is assumed to be lossfree, homogeneous, and at rest. For a complete description of the problem, boundary conditions on the surface of the radiator and at infinity are needed. If the body has edges corner conditions must be applied also, see > Sect. B.16. The most important Neumann boundary value problem describes a body,which vibrates with a given normal velocity v. Therefore, the pressure gradient ∂p = −j–0 v ∂n

on

S

(2a)

is prescribed on S. Here, 0 is the fluid density and ∂/∂n is the derivative in the direction of the outward normal n. Sometimes, the sound pressure p = p0

on

S

(2b)

is given,which is called the Dirichlet problem.The most general form of a local boundary condition is the Robin or impedance problem with ∂p R p := + j–0 Gp = f (S) ∂n

on

S,

(2c)


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1035

where f (S) is a function defined on S. R(p) is a linear boundary operator. G is the field admittance in the direction of the normal n. For G → 0, we obtain the Neumann problem, for G → ∞ the Dirichlet problem is obtained. If Z: = 1/G is a nonzero but finite quantity, the general impedance problem results, which describes locally reacting absorbing surfaces. In addition, the physical requirement that all radiated waves are outgoing leads to the Sommerfeld radiation condition ∂p lim R + jk0p = 0 , (3) R→∞ ∂R which can be interpreted as a boundary condition at infinity. Here, R = |x| = x12 + x22 + x32 denotes the distance from x to the origin, where points in space are denoted by simple letters such as x = (x1 , x2, x3 ). The Sommerfeld condition generally leads to the choice of functions among different mathematically possible alternatives (like the corner condition does) ; therefore it decides about the sign of exponents and of roots, and if a field function has branch points, it gives rules about which branch of multi-valued functions should be selected. Equations O.3.(1), O.3.(2a, 2b, or 2c), and O.3.(3) describe the radiation problem for the radiated pressure p. With the knowledge of p and v the effective sound intensity

I =

1 ∗ Re p v , 2

(4)

the effective sound power 1 ¢(p, v): = Re pv ∗ ds, 2

(5)

S

and the radiation efficiency = ¢ p, v /¢ 0 c0 v, v

(6)

can be easily calculated. Here, ¢ 0 c0 v, v is the power of a plane wave with p = 0 c0 v

(7)

on S [Junger/Feit (1972)]. The asterisk denotes the complex conjugate and Re{. . .} the real part of the quantity in brackets. Due to definition O.3.(6), the quantity ¢ 0 c0 v, v can be considered as a sound power with radiation efficiency = 1. Hence, the numerical computation of the radiated sound field allows a numerical sound intensity method to be used. Such a method can be used for the localisation of acoustical sources on the surface of the vibrating machine structure or for detecting cracks that cause changes of the acoustical surface intensity [Koopmann/Perraud (1981)].

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O.3.2 The Scattering Problem

See also:

>

Chap. E for scattering.

For the related scattering problem, an incident wave pin is impinging on the body B and causes a scattered wave ps . The scattering problem for the scattered pressure ps is again described by the Helmholtz equation ps + k02 ps = 0

in

Be

and the corresponding radiation condition ∂ps lim R + jk0ps = 0. R→∞ ∂R

(8)

(9)

Now, the boundary condition is homogeneous, since the body is at rest, and it must be formulated for the total pressure pT = pin + ps . Similar to the radiation problem, the Neumann problem for an acoustically rigid scatterer is given by: (10a) ∂pT ∂n = 0 on S. A perfectly (acoustically) soft scatterer leads to the Dirichlet condition pT = 0

on

S.

(10b)

The general form of the impedance boundary condition is described by: ∂pT + j–0 GpT = 0 , R pT : = ∂n where again G is the field admittance at the surface S.

(10c)

The scattering problem can be formulated as an equivalent radiation problem by the following procedure: considering the hard scatterer, the normal velocity vin of the incident pressure wave pin will be evaluated at the surface S where the scatterer is assumed (for the moment) to be sound transparent. If B is now vibrating with the negative normal velocity (−vin ), the radiated sound pressure is identical to the pressure ps scattered from B due to the incident wave pin . Hence, instead Eq. O.3.(10a), we simply have: ∂ps = −j–0 (−vin ) (11a) ∂n for the scattering problem, which again is an inhomogeneous boundary condition like Eq. O.3.(2a).Analogously, the impedance boundary condition O.3.(10c) can be written as R ps = f , (11b) where f is the known function f = −R pin . Equations O.3.(8), O.3.(9), O.3.(11a/11b) describe the scattering problem as an equivalent “radiation problem” for the scattered pressure ps . In conclusion, the radiation and the scattering problem can be treated by numerical methods in a uniform way. Having calculated the main quantities ps and v s , the scattered effective intensity (or intensity of return) is given by: 1

Is = Re ps v s∗ . 2

(12)


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Assuming that the incident wave is a plane wave with incident intensity Iin (in the direction of incidence), the target strength TS is defined by: TS = 10 lg

Is (r = 1 m) [dB] , Iin

(13)

where Is (r = 1 m) is the effective intensity of the sound returned by the scatterer at a distance of 1 m from its acoustical center in some specified direction. In Eq. O.3.(13), lg denotes the logarithm to the base 10. The target strength is often used in the context of underwater sound (see [Urick (1983)], for more details).

O.3.3 The Sound Field in Interior Spaces If a sound field in an interior space Bi is considered, the Helmholtz equation O.3.(1) must be satisfied in Bi . This is a classical problem of room acoustics. At the boundary S of the enclosure, there may exist a rigid part S1 with ∂p/∂n = 0, an acoustically soft part S2 with p = 0, and an absorbing surface S3 with ∂p/∂n + j–0 G p = 0, such that S = S1 ∪ S2 ∪ S3 . In contrast to exterior radiation or scattering problems, a finite volume is considered, and no radiation condition is necessary. If no acoustical sources such as vibrating walls or bodies are inside the room, the Helmholtz equation together with the boundary conditions represents an eigenvalue problem for the determination of the eigenfrequencies and eigenmodes of the enclosed fluid. On the other hand, sources such as vibrating parts of the boundary, point sources, etc. lead to forced vibrations of the fluid. Sometimes, interior and exterior acoustic problems can be coupled. Such a situation occurs when an interior acoustic space is connected to an exterior one through openings [Seybert/Cheng/Wu (1990)]. A simple example is a duct with an open end rising into the surrounding infinite space. The problem of a half space belongs to the class of exterior problems rather than to interior problems (see > Sect. O.5.5). Numerical methods based on the solution of the wave equation or Helmholtz equation (reduced wave equation) are only useful for the treatment of small enclosures at low frequencies, i. e. for small k0 a numbers, where a is a characteristic dimension of the room (for example the diameter or one side of the room). For problems with high k0 a numbers, it is more convenient to use methods of geometrical acoustics (see > Ch. M).

O.3.4 The Coupled Fluid–Elastic Structure Interaction Problem If the radiating or scattering structure B consists of an elastic material, the interaction between the body and the surrounding fluid must be taken into account. In addition, the structure can be coupled with an internal acoustic cavity. The problem is significantly simplified if the acoustic loading of sound inside the cavity can be neglected. Following Soize [Soize (1998)], the boundary value problem can be described as follows: B is the bounded domain occupied by the linearly elastic structure. The boundary S

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is divided into three parts: S = S0 ∪ S1 ∪ S2 . On S0 the boundary is fixed. This means that u = 0 on S0 where u = (u1 , u2, u3) denotes the displacement field. On S1 ∪ S2 , it is free (see the figure).

For simplicity, we only consider the coupling between a structure and an external fluid. Systems coupled with internal acoustic cavities are described in [Soize (1999)]. The structure is subject to a body force f = (f1 , f2, f3) and a surface force fS1 = fS1,1 , fS1,2 , fS1,3 . The steady state response of the linearly elastic structure is given by: −–2 s ui − ij,j = fi ij nj = −pni

on

ij nj = fS1 ,i

on

ui = 0

S0 ,

on

in

B,

S2 , S1 ,

(14a) (14b) (15) (16)

where the summation convention over repeated indices is used. s is the mass density of the structure and ij,j =

3 ∂ij j=1

∂xj

.

For a linear viscoelastic material, the stress tensor is given by: ij = aijkh (x, –) —kh (u) + bijkh (x, –) —kh j–u , where summation over indices k and h must be performed, and the linearized strain tensor is: ∂uk ∂uh —kh (u) = + /2. ∂xh ∂xk The coefficients aijkh (x, –) and bijkh (x, –) are real and depend on the properties of the elastic medium. The coupling term between the external fluid and the structure is the pressure p that acts like a fluid load. Introducing the velocity potential • by: grad • = v ,

(17)

the external fluid is described by the Helmholtz equation for the potential • + k02 • = 0.

(18)


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According to Newton’s law the external pressure is given by: p = −j–0 •.

(19)

At the outer boundary S2 the normal velocity of the structure ∂un ∂t = j–un and of the fluid vn = ∂• ∂n must be equal (20) ∂• ∂n = j–un . In addition, the potential • has to fulfill the Sommerfeld radiation condition O.3.(3). For solving the coupled fluid–structure problem, Eqs. O.3.(14), O.3.(15), O.3.(16), O.3.(18), O.3.(20) have to be solved. Clearly, the vibrating structure can be coupled to an external fluid and to an internal acoustic cavity leading to very similar equations.

O.3.5 The Transmission Problem

See also: Chap. I on sound transmission.

The transmission problem is characterized by the fact that the incident sound wave pin can penetrate into the body B, which is assumed to have acoustic constants (sound speed ci and density i ) different from those of the surrounding medium c0 and 0 . The total pressure p = pin + ps in Be and the interior pressure pi in Bi have to satisfy the Helmholtz equations p + k02 p = 0

in

Be ,

pi + ki2 pi = 0

in

Bi

with ki = –/ci , the radiation condition ∂p + jk0p = 0 lim R R→∞ ∂R

(21)

(22)

and the transmission conditions for the pressure and normal velocities p − pi = f

on

v − vi = g ⇔

S,

j ∂pi j ∂p − =g –0 ∂n –i ∂n

(23) on

S,

(24)

where f and g are given continuous functions on S. For f = g = 0 pressure and velocity are continuous at the boundary. It can be shown (see [Colton/Kress (1983), p. 101]) that the transmission problem has a unique solution.

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O.4 The Source Simulation Technique (SST) The source simulation technique is a general tool for calculating the sound radiation or scattering from complex-shaped structures into the three-dimensional space, [Kress/ Mohsen (1986); Ochmann, Acustica (1990); Ochmann (1995); Ochmann (1999); Ochmann (2000); Ochmann/Wellner (1991)]. Hence, it can be used for the numerical solution of the first and second standard problem (see > Sect. O.3, problem 1 and problem 2). It should be noted that many names are in use for the same or similar methods such as source simulation method, multipole method, superposition method, spherical wave synthesis , etc. [Attala/Winckelmans/Sgard (1999); Bobrovnitskii/Tomilina (1990); Bobrovnitskii/Tomilina (1995); Cremer/Wang (1988); Cunefare/Koopmann/Brod (1989); Fahnline/Koopmann (1991); Hwang/Chang (1991); Heckl (1989); Jeans/Mathews (1992); Johnson/Elliott/Baek/Garcia-Bonito (1998); Karageorghis/Fairweather (1998); Kress/Mohsen (1986); Koopmann/Song/Fahnline (1989); Masson/Redon/Priou/Gervais (1994); Ochmann,Acustica (1990); Ochmann (1995); Ochmann (1998); Ochmann (1999); Ochmann (2000); Ochmann/Homm (1994); Ochmann/Wellner (1991)]. The basic idea of the method consists in replacing the structure by a system of acoustical sources placed in the interior of the structure. By definition, these source functions have to satisfy the Helmholtz equation and the radiation condition. For solving the radiation or scattering problem completely, the source system also has to fulfil the boundary conditions on the surface of the body or, equivalently, a certain boundary equation. The better the system of sources satisfies the boundary condition on the surface of the structure, the closer is the agreement between the original and the simulated sound field. Spherical wave functions are often used as sources, since they can easily be calculated. Bodies of arbitrary shape are treated by taking into account spherical wave functions with different source locations. To solve the boundary equation, which minimizes the boundary error, the method of weighted residuals is applied. Depending on the choice of the weighting functions, different variants of the source simulation technique are obtained, for example, the null-field equations and the full-field equations. The full-field equations often lead to better conditioned sets of equations than the null-field equations. The null-field and the full-field equations can also be derived from the interior or exterior Helmholtz integral equation, respectively (see [Ochmann (1999); Ochmann (2000)]). This illustrates the close relationship between the source simulation technique and the boundary element method (see > Sect. O.5). This presentation follows the lines given in the review article [Ochmann (2000)], where more details and examples of calculations for several radiation and scattering problems can be found. In the list of references many different aspects of the SST can be found: for instance, the calculation of sound fields in the interior of enclosures containing scattering objects, [Johnson/Elliott/Baek/Garcia-Bonito (1998)], the treatment of scattering and radiation from bodies of revolution, [Stepanishen (1997)], a formulation in the time domain, [Kropp/Svensson (1995)], or the use of special surface sources weighted according to a Gaussian distribution, [Guyader (1994)]. In > Sect. O.3.2 it is shown that the scattering problem can be considered as an equivalent radiation problem. Thus, it is sufficient to treat only the radiation problem in the following. Exceptionally, only in this chapter, the time convention e−i–t is used, in order to be in agreement with most of the related literature about the SST.


O.4.1

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General Description of the Source Simulation Technique

The basic idea of the SST consists of replacing the vibrating body by a system of sources placed in the interior of the body. The sources are denoted by q(x, y) where x is an arbitrary point in space and y is the position of the singularity, i.e., the location of the source point. Now, for the SST in its most general form, it is assumed that the pressure can be presented in the form c(y)q(x, y)dy , (1) p(x) = Q

where Q is a region which is fully contained in Bi and embodies all sources; c(y) is the yet unknown source density, which gives every source a certain source strength. Every single source function (also called trial function) q(x, y) itself can consist of a finite or infinite sum of elementary sources such as monopoles, dipoles, etc.: q(x, y) = d0 q0(x, y)+d1 q1 (x, y)+. . . The volume integral in Eq.O.4.(1) reduces to a surface integral or a contour integral if the region Q is a surface or a line, respectively. The integral turns into a finite sum if isolated point sources are used or if the integral must be discretized for numerical reasons, since we cannot work with infinitely many sources. The system of functions c(y)q(y) with y ∈ Q will be called the source system. All functions of the source system have by definition to satisfy Eqs. O.3.(1) and O.3.(3) (with respect to x). The source system also has to satisfy the boundary condition on the surface S. As a consequence of the present time convention ei–t , the boundary conditions O.3.(2a) and O.3.(2c) must be written with a different sign ∂p ∂n

= i–0 v ,

R(p): =

∂p − i–0 Gp = f ∂n

(2a) on S .

(2b)

Hence,by substituting Eq.O.4.(1) into Eq.O.4.(2a) or into the general radiation boundary condition O.4.(2b), one gets for the rigid radiator ∂ c(y) q(x, y)dy − j–p0 v = 0 on S (2c) ∂n Q

or, in the general case ⎡ ⎤ R⎣ c(y)q(x, y)dy ⎦ − f = 0

on S .

(2d)

Q

These equations “live” on the boundary, and they are called the boundary equations of the SST. Thus by using sources as trial functions, the original domain problem can be transformed into a boundary problem. This illustrates that the SST can be considered as a counterpart to the finite element method (FEM, see > Sect. O.6), which solves the original domain problem. If the boundary equations can be solved exactly, the coefficients c(y) are determined such that the sound field generated by the vibrating

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structure is identical to the field produced by the source system. This follows from the unique solvability of the exterior problems described in > Sect. O.3 (see [Colton/Kress (1983)]). Such a source system is called an equivalent source system. Consequently, the exact solution of the radiation problem can be found if it is possible to construct an equivalent source system. However, such exact solutions are only known for special geometries in standard coordinate systems (e.g. in spherical coordinates). For nearly all relevant practical problems the surface S has a complicated shape. Therefore, we are only looking for approximate solutions of the boundary equations by minimizing the so-called boundary error or residual ∂ „(x) = q(x, y)dy − i–0 v on S c(y) (3a) ∂n(y) Q

or

⎡ ⎤ „(x) = R ⎣ c(y)q(x, y)dy ⎦ − f

on S .

(3b)

Q

The minimization process can be performed by means of the method of weighted residuals, which is a very general approach (see [Ochmann,Acustica (1990)]). It consists in choosing a complete family of weighting functions wn ; n = 1, 2, 3, . . . and demanding that ⎧ ⎡ ⎤ ⎫ ⎨ ⎬ „(x)w (x)ds = c(y)q(x, y)dy⎦ − f w (x)ds = 0; = 1, 2, 3 . . . . (4) R⎣ ⎩ ⎭ S

S

Q

The completeness ensures that the residual will go to zero if the number of weighting functions tends to infinity.Equation O.4.(4) is called the weighted residual equations of the SST for the determination of the source density c(y). It can be seen that different variants of the SST stem from different choices of sources and weighting functions. For example, the kind, number, and locations of source and corresponding weighting functions are important parameters of the method. Important source functions are the spherical wave functions, which will be introduced in the next section together with corresponding symmetry relations.

O.4.2

Spherical Wave Functions and Symmetry Relations

By definition, sources must be radiating wave functions. However, analytical solutions of the Helmholtz equation can only be constructed explicitly in separable coordinate systems, [Morse/Feshbach (1953), p. 494]. Inthree-dimensional space, the spherical wave functions present the simplest form of such solutions. Hence they are the type of sources most often used, and they are given by (1) m •c,s nm (x) = nm hn (k0 r)Pn (cos ˜ )

cos(mœ) sin(mœ)

,

(5)


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where the Pm n (cos ˜ ) are the associated Legendre polynomials, [Abramowitz/Stegun (1972)]. Here, spherical co-ordinates {r, ˜ , œ} are introduced by x = {r sin ˜ cos œ, r sin ˜ sin œ, r cos ˜ }. The superscript c (or s) in the equations below indicates that the cosine (or sine) is used. The cylindrical functions h(1) n (z) are the spherical Hankel functions of the first kind (with the choice e−i–t ), [Abramowitz/Stegun (1972)]. The normalizing factors 1; m = 0 —m (n − m)! 1/2 ; —m = (6) nm = (2n + 1) 4 (n + m)! 2; m > 0 are chosen in such a way that the spherical harmonics c,s ynm (x) = nm Pm n (cos ˜ ) ·

cos(mœ)

(7)

sin(mœ)

are orthonormal with respect to the integration over the unit sphere: 2 0

ymn · y‹Œ sin ˜ d˜ dœ =

0

1;

if = , ‹ = m, and Œ = n

0;

else

.

(8)

where and stand for c or s, respectively. Taking into account that there are only wave functions of cosine type for m = 0, the number of different spherical wave functions up to an index n0 is given by n0

(2j + 1) = (n0 + 1)2 .

(9)

j=0 c,s For simplicity, we denote the •c,s nm and ynm by • and y , respectively, where the index = 0, 1, 2, . . . runs through all combinations of m and n for c and s. The regular wave functions

”1 = Re{•1 }

(10)

present standing waves, where Re{} denotes the real part of the quantity in brackets. The regular wave functions contain spherical Bessel functions jn (z) instead of Hankel functions like the radiating wave functions, since h(1) n (z) = jn (z) + iyn (z)

(11)

with yn (z) spherical Neumann functions. For deriving the null-field and the full-field equations from the weighted residual Eq. O.4.(4), the following symmetry relations are very important ∂•m ∂• − •m ds = 0 , (12a) • ∂n ∂n S ∂•m ∂” i − •m ds = ƒm , ” (12b) ∂n ∂n k0 S ∂•m ∂•∗ 2i − •m ds = ƒm , •∗ (12c) ∂n ∂n k0 S

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where the asterisk denotes the complex conjugate, and ƒm is the Kronecker Delta. Equation O.4.(12a) is valid for all radiating wave functions as shown in [Ochmann, Acustica (1990)]. The proofs for Eqs. O.4.(12a)–O.4.(12c) can be found in [Ochmann (1999)] together with [Ochmann (1995)]. Since the bilinear form [u, v, ]: = uvdy (13) S

is symmetric, i.e. [u, v, ] = [v, u], all three symmetry relations O.4.(12) for the operator D = ∂/∂n can be extended to the more general boundary operator R (see Eq. O.4.(2b)): [• , R•m ] − [R• , •m] = 0 ,

(14a)

i , k0 2i [•∗ , R•m ] − [R•∗ , •m] = ƒm . k0 [” , R•m ] − [R” , •m] = ƒm

(14b) (14c)

It is well-known that the spherical wave functions can be interpreted as multipoles. Hence in related works the name multipole method or multipole radiator synthesis is used, especially in the case when a sum of multipoles is located at a few isolated points in the interior Bi.By combining the results of > Sects. O.2 and O.3,the SST with spherical wave functions is obtained.

O.4.3 Variants of the SST with Spherical Wave Functions In the following, spherical wave functions are used as equivalent sources. Only sparsely scattered attempts can be found in the literature, in which other types of source functions are used. For example, functions in prolate spheroidal coordinates were applied in [Hackman (1984)], and surface sources weighted according to a Gaussian distribution were employed in [Guyader (1994)]. The reason is that it is more difficult to deal with such functions than with spherical wave functions. Fortunately, working with spherical functions is adequate, since the use of several source locations distributed over the interior Bi enables us to treat complex non-spherical geometries. This is shown in > Sects. O.5 and O.6. In the present section, only spherical wave functions with respect to one source location (placed in the origin) as defined in Eq. O.4.(5) will be considered. Hence, the source area is simply Q = {0}, and the source function q(x, 0) (see Eq. O.4.(1)) consists of a series of spherical wave functions with increasing order in the most general case. Hence, in accordance with Eqs. O.4.(1) and O.4.(5) it can be written p(x) = c(0)q(x, 0) =

∞

cm •m .

(15)

m=0

By introducing expansion O.4.(15) into the weighted residuals O.4.(4) one gets the weighted residual equations with spherical wave functions as sources: ∞ m=0

cm

R[•m (y)]w (y)ds = S

f (y)w (y)ds; S

= 0, 1, 2, 3 . . . .

(16)


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These equations are called spherical weighted residual equations. Three important variants of the SST result from specifying the weighting functions in Eq. O.4.(16). All other parameters are fixed.

O.4.3.1

Null-Field-Like Equations

As mentioned in > Sect. O.3, the family of weighting functions has to form a complete system in the Hilbert space H = L2 (S) of square integrable functions over the surface S equipped with the usual scalar product uv ∗ds for u, v ∈ H . (17) (u, v) = S

Recall that a system of functions is complete if the only function orthogonal to all other functions is the null function (see for example [Higgins (1977), p. 15]). In [Vekua (1953)] Vekua has shown that the spherical wave functions are complete in the Hilbert space H for every wave number k if S is a Lyapunov surface as assumed in > Sect. O.2.1 (see Theorem 1 of [Ochmann (1995)]). Hence, the spherical wave functions are also admissible as weighting functions: w = • .

(18)

Such an approach is known as the Galerkin method where trial and weighting functions are identical. By introducing these weighting functions into the spherical weighted residual equations O.4.(16), we obtain ∞ cm R[•m (y)]• (y)ds = f (y)• (y)ds; = 0, 1, 2 . . . . (19) m=0

S

S

The symmetry relation O.4.(14a) yields ∞ cm •m (y)R[• (y)]ds = f (y)• (y)ds; S

m=0

(20)

S

or equivalently p(y)R[• (y)]ds = f (y)• (y)ds; S

= 0, 1, 2 . . .

= 0, 1, 2 . . .

(21)

S

by taking into account expansion O.4.(15). Choosing R = ∂/∂n and f = j–0 v yields exactly the null-field equations for Neumann data (see [Kleinman/Roach/Ström (1984)], Eq. O.3.(3)). Hence, the following result is obtained: the null-field equations for Neumann data use the spherical wave functions as weighting functions. If in addition the pressure p is expanded into spherical wave functions, then the null-field equations and the spherical SST are identical. Similarly, the null-field equations for Dirichlet data (see [Kleinman/Roach/Ström (1984), Eq. (2)]) are obtained if the weighting functions w1 = ∂•n /∂n are used. The null-field equations were described in many papers (see for example [Colton/Kress (1983); Kleinman/Roach/Ström (1984); Martin (1982); Stupfel/

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Lavie/Decarpigny (1988)]).Theorem 6 of [Ochmann (1995)] shows that the the null-field equations minimise the amplitude of the reactive power of the error field p =

N

cm • m − p ,

(22)

m=0

but not the sound power radiated by this error field (the error disappears if N → ∞, see Eq. O.4.(15)). For most radiation problems it would be more desirable to minimise the sound power of the radiated error wave directly. Unfortunately, to the author’s knowledge, such a variational principle has not yet been found (see [Ochmann (1995)]). In Sect. 6 of [Ochmann (2000)] the null-field equations were derived from the Helmholtz integral equation for the interior field, which is an ill-posed integral equation of the first kind. This shows that the null-field equations and the related T-matrix approach of Waterman, [Waterman (1969)], can lead to unstable systems of equations. Hence, alternative formulations of the SST can be found by choosing other sets of weighting functions.

O.4.3.2 The Full-Field Equations If the regular wave functions ” = Re{• } are selected as weighting functions and introduced into the spherical weighted residual equations O.4.(16) one obtains ∞ m=0

cm

R[•m (y)]” (y)ds = S

f (y)” (y)ds;

= 0, 1, 2 . . . .

(23)

S

By employing the symmetry relation O.4.(14b) we find ∞ i c + cm •m (y)R[” (y)]ds = f (y)” (y)ds; k0 m=0 S

= 0, 1, 2 . . . .

(24)

S

These equations are called full-field equations of the first kind. They are derived and discussed in [Ochmann (1999)]. In [Ochmann (1999); Ochmann (2000)], the full-field equations of the first kind are also derived from the exterior Helmholtz integral equation for the “full” outer space, which clarifies the term “full-field equations” in contrast to the “null-field equations”. It can be shown that the diagonal elements diag1 = i/k0 on the left-hand side of Eq. O.4.(24) have a stabilizing effect on the resulting system of equations. However, the full-filled equations of the first kind have the following disadvantage: The family of weighting functions {” } is not complete and hence does not constitute a basis for the Hilbert space L2 (S) whenever k0 is an eigenvalue of the interior Dirichlet problem (see [Waterman (1969),Appendix A]). This problem is closely related to the appearance of critical frequencies in the boundary element method (see [Schenck (1968)] and > Sect. O.5.4). However, the completeness of weighting functions is necessary for


O

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inverting the system of equations O.4.(16) (see [Kleinman/Roach/Ström (1984), Appendix 2]). To avoid such difficulties at certain critical frequencies, it is recommended to use the complex conjugate spherical wave functions (see [Ochmann (1999)]) w = •∗

(25)

as weighting functions. Since the • are complete, it follows that the •∗ are also complete and constitute a basis for L2 (S) at every wave number k0 . On the other hand, the symmetry relation O.4.(14c) also contains a diagonal element on the right-hand side similar to Eq. O.4.(14b). Hence, by inserting the functions O.4.(25) into Eq. O.4.(16) and applying Eq. O.4.(14c) one finds the equations ∞ 2i c + cm •m R[•∗ ]ds = f •∗ ds; k0 m=0 S

= 0, 1, 2 . . . .

(26)

S

As suggested in [Ochmann (1999)], Eq. O.4.(26) were called full-field equations of the second kind. Again, the diagonal elements diag2 = 2diag1 = 2

i k0

(27)

arising from the application of Eq. O.4.(14c), have a stabilizing effect. This was demonstrated in Sect. 10.3 of [Ochmann (2000)]. The idea of using the complex conjugate functions for constructing energy expressions goes back to Cremer and Wang [Cremer/ Wang (1988)].

O.4.3.3 The Least Squares Minimization Technique Finally, we consider the weighting functions w =

∂•∗ . ∂n

(28)

It is known that these functions are complete in L2 (S) (see [Ochmann (1995), p. 517]). By introducing these functions into the spherical weighted residual equations O.4.(16) with Neumann data (R = ∂/∂n, f = j–0 v) one gets ∞

cm

m=0

S

∂•m ∂•∗ ds = ∂n ∂n

f (y) S

∂•∗ ds; ∂n

= 0, 1, 2 . . . .

(29)

Equation O.4.(29) has the advantage that the corresponding matrix is Hermitian since am : =

∂•m ∂•∗ ∗ = am . ∂n ∂n

(30)

It can be shown that Eq. O.4.(29) minimise the surface velocity error E/i–0 with !2 !! ∞ ! ∂•m ! ! − i–v ! ds cm (31) E= ! ! ! ∂n S

m=0

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in the mean square sense (see [Ochmann (1995), Ch. 6.2]). Hence, Eq. O.4.(31) can be interpreted as the normal equations of the least squares method. It is also possible to consider the least squares method as an orthogonalization method and to employ the Gram-Schmidt technique as described briefly in [Ochmann (1995), p. 519].An approach which deals directly with orthonormalized functions can be found in [Wu/Yu (1998)]. For extending the least squares method to the mixed boundary conditions O.4.(2b) we suggest using the weighting functions w = (R• )∗ =

∂•∗ − G∗ •∗ . ∂n

(32)

According to Millar ([Millar (1983)], see also [Colton/Kress (1983), p. 97]), these functions are complete in the space L2 (S) if the assumption Im{G} ≤ 0

(33)

is satisfied. Then, the weighting functions O.4.(32) are admissible and again lead to a set of equations with a corresponding Hermitian matrix ∞

cm

m=0

∗

R•m (R• ) ds = S

f (R• )∗ ds;

= 0, 1, 2 . . . .

(34)

S

As before, Eq. O.4.(34) are the normal equations of the least squares method which now minimize the error functional !2 !! ∞ ! ! ! E= cm R•m − f ! ds . (35a) ! ! ! S

O.4.3.4

m=0

Summary: Weighting Functions and Corresponding Methods

> Table 1 provides an overview of all the variants of the spherical SST described, depending on the weighting functions chosen. In all cases only spherical wave functions with respect to one source location are used as source functions (cf. Eq. O.4.(15)).

Table 1 Methods and weighting functions Method

Weighting functions

Null-field equations for Neumann data

•`

Null-field equations for Dirichlet data

@•` =@n

Full-field equations of the first kind

”` = Ref•` g

Full-field equations of the second kind

•`

Least squares method

(@•` =@n) or (R•` )


O.4.3.5

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Extension of the SST by Using Equivalent Sources with Complex Source Points

It is possible to shift the coordinates of the singularities of the equivalent sources from real values into the complex plane. For example, if point sources (to the time factor e+j–t ) " (35b) g(x, y) = A exp(−jkR)/(4R); R = (x − xS )2 + (y − yS )2 + (z − zS )2 are used, the new sources are obtained by adding imaginary parts to the real source coordinate y = (xS , yS , zS ): , yc = (xS − ja, yS − jb, zS − jc) = y − j

= (a, b, c) .

(35c)

Here, A is the complex amplitude, x = (x, y, z) is the receiver position, and yc is the complex source position. Such a monopole with a complex source point is also a solution of the Helmholtz equation with respect to the spatial coordinates x = (x, y, z) for a fixed complex source position yc . The proof can be found in [Ochmann (2005)]. However, it must be taken in account that the singularities of the complex source are different from the point singularity ƒ(x, y) of the ordinary monopole.The complex distance R = Rr +jRi becomes zero on the circle C = {x ∈ E/|x − y| = },

| = |

that lies in the plane E = {x/(x − y) · = 0}) with center in y = (xS , yS, zS ). This is shown in the Fig. 1.

Figure 1 Complex source point and corresponding geometry By adding such complex source point solutions to the system of equivalent sources, strongly focused sound field similar to sound beams could be computed with promising accuracy. For example, the radiation from a circular piston in an infinite rigid baffle, which is vibrating in phase with the normal velocity v0 , was investigated in [Ochmann (2006)]. Only at the axis through the piston, an analytical solution for the sound pressure can be obtained by: # √ 2 2 $ paxis (x) = cv0 e−jk Rp +x − e−jkx , (35d)

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where Rp is the radius of the piston. In the far-field (kr 1), an approximate solution is given by pfar (r, ‡) =

jck 2 Rp2 v0 e−jkr 2kr

2J1 (kRp sin ‡) kRp sin ‡

,

(35e)

where ‡ is the angle between the field point r and the axis of radiation. J1 is the Bessel function of first order. As a numerical example, a circular piston with radius Rp = 0.1 m was imbedded in a square plate with a side length of 2 m. The elements are squares with a side length of 0.0167 m, so that six points per wavelength up to 3430 Hz are ensured. However, we have used this model also for 6860 Hz, and good results were still obtained. The total number of elements is 14 400. In order to simulate a circular piston, the elements whose centres lie within an area with radius 0.1 m are assumed to vibrate with the constant normal velocity amplitude of 1 m/s. A series of calculations shows that an appropriate source system consists of ten complex monopoles, which lie directly behind the piston in a distance of 0.005 m. The x-component of the source positions possesses a growing imaginary part which varies from 0.015 up to 0.15 m. To achieve a comparable accuracy,

Figure 2 Comparison of the analytical sound pressure with the results of the “complex SST” (CSST) and the “real SST” at the axis (left) and in the far-field (right)


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only ten complex monopoles are needed in contrast to 216 real multipoles (monopoles, dipoles and quadrupoles). This decreases the time of calculation on an ordinary PC about more than 20 times. The results are shown in the Fig. 2. In the figure, the amplitude of the exact sound pressure at the axis (see Eq. O.4.(35d)) and at a circle of radius 100 m in a plane perpendicular to the piston (see Eq. O.4.(35e)) is compared with the amplitude of the sound pressure calculated with the complex (abbreviation CSST) and real source systems (SST) for the two frequencies.The radiation patterns at the right are normalized with respect to the value at 0◦ . The agreement is excellent. The error of the sound pressure at the axis at 6840 Hz is slightly bigger when using the SST instead of the CSST. The radiation pattern of the SST is incorrect in the plane of the baffle, which indicates that this type of radiator is difficult to handle with the SST using only “real sources”. More details can be found in [Ochmann (2006)]. In [Piscoya/Ochmann (in preparation)], the radiation from a sphere cap and a vibrating wheel is simulated by using the SST with complex source points.

O.4.3.6

Extension of the SST to Bodies of Arbitrary Geometry

Up to now only spherical wave functions with a singularity at the origin of the coordinate system as defined in Eq. O.4.(5) are used. This kind of SST is called one-point multipole method, [Ochmann (1995)], and is well suited for sphere-like radiators. But the more the shape of the body deviates from a sphere the slower the rate of convergence of the one-point multipole expansion O.4.(15) will become. For this reason a more flexible source system can be constructed by using spherical wave functions with several source locations xq located in the interior Bi as source functions (multi-point multipole method, [Ochmann (1995)]): q

• (x) = • (x − xq );

q = 1, . . . , Q; = 0, . . . , N .

(36)

Such source systems allow the boundary conditions to be satisfied on complex-shaped structures. For example, by choosing N = 0 the source system consists of Q monopoles distributed over an interior auxiliary surface. Such an approach can be considered as derived from an acoustic single layer potential (see O.5.(1) and [Kress/Mohsen (1986)]). Also, it can be found in American [Koopmann/Song/Fahnline (1989); Fahnline/Koopmann (1991)], French [Guyader (1994)], and Russian papers [Bobrovnitskii/ Tomilina (1990); Bobrovnitskii/Tomilina (1995)] under the names superposition method or equivalent source method. Now, instead of Eq. O.4.(15) the pressure is expanded into the double series p(x) =

Q ∞

cqm •qm (x) .

(37)

q=1 m=0 q

First, the null-field equations are extended by using the • as source and weighting functions. Proceeding along the same lines as in > Sect. O.4.1, we obtain the generalized null-field equations Q ∞ S

q=1 m=0

cqm •qm (x)R •S (y) ds =

S

f (y)•S (y)ds;

s = 1, . . . , Q; = 0, 1, . . . (38)

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instead of the null-field equations O.4.(20). Here we have taken into account the symmetry relation O.4.(14a), which is valid for all kinds of radiating wave functions. However, in deriving generalized full-field equations some care is needed since the symmetry relations O.4.(14b, O.4.14c) are only valid for wave functions with the same source location. For example, instead of O.4.(14b) one gets ⎧ j ⎪ ⎪ ⎪ ⎪ k0 ⎨ S q q S ” R•m − •m R” ds = 0 ⎪ ⎪ ⎪ S ⎪ ⎩?

for l = m and s = q

(39a)

for l = m and s = q

(39b)

for s = q

(39c)

Equations O.4.(39a, 39b) follow directly from Eq. O.4.(14b), which does not depend on the special choice of the source location xq ∈ Bi . For wave functions with different source locations, the present author has not found analogous symmetry relations. This does not matter since the symmetry relations have only to be applied to the diagonal = m and s = q which yields the generalized full-field equations of the first kind Q

∞

j S q c + c k0 q=1 m=0 m

s,q

g,mds = S

f ”Sds;

s = 1, . . . , Q; = 0, 1, . . .

(40a)

S

with s,q

g,m

⎧

⎨•qm R ”S , = ⎩R •q ”S , m

for = m and s = q

(40b)

elsewhere .

Analogously, the weighting functions (•sl )∗ lead to the generalized generalized full-field equations of the second kind Q

∞

2j S q c + c k0 q=1 m=0 m

s,q h,m

s,q

h,m ds = S

⎧ ⎨•qm R •S ∗ , = ∗ ⎩ •S R[•q ]; m

∗ f •S ds;

s = 1, . . . , Q; = 0, 1, . . .

(41a)

S

for = m and s = q elsewhere .

(41b)

Equations O.4.(41) for Neumann data were derived in [Ochmann (1999)]. They possess the following advantages: the diagonal terms diag2 ensure an improved stability, no critical frequencies occur, and arbitrary surface geometries can be treated if the source locations are chosen in an appropriate manner. This will be considered in the next section. The analogous derivation of the generalized least squares method is straightforward and hence will be omitted.


O.4.4

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Position of Sources and Their Optimal Choice

In practical applications the important question arises of how to find optimal source locations in the interior Bi of the structure. Clearly, the main requirement is that the source locations have to be chosen such that the boundary error, for example the surface velocity error O.4.(31) for Neumann data, becomes minimal. No general rule exists for achieving this, but a rule of thumb was given in [Ochmann (1990)]: The structure B should be divided into Q substructures as indicated by broken lines in the figure. Then, one multipole has to be placed at the centroid of each substructure. The shape of every substructure Sq (q = 1, . . . , Q) should be as spherelike as possible. In addition, it has to be star-like with respect to its centroid, which means that every part of the surface can be seen from the position of the centroid. Such a choice seems to make sense from an intuitive point of view since the “influence area” of each multipole represents a sphere-like substructure. Many numerical calculations confirm that such source systems may lead to smaller boundary errors than other source configurations. However, no strong mathematical proof has been given up to now.

In [Ochmann (1992)] an attempt was made to optimize numerically the source locations. For this purpose a nonlinear least squares problem has to be solved, which can be done iteratively by applying the Levenberg-Marquardt algorithm. The investigation of idealized radiating structures like cubes and cylinders showed that the additional optimization of the multipole locations had a strong effect on the boundary error and did improve the accuracy of the sound field approximation remarkably, [Ochmann (1992)]. On the other hand, such an automatic optimisation needs a greater amount of computer time, which may be justified if the manual choice as described above does not lead to satisfactory results. After having presented the theory of the SST, numerical aspects and one example of the method will be considered.

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O.4.5


Numerical Aspects

O.4.5.1

Numerical Implementation q

In practical calculations, only a finite number NW = (N + 1)Q of source functions •m with m = 0, . . . , N and q = 1, . . . , Q can be used. For example, if only monopoles (N = 0) and two source locations (Q = 2) are used, the full-field equations of the second kind O.4.(41) for Neumann data O.4.(2a) lead to a system of equations Ac = f with ⎛

∂ •10

1 ∗ •0 ds ∂n

⎜2j/k0 + ⎜ ⎜ A=⎜ S 1 ⎜ ∂•0 2 ∗ ⎝ •0 ds ∂n S

⎞ ∂•20 1 ∗ •0 ds ⎟ ⎟ ∂n S 2 ∗ ⎟ ⎟ , ⎟ ∂ •0 ds⎠ 2j/k0 + •20 ∂n

(42a)

S

and c=

c10 c20

and

f = j–0

v(•1)∗ 0 S

v(•20)∗

.

(42b)

The solution of Eqs. O.4.(42) gives the unknown vector of coefficients c which determines the source strength of both monopoles. For performing the surface integration, the surface is divided into M boundary elements. The simplest integration scheme is obtained if one assumes that pressure and normal velocity are constant over a single surface element. Such constant elements are often used, but, as in BEM, linear, quadratic, or more sophisticated elements may lead to better numerical results (see > Sect. O.5.2). One main advantage of the SST can be seen from Eqs. O.4.(42): In general, the number of equations will be much smaller than the number of boundary elements, i.e. NW M, especially if fine-meshed grids are used for the purpose of high-frequency calculations. Hence, the SST may lead to faster numerical algorithms than boundary element techniques, which work with NW = M (if constant elements are used). On the other hand, the SST will give approximate solutions for NW < M, whereas the BEM normally provides exact solutions limited only by the discretization error (see [Makarov/Ochmann (1998)]). An example may illustrate this situation: let us consider a body that vibrates like a pulsating sphere. Then only one equation has to be solved for finding the exact solution if the equivalent monopole is placed at the right position. In contrast, the BEM has to solve as many equations as the body has boundary elements. However, a complex shaped body with a complicated vibration pattern may require that nearly NW ≈ M sources should be taken into account. The spherical wave functions involve the spherical Bessel and Hankel functions. Depending on order and argument, these functions may assume large numerical values. q q Hence, we recommend working with normalized functions • (x)/K instead of using the spherical wave functions directly. We have chosen the Hankel functions h(1) n (k0 a) as normalizing constants in most calculations with a being a typical dimension of the structure.


O.4.5.2

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Stability and Condition Number

The variants of the SST may be distinguished by their different stability behaviour, which is especially valid for the null-field equations and the full-field equations. The stability of a system of equations A · x = b can be investigated by considering the condition number, [Pärt-Enander/Sjöberg/Melin/Isaksson (1996); Press/Flannery/Teukolsky/ Vetterling (1990)] ‰ = cond(A) = A A−1 ,

(43)

where . . . is a matrix norm and ‰ is a real number greater than or equal to one. The condition number measures the sensitivity of the solution x with respect to perturbations in A or in b. An unstable or badly conditioned system has a large condition number, [Pärt-Enander/Sjöberg/Melin/Isaksson (1996); Press/Flannery/Teukolsky/ Vetterling (1990)]. The condition number depends on the chosen matrix norm. The Euclidean (or spectral) condition number ‰spec is defined as the ratio of the largest and the smallest singular value of A. As suggested by Tobocman, [Tobocman (1985)], the matrix norm ‰F =

1 A F A−1 F n

can also be used which is based on the Frobenius norm , - NW NW - A F = . |aij |2 .

(44a)

(44b)

i=1 j=1

Here again, A = (aij ) is the above mentioned NW × NW matrix. However, the calculation of ‰spec or ‰F may be very time consuming due to the calculation of the singular values or the inverse matrix, respectively. For this reason, it is advisable to use approximations for the condition number as given in [Pärt-Enander/Sjöberg/Melin/Isaksson (1996)]. Numerical examples show that the null-field equations may lead to large condition numbers, especially if the number of sources is large, [Ochmann (1999); Ochmann (2000)]. This drawback can be removed by applying the singular-value decomposition or by using the full-field equations instead of the null-field equations.

O.4.5.3

Calculation of Field Quantities and Sound Power

After having determined the coefficients ci , i.e. the source strengths of the equivalent sources by inverting the matrix A, c = A−1 f, all field quantities in the exterior space Be can easily be computed by a simple source superposition as indicated in Eqs. O.4.(1), O.4.(15), or O.4.(37). For example, the acoustic pressure p is obtained from Eqs. O.4.(1), O.4.(15), or O.4.(37) directly. Then, the velocity vector v = (vx , vy , vz ) can be obtained from the well-known formula (for a time factor e−i–t ) v = (i–0 )−1 grad p .

(45)

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From the knowledge of p and v the sound intensity and the sound power can be calculated (see > Sect. O.3.1). If the one-point multipole method is used, the radiated effective sound power ¢ is proportional to the sum of the source coefficients squared ⎫ ⎧ ⎪ ⎪ N ⎬ 1 ⎨ 1 ¢ = ¢(p, v) = Re p(x)v ∗ (x)dx = |cm |2 , (46) 2 ⎪ 2 ⎪ 2 c k 0 0 0 m=0 ⎭ ⎩ §Ra

where §Ra is a sphere surrounding the radiating body. Formula O.4.(46) can be derived from Eqs. O.4.(5), O.4.(8), and O.4.(15). If the multi-point multipole method is used, the sound power cannot be expressed in such a simple manner, since the source functions q •m are not orthogonal over an exterior sphere. Therefore, the integration over a closed surface has to be performed numerically. Some more useful definitions will be given for the Neumann problem where the normal velocity v is prescribed on the surface S. Clearly, the normal velocity w = vsim simulated by the source system may differ from the real v, since the SST will only give approximate solutions (see > Sect. O.2). Hence, the relative surface velocity error |v − w|2 ds/ |v|2ds (47) Frel = S

S

is a measure for the accuracy of the solution depending on the number and on the locations of the sources. For every calculation such a boundary error (Eq. O.4.(47) valid for Neumann data or a corresponding error for other boundary conditions) can be computed even if the solution is not known a priori. For Neumann data we can also differentiate between the sound power ¢(p, v), which is calculated, using the prescribed velocity v and the sound power ¢(p, vsim ) for the simulated velocity (see Eq. O.3.(5)). In both cases the pressure p is the simulated pressure since the true pressure is unknown.

O.4.6

A Numerical Example: Sound Scattering from a Non-Convex Cat’s-Eye Structure

To illustrate the theory of the SST numerical results for one specially chosen scattering problem will be presented. More results for additional radiation and scattering problems concerning idealized structures, a propeller, and a cylinder can be found in [Ochmann (2000)] and in [Homm/Ochmann (1996); Homm/Schneider (2000); Ochmann, VDIBerichte (1990); Ochmann/Heckl (1994); Ochmann/Homm (1996)]. In the example shown the SST is used for treating the scattering from complex-shaped, non-convex structures. In [Makarov/Ochmann (1998); Ochmann (1999); Ochmann/ Homm (1997)] such a structure was studied which consisted of a sphere where the positive octant, i.e. the part corresponding to x > 0, y > 0, z > 0, was cut out. The region of the missing octant is called “cat’s-eye”, since it acts like a three-dimensional reflector. As shown in Fig. 3, the finite element model of the cat’s-eye structure consists of 7911 boundary elements to achieve six elements per wavelength at k0a = 20.9 (a is the radius of the corresponding sphere). The incidence direction ni of the single frequency, plane wave is along the negative bisector of the angle between the x and y axes. Hence


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Figure 3 Finite element model for the cat’s eye structure with k0 a = 20:9 the incident wave illuminates the reflecting area of the cat’s-eye and leads to multiple reflections. According to the “rule of thumb” of > Sect. O.6 the source system was constructed as follows: multipoles of order N were placed in the middle of each of the seven octants. Only a few results are presented here. More details and results can be found in [Ochmann (1999)].In Fig.4 the directivity pattern of the target strength TS (see > Sect. O.3.2, Eq. O.3.(13)) is shown in the xy-plane. (The target strength was calculated in the farfield and then projected back to the distance of 1 m from the scatterer.) The surface is assumed to be rigid. The total number of sources was 700, since multipoles of order up to N = 9 at each of the seven source locations were taken into account. For the incidence direction ni the maximum of the forward and backward scattering can be seen in the xy-plane. Obviously, the data of the null-field equations, full-field equations of 2nd kind, and the plane wave approximation (see > Sect. O.3.1) agree well at forward scattering. However, the backscattering maximum is only predicted by the full-field equations of 2nd kind (‰spec = 4.6 · 104 , Frel = 23%). The null-field equations produce too large results since we have ‰spec = 1.1 · 108, Frel = 700%. The plane wave approximation could not find any backscattering since it principally neglects multiple reflections.

O.4.7

Concluding Remarks

A general assumption of the SST is that all sources must be located in the interior Bi of the structure. If the sources are placed on the boundary itself, the corresponding BEM is obtained (see > Sect. R.5). In addition, the SST as well as the BEM can be considered as methods of weighted residuals, [Ochmann,Acustica (1990)].Advantages and drawbacks of both methods are presented and compared in [Ochmann (1995)] with the following result: the application of the BEM is easier and more automatic than the SST, since no source system must be constructed explicitly. However, a BEM computation can become

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Figure 4 Directivity pattern of the target strength TS in the x; y plane for a rigid surface, direction of incidence along the negative bisector of the angle between the x and y axes; 700 sources were used; thick curve: plane wave approximation, thin curve: null-field equations, circles: full-field equations of the 2nd kind

Figure 5 Directivity pattern of the target strength TS in the x; y plane for an absorbing impedance, direction of incidence along the negative bisector of the angle between the x and y axes; 567 sources were used, thick curve: plane wave approximation, circles: least squares method


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extremely time-consuming for the treatment of complex structures involving a large number of elements. Consequently, the SST should be applied if the surface model of the structure is very finely discretized and the structural vibration shows a not too complicated pattern,such as a pulsating or oscillating body.This implies that the number of sources needed will be much smaller than the number of boundary elements, and a smaller system of equations has to be solved. Also, complex structures which vibrate in a complicated manner, can be treated by the SST in a very efficient way if one looks for an approximate solution with explicitly determined boundary errors. The main topic of the presented overview is the application of the SST to the calculation of sound radiation or scattering into the unbounded, three-dimensional space. In addition, the SST can be used for the treatment of several other acoustical problems. For example, two-dimensional sound fields, problems with axisymmetric or cyclic symmetry, sound fields in interior spaces or in half spaces, or scattering and radiation from elastic structures can be investigated by means of the SST. Moreover, Leviatan and his co-workers have analyzed various scattering problems by means of a source-model technique in a series of papers (see, e.g. [Erez/Leviatan (1993)] where more references can be found).

O.5 The Boundary Element Method (BEM) The BEM is mainly used for solving the radiation and scattering problem (see > Sect. O.3, standard problems 1 and 2). The interior problem (standard problem 3) can also be treated with the help of the BEM. The fluid-structure interaction problem can be solved by combining the BEM with the finite element method (see > Sect. O.6.3).

O.5.1

Boundary Integral Equations

The most frequently used integral equation formulation in acoustics is the well-known Helmholtz integral equation for exterior field problems.The Helmholtz integral equation is obtained by applying Green’s second theorem to the Helmholtz equation O.3.(1) (see for example [Schenck (1968)] or [Colton/Kress, Integral Equations in Scattering Theory (1983)]). Depending on the location of the field point x, the Helmholtz integral equation takes the form ⎧ p(x), ⎪ ⎪ ⎪ ⎪ ⎨ 1 ∂g(x, y) ∂p(y) − g(x, y) ds = p(x), p(y) ⎪ ∂n(y) ∂n(y) 2 ⎪ ⎪ S ⎪ ⎩ 0,

x ∈ Be

(1a)

x∈S

(1b)

x ∈ Bi ,

(1c)

where (for a time factor e+j–t ) g(x, y) =

1 −jk0 ˜r e 4˜r

with

/ / ˜r = ˜r(x, y) = /x − y /

(2)

is the free-space Green’s function, and y is a spatial point on the structural surface S. The geometrical notations are chosen as described in > Sect. O.3.1. Eqs. O.5.(1a), O.5.(1b),

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and O.5.(1c) are called exterior Helmholtz integral equation, the surface Helmholtz integral equation, and the interior Helmholtz integral equation, respectively. There also exists an analogous Helmholtz integral formulation for interior field problems, which will be considered in > Sect. O.5.5. The interior Helmholtz integral equation O.5.(1c) considered here gives a null-field in Bi which obviously does not represent a physical solution, and should not be confused with the Helmholtz integral formulation for interior problems. Solutions of the Helmholtz integral equation automatically satisfy the radiation condition O.3.(3). Many boundary element formulations in acoustics use the surface Helmholtz integral equation as a starting point since it is a second kind Fredholm integral equation for the familiar Neumann boundary condition with satisfactory numerical stability, [Schenck (1968)]. Fast numerical solvers for the discretized version of the surface Helmholtz integral equation can be obtained by using, for example, iterative algorithms, [Makarov/Ochmann (1998)], or multigrid methods, [Ochmann/Wellner (1991)]. The Helmholtz formula O.5.(1) is valid if the surface S is assumed to be closed and sufficiently smooth, i.e. there is a unique tangent to S at every x ∈ S. For the general case, where no unique tangent plane exists at x ∈ S (for example,when x is lying on a corner or an edge), the surface Helmholtz integral equation has to be modified slightly, [Seybert/ Soenarko/Rizzo/Shippy (1985)] S

∂g(x, y) ∂p(y) C(x) p(y) − g(x, y) ds = p(x) , ∂n(y) ∂n(y) 4

(3a)

where C(x) = 4 + S

∂ ∂n(y)

1 ds(y) ˜r(x, y)

(3b)

is the solid angle seen from x, [Peter (2000)]. For a smooth surface C(x) = 2 is in agreement with Eq. O.5.(1b). For calculating the quantities of the sound field,two steps are necessary.First,the surface Helmholtz integral equation O.5.(1b) has to be solved which gives pressure and normal velocity on the surface of the structure. This procedure requires the main effort, since a complex, fully populated, and unsymmetrical system of linear equations has to be solved. Second, the sound field in the whole outer space can be calculated with the help of the exterior Helmholtz integral equation by a simple integration over the surface S. The numerical treatment of the surface Helmholtz integral equation involves two characteristic difficulties. The equation possesses a weakly singular kernel, and it has no unique solution at the so-called critical frequencies. The interior Helmholtz integral equation does not suffer from these disadvantages. However, as an integral equation of the first kind, it provides a less satisfactory basis for numerical calculations, [Schenck (1968)]. Its numerical treatment needs extreme care, and regularisation methods should be applied, [Colton/Kress, Integral Equations in Scattering Theory (1983); Colton/Kress (1992)].


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Another formulation of acoustical boundary integral methods is the potential-layer approach. By representing the pressure p as a single-layer potential (i.e. a layer of monopoles) p(x) = (y)g(x, y)ds(y) ; x ∈ Be or x ∈ Bi (4a) S

one obtains the boundary integral equation (x) ∂g(x, y) − ds(y) = j–0 v ; x ∈ S (y) 2 ∂n(x)

(4b)

S

for the determination of the density , if the exterior Neumann problem with ∂p/∂n = −j–0 v

(5)

is considered (see Theorem 3.16 of [Colton/Kress, Integral Equations in Scattering Theory (1983)]). For the interior Neumann problem, one gets ∂g(x, y) (x) (y) + ds(y) = −j–0 v ; x ∈ S . 2 ∂n(x)

(6)

S

The double-layer potential (i.e. a layer of dipoles) g(x, y) •(y) p(x) = ds(y) ; x ∈ Be or x ∈ Bi ∂n(y)

(7a)

S

leads to the integral equation ∂g(x, y) •(x) + ds(y) = p0 (x) ; •(y) 2 ∂n(y)

x∈S

(7b)

S

for the exterior Dirichlet problem with p = p0 on S . It leads to •(x) ∂g(x, y) − ds(y) = −p0 (x) ; •(y) 2 ∂n(y)

(8)

x∈S

(9)

S

for the corresponding interior Dirichlet problem (Theorem 3.15 of [Colton/Kress, Integral Equations in Scattering Theory (1983)]). Whereas the single- or double-layer approach can be interpreted as a layer of monopoles or dipoles, respectively, the Helmholtz integral equation contains both layers of monopoles and dipoles. All the above presented layer approaches lead to boundary equations of the second kind. It is also possible to derive equations of the first kind. Following Colton and

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Kress, [Colton/Kress, Integral Equations in Scattering Theory (1983)], the single-layer potential O.5.(4a) solves the interior and exterior Dirichlet problem with boundary conditions O.5.(8) if the density is a solution of the integral equation (Theorem 3.28 of [Colton/Kress, Integral Equations in Scattering Theory (1983)]) (y)g(x, y)ds(y) = p0 ; x ∈ S . (10) S

The double-layer potential O.5.(8) solves the interior and exterior Neumann problem with boundary condition O.5.(6), provided the density • is a solution of the singular integral equation (Theorem 3.31 of [Colton/Kress, Integral Equations in Scattering Theory (1983)]) ∂ ∂g(x, y) ds(y) = −j–0 v ; x ∈ S . •(y) (11) ∂n(x) ∂n(y) S

Such integral equations of the first kind are improperly posed. The ill-posed nature of these equations is described in [Colton/Kress, Integral Equations in Scattering Theory (1983), p. 90]. However, there exist several attempts to deal with equations of the first kind too, and advances have been made in their numerical analysis. References can be found in [Colton/Kress, Integral Equations in Scattering Theory (1983); Colton/Kress (1992)]. In addition, Eq. O.5.(11) contains the normal derivative of the double-layer potential, which in general does not exist on the boundary. Even if it exists, Eq. O.5.(11) becomes strongly singular and a regularization is required. Table 2 provides an overview about the exterior integral equations considered above, where the Neumann boundary conditions, [Seybert/Soenarko/Rizzo/Shippy (1985)], are taken into account. Only slight modifications are necessary if other boundary data, such as Dirichlet or Robin data, are prescribed (see [Angell/Kleinman (1982), Kleinman/ Roach (1974)]). Table 2 shows the surface equation, which has to be solved first. The corresponding equation for the pressure in the exterior space is given in the right column. Table 2 Boundary integral formulations for the exterior domain; HIE = Helmholtz integral equation; SLP = Single-layer potential; DLP = Double-layer potential Surface equation on S

Exterior equation for p in Be

HIE

p = Kp + R j–0 v

p=

SLP

− K0 = 2j–0 v

p = 12 R

DLP

T• = −2j–0 v

p = 12 K•

1 2

Kp + R j–0 v


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Here, the following abbreviations for integral operators (after [Colton/Kress, Integral Equations in Scattering Theory (1983)]) are used ∂g(x, y) (Kœ)(x): = 2 ds(y) , (12) œ(y) ∂n(y) S

(K œ)(x): = 2

œ(y) S

∂g(x, y) ds(y) , ∂n(x)

(13)

œ(y)g(x, y)ds(y) ,

(Rœ)(x): = 2

(14)

S

∂ (Tœ)(x): = 2 ∂n(x)

O.5.2

œ(y) S

∂g(x, y) ds(y) . ∂n(y)

(15)

Discretization of the Boundary Integral Equation

In the following,only the exterior radiation problem with Neumann data is considered as the model problem. For the numerical solution of one of the above three surface integral equations,the continuous equation is discretised and transformed into a system of linear equations. For this reason, the surface of the radiator is approximated by a finite element model consisting of N finite surface elements Fk (k = 1, . . . , N). The generation of such finite element grids for complex machine structures, which often consists of several thousand elements, requires an immense effort, even if special finite element model generators (“preprocessors”) will be used. The easiest approach to performing the surface integration in the Helmholtz integral equation O.5.(1b) is to consider pressure and normal velocity as constant over each single element. This approach is called collocation method, since it means that a certain value of pressure and normal velocity is assigned to the centroid of each element. These simple and often used approaches are described by Rao and Raju, [Rao/Raju (1989)], under the name “method of moments”. The collocation method transforms the Helmholtz integral equation into the following system of N × N linear equations DP + MV = P/2 , where the matrix D = (dik ) consists of the dipole terms ∂g(xi , y) dik = ds(y) ∂n(y)

(16)

(17a)

Fk

and the matrix M = (mik ) consists of the monopole terms g(xi , y)ds(y) . mik = j–0 Fk

(17b)

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P is the N-dimensional vector of pressure values in the centroids of the N elements, and V is the corresponding normal velocity vector. For practical purposes, often plane triangular or rectangular elements are used. Hence, the discretised surface will be not smooth, but it will be equipped with several edges and corners. Nevertheless, it is not necessary to use modification O.5.(3), since the pressure is only evaluated in the element centroids.In addition,corners and edges can be considered as slightly rounded, which will not influence the sound radiation remarkably. Higher order elements such as linear, quadratic or cubic elements are used, too. A quadratic isoparametric element formulation with six nodes for the triangular and eight nodes for the quadrilateral curvilinear element was suggested in [Seybert/Soenarko/ Rizzo/Shippy (1985)], in which both the surface elements and the acoustic variables are represented by second-order shape functions. Often, complex structures consisting of a large number of plane triangular and rectangular elements occur in practical industrial problems. For such finite element models it is recommended to perform the integrations appearing in Eq. O.5.(17) as follows: choose variables that are constant over a single element and transform each element Fk on to the unit triangular or rectangular element (see [Chen/Schweikert (1963); Schwarz (1980)]). Then use Gaussian integration rules of the desired order of accuracy. The average size of the elements of a boundary grid determines the highest frequency that can be treated. A famous rule of thumb is the “six elements per wavelength rule”. It means that at least six elements per wavelength should be taken into account if constant or linear elements are used. This is approximately valid for k0d ≤ 1 where d is a typical dimension of the element. A detailed discussion about this rule can be found in [Marburg (acc. for publ.)]. Figure O.4.3 shows a boundary element mesh consisting of 7869 rectangular and 42 triangular elements, which can be used up to k0a ≤ 21, where a is the radius of the corresponding sphere.

O.5.3

Solution of the Linear System of Equations

One of the main problems of the BEM is that the matrix A of the system O.5.(16) written in the form AP = F

(18)

with A = D − 0.5 I (I = unity matrix) and F = −MV (V = prescribed normal velocity vector) is fully occupied, complex and unsymmetrical. This is an essential disadvantage of the BEM in comparison with the FEM, which leads to symmetrical and weakly populated matrices with small bandwidth (see > Sect. O.6.2). However, many technical structures consist of several thousands of elements leading to huge systems of equations. The numerical solution of such systems needs much computer time. Often, it is not possible to store the whole matrix A on the disk. In addition, if complete spectra should be calculated, a full system of equations has to be solved for every single frequency. For this reason, direct solvers such as the Gaussian elimination or the LU decomposition with backsubstitution, [Press/Flannery/Teukolsky/Vetterling (1990)], should only be used for systems up to a few hundreds of equations, [Stummel/Hainer (1971), p. 147], since the numerical effort is of order N3 .


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For vibrating structures with symmetry, simplified boundary integral equations with a reduced number of unknowns can be derived leading to reduced computer costs. For example, the sound radiation from axisymmetric bodies is treated in [Akyol (1986); Seybert/Soenarko/Rizzo/Shippy (1986)]. Under the condition that the boundary conditions are axisymmetric too, the introduction of elliptic integrals leads to further simplifications, [Seybert/Soenarko/Rizzo/Shippy (1986)]. For larger systems without special symmetry properties, iterative solvers should be preferred, since they lead to a numerical cost of order N2 approximately. The shape of the system O.5.(16) suggests the iteration scheme (called Picard iteration) 1 (i+1) P = DP(i) + MV 2

(19)

for a prescribed normal velocity vector V. Starting with an initial guess for the pressure vector P(0) , a sequence of iteration vectors P(i) is obtained. Such an iteration scheme corresponds to the basic iterative method of Jacobi, [Stummel/Hainer (1971), p. 148]. Unfortunately, the Jacobi iteration O.5.(19) does not converge generally. Convergence only takes place if the powers of the matrices D converge to a matrix of zeros, [Stummel/ Hainer (1971), p. 148], or if the eigenvalues of the corresponding integral operators do not lie inside of the unit circle, [Chertock (1968)]. For example, this is not the case for the spheroid investigated by Chertock, [Chertock (1968)]. However, by introducing a suitable chosen relaxation parameter into Eq. O.5.(19) and using

1 (i+1) 1 (i) P = P + (1 − ) DP(i) + MV . 2 2

(20)

Chertock achieved convergence in some cases. The iteration scheme O.5.(20) corresponds to the Successive Overrelaxation (SOR) method. But, Kleinman and Wendland, [Kleinman/Wendland (1977)], showed that a convergent series of successive approximations for the Helmholtz integral equation is only obtained by O.5.(20) if the wave number is sufficiently small. Hence, method O.5.(20) is not a satisfactory basis for practical applications. In [Kleinman/Roach (1988)] Kleinman and Roach presented an iterative method, which is convergent for all wave numbers k0. The key point of the method is a self-adjoint formulation of the Helmholtz integral equation which is obtained by multiplying Eq. O.5.(18) with the complex conjugate matrix A∗ A∗ AP = A∗ F .

(21)

Then again, a successive overrelaxation method is applied to Eq. O.5.(21) leading to P(i+1) = P(i) − A∗ AP(i) + A∗ F (22) with relaxation parameter for the radiation problem considered in this chapter. Makarov and Ochmann, [Makarov/Ochmann (1998)], applied a variant of this method to the high-frequency acoustic scattering from complex bodies consisting of up to 60 000 surface elements. One major problem arising from method O.5.(22) is to find the optimal parameter . Hence, it is more convenient to use parameter-free iterative methods such as the Generalized Minimum Residual Method, which was successfully applied to

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acoustic and medium-to-high electromagnetic scattering in [Makarov/Ochmann (submitted); Makarov/Ochmann/Ludwig (submitted); Ochmann/Homm/Semenov/Makarov (2001)]. Another promising iterative method for the solution of the Helmholtz integral equation is the multigrid method, [Ochmann/Wellner (1991)]. It consists mainly of two steps, the smoothing step and the coarse grid correction. Starting with an initial value P0 , one Picard iteration O.5.(19) is performed for determining the part P1/2 of the solution vector P that is highly oscillating with respect to the spatial coordinate on the surface S. This is the so-called smoothing step. It forces the solution of the system of equations for the error P − P1/2 to be smoother, so that it can be determined on a coarser finite element grid without significant loss of accuracy. For obtaining a coarser grid, two neighbouring grid points are condensed into one. The calculated error is projected by linear interpolation from the coarser grid to the finer one. If only two grids are used, the approach is called the two-grid method. The key idea of the multigrid method is as follows: The system for the error is not solved directly. Instead, a two-grid method is used again for obtaining an approximate solution, where one has to go over to an even coarser grid. This procedure is repeated until one arrives at a grid that contains only a small number of elements. Thus, the corresponding system of equation is small enough too, and can be solved with little effort by a direct solver. A detailed description of the multigrid method and corresponding calculations for radiating structures can be found in [Ochmann/Wellner (1991)].

O.5.4

Critical Frequencies and Other Singularities

The integral equation formulation for the exterior problem involves a characteristic problem: At the so-called critical wave numbers kc the integral equation is not solvable or not uniquely solvable. This phenomenon has been known for a long time (see Kupradze, [Kupradse (1956)], or Smirnow, [Smirnow (1977)]). Copley, [Copley (1968)], considered it in connection with the acoustical radiation problem and pointed out that the integral equation for the single-layer potential O.5.(4b) does not possess a solution if the wave number kc is an eigenvalue of the interior Dirichlet problem p + kc2 = 0 p=0

in Bi ;

on S .

(23) (24)

Physically, such a behaviour can be interpreted as a resonance phenomenon of the interior space, [Copley (1968)]. The surface Helmholtz integral equation, however, possesses solutions at the critical wave numbers, but these solutions are not uniquely determined. This was shown by Schenk, [Schenck (1968)], in detail, based on the general theory of Fredholm integral equations. He suggested to use the combined integral equation formulation.

O.5.4.1

Combined Integral Equation Formulation (CHIEF)

The idea is as follows. At wave numbers kc the surface Helmholtz integral equation O.5.(1b) has infinitely many solutions (i.e. a nontrivial null-space). However, it can be


O

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shown that the physically relevant solution of O.5.(1b) is the only one that satisfies the interior Helmholtz integral equation O.5.(1c) simultaneously.Hence,the idea of CHIEF is to solve the surface Helmholtz integral equation on S and the interior Helmholtz integral equation at certain selected interior points xi (so-called CHIEF points) simultaneously: 1 ∂g(x, y) ∂p(y) − g(x, y) ds = p(x) ; x ∈ S (25a) p(y) ∂n(y) ∂n(y) 2 S

p(y) S

∂g(xi , y) ∂p(y) − g(xi , y) ds = 0 ; ∂n(y) ∂n(y)

xi ∈ Bi .

(25b)

The discretization of both integral equations leads to an overdetermined system of equations. If N surface elements and M CHIEF points are used, then a (N+M)×N system results. Such a system can be solved by a least-squares orthonormalising procedure. An alternative approach is that of Rosen et al., [Rosen/Canning/Couchman (1995)], who suggested to create a square matrix by introducing a vector Š of Lagrange multipliers: A B∗ P F = . (26) B I Š FC A (see Eq.O.5.(18)) and B are the matrices of coefficients resulting from the discretisation of Eqs. O.5.(25a) and O.5.(25b), respectively. F and FC are the corresponding right-hand sides, B∗ is the N × M conjugate transpose of B, and I is the M × M identity matrix. The Lagrange multipliers Š = Fc − BP

(27)

are the residuals in satisfying the M CHIEF constraint equations. The CHIEF method suffers from the fact that the interior points xi are not allowed to lie on the node surfaces of the interior standing wave field. However, these nodes are not known for general radiator geometries, and their number increases with increasing frequency. On the other hand, the higher the frequency the more CHIEF points are needed, and no rules are known how to choose the number and locations of these points optimally. For avoiding this problem, the following similar method was proposed.

O.5.4.2

Combination with the Null-Field Equation

Stupfel et al., [Stupfel/Lavie/Decarpigny (1988)], suggested the following method, originally based on an idea of Jones,[Jones (1974)].The surface pressure p has simultaneously to satisfy the surface Helmholtz integral equation and additional M null-field equations of the form ∂p(y) ∂• (y) ds(y) = • (y)ds(y) ; = 1, 2, . . . , p(y) (28) ∂n(y) ∂n(y) S

S

where the spherical wave functions • are defined in > Sect. O.4.3.In this way,all critical wave numbers k ≤ kM are suppressed, where the eigenvalues of the interior Dirichlet problem O.5.(23), O.5.(24) are ordered so that k1 ≤ k2 ≤ k3 ≤ . . .

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A detailed description of the null-field equations can be found in

> Sect. O.4.3.

The surface Helmholtz integral equation as an integral equation of the second kind is well-posed, and the null-field equations do not suffer from the non-uniqueness problem, [Colton/Kress, Q.J. Mech. Appl. Math. (1983); Martin (1988)]. Hence, the combination of both types of equations seems to be a promising approach – also for numerical calculations in the high-frequency regime.

O.5.4.3

Combination with the Differentiated Integral Equation

According to Table 2 the surface Helmholtz integral equation for the exterior Neumann problem in operator notation can be written as p = Kp + R(j–0 v) .

(29a)

A second relationship can be obtained by formally differentiating the surface Helmholtz integral equation in the direction of the normal. This results in the boundary integral equation −j–0 v =

∂p = Tp + K (j–0 v) ∂n

(29b)

of the first kind for the unknown boundary value p on S, where the integral operators T and K are defined below in Eqs. O.5.(13), O.5.(15). Combining both equations leads to the equation p − Kp + j†Tp = R(j–0 v) − j†(j–0 v + K (j–0 v))

(30)

of the second kind.Burton and Miller showed in reference [Burton/Miller (1971)] that the linear combination O.5.(30) has a unique solution for all real values of the wave number k, if the real coupling parameter † is not zero. This approach is called the Burton and Miller method or the Composite Outward Normal Derivative Overlap Relation (with acronym CONDOR). A drawback of the method is that Eq. O.5.(30) includes the derivative of the double-layer Helmholtz potential ∂ ∂g(x, y) ds(y) (Tp)(x): = 2 p(y) ∂n(x) ∂n(y) S

which is a hypersingular operator. This operator must be transformed to reduce the strength of the singularity, and two ways of regularisation are discussed in [Burton/ Miller (1971)]. In a similar way, the solution of the exterior Neumann problem can be sought in the form of a combined single- and double-layer potential ∂g(x, y) p(x) = g(x, y) + j† (y)ds(y) (31) ∂n(y) S

leading to the hypersingular integral equation (see Table 2 and [Colton/Kress, Integral Equations in Scattering Theory (1983), p. 92]) − K − j†T = 2j–0 v

(32)


O

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for the unknown density ,which is the adjoint of the combined Green’s formula integral equation O.5.(30). Again, it can be shown that the combined single- and double-layer integral equation O.5.(32) is uniquely solvable for all wave numbers if † = 0, [Colton/ Kress, Integral Equations in Scattering Theory (1983), Theorem 3.34]. Kress and Spassow, [Kress/Spassow (1983)], had analyzed how to choose the coupling parameter † appropriately in order to minimise the condition number of the integral operators appearing in Eq. O.5.(32).

O.5.4.4

Modified Green’s Functions

Another approach leading to uniquely solvable integral equations for exterior boundary value problems was developed by Jones, [Jones (1974)], Ursell, [Ursell (1973); Ursell (1978)], and Kleinman and Roach, [Kleinman/Roach (1982), Kleinman/Roach (1988); Kleinman/Roach/Schuetz/Shirron (1988)]. They suggested to add a series of radiating wave functions • (all definitions are given in > Sect. O.4.3) to the free field Green’s function g(x, y) resulting in a modifiedßindexmodified double-layer potential Green’s function ∞ c • . (33) gm (x, y): = g(x, y) + =0

Now, all surface integral equations from Table 2 can be modified by using the Green’s function gm (x, y) instead of g(x, y) in the definition of the operators O.5.(12)–O.5.(15). For example, the modified double-layer potential is of the form gm (x, y) ds(y) ; x ∈ / Be or x ∈ p(x) = ¦ (y) / Bi (34) ∂n(y) S

with density •. It can be shown that the corresponding surface integral equations are uniquely solvable for all wave numbers provided that the coefficients cl appearing in O.5.(33) satisfy certain inequalities (see [Colton/Kress, Integral Equations in Scattering Theory (1983), Theorem (3.35)] and [Kleinman/Roach (1988)]). In addition, it was shown by Jones, [Jones (1974)], (see also [Kleinman/Roach/Schuetz/Shirron (1988)]) that the integral equations will still be uniquely solvable for k0 ≤ kN , where kN is a certain critical wave number, if the ck are chosen to vanish for > N. Hence, the Green’s function has only to be modified with a finite number of terms. However, this shows a drawback of the method: the higher the frequency the more terms in the series must be included, which may lead to numerical problems if high-frequency radiation or scattering is considered. For a numerical analysis, it can be of advantage to get an estimate about the locations of the critical frequencies appearing in a certain frequency range. In [Stupfel/Lavie/ Decarpigny (1988), p. 928] it was noted that the inequalities Kn Kn ≤ kn ≤ (35) A a are valid, where Kn is the n-th eigenvalue of the interior Dirichlet problem O.5.(23), O.5.(24) for the unit sphere. A and a are the radii of spheres lying completely in the exterior Be and the interior Bi , respectively.

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O.5.4.5 Treatment of Singularities The integral equations considered involve singularities,since the Green’s function g(x, y) and its normal derivations become singular for x → y . First, the surface Helmholtz integral equation contains the weekly monopole terms mii , Eq. O.5.(17b). This / /singular −1 singularity is of order ˜r−1 = /x − y / and can be removed by introducing polar coordinates. Following Everstine, [Everstine/Henderson (1990)], it is assumed that v is constant over a small circular area of radius bi with centroid xi so that the monopole terms O.5.(17b) can be written as mii = j–0

2 bi g(xi , y)ds(y) = j–0

Fi

0

0

e−jk0 r rdrdœ 4r

(36)

where bi is such that b2i = Fi gives the total area of the element Fi assigned to the point xi . Hence, the following result is obtained " (37) mii = j–0 Fi /(2bi) with bi = Fi / . Second, the dipole self terms dii O.5.(17a) must be evaluated. It can be shown, [Koopmann/Brenner (1982)], that for plane elements dii = 0, [Koopmann/Benner (1982)], since ˜r⊥n and hence ∂˜r/∂n = 0. If the curvature ci of the radiating surface element is taken into account, it can be shown that approximately, [Everstine/Henderson (1990)], dii = −(1 + jk0 bi )(ci Fi )/(4bi ) .

(38)

O.5.5 The Interior Problem: Sound Fields in Rooms and Half-Spaces The numerical calculation of a sound field in an interior space Bi is the third standard problem of > Sect. O.3.3. If a problem without acoustical sources in the interior Bi is considered, the application of Green’s second theorem to the Helmholtz equation O.3.(1) (see for example [Colton/Kress, Integral Equations in Scattering Theory (1983); [Skudrzyk (1971)], or [Seybert/Cheng (1987)]) leads to the interior Helmholtz integral formula ⎧ 0, ⎪ ⎪ ⎪ ⎪ ⎨ 1 ∂g(x, y) ∂p(y) − g(x, y) ds = − p(x), p(y) ⎪ ∂n(y) ∂n(y) 2 ⎪ ⎪ S ⎪ ⎩ − p,

x ∈ Be

(39a)

x∈S

(39b)

x ∈ Bi

(39c)

for the pressure p.Again, the Green’s function g(x, y) is given by Eq. O.5.(2). The interior formula is very similar to the exterior Helmholtz integral formula O.5.(1). Only the sign and the role of the interior and exterior space have changed. Now, the BEM can be applied to the solution of the interior problem in just the same way as described for the radiation problem (e.g. [Seybert/Cheng (1987)]). Boundary conditions for the pressure or the pressure gradient have to be inserted into the surface integral Eq.O.5.(39b).Clearly, a radiation condition is not necessary. Discretisation of Eq. O.5.(39b) leads to a system of equations for the determination of the second acoustic surface variable as described


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in > Sects. O.2 and O.3. Fortunately, critical frequencies cannot occur, since the adjoint problem now is the exterior problem which does not possess any discrete eigenvalues. Hence, the regularisation procedures described in > Sect. O.3 are not needed. However, fully interior problems are mainly the domain of the finite element method, which can deal with very general interior fluid-structure interactions problems (see > Sect. O.6). The BEM is well suited for the calculation of sound radiation into a half-space,[Ochmann (2000)],too.In many applications the radiator is situated on a locally reacting plane Splane. Such an infinite plane can be taken into account by using a half-space Green’s function gH (x, y) =

1 −jk0 ˜r 1 −jk0 ˜r +R e e 4˜r 4˜r

with

/ / ˜r = /x − y /

and

/ / r˜ = /x − y /

(40)

instead of the free-space Green’s function g(x, y) in the surface Helmholtz integral equation O.5.(1b). Here, x is the image point of x behind the plane and R is the reflection coefficient of the plane. For a rigid plane R=1 and for a free surface R = −1. For other values of R, Eq. O.5.(40) is only an approximation. The exact representation of the Green’s function over an absorbing impedance plane is more complicated,and a detailed discussion of various formulas can be found in [Mechel (1989)]. Exact solutions and approximations are given in > Sects. D.14–D.20. In [Ochmann (2004)], the half-space Green’s function above an impedance plane is formulated as a superposition of point sources which are located at complex source points (see Eq. O.5.(42) in [Ochmann (2004)]). This Green’s function is suitable for all kind of surface impedances and hence can be used as a building block for a boundary element method. By using such half-space Green’s function, one only has to extend the integration in Eq. O.5.(1b) over the surface of the radiator S. The infinite impedance plane Splane has not to be taken into account! If the radiating body is in contact with the plane, a slightly modified version of the surface Helmholtz integral Eq. O.5.(3a) has to be used, where the coefficient C(x) is now given by ⎤ ⎡ 1 ∂ ds(y)⎦ . (41) C(x) = (1 + R) ⎣2 − ∂n(y) ˜r S0 ∪Sc

Here S = S0 ∪ Sc , where Sc is the part of the radiator surface S that is in contact with Splane . The derivation of C(x) and more details can be found in [Seybert/Wu (1989)]. Equation O.5.(41) is valid for a normal pointing into the body. The method of mirror sources for constructing the Green’s function as a kernel for an appropriate integral equation can be generalized to regard additional plane boundaries around the radiating body. For example, if the vibrating structure is situated in a rectangular room, the resulting sound field can be computed by using two different representations of the Green’s function for a rectangular enclosure with dimensions x × y × z , [Lam/Hodgson (1990)]. The first formula for the Green’ function can be

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constructed by mirror sources. Hence it consists of an infinite series of exponentials, and converges for small distances from the radiator. For larger distances, the approximate Green’s function GE (x, y) =

∞ n=0

¥n (x)¥n (y) Vn kn2 − k02 − j‘n

(42)

which is combined of the known eigenmodes of a rectangular room, [Lam/Hodgson (1990); Morse/Ingard (1968)], shows a better convergence behaviour. The eigenfunctions are given by ¥n (x) = cos(knx x) cos(kny y) cos(knz z) with knx =

nx , x

kny =

ny , y

knz =

nz . z

The damping factor of the n-th mode is given by yo + yl xo + xl zo + zl + —ny + —nz , ‘n = k0 —nx x y z isthe specific acoustic admittance at the wall at x = x . n is where xl (for example) given by n = 1/ —nx —ny —nz , where the Neumann symbol —n is defined by —n =

1, for n = 0 2, otherwise

;

V is the volume of the room.

This method seems to be superior to the FEM for complex structures in large rectangular rooms, since it only requires to divide the surface of the structure into elements, whereas the FEM has to discretise the whole interior volume of the room into finite elements. The solution of coupled interior–exterior acoustics problems with the help of the BEM was investigated by Seybert et al. [Seybert/Cheng/Wu (1990)].

O.5.6 The Scattering and the Transmission Problem As explained in > Sect. O.3.2, the scattering problem can be considered as an equivalent radiation problem with respect to the scattered pressure ps . However, sometimes it is more convenient to have an explicit boundary integral equation for the total pressure p = pT = ps + pin as the starting point for a numerical calculation. The scattered wave ps has to fulfil the exterior Helmholtz formula O.5.(1). The incident pressure pin is assumed to have no singularities in Bi , and hence it must satisfy the interior Helmholtz formula O.5.(39). By adding both Eqs. O.5.(1) and O.5.(39), one gets the Helmholtz formula ⎧ p(x), ⎪ ⎪ ⎪ ⎪ ⎨1 ∂g(x, y) ∂p(y) − g(x, y) ds = p(x), p(y) pin + ⎪ ∂n(y) ∂n(y) 2 ⎪ ⎪ S ⎪ ⎩ 0,

x ∈ Be

(43a)

x∈S

(43b)

x ∈ Bi

(43c)


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for the total pressure p. Assuming that the surface of the scatterer is rigid, the pressure gradient ∂p/∂n = −j–0 v on the surface is zero.Hence,for a rigid scatterer the boundary integral equation O.5.(43b) can be written as ∂g(x, y) p(x) = 2 p(y) (44) ds(y) + 2pin . ∂n(y) S

For an arbitrary surface velocity distribution v, Eq. O.5.(43b), takes the form ∂g(x, y) ds(y) + 2 p(x) = 2 p(y) j–0 v(y)g(x, y)ds(y) + 2pi . ∂n(y) S

(45)

S

For the general impedance boundary value problem, the normal impedance Z = p/v is introduced at each point on the surface S of the scatterer. Substitution of the normalized impedance Z¯ = Z/(0 c0 ) into Eq. O.5.(45) gives the boundary integral equation ∂g(x, y) jk0 p(y) (46) p(x) = 2 ds(y) + 2 p(y) g(x, y)ds(y) + 2pin ¯ ∂n(y) Z(y) S

S

¯ for an impedance scatterer. It should be emphasised that the local impedance Z(y) can vary with the surface point y. Calculations for structures with varying normal surface impedance based on an iterative BE solver can be found in [Makarov/Ochmann (1998)]. The transmission problem is described in > Sect. O.3.5. The total pressure p1 : = p in the surrounding medium with constants c1 and 1 has to satisfy Eq. O.5.(43b) in the form ∂p1 (y) ∂g1 (x, y) ds(y) − 2 g1 (x, y) ds(y) + 2pin p1 (y) (47) p1 (x) = 2 ∂n(y) ∂n(y) S

S

with g1 (x, y) =

1 −jk1 ˜r e 4˜r

with

/ / ˜r = /x − y /

and

k1 =

– . c1

(48)

The pressure p2 : = pi inside of the scattering body with constants c2 and 2 must satisfy Eq. O.5.(39b) ∂p2 (y) ∂g2 (x, y) p2 (x) = −2 ds(y) + 2 g2 (x, y)ds(y) p2 (y) (49) ∂n(y) ∂n(y) S

S

with g2 (x, y) =

1 −jk2 ˜r e 4˜r

with

/ / ˜r = /x − y /

and

k2 =

– . c2

(50)

For f = g = 0, the transmission Conditions O.3.(23) and O.3.(24) take the form p1 = p2 ;

∂p1 1 ∂p2 = ; ∂n › ∂n

› = 2 /1 .

(51, 52)

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By introducing Eqs. O.5.(51) and O.5.(52) into Eqs. O.5.(47) and O.5.(48), two coupled integral equations are obtained ∂p1 (y) ∂g1 (x, y) p1 (x) = 2 ds(y) − 2 g1 (x, y)ds(y) + 2pin , p1 (y) (53) ∂n(y) ∂n(y) S

S

p1 (x) = −2 S

∂g2 (x, y) ds(y) + 2› p1 (y) ∂n(y)

S

∂p1 (y) g2 (x, y)ds(y) ∂n(y)

(54)

for the determination of the surface variables p1 and ∂p1 /∂n. For equal sound velocities c1 = c2 ,the Green’s functions also are equal g1(x, y) = g2 (x, y), and one single integral equation can be derived by combining Eqs. O.5.(53) and O.5.(54) in a suitable way › 2(› − 1) ∂g1 (x, y) ds(y) + 2 pin . p1 (y) (55) p1 (x) = ›+1 ∂n(y) ›+1 S

The as yet unpublished boundary integral Eq. O.5.(55) was derived by S. Makarov (Worcester Polytechnic Institute, MA, USA) and communicated to the author. For › → ∞, i.e. 2 1 , the integral Eq. O.5.(44) for the rigid scatterer is obtained as expected for physical reasons. Additional recent formulations and numerical implementations of the BEM can be found in the book [Estorff (2000)].

O.6 The Finite Element Method (FEM) O.6.1

Introduction

The finite element method (FEM) is especially suited for the numerical calculation of sound fields in irregularly formed inner spaces, since such spaces are of finite dimensions. Originally, the FEM was developed for predicting the static or dynamical response of structures under certain loads in mechanical engineering. Several high-developed FEM packages exist which are commercially available. Some of these programs contain modules for acoustical computations.In principle,acoustical calculations can be directly performed with programs which were originally developed for structural computations, since a mechanical analogy for fluid motion can be used, [Gockel (1983)]. Consider the acoustical wave equation for the pressure with spatial varying fluid density in Cartesian coordinates ∂ 1 ∂p ∂ 1 ∂p ∂ 1 ∂p 1 ∂ 2p + + = , (1) ∂x ∂x ∂y ∂y ∂z ∂z ” ∂t2 where ” = c2 is the bulk modulus, and the equation for the equilibrium of stresses in a particular fixed direction x ∂xx ∂‘xy ∂‘xz ∂ 2 ux + + = s 2 , ∂x ∂y ∂z ∂t

(2)


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where ux is the structural displacement in the x direction, xx , ‘xy , ‘xz are stress components, and s is the structural mass density. The acoustic-structural analogy can be established by comparing Eqs. O.6.(1) and O.6.(2) and taking ux = p;

s = 1/”,

xx =

∂ 2 wx 1 ∂p =− 2 , ∂x ∂t

‘xy =

∂ 2 wy 1 ∂p =− 2 , ∂y ∂t

‘xz =

∂ 2 wz 1 ∂p =− 2 . ∂z ∂t

Here, the acoustic equilibrium equation ∇p +

∂ 2w =0 ∂t2

(3)

was taken into account, where w

is the particle displacement within the fluid. For completing the analogy, the structural displacement components uy and uz must be set equal to zero and the general stress-strain relationship must be modified in a suitable way (details can be found in [Gockel (1983)]). Hence, all the tools of classical FEM programs such as the variety of element types, solution methods etc. are available for the acoustical analysis, too. Typical areas of application of the acoustical FEM are sound fields in small rooms at low frequencies as mentioned in > Sect. O.3.3 where aspects of wave propagation play an essential role. Also, all problems involving fluid-structure interaction are treated by the FEM with preference. At first, the principle of FEM will be explained for the simple example of an air-filled enclosure with rigid boundaries.Afterwards, more advanced applications will be shortly discussed.

O.6.2 The Sound Field in Irregular Shaped Cavities with Rigid Walls This problem belongs to the standard problems described in > Sect. O.3.3, where the notation is explained. The Helmholtz equation O.3.(1) has to be satisfied in the interior Bi of the cavity. The whole boundary S is assumed to be rigid with ∂p/∂n = 0. For simplicity, only the two-dimensional case is considered. Since no acoustical sources are specified, the eigenmodes and eigenfrequencies of the cavity are quantities which should be calculated. Following the presentation given in [Schwarz (1980)], the starting point of the FEM calculation is a variational principle for the Helmholtz equation, which must be minimized. In the present case, the functional [Gladwell/Zimmermann (1966); Petyt (1982); Petyt (1983)]

1 L= (grad p)2 − k02 p2 dv (4) 2 B

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must be minimised, where the integral has to be extended over the volume (or area in two dimensions) B, and the gradient is denoted by grad. The functional L can be interpreted as a Lagrange function, i. e. as the difference between kinetic and potential energy of the vibrating fluid. Hence, the minimisation of L corresponds to Hamilton’s principle. Therefore, the Helmholtz equation is the Euler-Lagrange equation of the functional O.6.(4). The rigid boundary conditions are so-called natural boundary conditions and will be automatically satisfied. The second step consists in dividing the cavity B into simple elements. In the following, only triangular elements are used. It is important to note that the process of triangularization should be adapted to the particular problem. This means, for example, that parts of the area, in which the solution changes more rapidly, should be modelled with more and smaller elements than other parts, where the change of the solution is slower. In addition, the triangular elements should not have too acute angles. Such requirements will be checked automatically in most commercial FE programs. If B is divided into N triangular elements En , the discretized functional is given by: N

1 LD = (grad p)2 − k02 p2 dv . (5) 2 n=1 En

The third step is to choose approximate functions for the sound pressure at each single element. Frequently, polynomials are used satisfying certain continuity conditions between adjacent elements. For fulfilling such continuity conditions in selected points of the element, named nodal points or nodes, the approximate function p(e) for the e-th element is represented by: p(e) (x, y) =

Ke

(e) (e) p(e) N(e) (x, y) , k Nk (x, y) = P T

(6)

k=1

where the so-called nodal variables p(e) k are the sound pressure values in the nodes (for example, in the three corner points (Ke = 3) of the triangular element), and the N(e) k are the shape functions. Moreover, the vectors $ # T (e) (e) P(e) = P(e) (7a) 1 , P2 , . . . , PKe , # $ T (e) (e) (7b) N(e) (x, y) = N(e) 1 , N2 , . . . , NKe were introduced, where the superscript T denotes transposition. The shape functions have to satisfy the interpolation property # $ 1 for i = k (e) (e) (e) (8) Ni xk , yk = 0 for i = k # $ in the nodal points Qi(e) = xi(e) , yi(e) of the e-th element. The global representation of pressure p in the whole area B is composed of all element pressures p(e) . By numbering all nodes lying in B from 1 to K successively, one obtains: p(x, y) =

K k=1

pk Nk (x, y) ,

(9)


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where the so-called global shape functions Nk (x, y) consist of the union of all element shape functions N(e) k (x, y) which possess the value 1 at the nodal point Qk . It follows from O.6.(8) that the global shape functions are different from zero only in a small part of B, i. e. they are functions with local support, which is a key property of the FEM. For example, if the linear substitution p(x, y) = c1 + c2 x + c3 y

(10)

is used in the unit triangle, the shape functions N1 (x, y) = 1 − x − y;

N2 (x, y) = x;

N3 (x, y) = y

(11)

are obtained (see [Schwarz (1980), p. 90]). The fourth step is to introduce the approximation O.6.(9) into the discretized functional O.6.(5). The integrals have to be performed element by element and can be solved for polynomial shape functions. For a given triangle Tn with nodes analytically Qk xk , yk , k = 1, 2, 3 and the linear substitution O.6.(10) the following results are obtained (see [Schwarz (1980), p. 71]): 0 2 2 1 ∂p ∂p T dx dy = P(e) Se P(e) , + (12) ∂x ∂y Tn

T

p2 dx dy = P(e) Me P(e) ,

(13)

Tn

where the stiffness matrix Se and the mass matrices ⎞ ⎛ ⎛ 1 −1 0 2 −1 1⎝ 1 −1 1 0⎠ ; S2 = ⎝−1 0 S1 = 2 0 2 −1 1 0 0 ⎛ 1 1⎝ 0 S3 = 2 −1

⎞ 0 −1 0 0 ⎠; 0 1

⎛ 2 1 1 ⎝ 1 2 S4 = 24 1 1

matrix Me are composed of the four basis ⎞ −1 1 ⎠; 0 ⎞ 1 1⎠ 2

(14)

in the following way Se = aS1 + bS2 + cS3

and

Me = J S4 ,

(15)

where the constants 2 2 3 a = (x3 − x1 )2 + y3 − y1 /J,

(16a)

b = − (x3 − x1 ) (x2 − x1 ) + y3 − y1 y2 − y1 /J,

(16b)

2 2 3 c = (x2 − x1 )2 + y2 − y1 /J,

(16c)

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J = (x2 − x1 ) y3 − y1 − (x3 − x1 ) y2 − y1

(16d)

only depend on the geometry of the triangle considered. Hence, the contribution of a single triangular element to the whole functional is given by: L(e) D =

1 (e)T 1 T P Se P(e) − k02 P(e) Me P(e) . 2 2

(17)

Summing up all contributions element by element leads to the discretized global functional of quadratic form: 1 1 LD = PT SP − k02 PT MP , 2 2

(18)

where P is the vector of all nodal pressure variables. S and M are the global stiffness and mass matrix, respectively. The requirement that LD takes a minimum leads to the equation: SP = k02 MP ,

(19)

which is a generalised eigenvalue problem for the eigenvalue parameter Š = k02. Such a problem was solved in [Schwarz (1980)] with three different numerical methods for an idealised automobile passenger compartment. For this specific example, it was shown that the simultaneous inverse vector iteration was the most efficient method. Details about numerical methods for solving the generalised eigenvalue problem can be found in [Schwarz (1980)]. A great advantage of the FEM is the fact that the global matrices S and M are symmetric and weakly populated, in contrast to the BEM. By using optimal numbering procedures such as the algorithms of Rosen or Cuthill-McKee (see [Schwarz (1980)]) for the nodal variables,the bandwidth of such sparse matrices can be minimised which leads to a large reduction of computer time and storage capacity.

O.6.3

Supplementary Aspects and Fluid–Structure Coupling

The simple model problem of a rigid two-dimensional cavity, which is discretised with triangular elements, can be extended into different directions. For example, elements or shape functions of higher orders can be used. Also, the generalization to the threedimensional case can be easily done. Especially, the sound field in enclosures and automobile passenger compartment was investigated by several authors: Craggs [Craggs (1972)] used tetrahedral and cuboid finite elements with a linear variation of the pressure between the nodes in order to find the eigenfrequencies and modes of a threedimensional enclosure. Shuku and Ishihara [Shuku/Ishihara (1973)] used triangular elements with cubic polynomial functions for the pressure. Petyt et al. [Petyt/Lea/ Koopmann (1976)] developed a twenty-node, isoparametric acoustic finite element for analysing the acoustics modes of irregular shaped cavities. Richards and Jha [Richards/ Jha (1979)] preferred quadratic triangular elements with six nodes.


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If only the part S0 of the surface S is rigid, whereas the normal velocity v is prescribed on S1 and an absorbing material with impedance Z is specified on S2 , the following boundary conditions are obtained: ∂p =0 ∂n

on

S0 ;

∂p = −j– 0 v ∂n

on

S1 ;

∂p p = −j– 0 ∂n Z

on

S2 .

(20)

Instead of Eq. O.6.(4) the corresponding variational principle is now given by [Petyt (1983)]: 2 3 2 1 p2 1 2 2 grad p − k0 p dv + L= (21) j– 0 pv ds + j– 0 ds , 2 2 Z B

S1

S2

which is minimised according to the same rules as described in > Sect. O.1.The resulting system of equations can be used for the investigation of passenger compartments or silencers which are partially lined with absorbing material (see [Petyt (1983)] and [Munjal (1987)],where more references can be found).In Eq.O.6.(21),the three-dimensional case is considered, and hence the first integral is extended about the volume B. In [Schulze Hobbeling (1989)], wave propagation within the porous absorber is taken into account. The coupled fluid-structure interaction problem is described in > Sect. O.3.4 as the fourth standard problem. A mathematically rigorous description of the corresponding finite dimension approximation, which can be associated with a finite element mesh, is given by Soize [Soize, Eur.J. Mech. A/Solids (1998); Soize, J. Acoust. Soc. Amer. (1998); Soize (1999)]. Instead, the method of Everstine and Henderson [Everstine/Henderson (1990)] will be described here, which is based on a coupled FE/BE approach. A very similar formulation was given by Smith, Hunt, and Barach [Smith/Hunt/Barach (1973)]. The structure is assumed to be modeled with finite elements. This leads to a matrix equation of motion for the structural degrees of freedom: Zs Vs = Fs − GAs P ,

(22)

where Zs is the structural impedance matrix, Vs is the global velocity vector, Fs is the vector of mechanical forces applied to the structure, G is a transformation matrix in order to transform a vector of outward normal forces at the so-called wet points (which are in contact with the fluid) to a vector of forces at all points in the selected coordinate system, As is the diagonal matrix of areas for the wet surface, and P is the vector of total acoustic pressures. Equation O.6.(22) can be derived in a similar way as described in the last section for the acoustical case. The structural impedance matrix Zs is given by: Zs =

1 2 −– Ms + j– Ds + Ks , j–

(23)

where Ms ,Ds and Ks are the structural mass,damping and stiffness matrices,respectively. The integral equation of the scattering problem for an incident pressure wave pi was derived in > Sect. O.5.6 and is given by (see Eq. O.5.(45)): ∂g(x, y) ds(y) + 2 p(y) j– 0 v(y) g(x, y) ds(y) + 2pi . (24) p(x) = 2 ∂n(y) S

S

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Corresponding to the BE approach described by Eqs. O.5.(16)–O.5.(18), the discretised version of the integral equation O.6.(24) is −AP = MV + Pi

with

A: = D − 0.5I,

(25)

where Pi is the vector of incident pressures, V is the vector of normal velocities, and D and M are the dipole and monopole matrices, respectively, defined in Eqs. O.5.(17). According to Everstine and Henderson [Everstine/Henderson (1990)] the vector of normal velocities V is transformed into the vector Vs of total structural velocities by applying the transposed matrix GT to Vs : V = GT V s .

(26)

Now, by combining Eqs. O.6.(22), O.6.(25), and O.6.(26), the velocity vectors V and Vs can be eliminated, which leads to the coupled fluid–structure equation HP = Q + Pi

(27)

with H : = −A + MGT Z−1 s GAs

and

Q = MGT Z−1 s Fs .

(28, 29)

Having solved system O.6.(27) for the pressure P, the vector Vs of structural velocities is obtained from Eq. O.6.(22) by solving the equation −1 Vs = Z−1 s Fs − Zs GAs P.

(30)

With the knowledge of the fluid surface variables P and V all acoustics quantities in the exterior field can be calculated by evaluating the exterior Helmholtz formula O.5.(43a). In [Hunt/Knittel/Barach (1974); Kirsch/Monk (1990); Masmoudi (1987)], a combined finite element and spectral approach is proposed: the FEM is used for calculating the vibration of the elastic structure and the acoustic field inside of a finite fluid sphere which totally surrounds the vibrating structure.At the boundary of the sphere, a perfect absorption condition is given explicitly in terms of the spherical wave functions O.4.(5). This method was generalised and implemented into the structural analysis code NASTRAN resulting in a commercially available FEM package. Some recent results of this technique can be found in [Zimmer/Ochmann/Holzheuer (2001)]. Another interesting idea is to use absorbing boundary condition operators [Gan/Levin/ Ludwig(1993)] on the artificial boundary of the finite domain instead of coupling with analytical wave functions. The perfect absorbing boundary condition is modeled by the so-called BGT operator named after Bayliss, Gunzburger and Turkel [Bayliss/ Gunzburger/Turkel (1982)]. The first-order operator B1 is defined by: 1 ∂ B1 p : = + jk0 + p, (31) ∂R 2R where p is the radiated sound pressure. R and k0 are defined as in the Sommerfeld radiation condition O.3.(3) which is approximated with increasing accuracy for increasing order of the BGT operator. The second operator B2 is given by: 5 1 ∂ ∂ + jk0 + + jk0 + p, (32) B2 p : = ∂R 2R ∂R 2R


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and, in general, the m-th order operator can be introduced by [Bayliss/Gunzburger/ Turkel (1982), Gan/Levin/Ludwig(1993)]: m 4 2 − 32 ∂ Bm : = + + jk0 . (33) ∂R R =1 It can be shown [Bayliss/Gunzburger/Turkel (1982), Gan/Levin/Ludwig(1993)] that the operator Bm annihilates terms in 1/r of the asymptotic far field solution ps e−jk0 R a1 (œ) a2 (œ) ps = √ + ... (34) + a0 + R R2 R such that the accuracy in approximating the radiation condition is 1 Bm ps = O 2m+1+1/2 . r

(35)

Here, a0 is a constant, and a1 (œ), a2 (œ), . . . are functions of the polar angle œ. These formulas are given for the two-dimensional case. The incorporation into a finite element model is described in [Gan/Levin/Ludwig(1993)]. Some more special examples from the huge number of papers dealing with the application of the FEM to acoustical problems should be mentioned: The formulation of the FEM for cavities with fluid-structure interaction can be found in [Nefske/Wolf JR/Howell (1982); Soize (1999)], for example. The sound transmission between enclosures using plate and acoustic finite elements is studied in [Craggs/Stead (1976)]. A comparison between FEM and BEM for the calculation of sound fields is performed in [Becker/Waller (1986)].

O.7 The Cat’s Eye Model Many variations and improvements of numerical methods are studied in the literature.It is helpful to have benchmark models available for which analytical solutions are known, and which are not over-simplified, so that comparisons between different variants of numerical methods can be performed. One favourite benchmark model is the Cat’s Eye model for which the analytical solution of the sound field will be described in this Section. An other model, with somewhat more simple geometryis the “Orange” model which will be described in > Sect. O.8. Both Sections treat the radiation problem for the special case of a modal surface velocity pattern of the sphere (radiation problem). It will be explained at the end of this Section, how multi-mode radiation problems and scattering problems can be treated.

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O.7.1

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Cat’s Eye Model and General Fundamental Solutions)

The Cat’s Eye model is a sphere with one octant of the sphere taken away.

An octant with cuts at œ = ±œ0 ; ˜ = /2 is taken away of a sphere with radius r0 . The walls of the cuts are hard. The remainder of the spherical surface oscillates with a velocity pattern of a spherical mode m0 , n0 . Radiation problems with a more general vibration pattern and scattering problems can be reduced to a repeated solution of the single-mode radiation problem. Target quantities are : • the sound field in the outer zone (I); • the sound field in the inner zone (II). A spherical co-ordinate system r, ˜ , œ is used, with the co-ordinate axis œ = 0 in the middle of the sector. A common time factor ej–t is dropped. The particle velocity of a sound pressure field p is given by: j grad p , k0Z0 ∂p j ∂p j ; v˜ = ; vr = k0Z0 ∂r k0 Z0 r · ∂˜

v =

∂p j ∂p j vœ = . − −−−−−→ ˜ →/2 k0 Z0 r sin ˜ · ∂œ k0 Z0 r · ∂œ

(1)

The wave equation separates in spherical co-ordinates, therefore the fundamental solutions can be written as a product: p(r, ˜ , œ) = R(k0r) · Ÿ(˜ ) · ¥ (œ) , ∗)


(2)


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in which the factors are linear combinations of elementary solutions: (1)

(2)

R(k0 r) = A · Œ (k0 r) + B · Œ (k0r) , ‹

‹

(3)

Ÿ(˜ ) = a · PŒ (cos ˜ ) + b · QŒ (cos ˜ ) , ¥ (œ) = · sin(‹œ) + · cos(‹œ) . Therein are: (i) Œ (k0 r) ‹ PŒ (x);

2 linear independent spherical Bessel functions; ‹ QŒ (x)

Legendre functions of the first and second kind, with x = cos ˜ .

To simplify the formulations, we suppose the excitation to be symmetrical relative to œ = 0; i. e., we suppose ¥ (œ) = cos(‹œ). Outer field in (I): From Sommerfeld’s condition follows: Œ (k0r) = h(2) Œ (k0 r) ‹

(spherical Hankel function of second kind). The QŒ (z) have logarithmic singularities at z = ±1, i. e., at ˜ = 0, ˜ = ; the sound field should be regular there, consequently the ‹ QŒ (z) are dropped. The outer field is periodic in ˜ , œ with a period 2; consequently the Œ, ‹ = n, m = 0, 1, 2, . . . are integer numbers. Thus the field in (I) can be formulated as a mode sum: m p(I) (r, ˜ , œ) = an,m · h(2) n (k0 r) · Pn (cos ˜ ) · cos(mœ) ,

(4a)

n,m≥0

Z0 vr(I) (r, ˜ , œ) = j

m an,m · h(2) n (k0 r) · Pn (cos ˜ ) · cos(mœ)

(4b)

n,m≥0

(a prime indicates the derivative with respect to the argument). Pm n (cos ˜ ) = 0

To be noticed: which follows from:

for

m > n, m m 2 m/2 d Pn (x) Pm (x) = (−1) (1 − x ) n dxm

(5) (6)

for m, n = integers, with Legendre polynomials of n-th degree Pn (x); i. e., m > n is excluded. The strength of the excitation by the modal pattern with indices n0 , m0 either can be described by a sound pressure amplitude Pe or by the particle velocity V at the surface of the sphere. They are interrelated by: m0 Z0 ver (r0 , ˜ , œ) = j Pe · h(2) n0 (k0 r0 ) · Pn0 (cos ˜ ) · cos(m0 œ) 0 : = Z0 V · Pm n0 (cos ˜ ) · cos(m0 œ) ,

(7a)

whence the reference velocity V follows as: Z0 V = j Pe · h(2) n0 (k0 r0 ).

(7b)

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Another kind of reference is the sound pressure which a full sphere with the exciting mode n0 , m0 would produce in the field point P = (r, ˜ , œ): m0 pref (r, ˜ , œ) = Pe · h(2) n0 (k0 r) · Pn0 (cos ˜ ) · cos(m0 œ)

= −jZ0 V

h(2) n0 (k0 r) h(2) n0 (k0 r0 )

(7c)

0 · Pm n0 (cos ˜ ) · cos(m0 œ).

Interior field in (II): From regularity at r = 0: Œ (k0r) = jŒ (k0 r)

(8)

with spherical Bessel functions. From symmetry in œ:

¥ (œ) = cos(‹œ).

Regularity at ˜ = 0 again excludes the

(9)

‹ QŒ (z).

The condition vœ −−−−−−→ 0 has the consequence: œ→±œ0

!

sin(‹œ0 ) = 0

⇒

‹œ0 = m;

m = 0, 1, 2, . . .

⇒

‹=

m −−−−−−→ 4m. (10) œ0 œ0 →/4

The angle œ0 is restricted henceforth to an integer part of , œ0 = /N; N = 2, 3, 4, . . ., from which condition follows: ‹ = m · N.

(11)

The condition v˜ −−−−−−→ 0 implies: ˜ →/2

! ! ‹ Ÿ (˜ )!˜ =/2 ∼ PŒ (0) = 0 ,

(12)

from which, with the relation (Œ/2 + ‹/2 + 1) 2‹+1 ‹ PŒ (0) = √ sin (Œ + ‹)/2 (Œ/2 − ‹/2 + 1/2) follows: ! sin (Œ + ‹)/2 = 0.

(13)

(14)

This is satisfied for integer ‹ with Œ = integer;

!

‹ + Œ = even.

(15)

Thus the sum of the mode indices in (II) must be an even number, Œ + ‹ = even, i. e., Œ, ‹ both are either even or odd. The relation ‹ 2 ‹ ‹ ‹ PŒ (−x) = PŒ (x) · cos (Œ + ‹) − sin (Œ + ‹) · QŒ (z) −−−−−−−→ PŒ (x) Œ+‹=even

(16)


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guarantees, because of Œ + ‹ = even, a sound field in (II) which is symmetrical relative to the hard cut at ˜ = /2. One consequently has the following sound field formulation in the interior zone (II) which satisfies the conditions of regularity and the boundary conditions at the hard walls of the cut: ‹ p(II) (r, œ, ˜ ) = bŒ,‹ · jŒ (k0 r) · PŒ (cos ˜ ) · cos(‹œ); Œ,‹ (17a) ! Œ = 0, 1, 2, . . . ; ‹ = † · N; † = 0, 1, 2, . . . ; ‹ + Œ = even; Z0 vr(II) (r, œ, ˜ ) = j

Œ,‹

O.7.2

‹

bŒ,‹ · jŒ (k0r) · PŒ (cos ˜ ) · cos(‹œ).

(17b)

Mode Orthogonality

Because of symmetry in œ only the range 0 ≤ œ ≤ must be considered. In œ direction: In the outer zone (I): 1 1 sin ((m + m )) sin ((m − m )) + cos(mœ) · cos(m œ) dœ = 2 (m + m ) (m − m ) 0

⎧ ⎨1; = 1/2; ⎩ 0;

m = m = 0 m = m = 0 m = m

(18a) ƒm,m = ƒm

with Kronecker’s symbol ƒn,m = 0; n = m; ƒn,m = 1; n = m and Heaviside’s symbol ƒn=0 = 1; ƒn=0 = 2. In the interior zone (II), where ‹œ0 = m; ‹ œ0 = m the orthogonality integral holds: 0 1 œ0 sin (‹ − ‹ )œ0 1 1 sin (‹ + ‹ )œ0 cos(‹œ) · cos(‹ œ) dœ = + œ0 2 (‹ + ‹ )œ0 (‹ − ‹ )œ0 (18b) 0 ƒm,m 1 sin ((m + m )) sin ((m − m )) + = . = 2 (m + m ) (m − m ) ƒm The orthogonality in ˜ -direction in the outer zone (I) is a consequence of:

1 0; Œ = Œ ‹ ‹ ‹ PŒ (x) · PŒ (x) dx = = ƒŒ,Œ NŒ ; Œ, Œ , ‹ = integer (19) ‹ NŒ ; Œ = Œ

−1

with the norms: 1 i 2 i Pk (x) dx = Nk = −1

2(k + i)! . (2k + 1)(k − i)!

(20)

In the interior zone (II) is Œ + ‹ = even; the Legendre functions are symmetrical with respect to x = 0; therefore the orthogonality O.7.(19) also holds in (II) over x = (0, 1), i. e., ˜ = (0, /2) with a halved value of the norm.

O

1086

O.7.3


Remaining Boundary Conditions

One is left, after the above preparations, with two sets of unknown mode amplitudes an,m , bŒ,‹ and has for their determination two boundary conditions of field matching: !

p(II) (r0 , ˜ , œ) = p(I) (r0 , ˜ , œ) !

Z0 vr(I) (r0 , ˜ , œ) =

in

˜ = (0, /2) & œ = (−œ0 , +œ0 ),

Z0 vr(II) (r0 , ˜ , œ)

in

˜ = (0, /2) & œ = (−œ0 , +œ0 ) , .

Z0 ver (r0 , ˜ , œ)

in

rest of the sphere.

(21) (22)

One uses in O.7.(21) the mode orthogonality in (II) and in O.7.(22) the mode orthogonality in (I). In doing that, one multiplies both sides of the matching conditions at the zone limit with a mode of the relevant zone with arbitrary but fixed mode indices Œ, ‹ or n, m, respectively, and integrates over the range of orthogonality which agrees with the range of definition of the boundary condition. One will obtain two linear systems of equations for the an,m , bŒ,‹ ; after their solution the sound fields will be determined. Beginning with the matching condition O.7.(21) for the sound pressures to each other, with: ‹ p(II) (r, œ, ˜ ) = bŒ,‹ · jŒ (k0r) · PŒ (cos ˜ ) · cos(‹œ), Œ,‹

p(I) (r, ˜ , œ) =

m an,m · h(2) n (k0 r) · Pn (cos ˜ ) · cos(mœ)

n,m≥0

apply on both sides the integral 1 œ0

œ0

1 dœ

0

. . . · Pˆ‰ (x) · cos(ˆœ) dx

0

with ˆ, ‰ from the range of values of ‹, Œ. This will give on the left-hand side: Œ,‹

ƒ‹,ˆ bŒ,‹ · · jŒ (k0r0 ) ƒˆ =

Œ

1 0

jŒ (k0r0 ) bŒ,ˆ · ƒˆ

‹

PŒ (x) · Pˆ‰ (x) dx

1

PˆŒ (x)

·

Pˆ‰ (x) dx

0

Nˆ = ‰ · b‰,ˆ · j‰ (k0 r0 ) , 2ƒˆ

and on the right-hand side: m,ˆ an,m · h(2) n (k0 r0 ) · Kn,‰ · Im,ˆ ,

(23a)

(23b)

n,m≥0

where the following integrals are introduced: Im,ˆ

1 = œ0

Km,ˆ n,‰ =

1 0

œ0 cos(mœ) · cos(ˆœ) dœ,

(24)

0 ˆ Pm n (x) · P‰ (x) dx.

(25)


O

1087

Thus the matching condition for the sound pressures leads to: Nˆ‰ m,ˆ · b‰,ˆ · j‰ (k0 r0 ) = an,m · h(2) n (k0 r0 ) · Kn,‰ · Im,ˆ . 2ƒˆ n,m≥0

(26)

This is a linear homogeneous system of equations for the vector {b‰,ˆ , an,m } of solutions. Next we apply on the matching condition O.7.(22) for the particle velocities, with m an,m · h(2) Z0 vr(I) (r, ˜ , œ) = j n (k0 r) · Pn (cos ˜ ) · cos(mœ), n,m≥0

Z0 vr(II) (r, œ, ˜ )

=j

Œ,‹

‹

bŒ,‹ · jŒ (k0 r) · PŒ (cos ˜ ) · cos(‹œ),

m0 Z0 ver (r0 , ˜ , œ) = jPe · h(2) n0 (k0 r0 ) · Pn0 (cos ˜ ) · cos(m0 œ) 0 : = Z0 V · Pm n0 (cos ˜ ) · cos(m0 œ),

the integral 1

1 dœ

0

. . . · Pik (x) · cos(iœ) dx ,

−1

in which i, k are from the range of values of m, n. The left-hand side will give:

ƒm,i (2) j an,m · · hn (k0r0 ) ƒi n,m≥0

=j =j

n≥0 Nik

ƒi

an,i · ·

hn(2) (k0r0 ) ƒi

(2) hk (k0 r0 )

1

i Pm n (x) · Pk (x) dx

−1

1

Pin (x) · Pik (x) dx

(27a)

−1

· ak,i ,

and the first line of the right-hand side of O.7.(22) by the integral 1

œ0

1 dœ

0

Z0 vr(II) (r0 , œ, ˜ ) · Pik (x) · cos(iœ) dx

(28)

0

will contribute: i,‹ bŒ,‹ · jŒ (k0 r0 ) · Lk,Œ · Ji,‹ , j

(27b)

Œ,‹

where the integrals are introduced: Ji,‹

1 =

œ0 cos(iœ) · cos(‹œ) dœ = 0

œ0 Ii,‹ ,

(29)

O

1088

i,‹ Lk,Œ

1


‹

i,‹

Pik (x) · PŒ (x) dx = Kk,Œ .

=

(30)

0

The second line of the right-hand side of O.7.(22) takes the form: 1

1

Z0 ver (r0, ˜ , œ) · Pik (x) · cos(iœ) dx

dœ 0

1 −

−1

œ0

1 dœ

0

(31) Z0 ver (r0 , ˜ , œ) · Pik (x) · cos(iœ) dx ,

0

whereof the first line contributes: ƒm ,i 0 Z0 V · 0 ƒn0,k Nm n0 , ƒm0

(27c)

and the second line gives the contribution:

i,m0 Z0 V · Ji,m0 · Kk,n 0

(27d)

(the apostrophe at J, K shall indicate that m0 , n0 generally are not members of ‹, Œ). In total, the boundary condition for vr gives: j

Nik · h(2) k (k0 r0 ) · ak,i ƒi ƒm ,i · ƒn0,k m0 i,‹ 0 =j bŒ,‹ · jŒ (k0 r0 ) · Lk,Œ · Ji,‹ + Z0 V · 0 Nn0 − Z0 V · Ji,m0 · Ki,m k,n0 . ƒ m0 Œ,‹

(27)

This is a second linear inhomogeneous system of equations for {ak,i , bŒ,‹ }. Either one packs the systems O.7.(26), O.7.(27) together to a compound system for the {an,m , bŒ,‹ } or, preferably, one next tries a reduction of the systems of equations (see below). Before doing that, the mode coupling integrals shall be discussed.

O.7.4

Mode Coupling Integrals

In principle, the coupling integrals can be evaluated numerically, since the integration interval is finite. But because the integrands oscillate, analytical solutions are preferable. The integrals from O.7.(24) and O.7.(29), with ‹ = † ·N; i, † = 0, 1, 2, . . . have the values: 1 Ii,‹ = œ0 1 = 2

œ0

0

cos(iœ) · cos(‹œ) dœ = 0

Ji,‹ œ0

1 sin (i + †N)œ0 sin (i − †N)œ0 + (i + †N)œ0 (i − †N)œ0

− −−−−−→ 1; i=†N=0

0 0 1 1 sin 2†Nœ0 sin 2† 1 1 1 1+ 1+ = . = − −−−−−→ i=†N=0 2 2†Nœ0 2 2† 2

(32)


O

The integrals from (27d), with i, m = 0, 1, 2, . . . are: œ0 sin (i − m)œ0 1 œ0 sin (i + m)œ0 cos(iœ) · cos(mœ) dœ = + Ji,m = 2 (i + m)œ0 (i − m)œ0 0 sin 2mœ0 œ0 œ0 1+ . −−−−−→ −−−−−→ i=m=0 2 i=m=0 2mœ0

1089

(33)

There still remain the integrals from O.7.(25), O.7.(30): m,‹ Ln,Œ

1 =

‹

m,‹

Pm n (x) · PŒ (x) dx = Kn,Œ ,

0

Solutions can be found in integral tables only for a few special values of the indices. An algorithm for the integration starts from the representation of Legendre functions for integer n, m and real x (with |x| < 1): m 2 m/2 Pm n (x) = (−1) (1 − x )

dm Pn (x) . dxm

(34)

The Legendre polynomials Pn (x) are polynomials of n-th degree in x. Therefore Pm n (x) = 0 for m > n. The Legendre polynomials have the form: [n/2]

Pn (x) =

k · xn−2k =

k=0

[n/2] (−1)k (2n − 2k)! 1 xn−2k , 2n k! · (n − k)! · (n − 2k)!

(35)

k=0

where [n/2] = largest integer ≤ n/2. The m-th derivative thereof is (m ≤ n): dm Pn (x) = dxm

[(n−m)/2]

k

(n − 2k)! · xn−2k−m (n − 2k − m)!

k=0 [(n−m)/2]

1 = n 2

k=0

(−1)k (2n − 2k)! xn−2k−m . k! · (n − k)!(n − 2k − m)!

(36)

Thus the Legendre functions become: Pm n (x) =

(−1)m (1 − x2 )m/2 2n

[(n−m)/2]

k=0

(−1)k (2n − 2k)! xn−2k−m . k! · (n − k)! (n − 2k − m)!

(37)

If one has available a computer program for mathematics which can perform symbolic operations (like Mathematica or Mapple) one can imagine to have the factor (1 − x2 )m/2 multiplied under the sum and expanded the sum terms with x remaining symbolic. After multiplication two sums for two Legendre functions (and expanding again) the program can integrate term-wise. This indeed is an easy method, but it will take long computing times, because the number of integrals to be evaluated roughly increases with the fourth power of the upper limits of the used mode orders (and the number of terms in the sums increases linearly with those limits). Therefore explicit solutions for the integrals shall be given here, even if the result becomes lengthy.

1090

O


One starts from the representation: Pm n (x) =

(−1)m (1 − x2 )m/2 2n

[(n−m)/2]

k=0

(−1)k (2n − 2k)! xn−2k−m k! · (n − k)! (n − 2k − m)!

and similarly after the substitutions m → ‹; n → Œ; k → i. The indefinite integral of their product then becomes (by multiplication of the sums, expanding the product, and term-wise integration): x

‹

Pm n (x) · PŒ (x) dx =

(−1)1+m+‹ x1+n+Œ−m−‹ · 2n+Œ

[(n−m)/2] [(Œ−‹)/2]

k=0

i=0

(−1)i+k (2n − 2k)! · (2Œ − 2i)! · x−2i−2k (−1 + 2i + 2k + m + ‹ − n − Œ) . . . ... . . . i! · k! · (n − k)! · (Œ − i)! · (n − m − 2k)! · (Œ − ‹ − 2i)! · 2 F1 (1 − 2i − 2k − m − ‹ + n + Œ)/2, − (m + ‹)/2;

(38)

1 + (1 − 2i − 2k − m − ‹ + n + Œ)/2; x2 , where 2 F1 (a, b; c; z) are hypergeometric functions. The definite integral is: 1

‹

Pm n (x) · PŒ (x) dx =

0

(−1)m+‹ · 2n+Œ

[(n−m)/2] [(Œ−‹)/2]

k=0

i=0

(−1)1+i+k (2n − 2k)! · (2Œ − 2i)! 1 + n + Œ − m − ‹ − 2i − 2k i! · k! · (n − k)! · (Œ − i)! . . . ... · ...

... · (n − m − 2k)! · (Œ − ‹ − 2i)! ((3 + n + Œ − 2i − 2k)/2) − (2 + m + ‹)/2 . . . ((3 + n + Œ − 2i − 2k)/2)

· (3 + n + Œ − m − ‹ − 2i − 2k)/2 ...

with the Gamma function (z).

(39)


O.7.5

O

1091

Reduction of the System of Equations

Eliminate the b‰,ˆ from O.7.(26) and to insert them into O.7.(27): bŒ,‹ =

‹ NŒ

2ƒ‹

· jŒ (k0r0 ) n,m≥0

m,‹

an,m · h(2) n (k0 r0 ) · Kn,Œ · Im,‹ .

(40)

After insertion in O.7.(27) and rearrangement that equation becomes: j

0

n,m≥0

− ƒm,i · ƒn,k ·

Nik (2) · hk (k0r0 ) ƒi

2ƒ‹ · jŒ (k0 r0)

⎤

œ0 (2) m,‹ i,‹ h (k0r0 ) · Im,‹ · Ii,‹ · Kn,Œ · Kk,Œ ⎦ · an,m ‹ n N · j (k r ) Œ Œ 0 0 Œ,‹ ƒm0 ,i · ƒn0 ,k m0 0 = Z0 V · Ji,m0 · Ki,m − N ; k ∈ {n}; i ∈ {m} ≤ k. n0 k,n0 ƒ m0 +

(41a)

This is a two-dimensional system of equations for the an,m ; in every sub-system the i, k have fixed values 0, 1, 2, . . .. By the variation of n, k = 0, 1, 2, . . ., nhi every line of the system contains the unknowns an,m , because of the requirements i ≤ k, m ≤ n, in an arrangement like:

an,m = {a0,0 }, {a1,0 , a1,1,}, {a2,0 , a2,1 , a2,2 }, {a3,0 , a3,1 , a3,2, a3,3 }, . . . .

The system O.7.(41a) of equations in principle has an infinite size. It must converge if a truncation shall be possible. A sufficient condition for convergence is a decrease (or constant value) of the elements on the main diagonal, together with a decrease of the other elements with increasing distance to the main diagonal, and a decrease of the right-hand side elements. The first condition is violated by the norms Nik in the first term in the brackets, because they assume huge values for larger indices. Therefore the lines are divided by Nik . Introducing the reference amplitude Pe instead of Z0 V one gets the system of equations:

0 −

n,m

ƒm,i · ƒn,k (2) · hk (k0 r0) ƒi

⎤ 2ƒ‹ · jŒ (k0 r0 ) œ0 h(2) (k r ) 0 0 m,‹ i,‹ n + · Im,‹ · Ii,‹ · Kn,Œ · Kk,Œ ⎦ · an,m ‹ Nik N · j (k r ) Œ Œ 0 0 Œ,‹

= Pe ·

h(2) n0 (k0 r0 )

k ∈ {n}; m ≤ n ;

Nik

ƒm0 ,i · ƒn0,k m0 œ0 0 · Ii,m0 · Ki,m ; − N n0 k,n0 ƒ m0

i ∈ {m} ≤ k . ‹≤Œ

(41b)

1092

O


i,m If one symbolises the matrix elements with Xk,n and the elements of the right-hand side with Yik the system O.7.(41b) at its beginning has the form: 0,0 0,0 0,1 0,0 0,1 0,2 X0,0 · a0,0 + (X0,1 · a1,0 + X0,1 · a1,1 ) + (X0,2 · a2,0 + X0,2 · a2,1 + X0,2 · a2,2 ) + . . . = Y00 , 0,0 0,0 0,1 0,0 0,1 0,2 · a0,0 + (X1,1 · a1,0 + X1,1 · a1,1 ) + (X1,2 · a2,0 + X1,2 · a2,1 + X1,2 · a2,2 ) + . . . = Y01 , X1,0 1,0 1,0 1,1 1,0 1,1 1,2 X1,0 · a0,0 + (X1,1 · a1,0 + X1,1 · a1,1 ) + (X1,2 · a2,0 + X1,2 · a2,1 + X1,2 · a2,2 ) + . . . = Y11 , 0,0 0,0 0,1 0,0 0,1 0,2 X2,0 · a0,0 + (X2,1 · a1,0 + X2,1 · a1,1 ) + (X2,2 · a2,0 + X2,2 · a2,1 + X2,2 · a2,2 ) + . . . = Y02 , 1,0 1,0 1,1 1,0 1,1 1,2 X2,0 · a0,0 + (X2,1 · a1,0 + X2,1 · a1,1 ) + (X2,2 · a2,0 + X2,2 · a2,1 + X2,2 · a2,2 ) + . . . = Y12 , 2,0 2,0 2,1 2,0 2,1 2,2 X2,0 · a0,0 + (X2,1 · a1,0 + X2,1 · a1,1 ) + (X2,2 · a2,0 + X2,2 · a2,1 + X2,2 · a2,2 ) + . . . = Y22 . (41c)

An important aspect, both for the precision and the computing time, are the upper limits of the mode orders nhi and ‹hi = †hi · N. In the outer zone (I) at least the orders n0 , m0 of the exciting mode must be exceeded. In the sector (II) a rather low number of modes often may be sufficient. In turn, the mode index m in (I) should reach the order ‹hi = †hi · N; i. e., nhi ≥ ‹hi = †hi · N. After solution of this system of equations for the an,m , the bŒ,‹ are obtained by insertion in O.7.(40). Thus the sound field in and around the Cat’s Eye is known. The formulas above are derived for the radiation problem in the simple case of a monomodal excitation. For any given distribution V(˜ , œ) first expand that distribution as sum of multipole sources. Then solve the multipole radiation task for each sum term and add the results.

Sound pressure magnitude |p(œ)/Pe | on a œ-orbit around a cat’s eye with higher mode excitation m0 = 1; n0 = 1 at elevation ˜ = 90◦ . k0r0 = 4; N = 3; œ0 = 60◦ ; n0 = 1; m0 = 1; k0 r = 4; ˜ = 90◦ ; nhi = 12; †hi = 4


O

1093

As above, but elevation ˜ = 45◦ In a scattering task, e. g. for an incident plane wave, first expand the plane wave in spherical harmonics: e−jk·r =

∞ n=0

·

(−j)n (2n + 1)

∞ m=0

ƒm

(n − m)! cos m(œ − v) (n + m)!

(42)

m Pm n (cos u) Pn (cos ˜ ) jn (k0 r) ,

where the vector r shows from the origin to the field point P = (r, œ, ˜ ), and the wave number vector k has the length k0 and has the spherical angles ˜ = u, œ = v. The particle velocities of the terms at the surface r = r0 are considered as given velocities of a multipole radiation task; these are solved in turn, and the fields are added. The presented numerical examples show |p(œ)/Pe | (as full lines) on orbits with varying œ and fixed r, ˜ . For orientation, the sound pressure |pe (œ)/Pe | is also shown (as dashed curve) which the exciting mode n0, m0 on a full sphere would produce at r, and the flanks of the cut out at ±œ0 are also indicated as dashed straight lines. Both |p(I) (œ)/Pe | (full line) and |p(II) (œ)/Pe | (with short dashes) are plotted.

1094

O


O.8 The Orange Model For tests of some numerical methods the Orange model with a simpler geometry may be sufficient also. This model consists of a sphere with “orange slices” taken away.

A sphere with radius r0 has at its surface the radial velocity distribution V of a spherical mode with mode indices n0 , m0 .A sector is cut at œ = ±œ0 . The walls of the cut are hard. Target quantities are: • outer sound field (in (I)); • sound field in the cut ((II)). The time factor ej–t dropped. The problem be symmetrical in œ relative to œ = 0.

O.8.1

Elementary Solutions and Field Formulations

The sound field in the outer zone (I) is formulated as a sum of spherical modes with spherical Hankel functions of second kind h(2) n (k0 r) as radial functions and tesseral Legendre functions Pm n (cos ˜ ) in the polar direction ˜ , and the even azimuthal functions cos(mœ) along œ. With integer indices n, m = 0, 1, 2, . . . the sum terms satisfy the wave equation, Sommerfeld’s far field condition and the periodicity in œ with period 2. m p(I) (r, ˜ , œ) = an,m · h(2) (1a) n (k0 r) · Pn (cos ˜ ) · cos(mœ), n,m≥0

Z0 vr(I) (r, ˜ , œ)

=j

m an,m · h(2) n (k0 r) · Pn (cos ˜ ) · cos(mœ).

(1b)

n,m≥0

The sound field which the exciting mode with mode indices n0, m0 on a full sphere would produce in (I) is, with arbitrary amplitude Pe : m0 pe (r, ˜ , œ) = Pe · h(2) n0 (k0 r) · Pn0 (cos ˜ ) · cos(m0 ˜ ).

(2)


O

1095

A corresponding radial reference velocity V may be defined by the relations: m0 Z0 ver (r0 , ˜ , œ) = jPe · h(2) n0 (k0 r0 ) · Pn0 (cos ˜ ) · cos(m0 œ) 0 : = Z0 V · Pm n0 (cos ˜ ) · cos(m0 œ),

(3)

Z0 V = jPe · h(2) n0 (k0 r0 ). The special case of constant surface velocity belongs to the mode indices n0, m0 = 0, 0, because of P00 (x) = 1. A formulation of the field in (II) as sum of modes satisfying the symmetry in œ around œ = 0, and being regular at r = 0 reads: ‹ p(II) (r, œ, ˜ ) = bŒ,‹ · jŒ (k0 r) · PŒ (cos ˜ ) · cos(‹œ), (4a) Œ,‹

Z0 vr(II) (r, œ, ˜ ) = j

Œ,‹

‹

bŒ,‹ · jŒ (k0r) · PŒ (cos ˜ ) · cos(‹œ)

(4b)

with spherical Bessel functions jŒ (k0 r). The boundary condition of zero azimuthal particle velocity at the flanks of the cut-out sector leads to the conditions: vœ −−−−−−→ 0 œ→±œ0

⇒

!

sin(‹œ0 ) = 0

⇒

‹œ0 = n;

n = 0, 1, 2, . . . .

(5a)

A restriction of œ0 to integer fractions of , œ0 = /N; N = 2, 3, 4, . . .; makes also ‹ an integer: ‹ = n · N.

(5b)

The modes in (II) are symmetrical or anti-symmetrical in ˜ relative to ˜ = /2 if Œ + ‹ is even or odd, respectively. This requires an integer Œ for integer values of ‹.

O.8.2

Orthogonality of Modes

The mode terms of the fields in (I) and (II) are orthogonal in both directions œ and ˜ . This follows in œ direction for the zone (I) from: 1 sin ((m + m )) sin ((m − m )) 1 + cos(mœ) · cos(m œ) dœ = 2 (m + m ) (m − m ) 0 (6) ⎧ ⎨1; m = m = 0 ƒm,m = = 1/2; m = m = 0 ⎩ ƒm 0; m = m with Kronecker symbols ƒn,m = 0; n = m; ƒn,m = 1; n = m; and Heaviside symbols ƒn=0 = 1; ƒn=0 = 2. ‹ œ0 = n ; n, n = 0, 1, 2, . . . (see O.8.(5a)): 0 1 œ0 sin (‹ − ‹ )œ0 1 sin (‹ + ‹ )œ0 1 cos(‹œ) · cos(‹ œ) dœ = + œ0 2 (‹ + ‹ )œ0 (‹ − ‹ )œ0 0 ƒn,n 1 sin((n + n )) sin((n − n )) + = . = 2 (n + n ) (n − n ) ƒn

In the zone (II), with ‹œ0 = n;

(7)

O

1096

Analytical and Numerical Methods in Acoustics ‹

‹

In ˜ direction, with PŒ (cos ˜ ) = PŒ (x), and x = cos ˜ , exists orthogonality in ˜ direction for integer orders ‹, Œ → m, n only when the polar orders m, m agree; this follows from the relation: 1

‹

‹

PŒ (x) · PŒ (x) dx = −1

0; Œ = Œ ‹ = ƒŒ,Œ NŒ ; ‹ NŒ ; Œ = Œ

Œ, Œ , ‹ = integer ,

(8)

‹

which defines the mode norms NŒ (evaluation see below).

O.8.3

Field Matching

Matching of the fields in (I) and (II) of sound pressure and radial velocity at r = r0 to each other and to the excitation on the “rest of the sphere” implies the two boundary conditions: p(II) (r0 , ˜ , œ)

!

= p(I) (r0 , ˜ , œ) !

Z0 vr(I) (r0 , ˜ , œ) =

in ˜ = (0, ) & œ = (−œ0 , +œ0 ),

Z0 vr(II) (r0, ˜ , œ)

in ˜ = (0, ) & œ = (−œ0 , +œ0 )

Z0 ver (r0 , ˜ , œ)

in rest of the sphere

.

(9) (10)

Apply in O.8.(9)the orthogonality in (II),and in O.8.(10)the orthogonality in (I); then the ranges of orthogonality will agree with the ranges of definition of the relevant boundary condition. For doing that multiply on both sides of a boundary condition with a mode function of the respective co-ordinate for arbitrary fix values Œ, ‹, or n, m, respectively, and then integrate over the range of orthogonality in œ, ˜ , i. e., in O.8.(9) over the surface of the sector, and in O.8.(10) over the sphere (or, because of the œ-symmetry over 0 ≤ œ ≤ ). So one will get two linear inhomogeneous systems of equations for the mode amplitudes an,m , bŒ,‹ . p-matching in the sector: Apply the sound pressure formulations (at r = r0 ): ‹ bŒ,‹ · jŒ (k0 r0) · PŒ (cos ˜ ) · cos(‹œ) , p(II) (r0 , œ, ˜ ) =

(11)

Œ,‹

p(I) (r0, ˜ , œ) =

m an,m · h(2) n (k0 r0 ) · Pn (cos ˜ ) · cos(mœ)

(12)

n,m≥0

in O.8.(9) and perform on both sides of that equation the integrals: 1 œ0

œ0

1 dœ

0

. . . · Pˆ‰ (x) · cos(ˆœ) dx

(13)

−1

with ˆ, ‰ from the range of values of ‹, Œ. Using the definitions of the mode coupling integrals œ0 1 1 m,ˆ ˆ Im,ˆ = cos(mœ) · cos(ˆœ) dœ , Kn,‰ = Pm (14, 15) n (x) · P‰ (x) dx , œ0 0

−1


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one gets from the p-condition the linear, homogeneous system of equations for the combined vector of mode amplitudes {b‰,ˆ , an,m }: Nˆ‰ m,ˆ · b‰,ˆ · j‰ (k0 r0 ) = an,m · h(2) n (k0 r0 ) · Kn,‰ · Im,ˆ . ƒˆ n,m≥0

(16)

vr -matching on the whole sphere: Apply in O.8.(10) the formulations of the radial particle velocities: Z0 vr(I) (r0 , ˜ , œ) = j

m an,m · h(2) n (k0 r0 ) · Pn (cos ˜ ) · cos(mœ),

(17)

n,m≥0

Z0 vr(II) (r0 , œ, ˜ ) = j

Œ,‹

‹

bŒ,‹ · jŒ (k0r0 ) · PŒ (cos ˜ ) · cos(‹œ) ,

(18)

and of the exciting mode: m0 Z0 ver (r0 , ˜ , œ) = jPe · h(2) n0 (k0 r0 ) · Pn0 (cos ˜ ) · cos(m0 œ) 0 : = Z0 V · Pm n0 (cos ˜ ) · cos(m0 œ).

(19)

Perform on both sides of O.8.(10) (with k, i = integers from the range of n, m) the integration: 1

1 dœ

0

. . . · Pik (x) · cos(iœ) dx.

(20)

−1

With the definitions of the mode coupling integrals

Ji,‹

1 =

Qi,m0

œ0 cos(iœ) · cos(‹œ) dœ = 0

1 =

œ0 Ii,‹ ,

(21)

cos(iœ) · cos(m0 œ) dœ ,

(22)

œ0

one gets the linear inhomogeneous system of equations for the {b‰,ˆ , an,m }: j

Nik (2) i,‹ 0 · hk (k0 r0 ) · ak,i = j bŒ,‹ · jŒ (k0 r0 ) · Kk,Œ · Ji,‹ + Z0 V · Qi,m0 Ki,m k,n0 ƒi Œ,‹

(23)

0 (The prime at Ki,m k,n0 shall recall that n0 , m0 in general are not in the range of ‰, ˆ). Both systems of equations O.8.(16), O.8.(23) can be solved for the mode amplitudes an,m , bŒ,‹ .

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O.8.4


Mode Coupling Integrals and Mode Norms

In O.8.(14) and O.8.(21) with ‹ = † · N; i, † = 0, 1, 2, . . .: Ii,‹

œ0 1 = cos(iœ) · cos(‹œ) dœ = Ji,‹ œ0 œ0 0 1 sin((i + †N)œ0 ) sin((i − †N)œ0 ) = + 2 (i + †N)œ0 (i − †N)œ0 sin(2†Nœ0 ) sin(2†) 1 1 1 1+ 1+ = . = −−−−−−→ 1; − −−−−−→ i=†N=0 i=†N=0 2 2†Nœ0 2 2† 2

(24)

In O.8.(22) with i, m = 0, 1, 2, . . .: Qi,m

1 =

cos(iœ) · cos(mœ) dœ œ0

1 sin((i − m)) sin((i + m)) œ0 sin((i − m)œ0 ) sin((i + m)œ0 ) + − + 2 (i − m) (i + m) (i − m)œ0 (i + m)œ0 ⎧ œ0 (25) ⎪ ⎪1 − ; i = m = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 œ0 sin(2mœ0 ) ; i = m = 0 − 1+ . = 2 2 2mœ0 ⎪ ⎪ ⎪ ⎪ ⎪ œ ⎪ sin((i − m)œ0 ) sin((i + m)œ0 ) ⎪ ⎩ − 0 ; i = m + 2 (i − m)œ0 (i + m)œ0

=

In O.8.(8) with i, k = 0, 1, 2, . . .: Nik

1 =

2 Pik (x) dx =

−1

2(k + i)! . (2k + 1)(k − i)!

(26)

In O.8.(15) with k, i, Œ, ‹ = integer: i,‹ Kk,Œ

1 =

‹

Pik (x) · PŒ (x) dx.

(27)

−1

This integral can be evaluated with the help of Eq. O.7.(38) in

O.8.5

> Sect. O.7.4.

Reduction of the Systems of Equations

Eliminate the bŒ,‹ from O.8.(16): bŒ,‹ =

‹ NŒ

ƒ‹

· jŒ (k0r0 ) n,m≥0

m,‹

an,m · h(2) n (k0 r0 ) · Kn,Œ · Im,‹

(28)


and insert in O.8.(23), leading to the system of equations for the an,m : 0 Ni −ƒm,i · ƒn,k · k · h(2) k (k0 r0 ) ƒi n,m≥0 ⎤ ƒ · j (k r ) œ0 ‹ Œ 0 0 m,‹ i,‹ · Im,‹ · Ii,‹ · Kn,Œ · Kk,Œ ⎦ · an,m + h(2) (k0r0 ) ‹ n N · j (k r ) Œ 0 0 Œ Œ,‹ 0 = jZ0 V · Qi,m0 Ki,m k,n0 ;

k ∈ {n};

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(29)

i ∈ {m} ≤ k.

With Pe instead of Z0 V (from O.8.(3)) as reference amplitude, O.8.(29) may be written as: ƒm,i · ƒn,k − ƒi n≥0,m≤n ⎤ ƒ‹ · jŒ (k0 r0 ) (k r ) œ0 h(2) 0 0 m,‹ i,‹ n · Im,‹ · Ii,‹ · Kn,Œ · Kk,Œ ⎦ · an,m + ‹ (30) Ni · h(2) (k r ) N · j (k r ) k

= −Pe ·

k

0 0

h(2) n0 (k0 r0 ) h(2) k (k0 r0 )

Œ,‹≤Œ

· Qi,m0

Œ

0 Ki,m k,n0

Nik

Œ

0 0

;

k ∈ {n};

i ∈ {m} ≤ k.

This is a two-dimensional system of linear, inhomogeneous systems of equations for the an,m ; in each subsystem i, k have fix values 0, 1, 2, . . .. Through the variations n, k = 0, 1, 2, . . ., nhi each equation contains the unknown an,m , because of the requirement i ≤ k, m ≤ n, in an arrangement: an,m = {a0,0 }, {a1,0 , a1,1 }, {a2,0 , a2,1, a2,2 }, {a3,0 , a3,1 , a3,2, a3,3 , . . . . nhi

(1 + kk) =

kk=0

(nhi + 1)(nhi + 2) lines of equations will arise by the variation of k, i ≤ k. 2

In total, there will arise a system with a square matrix having this number as side length. The upper limit of the ‹, ˆ = 0, 1 · N, 2 · N, . . ., †hi · N may be selected independently.

O.8.6

Numerical Examples

! ! The numerical examples will display !p(œ)/Pe ! on œ-orbits (as full curves, represented by the radial distance from the origin) for r, ˜ = const. The diagrams also indicate as dashed straight lines the flancs of the sector at ±œ0 , and as dashed curves the sound pressure which a full sphere with the vibration pattern of the n0 ,m0 mode would generate in the field point P = (r, ˜ , œ). If r = r0 the sound pressure orbit in zone (II) is shown with short dashes.

1100

O


Hemisphere, excited with constant radial velocity, n0 = 0; m0 = 0. k0r0 = 8; N = 2; œ0 = 90◦ ; k0r = 8; ˜ = 90◦ ; œ = 6◦ ; n0 = 0; m0 = 0; nhi = 6; mhi = 6

Sphere with sector 2œ0 = 90◦ , excited with constant radial velocity, n0 = 0; m0 = 0; œorbit at sphere radius r = r0. k0r0 = 8; N = 4; œ0 = 45◦ ; k0r = 8; ˜ = 90◦ ; œ = 6◦ ; n0 = 0; m0 = 0; nhi = 12; mhi = 3


O

1101

Sphere with sector 2œ0 = 90◦ , excited with constant radial velocity, n0 = 0; m0 = 0; œorbit at double sphere radius r = 2r0. k0r0 = 8; N = 4; œ0 = 45◦ ; k0r = 16; ˜ = 90◦ ; œ = 6◦ ; n0 = 0; m0 = 0; nhi = 12; mhi = 3

Sphere with sector 2œ0 = 90◦ , excited with mode pattern n0 = 2; m0 = 2. k0 r0 = 4; N = 4; œ0 = 45◦ ; k0 r = 4; ˜ = 90◦ ; œ = 6◦ ; n0 = 2; m0 = 2; nhi = 12; mhi = 3

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Schwarz, H.R.: Methode der finiten Elemente, Teubner, Stuttgart (1980) Seybert, A.F., Cheng C.Y.R.: Applications of the boundary element method to acoustic cavity response and muffler analysis. J. Vib. Acoust. Stress Reliab. Design 109, 15–21 (1987) Seybert, A.F., Cheng, C.Y.R., Wu, T.W.: The solution of coupled interior/exterior acoustic problems using the boundary element method. J. Acoust. Soc. Amer. 88, 1612–1618 (1990) Seybert, A.F., Soenarko, B.: Radiation and scattering of acoustic waves from bodies of arbitrary shape in a three-dimensional half space. ASME Trans.J.Vib.Acoust.Stress Reliab.Design.110, 112– 117 (1988) Seybert, A.F., Soenarko B., Rizzo, F.J., Shippy, D.J.: An advanced computational method for radiation and scattering of acoustic waves in three dimensions. J. Acoust. Soc. Amer. 77, 362–368 (1985) Seybert, A.F., Soenarko B., Rizzo, F.J., Shippy, D.J.: A special integral equation formulation for acoustic radiation and scattering for axisymmetric bodies and boundary conditions. J. Acoust. Soc. Amer. 80, 1241–1247 (1986) Seybert, A.F., Wu, T.W.: Modified Helmholtz integral equation for bodies sitting on an infinite plane. J. Acoust. Soc. Amer. 85, 19–23 (1989)

Section O.6: Bayliss, A., Gunzburger, M., Turkel, E.: Boundary conditions for the numerical solution of elliptic equations in exterior regions. SIAM J.Appl.Math. 42, 430–451 (1982) Becker, P., Waller, H.: Vergleich der Methoden der Finiten Elemente und der Boundary-Elemente bei der numerischen Berechnung von Schallfeldern. Acustica 60, 21–33 (1986) Craggs, A.: The use of simple three-dimensional acoustic finite elements for determining the natural modes and frequencies of complex shaped enclosures. J. Sound Vib. 23, 331–339 (1972) Craggs, A., Stead, G.: Sound transmission between enclosures – a study using plate and acoustic finite elements. Acustica 35, 89–98 (1976) Everstine, G.C., Henderson, F.M.: Coupled finite element/boundary element approach for fluid– structure interaction.J.Acoust.Soc.Amer.87,1938– 1947 (1990) Gan, H., Levin, P.L., Ludwig, R.: Finite element formulation of acoustic scattering phenomena with absorbing boundary condition in the frequency domain. J. Acoust. Soc. Amer. 94, 1651–1662 (1993) Gladwell, G.M.L., Zimmermann, G.: On energy and complementary energy formulations of acoustic and structural vibration problems. J. Sound Vib. 3, 233–241 (1966)

Skudrzyk, E.: The foundations of acoustics. Springer, Wien, New York (1971)

Gockel, M.A. (ed.): Handbook for dynamic analysis. Ch. 7.3, (MSC/NASTRAN) Los Angeles: The Mac Neal-Schwendler Corporation (1983)

Smirnov, W.I.: Lehrgang der Höheren Mathematik, Teil IV, VEB Deutscher Verlag der Wissenschaften, Berlin (1977)

Hunt, T.J., Knittel, M.R., Barach, D.: Finite element approach to acoustic radiation from elastic structures. J. Acoust. Soc. Amer. 55, 269–280 (1974)

Stummel, F., and Hainer, K.: Praktische Mathematik, Teubner, Stuttgart (1971)

Kirsch,A., Monk, P.: Convergence analysis of a coupled finite element and spectral method in acoustic scattering. IMA J. Numer. Anal. 9 425–447 (1990)

Stupfel,B.,Lavie,A.,Decarpigny,J.N.: Combined integral equation formulation and null-field method for the exterior acoustic problem. J. Acoust. Soc. Amer. 83, 927–941 (1988) Ursell, F.: On the exterior problems of acoustics. Proc. Camb. Phil. Soc. 74, 117–125 (1973) Ursell, F.: On the exterior problems of acoustics: II. Proc. Camb. Phil. Soc. 84, 545–548 (1978)

Masmoudi, M.: Numerical solution for exterior problems. Numer. Math. 51, 87–101 (1987) Munjal, M.L.: Acoustics of ducts and mufflers New York, John Wiley (1987) Nefske, D.J., Wolf, J.A. JR, Howell, L.J.: Structural– acoustic finite element analysis of the automobile passenger compartment: a review of current practice. J. Sound Vib. 80, 247–266 (1982)


Petyt, M., Lea, J., Koopmann, G.H.: A finite element method for determining the acoustic modes of irregular shaped cavities. J. Sound Vib. 45, 495–502 (1976) Petyt, M.: Finite element techniques for structural vibration and acoustics, Chap. 15, 16 in: Noise and vibration, (ed. R.G. White, J.G. Walker) New York, John Wiley (1982) Petyt, M.: Finite element techniques for acoustics, Ch. 2 In: Theoretical acoustics and numerical techniques, (ed. P. Filippi) Wien, New York, Springer (1983) Richards, T.L., Jha, S.K.: A simplified finite element method for studying acoustic characteristics inside a car cavity. J.Sound Vib. 63, 61–72 (1979) Schwarz, H.R.: Methode der finiten Elemente. Stuttgart, Teubner (1980) Schulze Hobbeling, H.: Berechnung komplexer Absorptions-/ Reflexions-Schalldämpfer mit Hilfe der Finite-Element-Methode. Acustica 67, 275–283 (1989) Shuku, T., Ishihara, K.: The analysis of the acoustics field in irregularly shaped rooms by the finite element method. J.Sound Vib. 29, 67–76 (1973) Smith, R.R., Hunt, J.T., Barach, T.: Finite element analysis of acoustically radiating structures with applications to sonar transducers. J. Acoust. Soc. Amer. 54, 1277–1288 (1973)

O

1107

Soize C.: Reduced models in the medium frequency range for general dissipative structural-dynamics systems. Eur. J. Mech. A/Solids 17, 657–685 (1998) Soize C.: Reduced models in the mediumfrequency range for general external structural– acoustic systems. J. Acoust. Soc. Amer. 103, 3393– 3406 (1998) Soize C.: Reduced models for structures in the medium-frequency range coupled with internal acoustic cavities. J. Acoust. Soc. Amer. 106, 3362– 3374 (1999) Zimmer, H., Ochmann, M., Holzheuer, C.: A finite element approach combined with analytical wave functions for acoustic radiation from elastic structures. Proc. 17th Int. Congress on Acoustics (ICA), Rome, Italy (2001), to appear

Sections O.7 & O.8: Mechel,F.P.: Das Orangen-Modell.Acta Acustica 90, 564–572 (2004) Mechel, F.P.: The Cat’s Eye Model. Acta Acustica 91, 653–660 (2005)

P Variational Principles in Acoustics with A. Cummings

Introduction In this chapter, some applications of variational principles in acoustics are discussed and several examples are given, mainly in duct acoustics. This subject area is not necessarily restrictive, since the reader will be able to see how, by extension, the ideas may be applied to other types of problems. In > Sect. B.11 of this book, Hamilton’s Principle is described. In its application to particles, the time average of the Lagrangian of a system, L = Ekin − Epot , is minimised and so its first variation is equated to zero, viz. ƒ L = 0 if there is no external work input. In spatially distributed systems, it is the space-time average of the Lagrange density that is minimised. Hamilton’s Principle gives rise to Lagrange’s equations, otherwise known as the Lagrange-Euler Eqs. or the Euler Eqs. (see, for example, chapter 3 of the book by Morse and Feshbach (1953)). One may apply Hamilton’s Principle to sound waves in the absence of dissipation. Here, the Lagrange density is given by: = (0 /2) |u|2 − (‰/2)p2, where 0 is fluid density, u is acoustic particle velocity, p is sound pressure and ‰ is isentropic fluid compressibility, 1/0c20 , where c0 is the sound speed. To proceed on the basis of the classical Lagrange equations, one must express both p and u in terms of the velocity potential “ , where p = 0 ∂“/∂t, u = −∇“ (see chapter 6 of the book by Morse and Ing˚ard (1968)). Substitution of into the appropriate Lagrange equation then yields the acoustic wave equation: ∇ 2 “ − (1/c20 )∂ 2 “/∂t2 = 0 2

∇ p

− (1/c20 )∂ 2 p/∂t2

or, in terms of sound pressure,

= 0.

Hamilton’s Principle is not in fact restricted to conservative systems, and may also include a potential related to dissipative forces (see, for example, the book by Achenbach (1973)). Hamilton’s Principle and Lagrange’s Eqs.will yield the governing differential equation(s) of a physical system, but not their solution.Variational methods can be employed to find approximate solutions, by the use of trial functions. A trial function is one that contains a number of arbitrary coefficients, which can be altered to change the shape of the function. Values of these coefficients can be found by substituting the trial function into a functional (such as the space-time average of the Lagrange density), and then finding a stationary value of this with respect to each of these coefficients in turn.

1110

P

Variational Principles in Acoustics

One can employ variational methods to solve eigenvalue problems too. Such methods will be illustrated in the various examples given in this chapter. Rather than using Lagrange’s Eqs. as the starting point of the analysis, an alternative technique is to find a variational principle such that the Euler Eqs. are the governing differential Eqs. of the problem and as many as possible of the prevailing physical boundary conditions. This approach is adopted here in the case of simple harmonic time dependence. The comments of Morse and Feshbach (1953) (see p. 1107), on variational methods vis-àvis perturbation techniques are worth repeating,“the variational method. . . permits the exploitation of any information bearing on the problem such as might be available from purely intuitional considerations”. There is considerable versatility in the application of variational techniques in engineering acoustics particularly and, indeed, intuition can provide valuable information in the solution of problems, as will be seen here. It should be noted that the trial functions employed in each of the various examples described here could, in principle, be extended to a full finite element discretization. This would give greater numerical accuracy, but at the expense of considerably increased computational effort. In the present context, it is preferable to maintain a level of relative simplicity in the analysis.

P.1 Eigenfrequencies of a Rigid-Walled Cavity and Modal Cuton Frequencies of a Uniform Flat-Oval Duct with Zero Mean Fluid Flow The problem of finding the eigenfrequencies of a rigid-walled cavity will first be considered, together with the related problem of modal cut-on frequencies in a rigid-walled “flat-oval”duct,a problem of some importance in the acoustics of air-conditioning ducts. In Morse and Feshbach (1953) (see p. 1112) it is shown that, for the scalar Helmholtz equation, ∇ 2 • + k 2 • = 0 (• being a scalar field variable in a volume § satisfying homogeneous Dirichlet or Neumann boundary conditions on the bounding surface, and k being the wavenumber), a variational principle exists such that ƒk 2 = ƒ[−

•∇ 2•d§/

§

•2 d§] = 0.

(1)

§

By the use of Green’s theorem, it is shown, Morse and Feshbach (1953), that 2

k =

2

(∇•) d§/ §

•2 d§.

(2)

§

This expression may be applied to the acoustic eigenmodes of a rigid-walled cavity, in which case p replaces • and k02 (k0 = –/c0 being the acoustic wavenumber, where – is the radian frequency) replaces k 2. If the cavity is of irregular shape, then a trial function for the sound pressure amplitude distribution, p˜ (x) = p(x) + —†(x) (x being a position vector, — being a small parameter and † being an arbitrary function), may be used to


P

1111

replace the (presumably unknown) exact form of sound pressure amplitude p(x) in this expression. One then has: ˜k 2 = k 2 + O(—2 ) = ∇ p˜ · ∇ p˜ d§/ p˜ 2 d§ (3) 0 0 §

§

giving the approximate value of the acoustic wavenumber of a particular eigenmode. If a suitable function for p˜ can be found for a particular mode, this expression may be used to find the approximate eigenfrequency. It is also possible to apply this formula to a uniform rigid-walled duct of flat-oval crosssection, see Cummings and Chang “Noise Breakout from Flat-Oval Ducts” (1986).

θ

This has two opposite flat sides (width 2a) and two opposite semi-circular sides (diameter 2b) as depicted. Only the (1,0),(0,1) and (1,1) modes will be discussed here,since higher modes than these present problems in the choice of trial functions. If the mode functions are represented as pi = Xi (x)Pi (y, z) exp(j–t), then the cut-on frequency for the i-th mode can be found from the acoustic wavenumber k˜ i corresponding to the trial function P˜ i for the crosssectional sound pressure pattern, given by: k˜ i2 = ∇t P˜ i · ∇t P˜ i dR/ (4) P˜ 2i dR , R

R

where R = R1 +R2 +R3 and the subscript“t” on ∇ signifies a gradient in two dimensions, on the duct cross-section. Suitably continuous trial functions for these modal pressure patterns, satisfying the rigid-wall boundary condition, were found in Cummings and Chang, “Sound Propagation in a Flat-Oval Waveguide” (1986) from intuition and experience (it was assumed that the nodal lines were straight, and this assumption was verified by experiment). These trial functions are as follows: ⎧ −A on R1 ⎪ ⎪ ⎪ ⎨ (5a) P˜ 10 = A sin(y/2a) on R2 , ⎪ ⎪ ⎪ on R3 ⎩A P˜ 01

on R1, 3 A 2r/b − r2 /b2 cos ‡ = 2 2 A 2z/b − sgn(z) · z /b on R2

(5b)

1112

P˜ 11

P


⎧ on R1 −A 2r/b − r2 /b2 cos ‡ ⎪ ⎪ ⎨ = A 2z/b − sgn(z) · z2/b2 sin(y/2a) on R2 . , ⎪ ⎪ ⎩ on R3 A 2r/b − r2/b2 cos ‡

(5c)

The acoustic wavenumbers at cut-on for these three modes are determined by inserting these trial functions into equation (4) and evaluating the two integrals. The cut-on frequencies (˜fmn = k˜ mn c0 /2) are then given as follows:

c c 1 5/4 + 16a/3b 0 0 f˜10 = ; f˜01 = ; 4a 1 + b/2a 2 11b2/30 + 32ab/15 (6a–c)

2 5/4 + 4 b/15a + 8a/3b c 0 f˜11 = . 2 11b2 /30 + 16ab/15 Experimental data were taken on a flat-oval cavity, to find the cut-on frequencies for a duct with a = b = 50 mm, and comparisons were also made with predictions from a finite difference numerical scheme (Cummings and Chang, “Sound Propagation in a Flat-Oval Waveguide” (1986)). The comparisons between experiment and numerical prediction are summarised in the table below (the % figure referring to the numerical prediction accuracy as compared to the measured cut-on frequency). Mode numbers

Measured f˜mn

f˜mn from FD

f˜mn from equs.

(m,n)

(Hz)

method (Hz)

(6a–c) (Hz)

(1,0)

959.9

970 (+1 %)

1084 (+13 %)

(0,1)

1829.9

1845 (+0.8 %)

1859 (+1.5 %)

(1,1)

2206

2232 (+1.2 %)

2257 (+2.3 %)

It can be seen that the prediction accuracy of the variational formulae is only modest for the (1,0) mode, but is much better for the (0,1) and (1,1) modes. In the case of the (1,0) mode, the assumed constant sound pressure distributions in the half-cylindrical parts of the duct would not be a particularly accurate representation of the actual pattern. As one would expect, the predicted frequencies are always too high as is the case, for example, in the use of Rayleigh’s method in vibration analysis.

P.2 Sound Propagation in a Uniform Narrow Tube of Arbitrary Cross-Section with Zero Mean Fluid Flow See also

> Sects. J.1, J.2, J.3 for sound propagation

in flat or circular capillaries.

This second example is not, strictly speaking, an acoustics problem at all (since the thermodynamic processes involved are not isentropic) but relates to wave propagation in narrow tubes, a topic connected with the acoustics of porous media. The procedure here will be to write a functional that has the correct Euler Eqs. and then to proceed to find suitable trial functions (see Cummings (1993)).


P

1113

The geometry of the problem is as depicted; x is the axial co-ordinate and y is a position vector in the transverse plane. The tube is assumed to have a uniform cross-section and rigid heat-conducting walls, which are at a constant temperature. Subject to the usual boundary-layer approximations, appropriate forms of the linearised Navier-Stokes and thermal energy Eqs. may be written, for simple-harmonic time dependence, as: (∇t2 − j–/Œ)u = (−j kx /‹) p

;

(∇t2 − j –0 Cp /K)T = (−j –/K) p.

(1)

Here, Œ, ‹, Cp and K are the fluid kinematic viscosity, dynamic viscosity, specific heat at constant pressure and thermal conductivity respectively, kx is the axial wavenumber of the fluid wave, T is the temperature perturbation and p is the sound pressure. It will be noted that the above two Eqs. are isomorphic and also have identical boundary conditions, namely zero axial velocity and temperature perturbations at the duct wall. Stinson (1991) wrote both these Eqs. in the form: (∇t2 − j –/†)• = −j –/†,

(2)

where • ≡ (–0 /kx p)u, † ≡ Œ in the velocity equation and • ≡ (0 Cp /p)T, † ≡ Œ/ Pr in the temperature equation, Pr being the fluid Prandtl number. The axial wavenumber can be found in Stinson (1991) as: kx = (–/c0 ){[‚ − (‚ − 1)F(Œ/Pr)]/F(Œ)}1/2, (3) where F(†) is defined as • , viz. the average of • over R, the cross-section of the tube, and ‚ is the ratio of principal specific heats. The problem now is to find an approximate solution to this equation that satisfies the Dirichlet boundary condition • = 0 on C. A variational principle for this problem may be obtained as follows, if • is expressed as • = ¦ (y) exp[j (–t − kx x)]. A functional may be defined, see Cummings (1993), 1 [∇t ¦ · ∇t ¦ + (j –/†)¦ 2 − (j 2–/†)¦ ] dR. (4) ¥ = 2 R

By putting ƒ¥ = 0 and using Green’s formula, it is easily shown that the Euler Eqs. for ¥ are the governing differential equation (2) and the “natural” Neumann boundary condition ∇¦ ·n = 0,where n is the outward unit normal to the tube surface.This natural

P

1114


boundary condition is not, in fact, the aforementioned physical boundary condition ¦ = 0 on C, and this latter condition must be imposed as a “forced” boundary condition with the proviso that ƒ¦ = 0 on C. This is done by choosing a trial function ¦˜ such that ¦˜ = 0 and ƒ¦˜ = 0 on C. For the sake of simplicity, two trial functions will be used here, the first intended to apply in the low frequency limit where the velocity and temperature perturbations are quasi-steady, and the second at high frequencies where both viscous and thermal boundary layers are thin. Low frequencies :

The tube cross-section is divided, approximately, into triangles as shown above.At point O, both velocity and temperature perturbations are assumed to have their maximum value. At sufficiently low frequencies in a narrow circular tube, both temperature and velocity fields exhibit a parabolic radial distribution of the field variable (as can be seen by solving equation (2)), and so an obvious choice for a low frequency trial function is: ¦˜ = ¦0(1 − …2 ),

(5)

where … = s/Li . It can be noted that ƒ¦ = ∂ ¦ /∂¦0 = 1 − …2 , which is zero everywhere on C, thus satisfying the aforementioned requirement for the forced boundary condition. The location of O is obvious in certain cases (e.g. tubes with circular or square crosssection) but may be less so in other cases. It is suggested in Cummings (1993) that, in general, O be located at the centre of the largest possible inscribed circle within R. For a single triangular area element, the contribution to ¥ is: ¥i =

1 2 [¦ C/Li + (j –/†)¦02 Ai /3 − (j 2–/†)¦0Ai /2], 2 0

(6)

and therefore ¥ =

i

⎛ ⎞ 1 ¥i = ⎝¦02 dC/L + j –¦02 R/3† − j –¦0 R/†⎠ . 2

(7)

C

Taking ƒ¥ = 0 is equivalent to putting ∂¥ /∂¦0 = 0, and this gives: ⎛ ⎞ ¦0 = (j –R/2†)/⎝ dC/L + j –R/3† ⎠,

(8)

C

and

⎛ ⎞ F(†) = ¦0/2 = (j –R/4†)/ ⎝ dC/L + j –R/3†⎠ . C

(9a)


P

1115

The integral over C can readily be evaluated in most cases of interest and an expression for kx found from eq. (3). For a regular polygon F(†) = (j – rh2 /4†) (2 + j – rh2 /3†) (9b) with rh = hydraulic radius ( = 2× area/perimeter). It can be assumed that the low frequency approximation is valid at frequencies where both viscous and thermal boundary layer thicknesses are greater than the largest value of L (Lmax , say). By using expressions for these√boundary layer √ thicknesses (see Morse and Ing˚ard (1968), p. 286), respectively ƒv = 2Œ/–, ƒt = 2Œ/– Pr, one may then express the upper limiting frequency for the low frequency model as: f1 = min(Œ/L2max , Œ/L2max Pr).

(10)

High frequencies:

At high frequencies, one may assume that the field variables have an approximately constant value in the central part of the tube, and that a boundary layer exists between this region and the wall. The most basic approximation for the boundary layer profile is linear, as shown above. Here, ƒ is the boundary layer thickness (viscous or thermal, as the case may be). Now the contribution to ¥ from a single element is: 1 ¥i = [¦02 Ai (2—i − —2i )/ƒ2 + (j –/†)¦02Ai (1 − 4—i /3 + —2i /2) 2 − (j 2–/†)¦0Ai (1 − —i + —2i /3)],

(11)

where —i = ƒ/Li , and so: ¥ =

1 2 [¦ I1 /ƒ2 + (j –/†)¦02 I2 − (j 2–/†)¦0I3 ], 2 0

where I1 = (ƒ − ƒ2/2L) dC; C

I3 =

I2 =

(12)

(L/2 − 2ƒ/3 + ƒ2 /4L) dC;

C

(13)

2

(L/2 − ƒ/2 + ƒ /6L) dC. C

Putting ƒ¥ = 0 now gives: ¦0 = (j –/†)I3 /[I1 /ƒ2 + (j –/†)I2 ];

F(†) = (j –/†R)I23 [I1 /ƒ2 + (j –/†)I2].

(14)

1116

P


In the case of a regular polygon, one may define — = ƒ/L = ƒ/rh and utilise the fact that ƒ = 2†/– to write: (15) F(†) = 2j (1 − — + —2 /3)2/[(2— − —2 ) + 2j (1 − 4—/3 + —2 /2)].

(16)

This expression should be valid at frequencies above a limit: f2 = max(Œ/L2min , Œ/L2min Pr). A circular section tube: (see also

(17) > Sect. J.3)

A circle is the limiting case of a regular polygon with an infinite number of sides. Accordingly, at low frequencies, the axial wavenumber here is immediately found by putting rh = a, the tube radius, in (9) and utilising (3). At high frequencies, kx is found by putting — = ƒ/a in (16) (with ƒ expressed in terms of † from (15)) and using (3). The predictions for both the real and imaginary parts of kx are in close agreement with the exact solution, given in Stinson (1991) or > Sect. J.3. The region f1 < f < f2 , in which neither the low frequency nor high frequency approximation is valid, is fairly narrow in most cases of practical interest, since f2 /f1 = Pr. It is easy to connect the two curves graphically. A parallel slit : (see also

> Sect. J.1)

Neither of the trial functions above is appropriate in the case of the slit, and equivalent trial functions in one dimension only – with a parabolic profile and a linear boundary layer profile respectively – may be employed in this case. A process analogous to that above yields: F(†) = (j –a2 /12†)/(1 + j –a2 /10†)

(18)

at low frequencies (where a is the width of the slit) and F(†) = j (1 − —)2 /[— + j (1 − 4—/3)]

(19)

(with — = ƒ/a) at high frequencies. The low frequency formula is in excellent agreement with the exact solution (Stinson (1991) or > Sect. J.1) and the high frequency formula yields good accuracy in this comparison, though this degenerates slightly as f2 is approached from above. Other geometries: In Cummings (1993), tubes with equilateral triangular, square, rectangular, hexagonal and semi-circular sections are examined and in all cases, the method described above yields predictions that are in at least reasonable agreement with other reported exact or numerical solutions. The detailed formulae for each case can readily by obtained from the above expressions.


P

1117

P.3 Sound Propagation in a Uniform, Rigid-Walled, Duct of Arbitrary Cross-Section with a Bulk-Reacting Lining and no Mean Fluid Flow: Low Frequency Approximation See also

> Ch. J

for circular and rectangular ducts.

Consider the above problem,in which a rigid-walled duct of uniform but arbitrary crosssectional geometry has an internal lining of “bulk-reacting” sound absorbent (initially considered to be isotropic), describable as an equivalent fluid, and characterised by a complex characteristic impedance and acoustic wavenumber.The duct contains no mean fluid flow. It is required to find an approximate expression for the axial wavenumber of the fundamental “coupled mode” in this duct, such that the axial wavenumbers for the sound fields in R1 and R2 (the central passage and lining respectively) are identical. Astley (1990) has (in a more general formulation) derived a variational principle with a functional 1 2 2 (∇t P · ∇t P − ‰ P )dR1 + (0 /2a ) (∇t P · ∇t P − ‰a2 P2 )dR2 , (1) ¥ = 2 R1

R2

where the sound pressure in the fundamental mode in R1 and R2 is expressed in the form p = P(y) exp j (–t − kx x) , and ‰2 = k02 − kx2, ‰a2 = ka2 − kx2 , ka being the (complex) effective acoustic wavenumber in the absorbent. The Euler Eqs. of this functional are the Helmholtz Eqs. in R1 and R2 (viz., (∇t2 + ‰2)P = 0 and (∇t2 + ‰a2)P = 0), together with the conditions of continuity of normal particle displacement on C1 and zero particle displacement on C2 . Note that continuity of sound pressure on C1 is not a natural boundary condition, and must therefore be satisfied by the trial function; the imposition of a forced boundary condition is not necessary here. The simplest trial function for the sound pressure is just a constant, representing a plane wave, viz. P˜ = P0 (this satisfies the requirement of continuity of sound pressure on C1 ). This function would normally be a reasonable approximation to reality at low frequencies. Inserting this expression into (1) and taking ƒ¥ = 0 (which simply involves taking ∂¥ /∂P0 = 0) then yields an explicit dispersion relationship:

1 + “a k02 /0 ka2 k x = ka , (2) 1 + “a /0

1118

P


where a is the (complex) effective density of the absorbent and “ = R1 /R2. Clearly, this formula gives the correct limiting behaviour as “ → 0 and “ → ∞ (namely kx → ka and kx → k0 respectively). Cummings (1991) has reported a version of this formula that is valid for an anisotropic absorbent, with acoustic wavenumber kax in the axial direction and kay in any transverse direction (independent of direction in the transverse plane), and effective density ay in any transverse direction: 2 1 + “ay k02 /0 kay kx = kax , (3) 2 / k 2 1 + “ay kax 0 ay which, again, can be seen to give the correct limiting behaviour as “ → 0 and “ → ∞.

P.4 Sound Propagation in a Uniform, Rigid-Walled, Rectangular Flow Duct Containing an Anisotropic Bulk-Reacting Wall Lining or Baffles The duct is depicted below. In its simplest form, the lining geometry involves one layer of bulk absorbent (treated as an equivalent fluid), placed against one wall of the duct. The bulk acoustic properties of the absorbent are assumed to be different in the x and y directions. A uniform mean gas flow (Mach number M) is assumed to be present in the remaining part of the duct cross-section. To treat a baffle silencer, one would assume the baffle width to be 2a and the “airway” width to be 2h . The following analysis would then be representative of the fundamental coupled acoustic mode in this arrangement. For the sake of simplicity, the sound field will be assumed to be two-dimensional. The extension to three dimensions is trivial. Cummings (1992) has reported a variational statement of this problem. If it is assumed that the sound fields in R1 and R2 can be expressed in the form p = P(y1,2) exp j (–t − kx x) , a variational functional may be defined: 1 ¥= {∇t P · ∇t P − k02 [(1 − MK)2 − K2]P2 }dR1 2(1 − MK)2 R1 (1) 0 2 2 2 + [∇t P · ∇t P − k02 (1 − K2 /‚ax )‚ay P ]dR2 , 2ay R2

where K = kx /k0, ‚ax = kax /k0 and ‚ay = kay /k0. The Euler Eqs. of this functional are: the convected wave equation in R1 , the Helmholtz equation in the lining, the rigid-wall boundary condition on the duct walls and continuity of normal particle displacement


P

1119

on the interface between the lining and the airway. Suitable trial functions for the sound pressure may be written separately for regions R1 and R2 : P˜ 1 = A +

M

m=1

Bm sin[(2m − 1)y1 /2h] ;

P˜ 2 = A +

N

Cn sin[(2n − 1)y2/2a]. (2a, b)

n=1

It is desirable to use separate functions here for P˜ 1 and P˜ 2 since the normal gradient of the sound pressure is discontinuous at y1 = y2 = 0, and this discontinuity cannot be represented by a finite number of terms.Only odd integers are included in the arguments of the sine functions because it is necessary for the trial functions to satisfy the rigid-wall boundary condition at y1 = h, y2 = a. Note that the boundary condition of continuity of sound pressure at y1 = 0, y2 = 0 (not one of the natural boundary conditions) is satisfied by these trial functions. By truncating these summations at appropriate values of M and N, solutions of the desired accuracy may be achieved. An example with M = 3, N = 3 (i.e. a seven degree-of-freedom trial function) will be discussed in 2 ˜ detail here. Equations (2a,b) are inserted into (1), with ∇t P˜ · ∇t P˜ ≡ (dP/dy 1,2) , and the appropriate integrations are carried out. Next, putting ƒ¥ = 0 involves taking ∂¥ /∂A = 0, ∂¥ /∂Bi = 0, ∂¥ /∂Ci = 0 (with i = 1, . . . 3). This gives rise to a homogeneous system of simultaneous linear Eqs. in A, Bi , Ci : 2 2 A{[k0 h/(1 − MK)2][(1 − MK)2 − K2 ] + k0a(0 /ay )[(1 − K2/‚ax )‚ay ]} 2 2 2 + B1 {[k0 h/(1 − MK) ](2/)[(1 − MK) − K ]}

+ B2 {[k0 h/(1 − MK)2 ](2/3)[(1 − MK)2 − K2 ]} + B3 {[k0 h/(1 − MK)2 ](2/5)[(1 − MK)2 − K2 ]} 2 2 2 2 + C1 {k0 a(0 /ay )(2/)(1 − K2 /‚ax )‚ay } + C2 {k0 a(0 /ay )(2/3)(1 − K2 /‚ax )‚ay } 2 2 2 + C3 {k0 a(0 /ay )(2/5)(1 − K /‚ax )‚ay } = 0,

(3a)

A{−(4/)[(1 − MK)2 − K2]} + B1 {(/2k0h)2 − [(1 − MK)2 − K2 ]} + B2 {0} + B3 {0} (3b) + C1 {0} + C2{0} + C3 {0} = 0, A{−(4/3)[(1 − MK)2 − K2 ]} + B1 {0} + B2 {9(/2k0h)2 − [(1 − MK)2 − K2 ]} + B3 {0} + C1 {0} + C2 {0} + C3 {0} = 0,

(3c)

A{−(4/5)[(1 − MK)2 − K2 ]} + B1 {0} + B2 {0} + B3 {25(/2k0h)2 −[(1 − MK)2 − K2 ]} + C1{0} + C2{0} + C3{0} = 0,

(3d)

2 2 )‚ay } + B1 {0} + B2 {0} + B3 {0} + C1 {(/2k0a)2 A{−(4/)(1 − K2 /‚ax 2 2 2 − (1 − K /‚ax )‚ay } + C2 {0} + C3{0} = 0,

(3e)

2 2 A{−(4/3)(1 − K2 /‚ax )‚ay } + B1 {0} + B2 {0} + B3 {0} + C1{0} + C2 {9(/2k0a)2 2 2 2 − (1 − K /‚ax )‚ay } + C3{0} = 0,

(3f)

1120

P


2 2 A{−(4/5)(1 − K2 /‚ax )‚ay } + B1 {0} + B2 {0} + B3 {0} + C1{0} + C2 {0} 2 2 2 )‚ay } = 0. + C3 {25(/2k0 a) − (1 − K2 /‚ax

(3g)

Equations (3a–g) may be written in the form: [Aij ]( A B1

B2

B3

C1

C2

C3 )T = {0},

and a dispersion relation follows as: [Aij ] = 0.

(4a)

(4b)

The elements of the (7 × 7) square matrix in (4a, b) are the coefficients of A, B1 , . . . C3 in Eqs. (3a–g). The solution of (4b) for kx is readily accomplished by standard numerical techniques such as Newton’s method or Muller’s method, given a suitable starting value for kx . This can be done for the fundamental mode or higher modes of propagation, though the most accurate results will be obtained for the fundamental mode. The seven degree-of-freedom trial function yields accurate results for the fundamental mode, Cummings (1992), even up to frequencies of several kHz. Better accuracy could be obtained by taking more terms in the summations in (2a, b). Of course, there is an exact dispersion relationship for this problem, involving circular functions (see, e.g., Cummings (1976)), and therefore the Rayleigh-Ritz method described here is to be regarded as an alternative method rather than one to be used out of necessity in this particular case. The mode shape (e.g., from B1 /A, B2 /A, . . . and Eqs. (2a, b)) may readily be found from Eqs. (3a–g), once kx has been determined.

P.5 Sound Propagation in a Uniform, Rigid-Walled, Flow Duct of Arbitrary Cross-Section, with an Inhomogeneous, Anisotropic Bulk Lining This is a more difficult problem than that described in > Section P.4, and is one which does not, in general, give rise to an exact dispersion relationship. General formulation:


P

1121

The duct geometry is shown in the diagram. The duct is of uniform but arbitrary crosssection, as is the lining, which is assumed to behave as an equivalent fluid. The bulk acoustic properties of the lining are assumed different in the x direction and in any transverse direction, independent of direction, in the transverse y plane. The lining is divided into sections 1,2, . . . , N.Within each of these sections,the properties are assumed uniform, but may vary between sections. The outer duct wall is rigid and the central passage carries a uniform mean gas flow. As in > Sections P.3 and P.4, coupled mode solutionswill be sought for the sound field, which will be taken to have the form p = P(y) exp j (–t − kx x) , where the acoustic pressure amplitude is defined piecewise in the various parts of the cross-section. It will prove convenient to find a variational functional that will have, as its Euler equations, not only the governing wave Eqs. in R0 and Ri together with the rigid-wall boundary (I) condition on C(i) and continuity of normal particle displacement on C0 , but also continuity of normal particle displacement and sound pressure on Ci , since the last two boundary conditions are not readily satisfied by the trial function in what follows. Such a functional is (see Cummings (1995)): 1 ¥= {∇t P · ∇t P − k02 [(1 − MK)2 − K2 ]P2 } dR0 2(1 − MK)2 R0 N

0 2 2 2 + [∇t Pi · ∇t Pi − k02(1 − K2 /‚axi )‚ayi Pi ] dRi (1) 2ayi i=1 Ri N

+ 0 –2 i (Pi+1 − Pi ) dCi , i=1 C i

where (again) K = kx /k0 and the summations are over the N sub-regions Ri of absorbent, separated by the boundaries Ci . The last summation ensures that the aforementioned two boundary conditions on Ci are natural boundary conditions. Application to a duct with two cross-sectional lines of symmetry: One important application of this formulation is to dissipative vehicle exhaust silencers of oval cross-section. These almost invariably have a central circular-section gas flow passage and a cross-section with two lines of symmetry at right angles. The lining material can be not only anisotropic, but also inhomogeneous, in its bulk properties.

P

1122


For the fundamental acoustic mode, it is clear that only one-quarter of the duct crosssection need be considered, and the appropriate geometry is shown in the figure. The portion Cw of the boundary is effectively rigid, from considerations of symmetry. At fairly low frequencies, it should be reasonable to assume a purely radial variation is acoustic pressure in both R0 and in each segment of lining Ri (though without any oscillatory behaviour), together with a circumferential variation between the lining segments. Accordingly, the lining is segmented radially, as shown. The functional in (1) still applies, except that the second summation is taken to N − 1 rather than N, and a lower limit of 2 is thus imposed on N.The Euler Eqs.now include the rigid-wall boundary condition on Cw . A composite trial function for this geometry is appropriate, having the following form: P˜ 0 = a + b cos(r/2R)

in

R0 ,

P˜ i = a + ci sin(yi /2Yi ) in

Ri

(i = 1, . . ., N),

˜ i = di sin(si /2Ci)

Ci

(i = 1, . . ., N − 1).

on

(2a–c)

Here, a, b, ci and di are (complex) arbitrary constants, r, yi and si are co-ordinates (as shown) and R, Yi and Ci are the lengths shown in the diagram. Continuity of pressure on C(i) 0 (not a natural boundary condition) is satisfied by (2a,b). It is convenient to divide the quarter of the absorbent into N equal segments. The next step is to insert the trial functions (2a–c) into (1) and carry out the integrations. The area integrals over the sub-regions of absorbent as depicted above are not easily found, and so it will be assumed that these radial segments are equivalent to segments of circular annuli,having an inner radius R and an outer radius R+Yi,with the radial sound pressure distribution as specified in (2b). This greatly simplifies these area integrals. The integrals over Ci are readily found exactly for the geometry above. The variational functional may now be written: ¥ = a2 f1 + b2f2 + abf3 +

N

i=1

c2i gi + a

N

ci hi +

i=1

N−1

di ci+1 ui −

N−1

i=1

di ci vi ,

(3)

i=1

where f1 = −(/8)[k0 R/(1 − MK)]2[(1 − MK)2 − K2 ] − (/4N)

N

2 2 (0 /ayi )(1 − K2 /‚axi )‚ayi [k02 RYi + (k0 Yi )2 /2],

(4a)

i=1

f2 = [/4(1 − MK)2 ]{(2 /4)(1/4 + 1/2) − (k0 R)2 [(1 − MK)2 − K2 ](1/4 − 1/2)}, f3 = −[k0 R/(1 − MK)]2 [(1 − MK)2 − K2 ](1 − 2/), 2 2 )‚ayi [k02 RYi /2 gi = {(2 /4)(R/2Yi + 1/4 − 1/2) − (1 − K2 /‚axi

+ (k0 Yi )2 (1/4 + 1/2)]}(0 /4Nayi ),

(4b) (4c) (4d)


P

2 2 hi = −(1 − K2 /‚axi )‚ayi (k0Yi /)(2k0Yi / + k0R)(0 /Nayi ),

ui =

vi =

(4e)

[20 –2 Yi+1 /(Y2i+1 /C2i − 1)] cos(Ci /2Yi+1 ), Yi+1 = Ci 0 –2 Ci /2, [20–2 Yi /(Y2i /C2i − 1)] cos(Ci /2Yi ),

Yi+1 = Ci Yi = Ci

0 –2 Ci /2,

Yi = Ci

1123

,

.

(4f)

(4g)

Now ƒ¥ = 0 is equivalent to taking ∂¥ /∂a = 0, ∂¥ /∂b = 0 ; ∂¥ /∂ci = 0 (i = 1, . . . , N) ; ∂¥ /∂di = 0 (1 = 1, . . . , N − 1). This gives rise to a system of linear equations: 2af1 + bf3 +

N

ci hi = 0; 2bf2 +af3 = 0; 2c1 g1 +ah1 −d1 v1 = 0 ;

i=1

2ci gi +ahi +di−1 ui−1 − di vi = 0 ci+1 ui − ci vi = 0

(i = 2, . . . , N − 1); 2cN gN +ahN +dN−1 uN−1 = 0 ;

(5a–f)

(i = 1, . . . , N − 1).

There are 2N +1 of these Eqs. in the 2N +1 unknowns, a, b,. . . , and the system of Eqs. may be written: [Aij ] (a

b

c1 ...

d1 ...)

T

= {0},

(6a)

where Aij are the coefficients in (5a–f) and, as in be written with the determinant: [Aij ] = 0.

> Sect. P.5, a

dispersion relation may (6b)

This may be solved numerically for kx , as in the example in

> Sect. P.5.

From Eqs.(5a–f),expressions for the ratios of coefficients,b/a, ci /a and di /a,may easily be found, from which the mode shape may be determined via Eqs. (2a–c): ⎧ ⎫ ⎤ ⎡ N−1 i ⎨ ⎬

⎣ (vj /uj)⎦ hi+1 ; (7a–b) b/a = −f3 /2f2 ; c1 /a = (f32/2f2 − 2f1)/ h1 + ⎩ ⎭ i=1

ci /a = (vi−1 /ui−1 )ci−1 /a ,

(i = 2, . . . , N)

di /a = (2gi ci /a + hi + ui−1 di−1 /a)/vi , dN−1 /a = −(2gN cN /a + hN )/uN−1 .

;

j=1

d1 /a = (2g1c1 /a + h1 )/v1 ;

(i = 2, . . . , N − 2);

(7c–d) (7e–f)

Clearly, some degree of substitution is required in implementing these formulae, e.g. c1 /a in (7d) has first to be found from (7b). This method has been shown by Cummings (1995), to yield predictions that are in good agreement with measured data.Where mean flow is present, it may be necessary to account for the effects of mean gas flow,within the absorbent itself,on the bulk properties

1124

P


of the absorbent (see Cummings (1995)). This aspect of the problem is, however, beyond the scope of this chapter and in any case does not affect the details of the analysis. The effects of a perforated tube, separating the gas flow passage from the absorbent, may – if desired – also be incorporated in the formulation provided a suitable model for the perforate impedance is available. This feature is, however, omitted from the above formulation for the sake of simplicity. In the case of a circular duct with a uniform (isotropic or anisotropic) lining of constant thickness and a circular gas flow passage, the above formulation may be used, with N = 2. Otherwise it may be more convenient to treat this case separately. The method of > Sect. P.5 may be applied here, with the same functional and a similar trial function. The area integrals have a different form, of course, but the formulation is simple and straightforward. This approach has been shown in Cummings (1992) to yield excellent results, and it has the possible advantage over the exact solution that Bessel functions do not have to be computed.

P.6 Sound Propagation in a Uniform Duct of Arbitrary Cross-Section with one or more Plane Flexible Walls, an Isotropic Bulk Lining and a Uniform Mean Gas Flow The duct is depicted below. It is shown as having one flexible wall, consisting of a flat elastic plate, though the analysis that follows is equally valid for an arbitrary number of such walls. For simplicity, it will be assumed that there is just one flexible wall; the extension of the treatment to the case of multiple flexible walls is obvious. The other parts of the duct wall are rigid. There is a uniform gas flow in the central passage.

In this problem, there is wave motion not only in the fluid in the duct and the lining, but also in the plate forming the flexible wall. Coupled mode solutions are sought, and accordingly the axial wavenumbers in the central gas flow passage (R1 ), the lining (R2 ) and the plate (C3) are all identical. General formulation:

It will be assumed that the sound pressure can be represented as p = P(y) exp j (–t −kx x) and the outward plate displacement as u = U(s) exp j (–t − kx x) . Then the three governing differential Eqs. in R1 , R2 and the plate are, respectively: ∇t2 P + k02 [(1 − MK)2 − K2 ]P = 0; g[(d2 /ds2 − kx2 )2 U − kp4 U] = Pp (s),

∇t2 P + (ka2 − kx2)P = 0;

(1a–c)


P

1125

where Pp (s) is the transverse factor in the acoustic pressure difference (inside-outside) across the plate, forcing its motion (it is assumed the axial dependence of this pressure difference is the same as that of p and u ), g is the flexural rigidity of the plate and the plate wavenumber is kp = (m–2 /g)1/4 (m being the mass/unit area of the plate). Other notation here is as in > Section P.3. A variational statement of this problem has been made by Astley (1990), and a functional defined: 2 {(g/2)[(d2 U/ds2 )2 + 2kx2 (dU/ds)2 + (kx4 − kp4 )U2 ] − UPp }ds ¥ = 0 – 1 {∇t P· ∇t P − k02[(1 − MK)2 − K2 ]P2 }dR1 2(1 − MK)2 R1 0 + [∇t P · ∇t P − (ka2 − kx2)P2 ]dR2 . 2a C3

+

(2)

R2

The Euler Eqs. of this functional are obtained by taking variations of ¥ with respect to P and U: (1a, b) (provided P is continuous on C1 ), (1c) (with the constraint of zero displacement at the edges of the flexible wall), the rigid-wall boundary condition on C2 , equality of the normal plate displacement and the normal acoustic particle displacement in the internal sound field on C3 (provided that the normal gradient of sound pressure is allowed to vary freely at the outer surface of the flexible wall) and continuity of normal particle displacement on C1 (provided that the normal gradient of sound pressure is allowed to vary independently on C1 in R1 and R2 ). Low frequency approximation: As in > Section P.3, the trial function for the internal acoustic field embodies a uniform transverse sound pressure distribution in R1 and R2 , i.e. P(y) = P ≡ const., representative of the fundamental mode. Furthermore, one may write U(s) = Pu U∗ (s, kx ), where U∗ (s, kx ) is the solution of (1c), with a unit pressure on the right-hand side, subject to the prevailing boundary conditions, and Pu is defined by the foregoing equation. With this trial function, the functional may be written (after some manipulation): 1 ¥ = j k0 (kx ) S(PPu − P2 ) 2 "# % (3) # $ $2 0 K 1 2 2 2 2 [K − (ka /k0) ] , + k0 P R1 − 1 + R2 2 1 − MK a where (kx ) is the space-average (over C3 ) of the dimensionless admittance of the flexible wall, j –0 c0 U∗ (s, kx ), and S is the width of the flexible duct wall. This expression is now minimised with respect to P and Pu , and P is equated to Pu on the basis that the external radiation load on the flexible duct wall is negligibly small, to give the dispersion equation: "# % $2 # $ K 0 2 2 2 j k0 (kx ) S + k0 R1 [K − (ka /k0) ] = 0. − 1 + R2 (4) 1 − MK a

1126

P


This can be solved by an appropriate standard root-finding method. Certain special cases are of interest in Astley (1990), as follows. (i) A duct with rigid walls and mean gas flow: In this case, (kx ) = 0 and equation (4) becomes: % "# $2 # $ 0 K R1 − 1 + R2 [K2 − (ka /k0)2 ] = 0. 1 − MK a

(5)

(ii) A duct with rigid walls and no mean gas flow: The result here is identical to that in

> Section

P.3.

(iii) A central region of gas flow in a rigid-walled duct, surrounded by a stagnant region: An example of this is an enclosed gas jet in a duct, such as that which is formed down stream of an abrupt area expansion in a flow duct.Here,ka = k0 , a = 0 and (kx ) = 0. The dispersion equation now becomes: "# % $2 K R1 − 1 + R2 (K2 − 1) = 0. (6) 1 − MK (iv) A duct with a flexible wall, no mean flow and no lining: In this case, M = 0 and R2 = 0, and equation (4) yields: & kx = k0 1 − jS (kx ) /k0R1 ,

(7)

which is identical to the expression obtained by Cummings (1978). If the flexible wall is clamped along both edges, an exact solution of equation (1c) exists: ' A2 A3 A1 (kx ) = j –0 c0 sin(1 S) − [cos(1S) − 1] + sinh(2 S) 1 S 1 S 2 S (8) A4 1 + [cosh(2S) − 1] + , 2 S g(kx4 − kp4 ) & & where 1 = kp2 − kx2, 2 = kp2 + kx2 and A1 = {1 [1 + cos(1 S) − cosh(2 S) − cos(1 S) cosh(2S)] + 2 sin(1 S) sinh(2 S)}/g(kx4 − kp4 )[21 cos(1 S) cosh(2 S) − 21 + (21 /2 − 2) sin(1 S) sinh(2S)]; A2 = {1 sin(1 S)[1 − cosh(2 S)] + 2 sinh(2S)[1 − cos(1S)]}/ g(kx4 − kp4 )[21 cos(1S) cosh(2 S) − 21 + (21 /2 − 2) sin(1 S) sinh(2S)]; A3 = −A1 − 1/g(kx4 − kp4 );

A4 = −A2 1/2.

(9)


P

1127

Equation (7) has been shown in Cummings (1978) to give predictions of axial phase speed for the fundamental coupled structural/acoustic mode that are in excellent agreement with measured data, for a duct of square cross-section (and, therefore, effectively having four flexible walls, clamped along their edges, with no rigid walls). Transverse structural resonance effects in the wall are very prominent in the wall admittance expression (8) and, of course, in the axial wavenumber in the duct.

P.7 Sound Propagation in a Rectangular Section Duct with four Flexible Walls, an Anisotropic Bulk Lining and no Mean Gas Flow The duct geometry and co-ordinate systems are shown in the graph. The open central channel R0 is surrounded by layers of bulk absorbent, all of thickness t, placed against the four flexible walls. These layers are denoted R1 , . . . , R4 . The perimetral co-ordinates s are local to each of the four walls of the duct, C1 , . . . , C4, and C0 is the interface between the central channel and the lining.

′

′ ′ ′

A global co-ordinate system x, y, z is centred on the duct axis, and local co-ordinate systems x, y , z are used to define position in the four layers of absorbent. The bulk acoustic properties of the absorbent are different in the y and x, z directions, and the properties of all layers are identical. Coupled eigenmodes are sought, for the sound pressure in the duct and the outward wall displacement, having the form: p(x, y, z; t) = P(y, z) · exp j (–t − kx x) ; u(x, s; t) = U(s) · exp j (–t − kx x) .

(1a–b)

The acoustic wave Eqs.in R0 and in the absorbent (R1 , . . ., R4 ) yield Helmholtz equations, respectively having the forms: ∂ 2P ∂ 2 P + + (k02 − kx2 )P = 0 ∂y 2 ∂z2

;

) ( 2 k2 ∂ 2 P kay ∂ 2 P 2 2 ay + 2 + kay − kx 2 P = 0, ∂y 2 kax ∂z 2 kax

(2a–b)

P

1128


where kax is the acoustic wavenumber of the absorbent in the x, z plane (independent of direction) and kay is the acoustic wavenumber in the y direction. The equation of motion in the duct walls may be written in the form: * 4 + 2 d U 2d U 4 4 g − 2k + (k − k )U = P on C1 , . . ., C4 , (2c) x x p ds4 ds2 where kp is defined in

> Section P.6.

A variational statement for this problem is reported by Cummings and Astley (1995), with a functional: ⎛ 1 ⎝ ¥= [(∂P/∂y)2 + (∂P/∂z)2 + (kx2 − k02 )P2 ] dR0 2 R0

+

4

(0 /ay )

i=1

2 2 [(∂P/∂y )2 + (kay /kax )(∂P/∂z )2

(3)

Ri

2 2 2 + (kx2 kay /kax − kay )P2 ] dRi

+

4

i=1

0 –2

⎞

{g[(d2 U/ds2 )2 + 2kx2 (dU/ds)2 + (kx4 − kp4 )U2 ] − 2UP} ds⎠ .

Ci

The notation here is essentially that of > Sections P.4 and P.5. The Euler Eqs. of this functional are the same as those outlined in > Section P.6 (except for the rigid-wall boundary condition), with the additional feature of anisotropy in the absorbent. A Rayleigh-Ritz approximation for the coupled eigenproblem may be found by the use of trial functions, having (for example) polynomial form, for the sound pressures in the central passage and lining, and for the wall displacement in Cummings and Astley (1995). These trial functions – intended to represent the fundamental coupled mode – are, respectively: P˜ = f1 (y, z)P1 + f2 (y, z)P2 + f3(y, z)P3

in R0 ,

P˜ = g1 (y , z )P1 + g2(y , z )P2 + g3 (y , z )P3

(4a)

in R1 , . . ., R4

(where f1 = [1 − (y/a1)2 ][1 − (z/b1)2 ], a1 = a − t, b1 = b − t, 2

2

(4b)

f2 = 1 − [1 − (y/a1 ) ][1−(z/b1) ], f3 = 0, g1 = 0, g2 = 1 − (y /t), g3 = y /t), ˜ = h1 (s)U1 + h2 (s)U2 + h3 (s)‡ U

(4c)

(where h1 = 1 − 3(s/b)2 + 2(s/b)3, h2 = 0, h3 = −s[(s/b) − (s/b)2 ] for 0 ≤ s ≤ b on the right-hand vertical side of the duct depicted above, with equivalent expressions for the other parts of the walls). In these equations, the constants P1 , P2 , P3 have the dimensions of sound pressure and are identified as follows: P1 is the sound pressure amplitude on the axis of the central


P

1129

passage, P2 is the (constant) sound pressure amplitude at the liner/air interface and P3 is the (constant) sound pressure amplitude on the inner surface of the duct wall, according to the assumed form of the trial function for the sound field. In the vibration field, constants U1, U2 are the displacement amplitudes at the mid-points of two adjacent sides of the duct wall and constant ‡ is the amplitude of the angle of rotation of the corner (assumed to remain right-angled) between these sides. The insertion of Eqs. (4a–c) into (3) yields (after some manipulation) an expression for the functional which may conveniently be written in matrix form, Cummings and Astley (1995) ¥ =

1 T 1 U [A + kx2 B + kx4 C]U − PT TU + PT [E + kx2G]P, 2 2

(5)

where P and U are column vectors containing the acoustic coefficients P1 , P2 , P3 and the structural coefficients U1, U2, ‡ respectively, and A, B, C, T, E, G are 3 × 3 matrices, the elements of which are given by: Ajk =

4

2

0 – g

i=1 4

Bjk =

4

0 –2 g

+

2(dhj /ds)(dhk /ds) ds,

(6b)

0 –2 g

hj hk ds;

Tjk =

4

0 –2

i=1

Ci

gj hk ds,

(6c–d)

Ci

[(dfj /dy)(dfk /dy) + (dfj /dz)(dfk /dz) − k02fj fk ] dydz

Ejk = R0

(6a)

Ci

i=1

[(d2 hj /ds2 )(d2 hk /ds2 ) − kp4 hj hk ] ds,

Ci

i=1

Cjk =

4

(0 /ay )

i=1

[(∂gj /∂y )(∂gk /∂y )

(6e)

Ri

2 2 2 /kax )(∂gj /∂z )(∂gk /∂z ) − kay gj gk ] dy dz , + (kay

Gjk =

fj fk dydz +

4

R0

i=1

(0 /ay )

2 2 (kay /kax )gjgk dy dz .

(6f)

Ri

The integrals in these Eqs.are readily evaluated analytically for the trial functions chosen, but could otherwise be evaluated numerically, for example by Gaussian quadrature. As before, putting ƒ¥ = 0 involves minimising ¥ with respect to the acoustic variables P1 , P2 , P3 and the structural variables U1 , U2, ‡. The former process gives the relationship: [E + kx2 G]P − TU = 0,

(7a)

1130

P


and the latter yields: [A + kx2B + kx4C]U − TT P = 0.

(7b)

These two Eqs. constitute a coupled eigenvalue problem in kx , the coupling occurring via the matrix T. It is worth noting that, if T is removed from Eqs. (7a, b), two uncoupled eigenvalue problems result: [E + kx2 G]P = 0;

[A + kx2B + kx4 C]U = 0.

(8a, b)

The first of these relates to acoustic modes in a lined duct with rigid walls, and the second to structural modes in an elastic-walled duct in which the acoustic loading on the walls is neglected. Solution of the eigenproblem posed in Eqs. (7a, b) is not entirely straightforward, particularly since the coupled mode types are – broadly – divided into “acoustic” type modes, in which the power flow is predominantly in the fluid, and “structural” type modes, in which most of the power flow is in the elastic walls. Astley, who was responsible for the Rayleigh-Ritz formulation reported here, described, in Cummings and Astley (1995), a robust iterative method of solution for this problem, based on the foregoing arguments. Although a detailed description of this is not appropriate here, the interested reader is referred to Cummings and Astley (1995). The predictive accuracy of the method described here is surprisingly good (see Cummings and Astley (1995), considering the relative crudity of the acoustic and structural trial functions. Better accuracy could be obtained by the use of trial functions containing more degrees of freedom, but one should then also consider a full finite element discretization as an alternative.

References Achenbach, J.D.: Wave Propagation in Elastic Solids. North Holland, Amsterdam (1973) Astley, R.J.: Acoustical Modes in Lined Ducts with Flexible Walls: a Variational Approach. Proc. InterNoise 90, 575–578 (1990) Cummings, A.: Sound Attenuation in Ducts Lined on Two Opposite Sides with Porous Material, with Some Applications to Splitters. J. Sound Vibr. 49, 9–35 (1976) Cummings, A.: Low Frequency Acoustic Transmission through the Walls of Rectangular Ducts. J. Sound Vibr. 61, 327–345 (1978) Cummings, A.: Impedance Tube Measurements on Porous Media: the Effects of Air-Gaps around the Sample. J. Sound Vibr. 151, 63–75 (1991)

Cummings, A.: Sound Absorbing Ducts. Proc. 2nd Internatl. Conference on Recent Developments in Air- and Structure-Borne Sound and Vibration, Auburn University, USA, 689–696 (1992) Cummings, A.: Sound Propagation in Narrow Tubes of Arbitrary Cross-Section. J. Sound Vibr. 162, 27–42 (1993) Cummings,A.:A Segmented Rayleigh-Ritz Method for Predicting Sound Transmission in a Dissipative Exhaust Silencer of Arbitrary Cross-Section. J. Sound Vibr. 187, 23–37 (1995) Cummings, A., Astley, R.J.: The Effects of Flanking Transmission on Sound Attenuation in Lined Ducts. J. Sound Vibr. 179, 617–646 (1995) Cummings, A., Chang, I.-J.: Noise Breakout from Flat-Oval Ducts. J. Sound Vibr. 106, 17–33 (1986)


Cummings, A., Chang, I.-J.: Sound Propagation in a Flat-Oval Waveguide. J. Sound Vibr. 106, 35–43 (1986) Morse, P.M., Feshbach, H.: Methods of Theoretical Physics, McGraw-Hill, N.Y. (1953)

P

1131

Morse, P.M., Ing˚ard, K.U.: Theoretical Acoustics. McGraw-Hill, N.Y. (1968) Stinson, M.R.: The Propagation of Plane Sound Waves in Narrow and Wide Circular Tubes, and Generalization to Uniform Tubes of Arbitrary Cross-Sectional Shape. J. Acoust. Soc. Amer. 89, 550–558 (1991)

Q Elasto-Acoustics with W. Maysenhölder

Some fundamental relations, and relations concerning sound transmission through plates may be found also in the > Ch. “I. Sound Transmission”.

Q.1 Fundamental Equations of Motion

See also: Achenbach (1975); Maysenhölder (1994)

Used notation (including some quantities of later Sections): xi (i = 1, 2, 3) or x, y, z ui vi = ∂ ui ∂ t c C Šw k ki —ij ij p Š, ‹ K E Œ ij Cijkl ekin epot L Ii ƒij

Cartesian co-ordinates of position; displacement; velocity; phase velocity; group velocity; wavelength (various subscripts); wavenumber; wavevector; strain; stress; sound pressure; mass density; Lamé’s constants (‹ ≡ shear modulus); compression (bulk) modulus; Young’s modulus; Poisson’s ratio for isotropic media (no subscript); Poisson’s ratios for anisotropic media; elasticity tensor; kinetic energy density; potential energy density; Lagrangian density; intensity; Kronecker’s delta (identity matrix)

Sometimes vectors will be written with arrows, e. g. x for xi . The summation rule is applied: 3 3 3 Cijkl —kl —kk ≡ —kk Cijkl —kl ≡ k=1 l=1

k=1

(summation over repeated subscripts).

1134

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Unlike electrodynamics the theory of elasticity is genuinely non-linear, since the exact relation between strain field —ij and displacement field ui is non-linear. This chapter, however, is confined to the linearised theory, i.e. with ∂ uj 1 ∂ ui (1) —ij = + 2 ∂ xj ∂ xi and the generalised version of Hooke’s law ij = Cijkl —kl .

(2)

The decomposition of the strain tensor, —ij =

V ƒij + —˜ij , 3V

(3)

is invariant with respect to co-ordinate transformations and therefore physically essential. The first term involving the trace (—kk = V/V) represents a change of volume without change of shape, whereas the strain deviator —˜ij with zero trace describes pure shear deformations (change of shape without change of volume). The corresponding decomposition for the stress tensor is ij = −p ƒij + ˜ ij with the pressure p (kk = −3p) and the stress deviator ˜ ij with zero trace for the shear stresses. In isotropic media Hooke’s law decomposes into the part for (isotropic) compression, p = −K · V V with compression (bulk) modulus K, and the shear part ˜ ij = 2 ‹ —˜ij with shear modulus ‹. Both parts are combined in the convenient form ij = Š —kk ƒij + 2 ‹ —ij with the Lamé 2 constant Š = K − ‹. 3 If dissipation effects are ignored, the Lagrangian density is given by L = ekin − epot with ekin =

1 vi v i , 2

epot =

∂ uk 1 1 1 ∂ ui ij —ij = —ij Cijkl —kl = Cijkl , 2 2 2 ∂ xj ∂ xl

(4)

where the last equal sign is justified because of the symmetries of the elastic tensor, Cijkl = Cjikl ,

Cijkl = Cklij .

The equation of motion (for time-independent material properties) may be obtained from Hamilton’s principle, ƒ L dt dx1 dx2 dx3 = 0 (5) leading to the Lagrange-Euler equations: ∂L d ∂L ∂ ui d = 0∗) + ∂L ∂ − d t ∂ vi d xj ∂ xj ∂ui

(6)

which for the considered case result in the partial differential equations:

∂ Cijkl ∂ uk ∂ 2 ui ∂ 2 uk = C + . ijkl 2 ∂t ∂ xj ∂ xl ∂ xj ∂ xl ∗)


(7)

Elasto-Acoustics

Q

1135

The last term vanishes for homogeneous media with position-independent material properties. Alternative derivation: The right-hand side of (7) equals the divergence ∂ ij ∂ xj of the stress tensor, which is zero for local elastostatic equilibrium in the absence of external forces. According to D’Alembert’s principle, the combination of this balance of forces with the inertia term (left-hand side of (7)) again yields the above equation of motion. In the case of external body forces, like gravity, a volume density of forces [N/m3 ] must be added. In the special case of locally isotropic media one obtains with Cijkl = Š ƒij ƒkl + ‹ ƒik ƒjl + ƒil ƒjk

(8)

the simplified equation of motion:

∂ 2 uj ∂ 2 ui ∂ 2 ui ∂ Š ∂ u j ∂ ‹ ∂ ui ∂ ‹ ∂ u j = Š + ‹ +‹ + + + , 2 ∂t ∂ xi ∂ xj ∂ xj ∂ xj ∂ xi ∂ xj ∂ xj ∂ xj ∂ xj ∂ xi

(9)

where the last three terms vanish in homogeneous media.

Q.2 Anisotropy and Isotropy See also: Helbig, pp. 68–92 (1994); Jones, pp. 56–70 (1999); Lai et al., pp. 293–314 (1993)

For an explicit description of anisotropic elasticity the fourth-rank tensor Cijkl is often transformed to the symmetric 6x6-matrix cIJ with subscript relations ij → I :

11 → 1 , 22 → 2 , 33 → 3 , 23 → 4 , 31 → 5 , 12 → 6

(contracted or Voigt’s notation). The most general (triclinic) anisotropy is described by a fully occupied matrix (21 independent elastic constants), which is shown here with four-subscript entries in order to illustrate the above subscript relations: ⎞ ⎛ C1111 C1122 C1133 C1123 C1113 C1112 ⎜ C2222 C2233 C2223 C1322 C1222 ⎟ ⎟ ⎜ ⎜

C3333 C2333 C1333 C1233 ⎟ ⎟, ⎜ cIJ = ⎜ (1) C2323 C1323 C1223 ⎟ ⎟ ⎜ ⎝ sym C1313 C1213 ⎠ C1212 triclinic “sym” means symmetric completion. Since cIJ is not a tensor, transformation of Hooke’s law (see > Sect. Q1) is not trivial: {I} = 11 22 33 23 13 12 I = cIJ —J with (2) . —J = —11 —22 —33 2 —23 2 —13 2 —12 Similarly, with the compliance tensor Sijkl , which is the inverse of Cijkl , the contracted form is: —J = sIJ I ;

sIJ = c−1 IJ .

(3)

1136

Q

Elasto-Acoustics

The most general anisotropy admissible in thin-plate theory, which implies the middle plane of the plate to be a plane of symmetry, needs 13 independent elastic constants (monoclinic), whereas orthotropic anisotropy, characterised by three mutually perpendicular planes of symmetry, requires nine: ⎞ ⎛ ⎞ ⎛ 0 c16 0 0 c11 c12 c13 0 c11 c12 c13 0 ⎜ ⎜ c22 c23 0 c22 c23 0 0 c26 ⎟ 0 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ c33 0 c33 0 0 c36 ⎟ ⎜ 0 0 ⎟ ⎟. ⎜ (4) ⎜ ⎜ c44 0 c44 c45 0 ⎟ 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎝ sym c55 0 ⎠ ⎝ sym c55 0 ⎠ c66 c66 monoclinic orthotropic plane of symmetry: x3 = const ( ≡ orthorhombic) In “engineering notation” orthotropic anisotropy is expressed by three Young’s moduli Ei , six Poisson numbers Œ ij , and three shear moduli Gij . Their physical meaning may be deduced from the compliance representation: ⎞ ⎛ Œ21 Œ31 1 − − 0 0 0 ⎟ ⎜ E1 E2 E3 ⎟ ⎜ 1 Œ32 ⎟ ⎜ Œ12 ⎜ − − 0 0 0 ⎟ ⎟ ⎜ E E E 1 2 3 ⎟ ⎜ 1 Œ23 ⎟ ⎜ Œ13 ⎟ − 0 0 0 ⎜ −

⎟ ⎜ E E E 1 2 3 (5) sIJ = ⎜ ⎟, 1 ⎜ 0 0 0 ⎟ 0 0 ⎟ ⎜ ⎟ G23 ⎜ ⎟ ⎜ 1 ⎜ 0 0 ⎟ 0 0 0 ⎟ ⎜ G31 ⎟ ⎜ ⎝ 1 ⎠ 0 0 0 0 0 G12 which also provides for the three relations between the six Poisson numbers due to sIJ = sJI . In terms of the cIJ , Maysenhölder (1996): E1 =

N , c22 c33 − c223

E2 =

N , c11 c33 − c213

E3 =

N , c11 c22 − c212

N = c11 c22 c33 − c11 c223 − c22 c213 − c33 c212 + 2 c12 c13 c23,

(6) (7)

Œ21 =

c12 c33 − c13 c23 , c11 c33 − c213

Œ31 =

c13 c22 − c12 c23 , c11 c22 − c212

Œ32 =

c23 c11 − c12 c13 , c11 c22 − c212

Œ12 =

c12 c33 − c13 c23 , c22 c33 − c223

Œ13 =

c13 c22 − c12 c23 , c22 c33 − c223

Œ23 =

c23 c11 − c12 c13 . c11 c33 − c213

(8)

Elasto-Acoustics

Q

1137

Backward transformation: c11 =

1 − Œ23 Œ32 E1 ,

c12 =

Œ21 + Œ31 Œ23 E1 ,

c44 = G23 ,

c22 =

c55 = G31 ,

1 − Œ13 Œ31 E2 ,

c23 =

c33 =

Œ32 + Œ12 Œ31 E2 ,

1 − Œ12 Œ21 E3 ,

c13 =

Œ13 + Œ23 Œ12 E3 ,

(9)

c66 = G12 ,

= 1 − Œ12 Œ21 − Œ23 Œ32 − Œ31 Œ13 − 2 Œ21 Œ32 Œ13 . Elastic stability requires all diagonal elements of both cIJ and sIJ as well as N and to be positive. In addition, ŒIJ2 < EI EJ for I = J. For further constraints on Poisson’s ratios see Jones, p. 69 (1999). Further frequent cases with even lower anisotropies (i.e. higher symmetries) are transversely isotropic (equivalent to hexagonal) and cubic with five and three independent elastic constants, respectively: ⎛ ⎞ ⎛ ⎞ c11 c12 c12 0 0 0 0 0 c11 c12 c13 0 ⎜ ⎜ c11 c13 0 c11 c12 0 0 0 ⎟ 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ c c 0 0 0 0 0 0 ⎟ 33 11 ⎜ ⎟ ⎜ ⎟. (10) ⎜ ⎜ c44 0 c44 0 0 ⎟ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ sym c44 0 ⎠ ⎝ sym c44 0 ⎠ c66 c44 1 cubic with c66 = ( c11 − c12 ) 2 transverse isotropy (hexagonal) The engineering notation is also used for transversely isotropic materials (E1 = E2 ,G31 = G23 , Œ32 = Œ31 , G12 = E1 /[2(1 + Œ21)]; further Œ12 = Œ21, Œ13/E1 = Œ31 /E3 , Œ23 /E1 = Œ32 /E3; Œ13 = Œ23 ; therefore E1 , Œ21 , E3 , Œ31 and G23 provide a complete description. For cubic materials E1 , Œ21 and G23 suffice (however, G23 = E1 /[2(1 + Œ21 )], if not isotropic!). The elastic properties of cubic materials are conveniently expressed by one modulus of compression K (bulk modulus) for changes of volume with constant shape and two shear moduli ‹ and ‹’ for changes of shape with constant volume: K=

1 ( c11 + 2 c12 ) , 3

‹ =

1 ( c11 − c12 ) , 2

‹ = c44 .

(11)

This anisotropy of shear deformation may be characterised by the dimensionless measure: a=

‹ − ‹ with ‹ + ‹

− 1 < a < 1.

(12)

With a = 0 one proceeds from cubic anisotropy to isotropy, which is determined by two independent parameters. Several pairs are in common use, e.g. compression (bulk) modulus K and shear modulus ‹, Lamé’s constants Š and ‹, finally Young’s modulus

Q

1138

Elasto-Acoustics

E and Poisson’s ratio . The dependencies are summarised in the table below. See also Table 4 in > Sect. I.8. Elastic stability requires K > 0 and ‹ > 0, leading to: 2 Š >− ‹, 3

E > 0,

−1 <
0, − c11 < c12 < c11 . 2

(13)

Table 1 Interrelations between isotropy parameters. For a more extensive table including eight additional pair combinations see Thurston, p. 74 (1964) From: To:

K ,‹

Š, ‹

K=

K

Š +

‹=

‹

‹

Š = c12 =

K −

E=

2 ‹ 3 9K‹ 3K + ‹

=

3K − 2 ‹ 6K + 2 ‹

c11 =

K +

4 ‹ 3

2 ‹ 3

Š ( 3 Š + 2 ‹) ‹ Š + ‹

E ,

c11 , c12 ( c44 )

E 3 (1 − 2) E 2 (1 + ) E (1 + ) (1 − 2)

1 ( c11 + 2 c12 ) 3 1 ( c11 − c12) = c44 2 c12

E

(c11 + 2 c12 ) (c11 − c12 ) c11 + c12

Š 2 ( Š + ‹)

c12 c11 + c12

Š + 2‹

E (1 − ) (1 + ) (1 − 2)

c11

Moduli associated with elementary deformations and corresponding waves: Compression modulus (bulk modulus) K: V V = compressibility; no associated wave.

p = −K K−1

F = ‹ (: engineering shear strain) A Velocity of transversal waves (shear waves and torsional waves): cT = ‹/. Shear modulus ‹:

(14)

(15)

(16)

Elasto-Acoustics

Q

1139

α

Young’s modulus E and Poisson’s ratio (lateral contraction) : L F =E A L —yy =− —xx

(x-direction along L)

Velocity of (quasi-) longitudinal waves in bars:

(17) (18) cQL =

E/.

(19)

Modulus for longitudinal waves D: −p = xx = D —xx = c11 —xx

(20)

Velocity of (pure) longitudinal waves (with compression/dilatation and shear): cL = D/.

(21)

Q.3 Interface Conditions, Reflection and Refraction of Plane Waves

See also: Auld, Vol. II, pp. 1–62 (1990)

At the interface (boundary) between two elastic media I and II the requirements for tight contact are continuity of displacements, uiI = uiII, and balance of forces (1) ijI − ijII nj = 0 with unit normal vector ni on the interface. If the medium II is a fluid with sound pressure pII at the interface, then ijI nj = −pII ni , since ijII = −pII ƒij . With the usual slip assumption only the displacement component normal to the interface, ni ui , must be continuous.

1140

Q

Elasto-Acoustics

A plane elastic wave incident on a plane interface between two homogeneous elastic solids with different material properties (tight contact) generates up to three reflected waves and up to three refracted (transmitted) waves, since there are up to three different wave velocities in an anisotropic medium. The propagation directions of these scattered waves are given by Snell’s law, the relations for their amplitudes are called Fresnel equations. The boundary conditions require that the wavevector component tangent to the boundary is the same for all waves. This immediately leads to Snell’s law: 1 sin Ÿi sin Ÿ0 = = ci c0 ctrace

(2)

with phase velocities c and angles Ÿ (0 ≤ Ÿ < ) between wavevector k and normal n . Subscript i : reflected and refracted waves; subscript 0: incident wave. For sin Ÿi > 1, the i-th wave becomes evanescent (exponentially decaying perpendicular to the interface) and propagates with trace velocity ctrace along the interface (e.g. for total reflection). Since in anisotropic media the phase velocities are no longer independent from the propagation direction (see > Sect. Q.9.1), the evaluation of Snell’s law is more involved. A helpful geometrical technique makes use of slowness surfaces, which provide the magnitude of k/– as a function of the direction of k [Helbig, pp. 32–38 (1994); Auld, Vol. I, pp. 393–413 (1990)]. In anisotropic media, where the direction of energy propagation (see > Sect. Q.5) may be different from the wavevector direction, the incident wave should be characterised by the former rather than by the latter.The energy directions need not lie in the plane defined by the wavevectors! [Rokhlin/Bolldand/Adler (1986); Lanceleur/Ribeiro/De Belleval (1993)]. For Fresnel equations see e.g. [Achenbach, pp. 168–187 (1975) and Auld, Vol. II, pp. 21– 43 (1990)].

Q.4 Material Damping

See also: Gaul (1999)

The conventional viscoelastic generalisation of the relation ˜ ij (t) = 2 ‹ —˜ij (t) between stress and strain deviators for pure shear deformation in an isotropic solid (‹: shear modulus; see > Sect. Q.1) is given by: M k=0

N

pk

dk dk qk ˜ ij (t) = —˜ij (t) k dt dtk

(1)

k=0

with integer k and real coefficients pk and qk . Classical viscoelastic models are the Zener model: ˜ ij (t) + p1

d ˜ ij (t) d ˜—ij (t) = q0 ˜—ij (t) + q1 dt dt

(2)

Elasto-Acoustics

Q

1141

and – as its descendants – the Kelvin-Voigt model (p1 = 0) and the Maxwell model (q0 = 0). In the following, however, the further generalisation to time derivatives of fractional order will be considered: M N d k d k ˜ —˜ij (t), pk (t) = q (3) ij k d t k d t k k=0

k=0

which leads in many cases to improved curve-fitting of measured data with less parameters (0 = 0 = 0; 0 < k , k < 1). An extension to fractional orders beyond one is possible. An alternative formulation in terms of relaxation functions reads: t ˜ ij (t) = 2 −∞

with

d dt

d —˜ij (‘) d‘, G(t − ‘) d‘

t —˜ij (t) = 2

J (t − ‘)

−∞

d ˜ ij (‘) d‘ d‘

(4)

t 2 G(‘) J(t − ‘) d‘ = 1. 0

The relaxation modulus G(t) and the creep compliance J(t) describe the fading memory of the (linear viscoelastic) material with respect to the loading history. Any elastic modulus M may be generalised in the same manner.

˜—ij (t) = Re ˜—ij (–) exp j– t , For time-harmonic fields, one obtains: ˜ ij (–) = 2 ‹(–) —˜ij (–) with

2 ‹(–) =

N

qk (j –)

k=0

k

M

pk (j –) k .

(5)

k=0

The decomposition ‹(–) = ‹ (–) + j ‹ (–) = ‹ (–)[ 1 + j †(–) ] of the complex shear modulus introduces the storage modulus ‹’(–) (real part), the loss modulus ‹ (–) (imaginary part) and the loss factor †(–), which may be interpreted as the ratio of the dissipated energy D(–) per cycle to the 2-fold of the stored energy U(–): † (–) =

D(–) ‹ (–) 1 = = = = tan ƒ(–) 2 U(–) ‹ (–) Q(–)

(6)

with quality factor Q(–),logarithmic decrement (–) and loss tangent tan ƒ(–).The relationship between the complex modulus ‹(–) and the relaxation modulus G(t) amounts to: ∞ ∞ ˆ ˆ (‘) cos(–‘) d‘ G (‘) sin(–‘) d‘, ‹ (–) = – G (7) ‹ (–) = ‹0 + – 0

0

ˆ with real static or equilibrium modulus ‹0 = ‹(– = 0) = G(t = ∞) and G(t) = ˆ G (t) − ‹0, hence G (∞) = 0. Because of causality and linearity the real and imaginary parts of any modulus M = M + jM are connected via the Kramers-Kronig relations: 2 M (–) = P

∞ 0

§M (§) d§, §2 − –2

2 M (–) = − P

∞ 0

§ M (§) d§, §2 − –2

(8)

1142

Q

Elasto-Acoustics

where P denotes Cauchy’s principal value [Beltzer, p. 20 (1988)]. Regarding practical applications of these relations see, e.g. [Mobley et al. (2000)]. The often utilised (nonviscoelastic) model with M’ and M” independent of frequency (‘constant hysteresis damping model’,‘hysteretic model’ or ‘structural damping model’) violates causality. The elastic-viscoelastic correspondence principle states that the solution of a viscoelastic problem may be obtained from the solution of the corresponding elastic problem. In the case of time-harmonic problems this means a straightforward substitution of the elastic moduli by the corresponding frequency-dependent and complex viscoelastic moduli. Otherwise, Fourier transformations have to be performed before and after this substitution. (An alternative realisation uses Laplace transforms and impact response functions.) Example: Five-parameter fractional-derivative model (generalised Zener model ) The complex shear modulus for a highly damped polymer, ‹(–) = ‹0

1 + b (j –) 1 + a (j –)

(9)

with ‹0 = 87 kPa,a = 0.039, = 0.39,b = 0.38, = 0.64,yields a good fit of experimental data of ‹’(–) and †(–) from 1 Hz to almost 10 kHz. Special case: Four-parameter fractional-derivative model [Pritz (1996)] Thermodynamic constraints like non-negative rate of dissipated energy and the requirement of a finite viscoelastic wave speed impose the conditions ‹ > 0,

b > a > 0,

0 e ; Šw > e (Ioffe-Regel criterion). Energy propagation is slower than diffusive (D = 0) due to coherent interference effects. Vibrations decay exponentially with localisation length . The transmission factor of a plate is proportional to exp(−h/ ).

Q.7 Periodic Media

See also: Maysenhölder, Ch. 5 (1994)

A periodically inhomogeneous medium may be constructed by infinite repetition of a unit cell defined by linearly independent basis vectors a1 , a2 , a3 . Its volume is V0 = a1 · a2 × a3 . A position r in a unit cell and an equivalent position in another unit cell are connected by a lattice vector gn = n1 a 1 + n2 a2 + n3 a3 (ni : integer). Any function f with the periodicity of the lattice, f r + gn = f ( r ), may be Fourier expanded: M · r f ( r ) = f M exp j G (1) M

with Fourier coefficients: 1 M M · r d3 r. f = f ( r ) exp − j G V0

(2)

The sum is over all reciprocal lattice points, the integral over one unit cell. M = M1 A 1 + M2 A 2 + M3 A 3 G

(3)

(Mi : integer) is a vector of the reciprocal lattice with the basis vectors: 2 ( a2 × a3 ) V0 j = 2 ƒij. ai · A 1 = A

;

2 = A

2 (a3 × a1 ) V0

;

3 = A

2 ( a1 × a2 ) ; V0

(4)

Analogous to plane waves in homogeneous media the fundamental wave solutions for periodic media are Bloch waves: u ( r, t ) = p ( r ) e

j

– t −k · r

(5)

Elasto-Acoustics

Q

1147

with the periodic function p r + gn = p (r). With the Fourier coefficients of the reciprocal density, ˜ = −1 , the Fourier transform of the equation of motion (see > Sect. Q.1) for Bloch waves attains the usual form of an eigenvalue problem: L M L + G + G (6) k k ˜ N−M CM−L –2 pNi = l j ijkl l j pk . L, M

Capital Latin superscripts again run over all points of the reciprocal lattice. Local isotropy in terms of the Lamé constants Š and ‹ leads to [Sigalas/Economou (1992)]: ! –2 pNi = ŠM−L kj + GLj k i + GM ˜ N−M i L, M

" pLj + ‹ M−L ki + GLi k j + GM j

(7)

k j + GM pLi . + ‹ M−L kj + GLj j Analytical expressions for energy density and intensity of Bloch waves and their averages over a unit cell including low-frequency limits are given in [Maysenhölder, Ch. 5 (1994)]. Special case: Locally isotropic medium with one-dimensional periodicity along the xdirection (density , stiffness … = Š + 2‹, spatial period h). Consider longitudinal waves in x-direction (one-dimensional problem). a) Example with analytical solution: Unit cell: two homogeneous layers (n = 1, 2) of thicknesses hn with n and …n . For given frequency – the Bloch-wavenumber k follows from [Beltzer, pp. 216–219 (1988)]: cos ( k h ) = cos ( – t1 ) cos ( – t2 ) − sin ( – t1 ) sin ( – t2 ) with travel times tn = hn /cn , phase velocities cn = …n /n and =

1 …1 + 2 …2 . 2 1 2 …1 …2

| cos ( k h ) | ≤ 1:

(8)

(9) pass bands;

| cos ( k h ) | > 1:

stop bands.

b) Low-frequency limit (homogenisation):

Real formulation for …(−x) = …(x) with spatial averages … (x) = …0 and (x) = 0 . For k 2/h the displacement field of a Bloch wave may be approximated by a ’phasemodulated’ wave: u ( x , t ) ∝ cos ( k x − – t ) − k q ( x ) sin ( k x − – t ) ≈ cos k x + q ( x ) − – t

(10)

with anti-symmetric periodic ‘modulation function’ x q ( x ) = − x + …eff 0

dx , … ( x )

h q(0) = q( ) = 0. 2

(11)

Q

1148

Elasto-Acoustics

The effective properties are: #

eff = 0 ,

…eff

1 = … (x)

$−1 ,

ceff =

…eff . 0

(12)

Explicitly for the example a): …eff =

q(x) =

h , h1 h2 + …1 …2 ⎧ …eff ⎪ ⎪ −1 x ⎪ ⎪ ⎪ …1 ⎪ ⎪ ⎨

(13)

0≤x≤

⎪ ⎪ ⎪ ⎪ ⎪ h1 …eff 1 …eff 1 ⎪ ⎪ −1 x+ − ⎩ …2 2 …1 …2

h1 2

for

.

(14)

h h1 <x ≤ 2 2

For continuous variation of the stiffness, e.g. …(x) = …0 + …s cos(2x/h) > 0, one obtains: * + ) 1 − … /… h x s 0 …eff = …20 − …2s , q ( x ) = −x + arctan . (15) tan 1 + …s /…0 h

Q.8 Homogenisation

See also: Beltzer, pp. 187–202 (1988)

Homogenisation substitutes an inhomogeneous medium by an ‘equivalent’ homogeneous medium with effective material properties in the limit of low frequencies. The definition of effective elastic constants is by spatial averages: ij = Ceff ; in general Ceff (1) ijkl —kl ijkl = Cijkl . In general, the effective density may become complex [Viktorova/Tyutekin (1998)] or even a tensor [Helbig, pp. 298, 322–323 (1994)], however, in all cases listed below eff = .

Q.8.1 Bounds on Effective Moduli For an inhomogeneous elastic medium consisting of N phases with volume fractions “n and elastic constants C(n) ijkl the Voigt and Reuss averages, Voigt Cijkl

=

N n=1

“n C(n) ijkl

,

CReuss ijkl =

+−1 * N “n (n) n=1 Cijkl

,

Voigt

CReuss ≤ Ceff ijkl ijkl ≤ Cijkl ,

(2)

are rigorous bounds for the effective elastic constants, which hold for arbitrary structures. These bounds are useful for small “n and small differences between the C(n) ijkl .

Elasto-Acoustics

Q

1149

Narrower bounds are available for special cases, e.g. the Hashin-Shtrikman bounds for N = 2 statistically distributed isotropic phases (Kn , ‹n : compression and shear moduli; “2 ≡ “): ˜ “ K + K ˜ 1 K1 + K “ Keff − K1 ≤ , (3) ≤ ˜ + (K2 − K1) (1 − “) ˜ + (K − K ) (1 − “) K2 − K 1 K1 + K K1 + K 2 1 ‹1 + ‹˜ “ ‹1 + ‹˜ “ ‹eff − ‹1 ≤ , ≤ ‹ 2 − ‹1 ‹1 + ‹˜ + ‹2 − ‹1 (1 − “) ‹1 + ‹˜ + ‹2 − ‹1 (1 − “)

(4)

where, if (‹2 − ‹1)(K2 − K1 ) ≥ 0, then −1 −1 ≈ ≈ 10 10 3 1 ˜ = 4 ‹1 , K = 4 ‹2 , ‹˜ = 3 1 + , ‹= + , (5) K 3 3 2 ‹ 1 9 K1 + 8 ‹ 1 2 ‹ 2 9 K2 + 8 ‹ 2 while, if (‹2 − ‹1 )(K2 − K1) ≤ 0, then −1 −1 10 3 1 10 4 ‹ 2 ≈ 4 ‹1 3 1 ˜ ˜ + , ‹= + . (6) K= , K= , ‹˜ = 3 3 2 ‹ 1 9 K2 + 8 ‹ 1 2 ‹ 2 9 K1 + 8 ‹ 2 For ‹1 = ‹2 the bounds for Keff coincide and yield the exact Keff .

Q.8.2 Effective Moduli for Particular Structures a) Voigt and Reuss averages for polycrystals with statistical orientation of the grains (all of the same anisotropic material) are approximations for compression and shear moduli of an effective isotropic material: 1 1 1 KVoigt = Ciikk , Cikik + Ciikk . ‹Voigt = (7) 9 10 3 Reuss averages are obtained accordingly with the compliance tensor (inverse of Cijkl ). For a material with cubic grains: KVoigt =

1 (c11 + 2 c12 ) = K , 3

‹Voigt =

1 1 2‹ +3‹ . (c11 − c12 + 3 c44 ) = 5 5

(8)

In this case KReuss = KVoigt = Keff . b) Spherical inclusions (Ks , ‹s , radius r) randomly dispersed in a homogeneous matrix with Km and ‹m ; wavelengths much larger than r. Neglecting multiple scattering leads to the approximation for small volume fractions “ of the spheres: 3 K m + 4 ‹ m ( K s − Km ) “ , (9) Keff = Km + 3 K s + 4 ‹ m − 3 ( K s − Km ) “ ‹eff

‹m 15 Km + 20 ‹m ‹s − ‹m “ . (10) = ‹m + 6 ‹ s Km + 2 ‹ m + ‹ m 9 K m + 8 ‹ m − 6 K m + 2 ‹ m ‹s − ‹m “

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c) Composite sphere assembly: A sphere composed of two isotropic materials (Ks , ‹s up to inner radius a; Km , ‹m from a to outer radius b; “ = a/b) has an effective compression modulus Keff , which is equal to the above expression for spherical inclusions. An assembly of such composite spheres with different sizes, but common ratio “, possesses the same Keff in the long-wavelength limit (exact results). d) Periodically spaced fibres parallel to the x-axis in isotropic matrix; “ = volume fraction of the fibres. Both the fibre material and the effective medium are transversely isotropic. Approximation for the effective moduli in ’engineering notation’ according to Skelton/James, p. 151 (1997): Exx = “ Efxx + ( 1 − “) Em , Gxy = Gxz =

Gm √ , 1 − (1 − Gm /Gfxy ) “

f Œxy = Œxz = “ Œxy + ( 1 − “ ) Œm ,

Eyy = Ezz =

Em √ , 1 − ( 1 − Em /Efyy ) “

Gm √ , 1 − ( 1 − Gm /Gfyz ) “ Eyy . Œyz = −1 + 2 Gyz

Gyz =

(11) (12) (13)

(Em , Œm , Gm = Em /[2(1 + Œm )]: Young’s modulus, Poisson’s ratio and shear modulus of isotropic matrix; superscript f denotes fibre properties.) An alternative for isotropic fibre materials (with Ef , Œf and Gf = Ef /[2(1 + Œf )] are the Halpin-Tsai equations [Jones, pp. 151–158 (1999)]: Exx = “ Ef + ( 1 − “ ) Em ,

Œxy = Œxz = “ Œf + ( 1 − “ ) Œm ,

1 + †“ M = Mm 1−†“

†=

with

(Mf /Mm ) − 1 , (Mf /Mm) +

(14) (15)

where M is one of the quantities Eyy , Gxy or Œyz of the composite and Mm or Mf are the corresponding quantities E, G or Œ for the matrix and the fibre materials. The three ’s are adjustable parameters depending on fibre and packing geometry and can range from 0 to ∞. Example: Circular fibres in a square array: Use = 2 for Eyy and = 1 for Gxy . e) Symmetric stack of layers with unidirectional fibre reinforcement (for properties of transversely isotropic fibres see above). Fractional layer thickness hn , total thickness h; the fibre directions relative to the global x-axis may be different for different layers, but lie all in the x-y-plane of the layers. By symmetric is meant that the top half of the stack is a mirror image of the bottom half. After transformation to global co-ordinates the elastic constants (now ’monoclinic-like’) of the n-th layer are denoted by cnIJ , the constants of the monoclinic effective medium by cIJ (no summation convention): * N + −1 N hn c33 1 , (16) , c13 = hn cn13 1 + c33 = c 2 n=1 cn33 n=1 n33 N c33 1 1 + , hn cn23 c23 = 2 n=1 cn33

c36 =

N c33 1 1 + , hn cn36 2 n=1 cn33

(17)

Elasto-Acoustics

cIJ =

N

hn cnIJ

{I, J = 1, 2, 6 } ,

cIJ =

n=1

N hn cnIJ n n=1

+* N + * N +2 * N hn cn44 hn cn55 hn cn45 , − = n n n n=1 n=1 n=1

Q

{I, J = 4, 5 } ,

1151

(18)

n = cn44 cn55 − ( cn45 ) 2 . (19)

The formulae are valid for wavelengths greater than the total thickness of the stack [Skelton/James, pp. 159–160 (1997)]. f) Periodic stack of transversely isotropic layers. Notation as in the previous case (h = period). Note that in both e) and f) the constituent layers are transversely isotropic, however, in e) the axes of symmetry lie – possibly in different directions – in the x-yplane, whereas now the axis of symmetry is the z-axis for all layers: * N + −1 N hn cn13 c33 = , c13 = c33 hn , (20) c cn33 n=1 n33 n=1 N (c13 ) 2 (cn13 ) 2 + hn cn11 − , c33 cn33 n=1 * N + −1 N hn , c66 = hn cn66 . c55 = c n=1 n55 n=1

c11 =

(21)

The formulae are valid for wavelengths greater than the total thickness of the period [Helbig, p. 313 (1994)]. If all constituent layers are isotropic, then c55 cannot be greater than c66 (because of the Cauchy-Schwartz-Kolmogorov inequality).For the inverse problem, the determination of the cnIJ from the cIJ , with N = 2 or 3 see [Helbig, pp. 324–336 (1994)]. For further homogenisation results see (periodic media).

> Sect. I.13 (sandwich panels)

and

> Sect. Q.7

Q.9 Plane Waves in Unbounded Homogeneous Media

See also: Maysenhölder, pp. 50–51, 91–100 (1994); Beltzer, pp. 95–109 (1988)

Q.9.1 Anisotropic Media The equation of motion for homogeneous anisotropic media (see > Sect. Q.1) can be satisfied by the plane-wave ansatz ui = Ai exp[(j(–t − km xm )] with real amplitude Ai defining the polarisation, wavenumber k and wavevector direction ei , leading to Christoffel’s equation:

il − ƒil c2 Al = 0 with il = Cijkl ej ek = li ∗) (1) ∗)


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( il : Christoffel’s tensor). For a given direction ei this is a cubic eigenvalue problem for eigenvalues c2 and eigenvectors Ai (c = –/k: magnitude of phase velocity). Since without dissipative effects the symmetric il is also real, the polarisations are mutually orthogonal or can be chosen as such in case of degenerate eigenvalues. ’Pure modes’ are – according to one definition – waves with purely longitudinal or purely transversal polarisation. Another definition requires the group velocity or intensity to be parallel to the wavevector. A wave which is not a pure mode may be termed quasi-longitudinal or quasi-transversal, if its polarisation is close to one of the pure polarisation. The group velocity Ci is equal to the velocity of energy transport, which is in general not parallel to ei : Cj =

Ai Cijkl Al ek ∂c ; = ∂ ej A2m c

Time average of energy density:

Ci ei = c.

ekin + epot

(2) t

= 2 ekin t =

1 –2 A2i ; 2

(3)

–2 Ai Cijkl Al ek . (4) Ij = ekin + epot t Cj = 2c The slowness vector m , which is useful for reflection and refraction problems (see > Sect. Q.3) is defined by:

Intensity:

em km = : exp[(j(–t − km xm )] = exp[(j–(t − m xm )]. c – Two-dimensional example with cubic (i.e. quadratic) anisotropy: cos œ ; Wave vector direction: ei = sin œ m =

Polarisation angle:

• = arctan

A2 ; A1

Eigenvalues: , . 1 1 2 1 2 2 c11 + c44 ± c = (c11 − c44 ) 1+cos 4 œ + (c12 + c44 ) 1 − cos 4 œ ; 2 2 2 Polarisations:

A2 c2 − c11 cos2 œ − c44 sin2 œ . = A1 ( c12 + c44 ) cos œ sin œ

(5) (6)

(7)

(8)

The figures below are drawn for compression modulus K = 1, shear modulus ‹ = 9 and shear modulus ‹ = 1 (fictitious material, arbitrary units; anisotropy a = 0.8). The two eigensolutions (modes) are numbered according to increasing phase velocity.

Elasto-Acoustics

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1153

ψ

ϕ

a) Slowness diagram (polar diagram of the slowness); [10]: x1 -direction; [01]: x2 direction; dotted circles: isotropic case. b) Polarisation •. Dotted lines: longitudinal polarisation • = œ, and transversal polarisation • = œ − 90ı χ

ϕ

c) Group velocity (polar diagram). Dotted lines: phase velocity. d) Intensity direction ”. Dotted line: propagation direction œ

Q.9.2 Isotropic Media In isotropic materials Christoffel’s equation leads to two types of solutions: longitudinal and transversal plane waves with wave speeds Š + 2‹ ‹ and cT = . (9) cL = The displacement field of longitudinal waves is curl free (changes of shape and volume without rotations), that of transversal waves is divergence free (change of shape without volume change). The strain and stress amplitudes for waves propagating in the x-direction with velocity amplitude v0 are for longitudinal polarisation: ⎞ ⎛ ⎞ ⎛ 1 0 0 1 0 0 ⎟ ⎜ 0 v0 0 ⎟ (10) − ⎝ 0 0 0 ⎠ , − v0 cL ⎜ 1 − ⎠ ⎝ cL 0 0 0 0 0 1− and for transversal polarisation along the y-direction: ⎞ ⎞ ⎛ ⎛ 0 1 0 0 1 0 v0 1 ⎝ 1 0 0 ⎠ , − v0 cT ⎝ 1 0 0 ⎠ . − (11) 2 cT 2 0 0 0 0 0 0

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The formulas for energetic quantities are of the same form for both wave types: 1 |v0 | 2 cos2 ( kx − –t) , Sx = |v0 | 2 c cos2 ( kx − –t) , 2 1 1 Ix = |v0| 2 c = 2 ekin t c. ekin t = epot t = |v0|2 4 2 (Add subscript L or T to the wave number k and the wave speed c as required.). ekin = epot =

(12) (13)

All y- and z-components and the reactive intensity vanish.

Q.10 Waves in Bounded Media Q.10.1 Plate Waves

See also: Maysenhölder (1990) Plate waves may be classified into two groups: shear

waves and Lamb waves. Each group is further subdivided into a symmetric family (mirror symmetry of the displacement vector with respect to the plane z = 0) and an anti-symmetric family (sign change of the displacement vector after the mirror operation). The shear waves, sometimes called SH (’Shear-Horizontal’) waves, possess non-vanishing displacements in the y direction only: n z uy = A cos , n = 0, 2, 4, ... (symmetric family); (1) h n z , n = 1, 3, 5, ... (anti-symmetric family). (2) uy = A sin h (A: Amplitude; phase factor exp[(j(–t − kx)] omitted.) The expressions for phase veloc) ities c and group velocities C are valid for both families ( cT = ‹ ): 2

c =

c2T

/ 1+

n 2 0 kh

=

c2T n c 2 , T 1− 2f h

C · c = c2T .

Conversely, uy ≡ 0 for Lamb waves. With the abbreviations 2 2 2 1 2 c c , 2 = 1 − , x = , 1 = 1 − cL cT 1 + 22 z =

2 , 1 + 22

Rs =

sinh (1kh/2) , sinh (2kh/2)

Ra =

cosh (1 kh/2) , cosh (2 kh/2)

(3)

(4) (5)

the x- and z-components of the displacement fields are for the symmetric family: ux = j A [cosh (1 kz) − x Rs cosh (2 kz)] , uz = 1 A [sinh (1kz) − z Rs sinh (2kz)] , and for the anti-symmetric family: ux = j A [sinh (1 kz) − x Ra sinh (2kz)] ,

(6)

Q

Elasto-Acoustics

uz = 1 A [cosh (1kz) − z Ra cosh (2 kz)] .

1155

(7)

Since measurements are usually confined to the surfaces of the plate, the displacement ratios ux /uz at z = h/2 are of particular interest [Maysenhölder (1987)]: 1 1 + 22 coth(1kh/2) − 2 1 2 coth (2 kh/2) ux 11 = (symmetric family), (8) uz 1 z=h/2 −j 1 22 − 1 1 1 + 22 tanh(1kh/2) − 2 1 2 tanh (2 kh/2) ux 11 = (anti-symmetric family).(9) uz 1 z=h/2 −j 1 22 − 1 Usually, the phase velocities have to be determined numerically from the transcendental Rayleigh-Lamb frequency equations: / 0 tanh ( 2f h/c) ± 1 4 1 2 = (10) 2 tanh ( 1f h/c) 1 + 22 (f: frequency) with the plus sign for the symmetric family and the minus sign for the anti-symmetric family. Fordispersion diagrams see e.g. [Auld, pp. 76–87 Vol. II,“Acoustic fields and waves in solids” (1990); Cremer/Heckl, p. 143 (1996)]. The orthogonality relation for two Rayleigh-Lamb modes (1) and (2) with common frequency, but different wavenumbers, h/2

(2) (1) − uz(2) zx ux(1) xx dz = 0,

(11)

− h/2

holds even for the corresponding modes of a layered plate with z-dependent Lamé constants [Murphy/Li/Chin-Bing (1994)]. If c is known, the corresponding group velocity C may be obtained analytically [Maysenhölder (1992)] (with the same meaning of ± as above): C=

c f dc 1− c df

X+ = T21 X,

with

f dc ±Y = , c df X± ± Z

X− = T22 X,

4 c3 X= c2T f h N2

*

(12)

T1 = tanh ( 1f h/c),

4 21 22 − 21 − N

2 cT cL

2 +

T2 = tanh ( 2 f h/c),

,

N = 1 + 22 = 2 −

c cT

2

Y = 1 T1 22 K2 − 2 T2 21 K1,

Z = 1 T1 K 2 − 2 T2 K 1 ,

K1 = cosh−2 ( 1f h/c),

K2 = cosh−2 ( 2 f h/c).

,

The group velocity C together with Rayleigh’s principle for propagating waves (see > Sect. Q.5.4) may be used for the calculation of the average intensity of a plate wave:

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Ix = 2 wkin C, with the time average wkin of the kinetic energy density and the average over the plate thickness denoted by ... . For both families of shear waves: wkin = With

2 |A| 2 f 2. 2

Sm =

wkin =

sinh (m k h) m k h

(13) (m = 1, 2)

for symmetric Lamb waves :

2 |A| 2 f 2 (S1 + 1) + | x | 2 | Rs | 2 (S2 + 1) 2 + | 1| 2 | S1 − 1| + | z | 2 | Rs | 2 | S2 − 1| − 4 21 z S1 ,

(14)

(15)

and for anti-symmetric Lamb waves: wkin =

2 |A| 2 f 2 | S1 − 1| + | x | 2 | Ra | 2 | S2 − 1| 2 + | 1| 2 (S1 + 1) + | z | 2 | Ra | 2 (S2 + 1) − 4 | 1 | 2 z S1 .

(16)

(Note the little difference in the last term. It is essential for imaginary 1, i.e. for c > cL .) The y- and z-components of the intensity are everywhere zero for all wave families. Special case: Quasi-longitudinal mode: This is the fundamental symmetric Lamb wave, which exhibits predominantly longitudinal character at low frequencies.

Phase velocity c and group velocity C of the quasi-longitudinal mode for Poisson’s ratio = 0.3. Units: cT for velocities, cT/h for frequency √ are independent of Poisson’s ratio and anaAt f = cT / 2h phase and group velocity √ √ lytically known (Lamé wave): c = cT 2, C = cT / 2 . This implies kh = (wavelength = 2h) and

Elasto-Acoustics

ux = −A cos

wkin =

z h

,

1 1 2 11 A 11 2 ‹, 2 1h1

uz = j A sin

z h

Q

,

1157

(17)

1 1 2 1 A 1 2 Ix = √ 11 11 ‹ cT . 2 h

(18)

The strain field of this wave is pure shear and ux ≡ 0 at the plate surfaces! Low-frequency approximation: cQL = CQL = cT

2 , 1−

−j uz = k z, ux 1 −

wkin = 2 |A| 2 f 2.

(19)

For Poisson’s ratio = 0.3 the error of Ix = 2 wkin C in this approximation is smaller than 20% (10%; 5%) for fh/cT < 0.4 (0.3; 0.1). Low-frequency expansions up to the third non-vanishing order [Maysenhölder (1987)]: * c

2

= c2QL = c2QL

+ 2 6 − 10 − 7 2 2 2 4 1− (h k) − (h k) 12 ( 1 − ) 2 720 ( 1 − ) 4 * + 2 2 h f 2 4 2 6 − 10 − 2 2 hf 4 1− . − cQL 3 ( 1 − ) 2 cQL 45 ( 1 − ) 4

(20)

High-frequency approximation (≈ a Rayleigh wave on each plate surface):

c = C = cR ,

wkin =

j exp (1kz) − x exp (2 kz) ux , = uz 1 exp (1kz) − z exp (2 kz)

f cR 1 kh |A| 2 e 8 h

1 + 1 ( 1 − 4 z ) + (1 z ) 2 1

(21)

1 + 2 2

.

(22)

For Poisson’s ratio = 0.3 the error of Ix = 2 wkin C in this approximation is smaller than 20% (10%; 5%) for fh/cT > 1.5 (1.9; 3.0). Special case: Bending mode: This is the fundamental anti-symmetric Lamb wave, which coincides with the bending wave in thin plates at low frequencies.

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Phase velocity c and group velocity C of the bending mode for Poisson’s ratio = 0.3. Units: cT for velocities, cT/h for frequency Low-frequency approximation (lowest order; corresponding to thin plate theory): √ 2 f h cT 2f , (23) , CB = 2 cB , kB = 4 6 (1 − ) cB = √ 4 6 (1 − ) h cT √ 2 h cT ŠB = √ , 4 6 (1 − ) √f

ux = −j kz, uz

wkin = 2 |A| 2 f 2 .

(24)

For Poisson’s ratio = 0.3 the error of Ix = 2 wkin C in this approximation is smaller than 20% (10%; 5%) for fh/cT < 0.13 (0.065; 0.007). (Forformulas with bending stiffness B see > Sect. Q.10.3) Low-frequency expansions up to the third non-vanishing order [Maysenhölder (1987)]: / 0 c2QL 489 − 418 + 62 2 17 − 7 2 2 4 2 1− c = (h k) (h k) + (h k) 12 60 ( 1 − ) 5040 ( 1 − ) 2 * √ 3 (17 − 7 ) cQL hf = √ (h f ) 1 − (25) 30 ( 1 − ) cQL 3 + 2 3711 − 3362 + 211 2 hf 2 . + cQL 4200 ( 1 − ) 2 The high-frequency approximation (≈ a Rayleigh wave on each plate surface) is the same as with the quasi-longitudinal mode. For Poisson’s ratio = 0.3 the error of Ix = 2 wkin C in this approximation is smaller than 20% (10%; 5%) for fh/cT > 0.67 (0.77; 3.0). For additional formulas and diagrams including z-dependence of displacements, energy densities and intensities see [Maysenhölder (1990)].

Elasto-Acoustics

Q.10.2

Q

1159

Rayleigh Waves

The wave speed cR of Rayleigh waves on a force-free surface of an isotropic half-space is obtained from the positive solution of the equation for ‚ 2 : ) ( 2 − ‚ ) − 4 ( 1 − …2 ‚ 2 ) ( 1 − ‚ 2 ) = 0, 2 2

cR ‚= , cT

2

… =

cT cL

2 =

1− 2 . 2− 2

(26)

There is exact one such solution within the bounds 0 < cR < cT . Eliminating the square root leads to the more familiar form ‚ 6 − 8 ‚ 4 + 8 ( 3 − 2 …2 ) ‚ 2 − 16 ( 1 − …2 ) = 0, which, however, has additional extraneous solutions [Achenbach, pp. 189–191 (1975)]. The non-vanishing displacement components of Rayleigh waves propagating along the x-direction on a half-space z ≤ 0 may be obtained by superposition of the two fundamental symmetric and anti-symmetric Lamb modes (for the ’s with c = cR see > Sect. Q.10.1): ˜ exp (1 kz) − x exp (2kz) , ux = j A ˜ exp (1kz) − z exp (2 kz) . u z = 1 A

(27)

This describes elliptical trajectories at arbitrary depth. The sense of rotation changes at a depth of about 0.2 wavelengths, where ux = 0. 1 ux 11 j ( 1 − x ) Displacement ratio at the surface: = ; (28) 1 uz z=0 1( 1 − z ) Time average of kinetic energy per unit width in y-direction: 0 Wkin =

wkin −∞

1 12 1 1 2 1 1 ˜ dz = cR f A +1 ( 1 − 4 z ) + (1 z ) + 2 ; 4 1 2

(29)

Time average of energy flow per unit width in y-direction: 0 I dz = 2 Wkin cR .

(30)

−∞

There are various approximations to the velocity of Rayleigh waves for the range 0 < 1 < , some of which are discussed in [Mozhaev (1991)]. The Bergmann-Viktorov 2 equation: ‚=

cR 0.87 + 1.12 = cT 1+

(31)

is accurate to within 0.5 %. For surface waves on anisotropic half-spaces see also [Ting/ Barnett (1997)].

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Q.10.3 Waves in Thin Plates Wave equations for thin plates with thickness h (thickness direction = z-direction) and mass density ; excitation terms like force densities, moment densities or pressure differences are omitted. Solutions: Phase velocities c = –/k of waves propagating along the x-direction (u, w: displacements in x- and z-direction) with phase factor exp[j(–t − kx)]. Nota bene: The symbols cQL and B have different meanings for plates and beams! The intensity I has dimension Wm−2 ; the mean energy flow per unit width in a plate is I · h. Quasi-longitudinal waves (in-plane waves), [Cremer/Heckl, p. 86 (1996)]: For a homogeneous isotropic plate: 2 E ∂2 u 2 ∂ u = 0, cQL = ; − 2 + cQL ∂t ∂ x2 (1 − 2) Transversal contraction:

h wˆ = . uˆ 1− Š

(32) IQL = 2 2 |u0 | 2 f 2 cQL

(33)

(w, ˆ uˆ : maximum transversal and longitudinal displacements; u0: amplitude of u). For range of validity see > Sect. Q.10.1. Bending waves: Inhomogeneous, locally monoclinic plate (x ≡ x1 , y ≡ x2 ), [Maysenhölder (1998)]: h

∂2 w + ∂ t2

2 ,,‚,ƒ=1

∂2 ∂ x ∂ x

B ‚ ƒ

∂2 w ∂ x‚ ∂ xƒ

=0

with generalised bending stiffnesses: C 3 3 C‚ ƒ 3 3 h3 B ‚ ƒ = C ‚ ƒ − ; 12 C3 3 3 3

(34)

(35)

Cijkl : elasticity tensor. , h and B‚ƒ may depend on x1 and x2 . Inhomogeneous, locally isotropic plate, [Pierce (1993)] : ∂2 w ∂2 B ∂2 w ∂2 B ∂2 w B ∂2 w ∂2 h + + +2 ∂ t2 ∂ x2 ∂ x2 ∂x∂y ∂x∂y ∂ y2 ∂ y2 ∂2 B ∂2 w B ∂2 w ∂2 ∂2 B ∂2 w + −2 =0 + ∂ x2 ∂ y2 ∂x∂y ∂x∂ y ∂ y2 ∂ x2

(36)

with the usual bending stiffness: B=

‹ h3 ‹ ( Š + ‹ ) h3 E h3 = = . 2 12 ( 1 − ) 6 ( 1 − ) 3 ( Š + 2 ‹)

(37)

Elasto-Acoustics

Q

Homogeneous orthotropic plate, [Heckl (1960)]: ∂4 ∂4 ∂4 ∂2 w + B + 2 B + B w = 0. h x xy y ∂ t2 ∂ x4 ∂ x2 ∂ y 2 ∂ y2

1161

(38)

The bending stiffnesses are, [Haberkern, personal communication], in terms of the Voigt constants: / / 0 0 h3 c2 h3 c13 c23 Bx = c11 − 13 , c12 + 2 c66 − Bxy = , 12 c33 12 c33 (39) / 0 h3 c223 c22 − By = 12 c33 and with engineering constants (see > Sect. Q.2): 0 0 / / Ex Œyx h3 h3 Ex Bx = , Bxy = , 2 Gxy + 12 1 − Œxy Œyx 12 1 − Œxy Œyx / 0 Ey h3 By = . 12 1 − Œxy Œyx

(40)

A plane wave propagating at an angle ¥ with the x-direction experiences a bending stiffness: B (¥ ) = Bx ( cos ¥ ) 4 + 2 Bxy ( cos ¥ sin ¥ ) 2 + By ( sin ¥ ) 4 .

(41)

1 Bx + 2 Bxy + By ; phase velocity as for an isotropic plate with B = B(¥ )). 4 The extremal values of B(¥ ) from 0◦ to 90◦ are B(0◦ ) = Bx , B(90◦ ) = By and

(B (45◦ ) =

B(¥e ) =

Bx By − B2xy Bx − 2 Bxy + By

with

In order that B(¥ ) > 0 for all ¥ : Transition to isotropic case with

¥e =

− Bx + By 1 (for real ¥e ). arccos 2 Bx − 2 Bxy + By

Bx > 0, By > 0 , Bxy > − Bx By . Bx = Bxy = By = B =

Homogeneous isotropic plate – classical theory: 2 2 ∂2 w B √ ∂2 ∂ h +B + w = 0, cB = 4 – ∂ t2 ∂ x2 ∂ y 2 h ‹ h3 ‹ ( Š + ‹ ) h3 E h3 = = : 12 (1 − 2 ) 6 ( 1 − ) 3(Š +2‹) √ √ E – h cT cB = 4 , – h = √ 2 4 6 (1 − ) 12 ( 1 − ) E h 2 cT h/f 4 . = √ ŠB = 4 6 (1 − ) 3 (1 − 2 ) f

3

Eh . 12 (1 − 2)

(42) (43) (44)

(45)

B =

with

(46)

1162

Q

Elasto-Acoustics

The accuracy of cB is better than 10 %,if the bending wavelength ŠB > 6h (Cremer-Heckl limit [Cremer/Heckl, p. 162 (1996)]). The non-propagating solutions with imaginary speed ±jcB and imaginary wavenumber ±jkB are called nearfields. Amplitude ratio:

h uˆ = ; wˆ ŠB

(47)

Bending wave intensity:

IB = 2 2 |w0 | 2 f 2 CB ;

(48)

Group velocity:

CB = 2cB

(49)

(u, ˆ w: ˆ maximum longitudinal and transversal displacements; w0 : amplitude of w). Measurement of the intensity of bending waves along the x-direction (without nearfields) by two accelerometers according to : Bh ∂ a∗ ∂2 w ; (50) Re a dt , a = IB = − –h ∂x ∂ t2 ∂ a∗ ∂ x being approximated by a finite difference. The error of IB relative to the exact result Ix (see > Sect. Q.10.1): ƒ=

IB − Ix Ix

σ σ

σ σ

ƒ is shown in the diagram for various Poisson ratios (frequency in units of cT /h) [Maysenholder ¨ (1990)] Critical frequency f cr , where cB = c0 (phase velocity in an ambient fluid): c20 c20 h 3 ( 1 − 2) = fcr = 2 B h E

(51)

Elasto-Acoustics

Coincidence frequency

fc =

Q

fcr , sin2 Ÿ

1163

(52)

where the trace velocity c0/sin Ÿ of a plane wave incident on the plate with polar angle Ÿ equals the wave speed cB of free bending waves. – fcr 2 – 4 . (53) Wavenumber relations: kB = = = 6 (1 − ) = k0 ŠB cB h cT f Homogeneous isotropic plate – Timoshenko-Mindlin model, [Beltzer, p. 160 (1988)]: (including rotatory inertia and transverse shear effects) ∂2 w h3 ∂ 2 ∂2 ∂2 ∂2 2 2 2 h 2 + B∇ − − = + (54) ∇ w = 0, ∇ ∂t 12 ∂ t2 ‰2 ‹ ∂ t2 ∂ x2 ∂ y 2 with ‹: shear modulus; ‰: factor near unity. For a wave propagating in x-direction: ∂2 w B ∂4 w h2 ∂ 4 w B h2 ∂4 w + − + = 0. (55) + 2 4 2 2 2 ∂t h ∂x ‰ ‹ h 12 ∂ x ∂ t 12 ‰2 ‹ ∂ t4 The compact form of the dispersion relation: 2 cQL 12 c2 −1 = 1− 2 c‰ c2 (kh)2

(56)

with c2‰ = ‰2 c2T = ‰2 ‹/ may be transformed to quadratic equations for c˜2 = c2 /c2QL : 12 1 1 12 c2QL 1 c˜4 2 4 − 2 +1+ − c˜ + 1 = 0, c˜ − 2 + 1 c˜2 + 1 = 0 (57) c˜2‰ c˜‰ (kh)2 c˜2‰ (–h)2 c˜‰ with c˜‰ = c‰ /cQL . The smaller root c2 of the dispersion relation is the desired solution. The choice c‰ = cR (Rayleigh velocity) assures the correct value in the limit of high frequencies (‰ = 0.925 for steel).

Q.10.4 Waves in Thin Beams Wave equations for thin, straight, isotropic, homogeneous beams with cross-sectional area A. Young’s modulus E, Poisson’s ratio and mass density ; excitation terms like forces or moments are omitted. Solutions: Phase velocities c = –/k of waves propagating along the x-direction (= beam axis) with wavelengths ŠQL or ŠB and phase factor exp[j(–t − kx)]. Nota bene: The symbols cQL and B have different meanings for plates and beams! The intensity I has dimension Wm−2 ; the mean total energy flow in a beam is I · A. Quasi-longitudinal waves, [Cremer/Heckl, p. 82–85 (1996)]: 2 ∂ u E ∂2 u , IQL = 2 2 |u0 | 2 f 2cQL = 0, cQL = − 2 + c2QL ∂t ∂ x2

(58)

1164

Q

Elasto-Acoustics

u: longitudinal displacement; u0: amplitude of u. Valid for arbitrary cross-section, if greatest thickness d ŠQL . (For d >> ŠQL , c = cL .) Lateral contraction for quadratic or circular cross-section: d wˆ = uˆ ŠQL

(59)

(w, ˆ u: ˆ maximum lateral and longitudinal displacements). Torsional waves, [Cremer/Heckl, pp. 90–94 (1996)]: 2 T ∂2 ¥ 2 ∂ ¥ + c¥ = 0, c¥ = , I¥ = 2 2 |¥0 | 2 Ÿf 2c¥ ; − 2 2 ∂t ∂x Ÿ

(60)

¥ : angle of rotation of a cross-section; ¥0: amplitude of ¥ ; T: torsional stiffness (rigidity) defined by Mx = T · ∂ ¥ /∂x for arbitrary cross-section (Mx : torsional moment), Ÿ: moment of inertia per unit length about the x-axis. For a hollow circular cylinder with inner radius ri and outer radius ro : ‹ T = ‹ ro4 − ri4 , Ÿ = ro4 − ri4 , c¥ = = cT ; (61) 2 2 (‹: shear modulus). This is the exact solution for arbitrary frequency (lowest torsional mode) including the solid cylinder (ri = 0), Gazis (1959). Cross-sections without rotational symmetry do not remain plane (‘warping’). Therefore the above torsional wave solution is only approximate; it always yields c¥ < cT . Results for rectangular cross-sections with dimensions a ≥ b: a b 3 + a 3 b A2 a b b Ÿ = = + , T = s ‹ a b3 = s ‹ A 2 12 12 b a a

a/b

1

1.5

s

0.141

0.196

c¥

0.920 cT

2

3

0.229 0.263 3s b 2 cT a 1 + (a=b)−2

(62)

6

10

1

0.298

0.312

1/3 2

b cT within 7 % for a/b ≥ 6. a Bending waves – Bernoulli-Euler model, [Cremer/Heckl, pp. 95–99 (1996)] : B ∂4 w B √ ∂2 w + = 0, cB = 4 –, IB = 2 2 |w0 | 2 f 2 CB 2 4 ∂t A ∂x A

b cT a

Hence c¥ ≈ 2

(CB = 2cB).

(63)

Elasto-Acoustics

Q

1165

w: displacement along z-direction (displacement vector is in x-z-plane); w0 : amplitude of w; B = EJ: bending stiffness; J: second moment of cross-sectional area A about the neutral axis: 2 J = z dy dz with z dy dz = 0 A

A

(Select z = 0 accordingly for the neutral axis.). Hollow circular cylinder: A 2 4 ro − ri4 = ro + ri2 , J= 4 4 Rectangular cross-section: h3 b , J= 12

E (ri2 + ro2 ) √ cB = 4 –. 4 E h2 √ cB = 4 – 12

(64)

(65)

(h: thickness in the z-direction, b: thickness in the y-direction). For a solid cylinder (ri = 0) the Bernoulli-Euler model agrees with the exact solution only for ro /ŠB < 0.1 [Beltzer, pp. 155–156 (1988)]. Bending waves – Timoshenko model, [Junger/Feit, pp. 201–205 (1986)]: (including rotatory inertia and transverse shear effects) B ∂4 w J ∂4 w B J ∂2 w ∂4 w + − + 2 = 0, + 2 4 2 2 2 ∂t A ∂x ‰ ‹A A ∂x ∂t ‰ ‹ A ∂ t4

(66)

‰: factor near unity depending on the shape of the cross-section. The smaller root c2 of the dispersion relation: c2QL A 1 4 − − + 1 c2 + c2QL = 0 or c c2‰ J –2 c2‰ c2QL c4 A − + 1 − 2 c2 + c2QL =0 2 2 c‰ c‰ Jk

(67)

with c‰ = ‰ cT , is the desired solution. The Rayleigh velocity cR is a convenient choice for c‰ , which is the high frequency limit of c (‰ = 0.925 for steel). For rectangular crosssection (A/J = 12/h2 ) the dispersion relation is the same as for Timoshenko-Mindlin plates, however, with the different definition of cQL for plates (see > Sect. Q.10.3).

Q.11 Moduli of Isotropic Materials and Related Quantities Notice: Some notations in this and the following Sections (written by F.P. Mechel) are different from corresponding notations in the previous Sections of this chapter (by W. Maysenhölder).

Q

1166

Elasto-Acoustics

σ σ

σ

σ σ

σ σ

σ

σ

Co-ordinates of a point: displacement: strain: force: stress: pressure:

—ii = div u

;

i

r = {x, y, z} = {xi } u = {ui } = {ƒxi } — = {—ik } F = {Fi ) = { ik } p = −ii ∗) 1 ∂ui ∂uk —ik = + 2 ∂xk ∂xi

Lamé constants Š, ‹ : ∂ui ∂uk = 2‹ · —ik + ik = ‹ ∂xk ∂xi

;

(1)

i = k ,

(2)

ii = Š · div u + 2 ‹ · —ii . With losses:

Š → Š(1 + j†)

(3) ;

‹ → ‹(1 + j†)

Shear modulus S : = F/A S=‹=

;

S = / ,

(4)

E . 2 (1 + )

(5)

α

Free shear wave velocity ( = material density): ∗)


cS =

S/ .

(6)

Elasto-Acoustics

Q

1167

Young’s modulus E : (7)

− = F/A = E · L/L = E · sxx

Lateral contraction: = −—xx /—yy , E=

(8)

‹ (3Š + 2‹) = 2‹(1 + ) = 2S (1 + ) . Š+‹

(9) cE =

Free bar longitudinal wave velocity:

E/ .

(10)

Poisson’s lateral contraction (Poisson ratio): =−

—yy E Š = −1 = —xx 2(Š + ‹) 2S

;

−1 < < 0.5 ,

‹ 1 − 2 = . Š 2

(11)

Compression modulus K : K=−

p , dV/V

2 K=Š+ ‹=S 3

(12) /

0 2 E 1 2 + = = , 3 1 − 2 3 (1 − 2) C

(13)

C = compressibility.

Dilatation modulus D: (1-dimensional deformation) p = −D · —xx , D = Š + 2‹ = 2S

(14) 1− 1 1− 1− =E = K. 1 − 2 (1 + ) (1 − 2) 3 1 +

(15)

1168

Q

Elasto-Acoustics

Free dilatational wave velocity:

cD =

D/ .

(16)

Bar bending modulus Bst : BSt = E

b · h3 . 12

Free bar bending wave velocity:

(17) cBSt = 4 –2 BSt /m

(18)

(m = mass per bar length).

Plate bending modulus B : B=E

h3 h3 ‹ (Š + ‹) h3 = S = . 12 (1 − 2 ) 6 (1 − ) 3 Š + 2‹

Free plate bending wave velocity:

(19) cB = 4 –2 B/m .

Relations for free plate bending velocity cB : √ ) √ E 2 f cE h 4 4 2 = 1.391 f cE h = . cB = – B m = 1.347 f h 4 12 (1 − 2) (1 − 2)

(20)

(21)

Elasto-Acoustics

Q

Relations for free plate bending wave number kB = –/cB (for h≤ ŠB /6): – 2 √ – 4 12 (1 − 2) 4 12 (1 − 2) = 1 k h = = – 4 m /B = kB = E cB ŠB cE h h f = k0 fcr /f −−−−−→ 4.515 =0.35 cE h

1169

(22)

(f cr = critical frequency) Relations at coincidence: The free plate bending wave speed cB agrees with the trace speed c0/ sin ˜ of a plane wave incident on the plate with a polar angle ˜ . 2 12 (1 − 2) c20 (23) Coincidence frequency: fc = fcr sin ˜ = 2 E 2 h sin ˜ Critical frequency (at ˜ = /2; c0 = cB ): 2 2 c c c20 m 12 (1 − 2 ) 12 (1 − 2 ) = 0 = fcr = 0 2 B 2 h E 2 h cE

;

Šcr = c0 /fcr

) kB = k0 fcr f

(24) (25)

Effective bending moduli for sandwich panels : (sheets and boards with subscripts 1,2 ; adhesive layers without subscript) Table 1 Sandwich panels No.

Sandwich

Connection

1

fix connection

2

fix connection

3

δ

4

δ δ

connection with shear 3:5 10−3 G − 1:3 10−12 [m] ƒ h1 E1 E2 h2 2 107 [Pa m] connection with shear 0:25 10−3 G ƒ [m] h1 E1 hi in [m]; G, Ei in [Pa]

1170

Q

Elasto-Acoustics

Table 2 Effective bending moduli B for sandwiches from Table 1 No.

1

B = B2

Effective bending modulus * 2 3 + 2 4 h1 E1 h1 E1 h1 h1 +3 1+2 +2 2 + E2 h2 h2 h2 h2 E2 1+

h1 E1 h2 E2

h2 3 B B1 1 + + B2 h1 h2 E1 h1 E2 h2 (h1 =2 + h2 =2)2 g B B1 + B2 + 3Gƒ 1 + 4 E1 h1 + g (E1 h1 + E2 h2 )

2

3

E1 h21

+ 2gE2 h1 h2 h1 + h2 − 3Gƒ 4 E1 h1 + g (E1 h1 + E2 h2 ) (h1 =2)3 B = B2 + 2E1 12(1 − 12 )

4

Remark

ƒ h1 g=

G B=m ƒE1 h1 –

Q.12 Modes of Rectangular Plates

See also: Mechel, Vol. II, Ch. 27 (1995); Gorman (1982)

A rectangular plate in the x,y-plane has the dimensions a in x-direction and b in ydirection; its shape factor is ß = b/a. The non-dimensional co-ordinates are = x/a, † = y/b. Two oppo-site borders are supposed to be simply supported, the other borders may have different supports. Distinguish between the wave number kB of the “free bending wave” (in an infinite plate) and kb the bending wave number in the finite, supported plate. Used abbreviations (with integer m): (1) ‹m = m ; ‚m = |(m)2 − (kb a)2 | ; ƒm = (m)2 + (kb a)2 . Table 1 Classical boundary conditions for plates Fixation Simply supported S

Clamped C

Condition @ 2 u(; 1) u(; 1) = =0 @†2

u(; 1) =

Symbol

η

@u(; 1) =0 @†

η Free F

@2 u @2 u + ß2 2 = 0 2 @† @ @3 u @3 u + (2 − )ß2 =0 3 @† @†@2

η

Elasto-Acoustics

Q

Bending wave equation for the plate displacement u( , †): 0 / 4 4 ∂4 ∂ 4 ∂ 4 4 + 2ß 2 2 + ß − ß (kb a) u( , †) = 0 ∂† 4 ∂† ∂ ∂ 4

1171

(2)

Supposed the plate is simply supported at = 0 and = 1, the plate displacement field can be formulated as: u( , †) =

∞

Ym (†) · sin (m )

(3)

m=1

with the wave equation after insertion: 0 / 4 2 d 2 2 d 4 4 4 − 2ß ‹ + ß (‹ − (k a) ) Ym (†) = 0. b m m d† 4 d† 2

(4)

General solutions (with yet undetermined Am , Dm ): 2 > (kba)2 : ‹m

i.e.

m > a/(Šb /2) ,

Ym (†) = Am · cosh (ß †

2 + (k a)2 ) + B · sinh (ß † ‹ 2 + (k a)2 ) , ‹m b m b m

+ Cm · cosh (ß † 2 < (kba)2 : ‹m

i.e.

2 − (k a)2 ) + D · sinh (ß † ‹m b m

(5)

2 − (k a)2 ) , ‹m b

m < a/(Šb /2) ,

Ym (†) = Am · cosh (ß †

2 + (k a)2 ) + B · sinh (ß † ‹ 2 + (k a)2 ) , ‹m b m b m

+ Cm · cos (ß †

(6)

2 ) + D · sin (ß † (k a)2 − ‹ 2 ) . (kb a)2 − ‹m m b m

Eigenvalues must be found for kb a; they follow from the boundary conditions. These are: simply supported (S):

Ym (†) =

d2 Ym (†) = 0, d† 2

(7)

clamped (C):

Ym (†) =

dYm (†) = 0, d†

(8)

free (F):

2 ∂2u 2 ∂ u + ß =0 ∂† 2 ∂ 2

;

3 ∂ 3u 2 ∂ u + (2 − )ß = 0. ∂† 3 ∂† ∂ 2

(9)

The boundary conditions give a system of homogeneous equations for the amplitudes; for a non-trivial solution the determinant must vanish; this is the eigenwert equation for kb .

Q

1172

Elasto-Acoustics

Cases: (SSSS): Ym (†) = Am,n · sin(n†) .

(10)

Determinant equation: (‚m2 + ƒ2m) · cosh S S η

‚m ƒm · cos =0. 2 2

(11)

ξ S

S

Solutions: (kb a)2 = (m)2 +

(n)2 ß2

;

m, n = 1, 2, . . . .

(12)

(SCSS): Ym (†) = − S

(13)

ξ S

S η

sin ‚m · sin (‚m †) · sinh (ƒm †) . sinh ƒm

C

Determinant equation: ‚m · sinh ƒm · cos ‚m − ß cosh ƒm · sin ‚m = 0 .

(14)

The equation must be solved numerically for kb a. (SCSC): Symmetrical modes (m = 1, 3, 5. . .): Ym (†) = − C S η

cos (‚m /2) · cos (‚m †) · cosh (ƒm †) . cosh (ƒm/2)

ξ S

C

(15)

Elasto-Acoustics

Q

1173

Determinant equation: ‚m · cosh (ƒm/2) · sin (‚m /2) + ƒm sinh (ƒm/2) · cos (‚m /2) = 0 .

(16)

Anti-symmetrical modes (m = 2, 4, 6. . .): Ym (†) = −

sin (‚m /2) · sin (‚m †) · sinh (ƒm †) . sinh (ƒm /2)

(17)

Determinant equation: ‚m · sinh (ƒm/2) · cos (‚m /2) − ƒm cosh (ƒm /2) · sin (‚m /2) = 0 .

(18)

Both equations must be solved numerically for kb a . (SFSS): (kb a)2 > (m)2 : Ym (†) = sin (‚m †) +

S S η

(‚m2 + ß2 m2 2 ) sin ‚m · sinh (ƒm †) . (ƒ2m − ß2 m2 2 ) sinh ƒm

(19)

ξ S

F

Determinant equation ( = Poisson ratio): ‚m (‚m2 + (2 − )ß2 m2 2 )(ƒ2m − ß2 m2 2 ) · sinh ƒm · cos ‚m − ƒm (ƒ2m − (2 − )ß2 m2 2 )(‚m2 + ß2 m2 2 ) · cosh ƒm · sin ‚m = 0 ,

(20)

(kb a)2 < (m)2 : Ym (†) = sinh (‚m †) −

(‚m2 − ß2 m2 2 ) sinh ‚m · sinh (ƒm †) . (ƒ2m − ß2 m2 2 ) sinh ƒm

(21)

Determinant equation: ‚m (‚m2 − (2 − )ß2 m2 2 )(ƒ2m − ß2 m2 2 ) · sinh ƒm · cosh ‚m − ƒm (ƒ2m − (2 − )ß2 m2 2 )(‚m2 − ß2 m2 2 ) · cosh ƒm · sinh ‚m = 0 .

(22)

(CCCC): All sides clamped. Approximate resonance frequencies, after [Mitchel, Hazell (1987)]: * 2 2 + 2 4 B m + n + m n + (23) –2m,n = h a b

1174

Q

Elasto-Acoustics

with ‘edge effect factors’: ! 2 "−1 + 0.17/m m = 2 + na mb

;

! 2 "−1 n = 2 + ma nb + 0.17/n

(24)

and: h= thickness; a, b = dimensions; = mass density; B = bending stiffness. In the case of simply supported plates: m = n = 0. More combinations of boundary conditions in [Gorman (1982)]. C C

C C

Q.13 Partition Impedance of Plates


The partition impedance ZT is a useful quantity in boundary value problems. It displays its full usefulness if the plate is homogeneous, i.e., has no ribs etc., and is either infinite or at least so large that border effects can be neglected in the given task. Then the sound fields on both sides can be supposed to have the same distribution along the plate. Suppose a Cartesian co-ordinate system x, y, z with the plate in the plane x, y at the position z = …, and the z axis directed from the front side to the back side. The partition impedance for a plate is defined by: ZT =

pfront (x, y, …) − pback (x, y, …) . vplate (x, y)

(1)

Be pfront = pe + pr the sum of an incident wave pe and a reflected wave pr , and pback = pt the transmitted wave. All waves = e, r, t may have the distributions p (x, y, z) = P · X(x) · Y(y) · Z (z),

(2)

and also the plate velocity has the distribution vp (x, y) = Vp · X(x) · Y(y). Thus the profile X(x) · Y(y) cancels in ZT . It is supposed that the waves p satisfy the wave equation, Sommerfeld’s far field condition, the source condition (if a source exists) and, possibly, boundary conditions at other boundaries than the plate. There exist three boundary conditions at the plate: ! pe + pr − pt z=… = ZT · vp , (3) ! ! ve,z + vr,z z=… = vp =! vt,z z=… , wherein v,z , if the plate is in contact with air, follows from: v,z (x, y, …) =

j · gradz p (x, y, …) , k0Z0

(4)

Elasto-Acoustics

Q

1175

and if the plate is in contact with a porous absorber, from: v,z (x, y, …) =

−1 · gradz p (x, y, …) .

a Za

(5)

The plate has to satisfy the bending wave equation:

j– · ƒp, x,y x,y − kB4 vp = B

(6)

in which is the Laplace operator in the indicated co-ordinates, kB is the wave number of the free bending wave on the plate, B is the bending stiffness, and ƒp = pfront − pback is the driving sound pressure difference. With the relations k0 f 4 2m kB = – = ; , (7) B kB fcr in which – = 2f is the circular frequency, m the surface mass density of the plate, f cr the critical (coincidence) frequency, one immediately gets: * + 2 m 1 x,y x,y vp ZT f = j k0 · 1− · , (8) Z0 0 fcr vp k04 or alternatively: * + 2 m f ZT 4 = j k0 · 1− sin ” , Z0 0 fcr

(9)

where the last two fractions in the brackets are replaced by the sine function of an effective angle of sound incidence ” of the incident wave pe (defined below). Energy dissipation in the plate can be taken into account by a loss factor † introducing a complex modulus B → B · (1 + j†). This leads to: * 2 + 2 4 ZT f 2 f 4 2 = Zm F · † F sin ” + j 1 − F sin ” = Zm F · † +j 1− (10) Z0 fc fc with

Zm =

–cr m Z0

;

F=

f , fcr

(11)

where Zm is the normalised inertial impedance of the plate at the critical frequency (–cr = 2fcr ), and fc is the coincidence frequency at the incidence angle ” , with fcr = fc · sin2 ”. It remains to determine: sin4 ” =

1 x,y x,y vp 1 x,y x,y vz (x, y, …) , · = 4· 4 vp vz(x, y, …) k0 k0

(12)

where the last form makes use of the boundary condition that the pattern of the waves p at the plate agrees with that of vp . After this determination all wave equations and all boundary conditions are satisfied, therefore the waves p make up a solution of the task.

1176

Q

Elasto-Acoustics

Set x = x1 , y = x2 , X(x) = X1(x1 ), Y(y) = X2 (x2 ) and suppose the wave factors Xi (xi ) to have one of the forms: Xi (xi ) = e±j kxi xi

or

cos (kxi xi )

or

sin (kxi xi )

(13)

or a linear combination thereof. Then 2 2 2 k + k x y v (x, y, …) 1 x,y x,y z sin4 ” = 4 · = . vz(x, y, …) k0 k04

(14)

This corresponds to the possibility,which always exists,to transform the secular equation k02 = kx2 + ky2 + kz2 to a form: (15) 1 = (kx /k0)2 + (ky /k0)2 + (kz /k0)2 = sin2 ” + cos2 ”, where ” evidently is the polar angle of incidence of pe on the plate. This holds also, if some or all of the kx , ky , kz are complex. The plates need not be thin in the sense of “thin plate theory”; it is important, that the compressibility of the plate normal to its surface is negligible.For a Timoshenko-Mindlin plate (with shear stress and rotational inertia) the result for ZT is: ZT = j–m

kB4 − (ks2 − kL2 )(ks2 − kR2 ) kB4 − kR2 (ks2 − kL2 )

;

kR2 =

12 2 k 2 s

;

ks2 = kx2 + ky2

(16)

with the characteristic wave numbers kB , kL , kS of the free bending wave with the bending stiffness B, of the longitudinal wave with the plate dilatational stiffness D, and of the shear wave with the shear stiffness S . An equivalent form is: / 0 (kx2 + ky2 )2 h2 c2L –2 2 2 1− + + k ) 1 + (k − x y 12 kB4 c2T c2T (17) ZT = j–m 0 / h2 c2 –2 1+ (kx2 + ky2) 2L − 2 12 cT cT with the plate thickness h and the speeds cL = E/ of the longitudinal wave and cT = S/ of the shear wave. An approximation has the form: * + (kx2 + ky2 )2 , ZT = j–m 1 − kˆ B4 (18) 2 2 2 2 2 2 2 4.43 m 4.43 – – 0.26 – m h m h – m h − + −1 . kˆ B2 = 24 B 24 B B E

Q.14 Partition Impedance of Shells


For fundamental considerations about the partition impedance ZT see the previous > Sect. Q.13.

Elasto-Acoustics

Q

1177

Circular cylindrical shell: In cylindrical co-ordinates r, ˜ , z the sound field near the shell and the vibration velocity of the shell with radius r = a (if necessary after expanding the incident wave in cylindrical waves) be: p(r, ˜ , z) = R(r) · T(˜ ) · U(z)

;

vp (a, ˜ , z) = A · T(˜ ) · U(z).

(1, 2)

The axial function U(z) may be one of (or a linear combination) of the terms below. The factor R(r) may be one of the cylinder functions and T(˜ ) a trigonometric function (or a linear combination): ⎧ ⎪ Jm (kr r); i=1 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ e±j kz z ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i=2 ⎨ cos (m˜ ) ⎨ ⎨ Ym (kr r); cos (kz z) . (3) ; T(˜ ) = ; U(z) = Z(i) m (kr r) = (1) ⎪ ⎩ ⎪ H (k r) ; i = 3 ⎪ ⎪ r sin (m˜ ) ⎪ ⎪ ⎪ sin (k z) ⎪ m ⎩ ⎪ z ⎪ ⎪ (2) ⎪ H (kr r); i = 4 ⎩ m

The Laplace operators in cylindrical co-ordinates are: =

1 ∂2 1 ∂ ∂2 ∂2 + + + ∂r2 r ∂r r2 ∂˜ 2 ∂z2

;

˜ ,z =

1 ∂2 ∂2 + . a2 ∂˜ 2 ∂z2

(4)

The wave equation is satisfied by the above field factors, if the secular equation k02 = kz2 + kr2

;

1 = (kz /k0)2 + (kr /k0)2 = sin2 Ÿ + cos2 Ÿ

(5)

holds. The angle Ÿ is between the wave vector and the radius. The two-dimensional Laplace operator together with the Bessel differential equation for the Z(i) m (kr ) gives: 2 m ˜ ,z p(a, ˜ , z) = − + kz2 · p(a, ˜ , z). (6) a2 Therefore: 1 ˜ ,z ˜ ,z vp 1 sin ” = 4 = 4 vp k0 k0 4

m2 + kz2 a2

2

=

m2 + sin2 Ÿ (k0a)2

2 .

(7)

With this quantity the partition impedance ZT can be evaluated from the previous > Sect. Q.13, Eqs.(9), (10). Because T(˜ ) is orthogonal over 0 ≤ ˜ ≤ 2 for different values of m , the boundary conditions at the shell hold term-wise, if p(r, ˜ , z) is a sum of multi-pole terms. Spherical shell : Suppose spherical co-ordinates r, ˜ , œ and a shell with radius r = a. The field near the shell and the shell vibration velocity have the forms (if necessary after expanding the incident wave in spherical waves): p(r, ˜ , œ) = R(r) · T(˜ ) · P(œ)

;

vp (a, ˜ , œ) = A · T(˜ ) · P(œ)

(8)

Q

1178

Elasto-Acoustics

with spherical Bessel functions for R(r) and associated Legendre functions for T(˜ ) (or linear combinations thereof): ⎧ ⎪ jm (k0r) ⎪ ⎪ ⎪ ⎧ ⎪ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ cos (nœ) ⎨ Pnm (cos ˜ ) ⎨ ym (k0 r) ; P(œ) = . (9) ; T(˜ ) = R(r) = n ⎩ ⎪ ⎪ h(1) (cos ˜ ) ⎩ Qm ⎪ m (k0 r) sin (nœ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ h(2) m (k0 r) The Laplace operators are:

=

∂2 2 ∂ 1 ∂2 1 ∂2 1 ∂ + + 2 , + 2 + 2 2 2 2 ∂r r ∂r r ∂˜ r tan ˜ ∂˜ r sin ˜ ∂œ2

∂2 1 1 ∂2 1 ∂ ˜ ,œ = 2 + + a ∂˜ 2 a2 tan ˜ ∂˜ a2 sin2 ˜ ∂œ2

(10)

and, with the above separation: ˜ ,œ p(a, ˜ , œ) = −

m(m + 1) · p(a, ˜ , z). a2

Therefore: 1 ˜ ,z ˜ ,z vp sin ” = 4 = vp k0 4

m(m + 1) (k0a)2

(11)

2 .

(12)

With this quantity the partition impedance ZT can be evaluated from the previous > Sect. Q.13, Eqs.(9), (10). Because T(˜ ), P(œ) are orthogonal over (0, 2) for different values of m or n, respectively, the boundary conditions at the shell hold term-wise, if p(r, ˜ , œ) is a sum of multi-pole terms. Because of Pnm (cos ˜ ) ≡ 0; n > m, the angle of incidence is ” = 0 for the “breathing sphere” m = n = 0, which is plausible.

Q.15 Density of Eigenfrequencies in Plates, Bars, Strings, Membranes Be n the number of eigenfrequencies in an interval of 1 Hertz. √ String of length : n = 2 m/T

(1)

(m = mass per string length; T = string tension) Longitudinal wave on a bar of length : n = 2 /E

(2)

( = material density; E = Young’s modulus) Bending wave on a bar of length :

n = 4 m/(–2 B)

(m = mass per bar length: B = bar bending modulus)

(3)

Elasto-Acoustics

Q

1179

1 n = S m/B 2

Plate, simply supported:

(4)

(S = plate area; m = surface mass density; B = plate bending stiffness) n=

Circular membrane:

Sd · f T

(5)

(S = membrane area; d = membrane thickness; = material density; T = tension per unit length of circumference; f = frequency) Tube of length with outer diameter 2a and wall thickness d (simply supported at the ends; = material density; E = Young’s modulus): ⎧ ⎪ ⎪ 5 4 3 √ 3 ⎪ ⎪ –a ⎨ 2 E3 d n= ⎪ ⎪ ⎪ 3 a ⎪ ⎩2 E d

;

–<

;

–>

E/ a

(6)

E/ a

Q.16 Foot Point Impedances of Forces

See also: Fahy (1985); Cremer/Heckl (1996)

Foot point impedances Z of structures for external forces are defined as ratios of • force of a point source to structure velocity at the point of attack, • force per length of a line source to average structure velocity at the line of attack, • force per area of an area source to average structure velocity in the area. It is advantageous to introduce the foot point admittance G = 1/Z. Be F the force of a point source or the constant force of a line source, be v(0, 0) the structure velocity at the foot point (0, 0) of a point source, and v(0) the structure velocity at the line of excitation. The real part of the admittance can always be written as: Re{G} = Re {F0 /v(0, 0)} = Re{GL } = Re {F/v(0)} =

1 – Vq

1 – Sq

point force,

(1)

line force.

(2)

The quantity Vq with the dimension of a volume is called source volume, the quantity Sq with the dimension of an area is called source area. The Table 1 collects values of the foot point admittance G and the source volume Vq for a point force on several objects. The arrows indicate the direction of the force.

1180

Q

Elasto-Acoustics

h

= thickness of plate or membrane;

A m m

= = = =

cross section of bar; material density; mass per unit length of bar; mass per unit area of plate;

cL = longitudinal wave velocity; cB = bending wave velocity; cT = torsional wave velocity; ŠL = ŠB = ŠT = F = cF T B

= = = =

longitudinal wave length; bending wave length; torsional wave length; fluid density; fluid sound speed; tension of a membrane; plate bending stiffness; Poisson ratio

Table 1 Foot point admittances Re{G}

Im{G}

Vq

−1 4m0 cB −1 m0 cB

AŠL (2) 4AŠB (2) AŠB (2)

Eq. (3)

–

Plate, thin

Eq. (3) – (4T 0 ) 2 p 1 (8 Bm00 ) 2 p 1 (3:5 Bm00 )

Plate with shear stiffness

Object Bar Bar, thin Bar, thin Bar, Timoshenko

1 (cL A) 1 4m0 cB 1 m0 cB

0

0

hŠ2 2 2hŠ2B 2

0

0:9 hŠ2B 2

Eq. (5)

Eq. (5)

–

Plate with tangential force

–

hŠ2T (32 )

Plate, orthotropic

Eq. (7)

0

–

Plate on elastic bed

Eq. (10)

Eq. (10)

–

Membrane Plate, thin

1

2–/(Eh)

Elasto-Acoustics

Q

1181

Table 1 (continued) Re{G}

Object

Im{G}

Vq

Strip of plate

Eq. (11)

Eq. (11)

–

Tube

eqs. (12,13)

eqs. (12,13)

–

Elastic half space

eqs. (14,15)

eqs. (14,15)

(ŠT =)3

Fluid half space

–2 6F c3F

1

3Š3F =(42 )

Thick plate

eqs. (16, 17)

eqs. (16, 17)

–

Timoshenko bar with central excitation: G=

1 kT2 + kI kII 2–m kI + kII

2 kT2 + kL2 + kB2 , 2 2 2 kT2 + kL2 kT + kL2 2 + kB2 , − kII = 2 2

k 2 + kL2 kI2 = T + 2 with:

(3)

(4)

= 1 − kT2 h2 /12 . Plate with shear stiffness: /

G= 8–m

kB4 0 AR + j A I 1 kI2 + (kT2 + kL2 2

(see Eq. (4) for⎧kI , kII, ), and ⎪ + kT2 2kI2 − kL2 − kT2 /kB4 ; > 0 ⎪ ⎪ ⎨ , AR = ⎪ ; –

Elasto-Acoustics

Q

1183

Foot point admittance of a strip of plate with width ls and thickness h for a point force at z = 0, x = x0 : + * ∞ j 1 1−j 1 2 + − G= œn (x0 ) ; 2hls cB 2 1 − ‰n2 1 + ‰n2 n=1 ‰n = n/(kBls )

;

(11)

= (œ0 (x0 ))2;

œn (x) = cos (nx/ls )

;

n = 0, 1, 2, . . . .

Foot point admittance of a tube with outer radius a and wall thickness h: 0 / ) √ −1 ; Œ = –a cL < 0.77 h/a; G ≈ (1 − j) 2ah –cL a/ 2 ⎧ ⎪ ⎨

0.66 ) –a cL 2.3 cL h2 Re{G} ≈ ⎪ −1 ⎩ 2.3 cL h2

;

0.77 h/a < Œ < 0.6

;

Œ>2

.

(12)

(13)

Foot point admittance of a force acting in a small circle with radius a on an elastic half space: –kT (14) (1 − ) 0.19 + j 0.3/(kTa) . S Foot point admittance of a force acting in a strip of width b on an elastic half space:

– (15) G ≈ (1 − ) 0.463 + j 1.5 ln 1.9 − 15( − 0.25)2 kT b . S Foot point admittance of a force acting in a small circle with radius a on a thick plate: , . 2 –kT 0.063 1 0.001 ŠT H G≈ ; H = kT h 2 . (16) + + j 0.06 + 1.3 2 S H 8 1.6 + H H 2a G≈

Foot point admittance of a force acting in a strip of width b on a thick plate: , 2 H 1 – + 0.31 G≈ S 8 H1.5 1.6 + H . −1 +j + 0.16 ln(ŠT /b) ; H = kT h/2 8 H1.5

(17)

Foot point admittance of a point force acting on an isotropic, thin plate with bending stiffness B and membrane stress T: √ * + j 1 T 1 + ß2 + ß ln √ 1+ G= √ ; ß= . (18) √ √ 2 2 2– Bm 8 1 + ß Bm 1+ß −ß Foot point admittance of a point force acting in the centre of a bar of length , width w, thickness h, simply supported at both ends: G=

j– (tan (kB ) − tanh (kB )) 4E I kB3

;

I = wh3 /12 .

(19)

1184

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Q.17 Transmission Loss at Steps, Joints, Corners

See also: Cremer/Heckl (1996)

Bars or plates i = 1, 2, . . ., are joined to each other at steps (of material and/or cross section), joints, corners. The branch i = 1 is the side of excitation, either by longitudinal or bending waves. The other branches are anechoic.

The transmission coefficient ‘1 i = incident power.

¢i is the ratio of the effective powers ¢i , with ¢1 the ¢1

Definitions: 1/4 1/4 m2 B1 2 E1 K21 = , ‰ = 2 m1 B2 1 E2 K2 K2A2 2 E2 m2 B2 • = = , K1A1 1 E1 m1 B1 √ K1 = h1 / 12 for plates . hi = thickness; Ai = cross section (of bars); i mi Ei ci

= = = =

mass density; mass per length of bars; Young’s modulus; group velocity;

Ki = radius of gyration; c = compressional stiffness of interlayer; m = blocking mass; K SF IF si,k

= = = =

radius of gyration of blocking mass; shear modulus of interlayer; thickness of interlayer; hi /hk for plates; s = s12 ;

si,k = Ai /Ak for bars; s = s12

(1)

Elasto-Acoustics

Q

Table 1 Transmission coefficients at steps Object

Longitudinal wave

τ

Cross section change

Transmission coefficient !

−1=2 ‘ = 4 s1=2 12 + s12

*

τ ‘=4

Material change

Elastic interlayer

Blocking mass

−1=4 +−2

τ

τ

Material change

τ

‘=

/

c p A1 E1 1

s−5=4 + s−3=4 + s3=4 + s5=4 −2 2 + s−1=2 + 1 + s1=2 + s2 2 s p 02 2 ‰•(1 + ‰)(1 + •) ‰(1 + •)2 + 2•(1 + ‰2 )

−2 ‘ = 2 s−5=4 + s5=4

"−2 1 ! −5=4 s + s5=4 2 "−1 1 ! 1 + 2s5=2 + s5 = 2

‘12 =

τ12 τ13

Branching

E1 1 E2 2

p −1 A1 E1 1 ‘ = 1 + (f fu )2 ; fu = m *

Cross

+

−1 ‘ = 1 + (f fu )2 ; fu =

‘=

τ

1=4

τ

Cross-section change

Corner

E1 1 E2 2

"−2

‘13

p "−2 2s−5=4 + s5=4 = 2 −1 = 2 + 2s5=2 + s5 =2

‘12 =

τ12 τ13

‘13

!p

+2

1185

1186

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Elasto-Acoustics

Table 1 continued Object & Wave

Bending wave

Elastic interlayer

τ

Transmission coefficient −1 ‘ = 1 + (f fu )3 1=3 G2F fu = p 2 1:82 1 E1 1 h1 lF

Blocking masses

τ

‘ = 1 ; f < 0:5 fs −1 ‘ = 1 + f=fu ; f > 2 fs fs =

K1 2K2

p 21 A21 K1 E1 1 E1 ; fu = 1 m2

Q.18 Cylindrical Shell

See also: Dym (1973)

A cylindrical shell is simply supported at its ends. Used co-ordinates are: 0≤x≤L 0 ≤ y ≤ 2R.

= material density;

E S

= Young’s modulus; = shear modulus; = Poisson ratio;

u, v, w = elongations Abbreviations: 1 a = (1 − ) ; 2 = mR L

H=

1 (h/R)2 ; 12

1 − 2 2 2 R – ; K = E Mode shapes: with m, n = 0, 1, 2, . . . , ny mx ny mx · cos ; v(x, y) = B · sin · sin ; u(x, y) = A · cos L R L R mx ny w(x, y) = C · sin · cos . L R

(1)

2

(2)

Elasto-Acoustics

Q

1187

Special cases: n = 0 axial symmetry; n = 1 no deformation of cross section; m = 0 axial shear motion; m = 1 fundamental bending motion. General eigenvalue equation (for eigenvalues K): K6 − (Q3 + Q4 )K4 + (Q1 + Q2 )K2 − Q0 = 0

(3)

with coefficients: / 9 2 4 Q0 = a (1 − ) + H ( 2 + n2 )4 + (1 − 2) 4 4 + 4 2 n2 + n4 + 6 4 n2 − 8 2 n4 − 2n6 /

3 1 1 2 3 2 4 2 1 6 9 8 n − n − n + + 4 6 n2 + 4 n4 + n8 +H 4 2 2 4 2 4 + H3 a(1 − a) 4 n4 ,

0

2

Q1 = a (5 − 4a) 2 + n2 /

0 3 2 1+a 2 9 2 1 1 2 + + n2 − + 4a 2 n2 − n4 + + n2 4 a 4 a a a / 0 1 11 3 9 − a 4 n2 + − a 2 n4 + n6 , + H2 6 + 4 4 4 4 / 0 2 5 4 9 4 3 1 2 4 2 2 2 2 + + a + 1/a n + n + H n , Q2 = a + n + H 4 2 4 4 2 Q3 = 1 + H + n2 , 0 / 9 2 + (1 + a/4) n2 . Q4 = (1 + a) 2 + n2 + H 4 +H

Eigenfrequencies, from solutions, with Eq. (1) : E K . –= R (1 − 2)

(4)

(5)

Linear approximation to K (at low frequencies): K2 ≈

Q0 . Q1 + Q2

(6)

Approximation with quadratic correction: K2 ≈

Q0 Q2 (Q3 + Q4 ) + 0 . Q1 + Q2 (Q1 + Q2 )3

(7)

1188

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Elasto-Acoustics

Amplitude ratios: 0 / 9 A 1 2 2 2 = K − a 1 + H − (1 + H) n · (aHn2 − ) C 4 3 , − a + − aH n2 + (1 + a)H 3 n2 + H n4 4 / 0 B 1 1 2 2 = K − − a 1 + H n2 · n + (1 + a)H 2 n + Hn3 C 4 3 − aHn2 − a + − aH 2 n , 4 / 0 9 1 4 2 2 1 + a + aH + 1 + a + aH n2 =K −K 4 4 9 1 5 3 + a 1 + H 4 + a 1 + H + H2 n4 + 2a + H + a2H + aH 2 n2 . 4 4 2 2 Special case: Differential equation:

Modes:

Eigenvalues:

Special case: Differential equation:

Eigenvalues:

Special case: Differential equation: Modes:

torsion with axial symmetry u = w = 0 . 2 9 ∂ v 1 − ∂ 2v 1− 1+ H 2 . = 2 4 ∂x2 E ∂t mx ⎧ ⎪ ; simply supported ⎨sin L , v= ⎪ ⎩cos mx ; free L 1− 1 2 K = 1 + H 2 , 2 4 E 1 –2 = 1 + H (m/L)2 . 2(1 + ) 4 longitudinal vibration with axial symmetry

(9)

(10)

(11)

v=w=0.

∂ 2 u 1 − 2 ∂ 2 u 2 . = ∂x2 E ∂t mR 2 , K2 = 2 = L m 2 E –2 = . 2 (1 − ) L radial vibration with axial symmetry

(8)

(12)

(13)

u=v=0.

w h2 ∂ 4 w 1 − 2 ∂ 2 w ∂ 4w w 2 . + HR2 4 = 2 + =− 2 4 R ∂x R 12 ∂x E ∂t mx mx or ∼ cos . w ∼ sin L L

(14) (15)

Elasto-Acoustics

Q

1189

K2 = 1 + H 4 , Eigenvalues:

/ 0 E 1 2 m 4 1 – = + h . (1 − 2) R2 12 L

(16)

2

Special case:

ring-shaped vibration (1 + H)

Differential equations:

Modes:

Eigenvalue equation:

u=0.

∂ 3 w 1 − 2 ∂ 2 v ∂ 2 v 1 ∂w − 4R 2 , + = ∂y 2 R ∂y ∂y 3 E ∂t

w 1 ∂v ∂ 3v ∂ 4 w 1 − 2 ∂ 2 v − HR 3 + 2 + HR2 4 = 2 . R ∂y ∂y R ∂y E ∂t ny ny v = V · sin ; w = W · cos , R R K2 − (1 + H)n2 V = . W n + Hn3 2 K4 − K2 1 + n2 1 + Hn2 + Hn2 n2 − 1 = 0 .

(17)

(18)

(19)

Approximation for thin shells : Hn2 1: n2(n2 − 1)2 ; K22 = H(n2 + 1); n2 + 1 E E 1 n2 (n2 − 1)2 1 2 –21 = (n + 1) . ; –22 = 2 2 2 2 (1 − ) R n +1 (1 − ) R2

K21 = H

Special case:

ring-shaped vibration without dilatation

(20)

nV + W = 0 .

K2 = H n2 (n2 − 1)2 , Eigenvalues: –2 =

1 2 2 E n (n − 1)2 . 2 (1 − ) R2

(21)

Compare with plate in 0 ≤ x ≤ L; 0 ≤ y ≤ R; simply supported: K2 = H (n2 + 2 )2 , –2 =

/ 0 Eh2 m 2 n 2 + . 12(1 − 2 ) L R

Special case: with Eigenvalues:

(22)

ring-shaped bar, simply supported A = 2R h cross section, 3 I = R h (1 + 3H) moment of inertia. E 1 m 2 R2 + h2 . –2 = 2 4 L

(23) (24)

Q

1190

Elasto-Acoustics

Q.19 Similarity Relations for Spherical Shells

See also: Soedel (1973)

The shell be supported anyhow. The bending stiffness is defined as : B=

E h3 . 12 (1 − 2 )

(1)

h = shell thickness; s R E

= = = =

shell dimension; radius of curvature; material density; Young’s modulus;

B = bending stiffness; = Poisson ratio Free bending wave number kB∩ in the shell in relation to the free bending wave number kB|| in a plate: / 01/4 E 1 kB∩ = = kB|| 1 − . (2) h–2 − Eh R2 B –2 R2 Plates and shells with geometrically similar contours and equal supports have same mode solutions wn /a if they agree in:

Elasto-Acoustics

Q

1191

a3 h–3n − Eh R2 , B • .

•

Relation between the eigenfrequencies –n1 , –n2 of two spherical shells, i = 1, 2, which have: • similar contours, • equal supports E2 a1 3 E2 h2 2 1 − 12 1 2 E1 –n2 = + – − . (3) n1 2 2 a2 E1 h1 1 − 2 2 2 R1 2 R22 Relation between the eigenfrequencies –n2 of a spherical shell and –n1 of a plane plate having • similar contours, • equal supports a1 3 E2 h2 2 1 − 12 1 2 E2 –n2 = –n1 + . (4) 2 a2 E1 h1 1 − 2 2 2 R22 If material, contour, dimensions and support are equal: E . –n2 = –2n1 + R2

(5)

Q.20 Sound Radiation From Plates

See also: Cremer/Heckl (1996); Heckl (1964); Maidanik (1962)

The radiation efficiency £, presented below, is the real part of the normalised radiation impedance. See > Ch. F. “Radiation of Sound” for radiation impedances. A plate, infinite if not otherwise stated, be excited by a point or line force, or by a sound field. If the plate has a finite area A it is supposed to be mounted in an infinite hard baffle wall. The effective sound power ¢ radiated to one side will be given.

1192

Q

Elasto-Acoustics

0 = density of surrounding medium; c0 k0 Z0 h

= = = =

sound velocity in medium; –/c0 ; 0 c0 ; plate thickness;

m = surface mass density of plate; Feff = root mean square of force; A = plate area; p E †

= = = =

plate material density; Young’s modulus; Poisson ratio; plate bending loss factor;

kB = free plate bending wave number; cL = longitudinal wave speed in plate Plate excited by a point force: / 0 0 c0 0 –m 2 1− ¢= F arctg 2c0m2 eff –m 0 c0 −−−−−−−−−→ k0 m /0 1

−−−−−−−−−→ k0 m /0 1

0 F2eff 2c0 m2

k02 F2 . 60 c0 eff

(1)

Radius a of an equivalent piston radiator (i.e. piston radiator with radiation efficiency £ = 1 and same effective velocity): ) a = 8 3 · Šc = 0.286 · Šc (2) with Šc = c0 /fc and f c = coincidence frequency. Plate excited by a line source (¢ and Feff per unit length): ! −1/2 " 0 c0 0 F2eff 1 − 1 + (0 c0 –m )2 ¢= 2 2–m –m

(3)

Space- and time-averaged squared velocity for excitation of a finite plate by a point source: v 2 s,t =

kB2 F2 . 8 †A–2 m2 eff

(4)

Radiated power of a finite plate, excited by a point force, follows from the definition of radiation efficiency £: ¢ = A · 0 c0 · £ · v 2 s,t , ¢=

0 c0 · kB2 · £ 2 F . 8 †–2m2 eff

(5) (6)

Elasto-Acoustics

Q

1193

Velocity of a plate when excited by a diffuse sound field (p = effective sound pressure): c20 · kB2 · £ p 2 †m2 1 kB2 · £ 2 p v = 2 2 2 + –m 2 k02 †

v2 =

above coincidence frequency,

(7)

below coincidence frequency.

(8)

Ratio of radiated sound power by a point-excited plate to sound field excitation: • excitation by a point force

¢ = · F2eff ,

• excitation by a diffuse sound field

v 2 = ß · p2

follows:

0 c0 k02 0 –2 = = . ß 4 4c0

(9)

Similar relations for a line force (and 2-dimensional sound field): ¢L = L · F2Leff

;

vL2 = ßL · p2L

;

L 0 c0 k0 0 – = = . ßL 4 4

(10)

Sound power ¢m fed into a plate by a diffuse sound field: ¢m =

AkB2 · £ 2 p . 2 k02–m

(11)

Radiated sound power of a finite plate, with area A, periphery U, driven by a point force (approximation): U 0 2 F (12) 1+ ¢= ; f fc . 2c0m2 eff 2AkB† More general, if radiated power is small compared with internally lost power, i.e. 20 c0 · £ –m · †: 0 2c20 · £ 2 ¢= F 1+ . 2c0m2 eff 2.3 cL h – †

(13)

Approximations for radiation efficiency of point-excited, weakly damped, finite plates: ⎧ ) 2 ⎪ ⎪ /( A)· f fc ; f fc UŠ c ⎨ ; Šc = c0 fc . (14) £≈ 0.45 U/Š ; f = f c c ⎪ ⎪ ⎩ 1 ; f fc More precise approximation for f< fc (Maidanik) : 1+ 2 4 Š0 Šc 1 − 22 U Šc (1 − ) ln 1 − + 2 £= 2 + √ A 1 − 2 42 A (1 − 2)3/2

;

with Š0 = wave length of air-borne sound at frequency f. Sound pressure far field of a plate excited by a point force F = point of excitation; Ÿ = polar angle):

=

f /fc

(15)

√ 2 · Feff (R = radius from

1194

Q

Elasto-Acoustics

Loss-free plate: p(R, Ÿ) =

jk0 e−j k0 R F 2 R

p(R, Ÿ) =


cos Ÿ ; f < 0.7 · fc , (16) jk0m 1+ cos Ÿ · 1 − (f /fc)2 sin4 Ÿ 0 1 + œ(Ÿ) · cos Ÿ , * + .; 1− (cL sin Ÿ)2 jk0 m 1+ 1− · œ(Ÿ) 1 + œ(Ÿ) + 0 24 c20

f > 0.7 · fc with

f c = coincidence frequency; 2(k0h)2 2 œ(Ÿ) = 2 sin Ÿ − (c0 cL )2 . (1 − )

(17)

In the direction Ÿ = 0 normal to the plate: p(R, 0) =


cos Ÿ . jk0m 1+ 0

In the direction of the angle of coincidence ) jk0 e−j k0 R F p(R, 0) = 1 − (fc /f )2 . 2 R

(18)

Ÿc = sin−1

fc /f : (19)

Plate with bending loss factor † 1: Losses have negligible influence for f< fc . Define complex coincidence frequency:

p(R, Ÿc ) =

−j k0 R

jk0 e F 2 R

–c = 2fc =

) 1 − (fc/f )2 . k0m 1+† 1 − fc /f 0

√

12

c20 1 + j †/2 , cL h

(20)

(21)

References Achenbach,J.D.: Wave propagation in elastic solids. North-Holland, Amsterdam (1975)

Cremer, L., Heckl, M.: Körperschall. Springer, Berlin (1996)

Auld, B.A.: Acoustic fields and waves in solids. Vol. I and II, Krieger Publishing Company, Malabar, Florida (1990)

Dym, C.L.: Some new results for the vibration of circular cylinders. J.Sound and Vibr. 29, 189–205 (1973)

Beltzer, A.I.: Acoustics of solids. Springer-Verlag, Berlin (1988)

Fahy, F.: Sound and Structural Vibration.Academic Press, London (1985)

Elasto-Acoustics

Gaul,L.: The influence of damping on waves and vibrations. Mechanical Systems and Signal Processing 13, 1–30 (1999) Gazis,D.C.: Three-dimensional investigation of the propagation of waves in hollow circular cylinders. I. Analytical formulation. J. Acoust. Soc. Am. 31, 568–578 (1959) ‘ Gorman, D.J.: Free Vibration Analysis of Rectangular Plates Elesevier/North Holland Inc., N.Y. (1982) Haberkern, R.: Personal communication Heckl,M.: Untersuchungen an orthotropen Platten. Acustica 10, 109–115 (1960) Heckl, M.: Einige Anwendungen des Reziprozitätsprinzips in der Akustik. Frequenz 18, 299–304 (1964) Helbig, K.: Foundations of anisotropy for exploration seismics. Pergamon/Elsevier, Oxford (1994) Jones, R.M.: Mechanics of composite materials. Taylor & Francis, Philadelphia (1999) Junger, M.C., Feit, D.: Sound, structures and their interactions. MIT Press, Cambridge MA (1986) Lai, W.M., Rubin, D., Krempl, E.: Introduction to continuum mechanics. Pergamon Press, Oxford (1993) Lanceleur, P., Ribeiro, H., De Belleval, J.-F.J.: The use of inhomogeneous waves in the reflectiontransmission problem at a plane interface between two anisotropic media. Acoust. Soc. Am. 93, 1882– 1892 (1993) Lighthill, M.J.: J. Inst. Maths. Appls. 1, 1–28 (1965) Maidanik: Response of ribbed panels to reverberant acoustic fields. J.Acoust.Soc.Amer. 34, 809–826 (1962) Maysenhölder,W.: Some didactical and some practical remarks on free plate waves. J. Sound Vib. 118, 531–538 (1987)

Q

1195

Maysenhölder, W.: Proof of two theorems related to the energy of acoustic Bloch waves in periodically inhomogeneous media. Acustica 78, 246–249 (1993) Maysenhölder, W.: Körperschallenergie. Hirzel, Stuttgart (1994) Maysenhölder, W.: Low-frequency sound transmission through periodically inhomogeneous plates with arbitrary local anisotropy and arbitrary global symmetry. Acustica acta acustica 82, 628–635 (1996) Maysenhölder, W.: Sound transmission through periodically inhomogeneous anisotropic plates: Generalizations of Cremer’s thin plate theory. Acustica 84, 668–680 (1998) Mechel, F.P.: Schallabsorber, Vol. II, Ch. 27: Plate and Membrane Absorbers. Hirzel, Stuttgart (1995) Mechel, F.P.: About the Partition Impedance of Plates, Shells, and Membranes. Acta Acustica, submitted (1999) Mitchell,A.K., Hazell, C.R.: A simple frequency formula for clamped rectangular plates. J. Sound Vibr. 118, 271–281 (1987) Mobley, J., et al.: Kramers-Kronig relations applied to finite bandwidth data from suspensions of encapsulated microbubbles. J. Acoust. Soc. Am. 108, 2091–2106 (2000) Mozhaev,V.G.: Approximate analytical expressions for the velocity of Rayleigh waves in isotropic media and on the basal plane in high-symmetry crystals. Sov. Phys. Acoust. 37, 186–189 (1991) Murphy, J.E., Li, G., Chin-Bing, S.A.: Orthogonality relation for Rayleigh-Lamb modes of vibration of an arbitrarily layered elastic plate with and without fluid loading. J. Acoust. Soc. Am. 96, 2313–2317 (1994) Pavic, G.: Structural surface intensity: An alternative approach in vibration analysis and diagnosis, J. Sound Vib. 115, 405–422 (1987)

Maysenhölder, W.: Rigorous computation of platewave intensity. Acustica 72, 166–179 (1990)

Pierce, A.D.: Variational formulations in acoustic radiation and scattering. In: Physical Acoustics, Vol. XXII (Underwater Scattering and Radiation), A.D. Pierce, R.N. Thurston (eds.), Academic Press, Boston, 195–371 (1993)

Maysenhölder,W.: Analytical determination of the group velocity of an arbitrary Lamb wave from its phase velocity. Acustica 77, 208 (1992)

Pierce, A.D.: The natural reference wavenumber for parabolic approximations in ocean acoustics. Comp. & Maths. with Appls. 11, 831–841 (1985)

1196

Q

Elasto-Acoustics

Pritz, T.: Analysis of four-parameter fractional derivative model of real solid materials. J. Sound Vib. 195, 103–115 (1996) Rokhlin, S.I., Bolland, T.K., Adler, L.: Reflection and refraction of elastic waves on a plane interface between two generally anisotropic media. J. Acoust. Soc. Am. 79, 906–918 (1986) Sigalas, M.M., Economou, E.N.: Elastic and acoustic wave band structure. J. Sound Vib. 158, 377–382 (1992) Skelton, E.A., James, J.H.: Theoretical acoustics of underwater structures. Imperial College Press, London (1997) Soedel: A natural frequency analogy between spherically curved panels and flat plates. J. Sound and Vibr. 29, 457–461 (1973)

Sornette, D.: “Acoustic waves in random media. I. Weak disorder regime. II. Coherent effects and strong disorder regime. III. Experimental situations.Acustica 67, 199–215 (1989), 251–265; 68, 15– 25 (1989) Thurston, R.N.: Wave propagation in fluids and normal solids. In: Physical Acoustics, Vol. I (Methods and devices, Part A), W. P. Mason (ed.), Academic Press, New York (1964), 1–110 Ting, T.C.T., Barnett, D.M.: Classifications of surface waves in anisotropic elastic materials. Wave Motion 26, 207–218 (1997) Viktorova, R.N., Tyutekin, V.V.: Physical foundations for synthesis of sound absorbers using complex-density composites. Acoust. Phys. 44, 275–280 (1998)

R Ultrasound Absorption in Solids with W. Arnold

List of symbols used in this Chapter: a A b c cp d e f D E G H k kB K le M n P Q S SL,T T v vF V W x Z — ‚

transducer radius; area, amplitude; Burger’s vector; elastic constant; specific heat at constant pressure; thickness, grain size; piezoelectric constant; frequency; deformation potential; electric field, Young’s modulus; shear modulus; magnetic field; k-vector; Boltzmann’s constant; piezoelectric coupling factor, compressibility; mean free path for electrons; elastic moduli, deformation potential; volume density of scatterers; polarisation, power; Q-value; ultrasonic energy; scattering parameter; temperature; sound velocity; Fermi velocity; volume; energy; path length; acoustic impedance; attenuation, absorption; thermal expansion; strain; cross-section, heat conduction, Grüneisen constant;

1198

‰ Š Š, ‹ † ‘ Ÿ ’ –

R

Ultrasound Absorption in Solids

dielectric permittivity, k-vector; wavelength; Lamé constants; polarisability; displacement; density; relaxation time; angle, Debye temperature; stress, electrical conductivity; frequency; angular frequency

R.1 Generation of Ultrasound Surface excitation: In most cases ultrasound is generated by piezoelectric transducers. The principle can be easily seen by a one-dimensional consideration. Piezoelectric equations:

1 = c11 —1 − e11 E1 ;

P1 = e11 —1 + †11 E1 ,

(1, 2)

E is the electric field,— the strain,e the piezoelectric constant,† the dielectric suszeptibility, the stress and P the electric polarisation.Applying an electrical field E1 (t) = E0 ej –t across the surface of a piezoelectric crystal yields for the wave equation: 1 ∂2 ∂2 ∂E1 − 2 2 = d11 2 dx v ∂t ∂x

(3)

with d11 = e11 /c11 and the boundary condition that the surface is stress free: ∂/∂x = —1 = d11 E1 (t)

(4)

one obtains: = j (d11 E0 /k) ej–t e−j(x−x0 ).

(5)

Equation (3) shows that the gradient of the E-field is the source of ultrasound. This holds for all piezoelectric transducers. The radiated ultrasonic energy is, see figure below: 2 vAQ/–—r —0 —r Vg + Vr . S = Pin c11d11

(6)

A is the cross section of the rod, Q is the quality factor of the resonator, Pin is the input electrical intensity. Equation (6) is also valid correspondingly for placing the transducer into a capacitor.


R

1199

ε

Principle of surface generation of ultrasonic waves due to piezoelectricity Thin films and piezoelectric discs: In the case there is a thin film or a piezoelectric disc as a transducer, waves are generated at both surfaces of the transducer: Avl ‰K2 E02 (1 − cos(kd))2 Z S= 2 . (7) Z 2 sin (kd) + (Z /Z)2 cos2(kd) Here, d is the film thickness, Z and Z are the impedances of the transducer and material, respectively, K2 is the piezoelectric coupling factor, ‰ is the dielectric permittivity, and A is the area of the transducer, see figure below.

′

Principle of generation of ultrasonic waves due to piezoelectricity in a thin film

R.2 Ultrasonic attenuation General considerations: Strain wave: (x, t) = 0 ej (–t−kx)

(1)

R

1200


The relation between k-vector k and angular frequency – is k 2 v 2 = –2 with phase velocity v. If there is attenuation, either the velocity v, the k-vector or the frequency is complex: v = v1 + j v2 ;

k = k1 − j .

(2)

Hence: (x, t) = 0 e−x ej (–t−k1 x) ,

(3)

is the damping coefficient and vp is the sound velocity as a real quantity; vp = –/k1. With (2): =

–v2 –v2 = ; v12 + v22 |v|2

k1 =

–v1 –v1 = . v12 + v22 |v|2

(4a, b)

Likewise one may start with a complex elastic modulus: and obtains: –v2 – (c2 )1/2 = = . v v1 v (c1 )1/2

c = c1 + j c2

(5)

(6)

Less common is the assumption that the frequency is complex and k real, details see [Truell/Elbaum/Chick (1969)]. Definition of attenuation: (x) = 0e−x ; then =

=

1 log e x2 − x1

(x1 ) (x2 ) (x1 ) 1 loge = x2 − x1 (x2 ) 1 x2 − x1

(x1 ) , (x2 )

(7a, b)

20 log10

[dB/unit length];

(8a)

[nepers/unit length].

(8b)

Conversion factors: [dB/unit length] = 8.686 [nepers/unit length]

(9a)

[dB/unit time] = [dB/unit length] × sound velocity

(9b)

Logarithmic decrement:

ƒ = loge

n n+1

,

(10)

where two consecutive oscillations are considered. Hence: ƒ [nepers] = [nepers/cm] Š [cm];

ƒ=

[nepers/cm] v [cm/ sec]. ’ [1/ sec]

(11a, b)


R

1201

Therefore: [dB/‹ sec] = 8.68 × 10−6 v [cm/ sec] [nepers/cm] [dB/‹ sec] = 8.68 × 10−6 ’ [sec −1 ] ƒ [nepers]

(12)

[dB/‹ sec] = [dB/cm] × 10−6 v [cm/ sec]. Definition of Q-value from the bandwidth Œ and resonance Œr :

Q = Œr /’

(13)

equivalent to: Q = –r or

energy in the system energy dissipated per second

Q = –1

(14a)

W , dW/dt

(14b)

where W is the energy stored dissipating with dW/dt per second, hence: W = Wo e−(–1/Q) t .

(15)

There are close similarities to the behaviour of oscillators. Geometrical losses: Diffraction losses play an important role, particularly at low frequencies: g =

1.8 [dB/cm] 1.05a2 /Š

or

d = 1.7

’ [dB/cm] , a2 v

(16)

a = radius of transducer, Š = wavelength, ’ = frequency, and v = sound velocity. Non-parallelism is another source of geometrical attenuation: ∼ = 8.7x10−5 va‡,

(17)

‡ is the angle between non-parallel surfaces of the sample. Scattering losses: Scattering at single spheres: Scattering losses occur if there is a change in local mechanical impedance in polycrystalline and two- or multiphase materials. In case of cavities, diameter a , in elastic materials (denoted 1), this leads to a scattering cross-section for da 1 (Rayleigh approximation): ‚N =

4 gc (k1 a)4 , 9

2 + 3 (‰1 /k1)5 4 3 gc = + 40 2 − 2 3 2 4 − 9 (‰1/k1)

(18a)

‰1 k1

2

2 + 3

‰1 k1

3

9 + 16

‰1 k1

4 .

(18b)

R

1202


Here (and below): k = –

Š + 2‹

1/2 ,

1/2 ‰ = –/ ‹/

(19)

with k = longitudinal wavenumber, ‰ = transversal wavenumber, = density, Š, ‹ = Lamé constants. In case of elastic spheres (denoted 2) in matrix 1, incident longitudinal waves: 4 gel (k1 a)4 , 9

2 2 3 2 1 ‰1 ‰2 3 (‰1 /k1)2 ‹2 1+2 −1 gel = −1 + 3 k1 ‰1 ‹1 3 (‰2 /k2)2 −4 ‹2 /‹1 +4

2 5

‹2/‹1 − 1 ‰1 + 40 2 + 3 . k1 2 3 (‰1 /k1)2 + 2 ‹2/‹1 + 9 (‰1 /k1)2 − 4

‚N =

(20a)

(20b)

In case of elastic spheres (denoted 2) in matrix 1, incident transversal waves: ‚N =

gel,t

4 gel,t (‰1 a)4 , 9

(21a)

⎧ 2

⎫ ‰22 k22 ‰12 k22 k22 ⎪ ‰1 ⎪ ⎪ ⎪

⎪ 3 − 3 2 − 4 4 + 10 2 − 6 2 ⎪ 8 1 k13 ⎨ ‰22 ‰1 ‰2 ‰2 ‰1 ⎬ = 1+

2 ⎪ . 3 2 ‰13 ⎪ k2 k2 ‰22 ‰22 ‰1 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 1 − 10 + 6 − 6 +9 2 k1 k1 k1 ‰12 ‰1

(21b)

Scattering (neglecting multiple scattering) leads to an attenuation coefficient: 1 s = n0 ‚ 2

(22)

with n0 = density of scatterers. Scattering in polycrystalline materials: Here, scattering at the polycrystals boundaries arises due the anisotropy of the crystals described by the anisotropy factor:

5 5 83 Vf 4A2 63 Vf 4A2 vL vT · 2 8 2+3 · 2 8 3+2 ; T = . (23a, b) L = 375 0 vL vT 375 0 vT vL Attenuation can be written as: L,T = SL,T Vf 4 ≈ SL,T d3 f 4 ,

(24)

where: SL,T is the scattering parameter for longitudinal and transverse waves, respectively. The volume of the scatterers is V ≈ d3 , d grain size. The anisotropy factor is shown in the figures below. For cubic crystals:

A2 = (c11 − c12 − 2c44)2

Because (vL /vT )5 ≈ 32 transverse wave scattering is much stronger.


μ

α

v

μ

Scattering parameter for longitudinal waves

μ

α

v

Scattering parameter for transverse waves

μ

R

1203

1204

R


Ultrasonic backscattering: Ultrasonic backscattering may be used to characterise microstructures which are influenced by damage of various kinds, see Ref. [Goebbels (1980); Hirsekorn/Andel/Netzelmann (1980); Boyd et al. (1998)]. Neglecting multiple scattering, the backscattered signal is: Intensity: Amplitude:

IS (x) = I0 · 2L,T x exp (−2x) i

AS (x) = A0 (S x) · exp (−x)

(25a) (25b)

with = L,T + A the total attenuation coefficient, and A is the absorption coefficient which describes internal friction, see the following Sections.

R.3 Absorption and Dispersion in Solids Due to Dislocations Equation of motion: ∂ 2 ij ∂2 = —ij ∂xj2 ∂t2

(1)

Strain in the solid: Elastic strain is determined by Hooke’s law. The contribution of the dislocation to the strain is cast into —dis : — = —el + —dis ;

= G—el ,

(2a, b)

where G is the shear modulus. The dislocations can undergo oscillations much like a string: A

∂ ∂2 ∂ 2 + B − C 2 = b 2 ∂t ∂t ∂y

with

A = b2

and

(3) C = 2Gb2/ (1 − ’) .

Here, is the amplitude of the oscillating dislocation, A is their effective mass per unit length, B is the damping constant and is determined by viscous drag in the phonon and electron bath, C is the tension in the bowed-out dislocation, b is the Burger’s vector, is given above, and Œ is the Poisson ratio. Combining the above equations leads to: ∂ 2 xx ∂ 2 xx b ∂ 2 − = ∂x2 G ∂t2 l ∂t2

l

y dy.

(4)

0

The integral in (4) represents the average amplitude of the dislocation oscillating in ydirection with l being their length. Solving (4) leads for the absorption and dispersion v(–) to: 1 4Gb2 –2 d 2 2 (–) = – L , (5) 2 0 v 4 C –20 − –2 + (–d)2


v(–) = v0 1 −

where

v0 =

4Gb2 4 C

G/;

–20 L2

–20 − –2 , 2 –20 − –2 + (–d)2

√ –0 = /L C/A ;

R

1205

(6)

d = B/A.

(7)

L is the loop length of the dislocation, their density per area, and –0 is the resonance frequency of the dislocation. Two limits arise: resonance (a ≡ d/–0 1) and relaxation (a 1). In case of relaxation Eq. (5) reduces to:

4Gb2 (–/–m )2 2 = 8.68 × 10−6 L – m 4 C 1 + (–/–m )2

dB/‹s) .

(8)

For convenience Eq. (8) is written to give units dB/‹s. The constant B is of the order B ≈ 10−4 [s/cm]. There are many other absorption mechanisms possible, based on dislocations dynamics, see [Truell/Elbaum/Chick (1969); Gremaud/Kustov (1999); Gremaud (2001)] and references contained in [Gremaud/Kustov (1999); Gremaud (2001)].

ωω

Absorption due to dislocations with transition from resonant (a ≡ d/–0 1) to relaxation absorption (a 1). Dependence of y=

(–) (1/v)(4Gb2/2C) L2 –0

on –/–0 for various values of a = d/–0

1206

R


ωω Relaxation absorption from dislocations for a 1. Normalised attenuation and decrement as functions of –/–m for the case of large damping

R.4 Absorption Due to the Thermoelastic Effects, Phonon Scattering and Related Effects Thermoelastic effect: In all solids an absorption mechanism arises because a propagating ultrasonic wave entails a temperature modulation for longitudinal waves due to thermoelasticity. This temperature modulation tends to return to equilibrium by thermal conductivity. Phenomenological description for the stress-strain relation: ‘−1 + ˙ = (M1/‘) — + M0—˙ ,

(1)

‘ is the relaxation time, the stress and — the strain.The modulus defect M/M describes the microscopic coupling and has to be determined in each individual case: M M0 − M1 . = M M0 General solution: –2 ‘ M v=≈ 1− ; 2M0 1 + –2 ‘2 1/2 M/Mo – M0 M– 1/2 = 1− ≡ , v= k 1 + –2 ‘2

(2)

(3) (4)


vg =

M0

1/2

M 1 − –2 ‘2 1− . M0 (1 + –2 ‘2 )2

R

1207

(5)

Equations (4) and (5) indicate that there is dispersion, i.e., that the phase velocity v and group velocities vg are not the same. For the thermoelastic effect: M M0 − M1 Ead − ET 2 T 2 T = = = Ead ≈ ET , M M0 ET cp cp

(6)

where T is the temperature, is the thermal expansion, Ead and ET are the adiabatic Young’s modulus and at constant temperature, respectively, and cP is the specific heat at constant pressure. The relaxation time ‘ is given by the thermal diffusivity D and the Debye average for the sound velocity v: ‘≡

L2 L2 cp D ‚ = = 2 = . D ‚ v cp v 2

(7)

L is a length comparable to the wavelength,and ‚ the heat conductivity and cp the specific heat. Over the length L ∼ = Š the temperature difference generated by the ultrasonic wave due to the thermoelastic effect is equalised via thermal conductivity. In polycrystals, heat may flow from one grain to the next because they heat up differently due to their anisotropy. This leads to an absorption, important for all technical materials: =

Cp − Cv R R –2 ‘ Cv 2v 1 + –2 ‘2

(8)

with ‘ again the relaxation time, Cp , Cv are heat capacities for longitudinal and transversal sound. L is the mean grain diameter. R depends on the anisotropy for the strain energy and extends from R ≈ 10−6 for tungsten to R = 6.5 × 10−2 for lead [Beyer/Letcher (1969)]. Phonon interactions: Ultrasonic waves also modulate locally the thermal phonon distribution [Beyer/Letcher (1969)]. This holds both for longitudinal and shear waves and hence differs from the thermoelastic effect. For –‘th 1, the absorption for longitudinal waves is:

cjj 1 –2 ‘th 3U0 2 –2 ‘th 2 = (9) ‚j(i) − ‚ Cv Ÿ l = 3 2 3 2 2vlong 1 + – ‘th 2vlong N i 1 + –2 ‘2th with the modulus defect: 3U0 2 cjj = ‚j(i) − ‚ 2Cv Ÿ. N

(10)

i

For transverse wave: 3U0 2 ‚j(i) , cjj = N i

(11)

R

1208


cjj –2 ‘th 1 t = = 3 3 2vlong 1 + –2 ‘2th 2vlong

3U0 2 –2 ‘th ‚j(i) , N 1 + –2 ‘2th

(12)

i

‘th =

3k . Cp v02

Here,

2 ‚j(i) is an average Grüneisen constant for transverse and longitudinal pho-

(13)

i

nons and Cv and Cp are the corresponding heat capacities per volume. U0 is the total thermal energy. Note that this effect occurs in addition to the one described by the thermoelastic effect. Equations (9), (12) can be simplified for practical purposes: Eq. (9):

l =

Ÿ‰‚ 2–2 5 2vlong

and Eq. (12):

t =

Ÿ‰‚ 2 –2 . 2vt5

(14a, b)

If –‘th 1, a different view-point must be considered. The ultrasonic wave no longer modulates the phonon distribution, but ultrasonic phonon is scattered by the thermal phonons and corresponding conservation of momentums must be taken into account. This mechanism plays a role for low temperatures (< 40 K), i.e. much below the Debye temperature Ÿ [Tucker/Rampton (1972); Beyer/Letcher (1969); Dransfeld (1967)]. t =

3 kB4 F¯ 21 –1 T4 , 60 3 v0103

(15)

F1 is an average of second and third order elastic constants. Note the strong dependence on sound velocity which was verified experimentally. This mechanism also holds for longitudinal and transverse waves in various crystallographic orientations in crystals and the –1 T4 is retained,however,the pre-factor is different.The pre-factor also depends on the crystal class.

R.5 Interaction of Ultrasound with Electrons in Metals The ultrasonic wave leads to a spatial separation of the ions from the free electrons at the Fermi surface. This causes an electrical field which eventually leads to a redistribution of the electrons and hence to an absorption coefficient [Tucker/Rampton (1972)]: ⎤ ⎡1 2 2 −1 k l tan (kl ) e nm ⎢ 3 e ⎥ − 1⎦ . (1) = ⎣ 2vl ‘ kle − tan−1 (kle ) Here, n is the number of conduction electrons per volume, m is the electronic mass, le is the mean free path for electrons, vF is the Fermi velocity, k the wave vector of the ultrasonic wave, and vl the longitudinal sound velocity. In the limit of kle ≤ 1 this leads to: =

2 nmvF2‘ (1 − 9/35(kel)2 + . . .). 15 vl3

(2)


R

1209

In the limit kle 1: nmvF = –. 12vl2

(3)

The collision time ‘ is determined by the electrical conductivity = ne2 ‘/m. Transverse waves also cause absorption because they generate indirectly an internal electrical field via an internal magnetic field. Final results are similar to the equations above. The absorption is magnetic field dependent because the electrons follow curved trajectories in a magnetic field and are bound to the Fermi surface. For transverse waves and a magnetic field perpendicular to the polarisation vector and k-vector: (H) 1 = , (0) 1 + (2–c ‘)2 where –c is the cyclotron resonance frequency

(4) –c = (eH/mc)

(5)

similarly for H parallel to the polarisation vector and perpendicular to the k-vector: 1 (H) . = (0) 1 + (–c ‘)2

(6)

Various resonance phenomena may occur if the electrons can complete an orbit on the Fermi surface whose size is equal to an integer number of the wavelength. If H is perpendicular to k, geometrical resonance occurs. When the ultrasonic frequency – = n–c , n integer, and k parallel to H, temporal resonance may be induced. Finally quantum oscillations may be excited. Here the effect is due to the quantisation of the electron energy in a strong magnetic field applied in the z-direction.In all three cases the ultrasonic absorption becomes periodically field dependent with the periodicity (1/H): (1/H) = eŠ/ky vl ,

(7)

ky is the size of the orbit in k-space equivalent to the cross section of the Fermi surface. (1/H) = e/–mvl ,

(8)

−1 1 e 2 kf2 2 kz2 2e 1 , = − = H mv1 2m 2m cl A (Ef , kz)

(9)

m is the effective mass of the electron and A is the cross section of the Fermi surface at Ef and kz . The latter case is analogue to the so-called de Haas van Alphen effect. Exploitation of Eqs. (7)–(9) allow to measure the shape of the Fermi surface in metals and its anisotropy. If the metal becomes superconducting, the absorption drops. This can be exploited to measure the gap function (T) in the superconductor: 2 s . = n exp (/kT) + 1

(10)

1210

R


R.6 Wave Propagation in Piezoelectric Semiconducting Solids The generation of a stress field in a piezoelectric solids leads to an accompanying electric field which accelerates electrons and leads to absorption and dispersion because the electrons undergo inelastic scattering [Hutson/White (1962)]:

e2 1 + (–C /–D ) + (–/–D )2 v = v0 1 + , (1) 2cp 1 + 2 (–C /–D ) + (–/–D )2 + (–C /–)2

–C /– – e2 . (2) = v0 2cp 1 + 2 (–C /–D ) + (–/–D )2 + (–C /–)2 Here, –C is the so-called conductivity frequency and –D the diffusion frequency; –C = /—d (—d dielectric constant), –D ≈ (ev 2 /‹kT) (‹: mobility of the electrons,k: Boltzmann constant, T: temperature).

R.7 Absorption in Amorphous Solids and Glasses Disordered materials exhibit additional absorption mechanism, mostly due to relaxing units in the molecular or atomic structure. Similar to Eq. R.4.(3), the absorption is described as: –2 ‘ M 1− , ≈ 2M0 1 + –2 ‘2

(1)

however, with the modulus defect or relaxation strength M/2M0, orders of magnitude larger than in the thermoelastic regime, see figure below. The relaxation frequency 1/‘ is determined by an Arrhenius process: 1/‘ = kT/he−E/kT with E the activation energy, k the Boltzmann constant and T the temperature.


R

1211

For low temperatures, amorphous solids behave in a much different way than crystalline solids. Some of the structural units can tunnel between different local spatial co-ordinates leading to a resonant absorption which depends on the ultrasonic intensity S, Sc is a critical intensity, M is the coupling coefficient (M ≈ 0.5 eV) and n is the density of the tunnel units per energy and volume (n0 ≈ 1033 erg−1 cm−3 ): res = −

n0 –M2 tanh(–/kT) √ . v 3 1 + S/Sc

(2)

The corresponding change of sound velocity is for S Sc : v/v0 =

n0 M2 ln (T/T0 ) , v 2

(3)

where T0 is a reference temperature. There is also a relaxation absorption due to the coupling to the phonon bath of the tunnelling units. An ultrasonic wave modulates the thermal occupation leading to: 2M2tl 3n0 D2 T3 M2l rel = + . (4) 2 4 v 3 vl5 vt5 Also this absorption mechanism leads to dispersion, see [Hunklinger/Arnold (1976)] and [Enss/Hunklinger (2000)]. If crystals exhibit a certain disorder or are irradiated by electrons or neutrons, similar phenomena are observed.

R.8 Relation of Ultrasonic Absorption to Internal Friction Internal friction discusses the absorption mechanism of mechanical waves and oscillations at low frequencies usually below 20 kHz, i.e., in the audible range. The mechanism are mostly relaxation phenomena which are described by equations like R.7.(1), where the relaxation strength is adapted to the corresponding situation. Overviews can be found in [de Batist (1972); Nowick/Berry (1972); Schaller/Fantozzi/Gremaud (2001)].

R.9 Gases and Liquids Treatments of the ultrasonic absorption due to relaxation phenomena in gases and liquids can be found in [Bhatia (1967)].

R.10 Kramers-Kroning Relation Kramers-Kroning relation describe the interdependence between absorption and dispersion: 2 K1 = P

∞ 0

– K2 (– ) d– , –2 − –2

(1)

1212

R

2 K2 = P

∞ 0


– K1 (– ) d– . –2 − –2

(2)

Here, K1 is the real part and K2 the imaginary part of the compressibility. Analogue expressions hold for the k-vector k and the absorption coefficient. Practical simplification of eqs. (1) and (2) are: dv (–) 2 dK1 (–) =− , d– 0 v 3 (–) d–

(3)

dv (–) = 2v 2 (–) (–) /–2 , d–

(4)

dv (–) 2 (–) = d–, v 2 (–) –2

(5)

1 2 1 − = v0 v (–) (–) =

– –0

(– ) d– , –2

(6)

–2 dv (–) , 2v02 d–

2v 2 v = v (–) − v0 = 0

(7) –

(– ) d– . –2

(8)

0

Here v0 is the sound velocity at the frequency –0 .

References de Batist, R.: Internal Friction of Structural Defects in Solids. North-Holland Publishing Company, Amsterdam (1972) Bhatia, A.B.: Ultrasonic Absorption. Clarendon Press, Oxford (1967) Beyer, R.T., Letcher, S.V.: Physical Ultrasonics. Academic Press (1969) Boyd, B., Chiou, C.P., Thompson, B., Oliver, J.: Development of Geometrical Models of Hard-Alpha Inclusions for Ultrasonic Analysis in Titanium Alloys. Review of Progress in Quantitative Nondestructive Evaluation,Eds.D.O. Thompson,D.E.Chimenti, Plenum, New York, XVIII (1998), 823–830 Dransfeld, K.: J. de Physique C1 28, 157–162 (1967)

Enss, C., Hunklinger, S.: Tieftemperaturphysik. Springer Berlin (2000) Goebbels, K.: Structure Analysis by Scattered Ultrasonic Radiation. in Research Techniques in ND, Ed.R.S.Sharpe,Academic Press,London IV,87–150 (1980) Gremaud, G., Kustov, S.: Theory of dislocationsolute atom interaction in solid solutions and related nonlinear anelasticity. Phys. Rev. B 60, 9353– 9364 (1999) Gremaud, G.: Dislocation-point defect interaction. In: Mechanical Spectroscopy Q-1 2001, edited by R. Schaller, G. Fantozzi and G. Gremaud, Ch. 3.3, Materials Science Forum 366–368 (2001) 178–247, Trans Tech Publications, Switzerland


Hirsekorn, S., Andel, P.W., Netzelmann, U.: Ultrasonic Methods to Detect and Evaluate Damage in Steel. Nondestr. Test. and Evaluation 15, 373–393 (1980)

R

1213

O’Donell, M., Jaynes, E.T., Miller, G.: J. Acosut. Soc. 69, 696–701 (1969)

Hunklinger, S., Arnold, W.: Phys. Acoustics, Eds. W.P. Mason and R.N. Thurston, XII (1976) 156–215

Schaller, R., Fantozzi, G., Gremaud, G. (Eds.): Mechanical Spectroscopy Q-1 2001. Materials Science Forum 366–368, Trans Tech Publications, Switzerland (2001)

Hutson, A.R., White, D.L.: J. Appl. Phys. 33, 40–47 (1962)

Truell, R., Elbaum, C., Chick, B.B.: Ultrasonic Methods. In: Solid State Physics, Academic Press (1969)

Kino, G.S.: Acoustic Waves, Devices, Imaging, and Signal Processing. Prentice-Hall, Inc. (1987)

Tucker, J.W., Rampton,V.W.: Microwave Ultrasonic Methods. North-Holland (1972)

Nowick, A.S., Berry, B.S.: Anelastic Relaxation in Crystalline Solids. Academic Press (1972)

Chapter Index The material of the Index is arranged according to the Chapters, because it is supposed that the reader is searching within some context. Capitalised initials of entries (if not names) indicate Sections.

B. General Linear Fluid Acoustics A absorption coefficient 46 adiabatic exponent 6, 9, 10, 35,37 admittance relation 11, 42 admittance rule 11 averaged intensity 11 B balance – of entropy 36 – of heat 36 – of internal energy 36 bilinear concomitant 26 Bloch waves 44 Boundary Condition – at a moving boundary 38 – at liquids and solids 39 boundary conditions – adiabatic 8 – for particle velocity 5, 13, 23, 24, 39 – for sound pressure 5 – isothermal 8 C Christoffel symbols – of first kind 30 – of second kind 29 co-ordinate system 16, 20, 21, 26,30,32 – Cartesian 33 – curvilinear 26 – cylindrical 21 – reciprocal 27 – spherical 34

co-ordinates – contravariant 27 – covariant 27 – mixed 27 – transformation 28 compressibility – isothermal 9, 36 – isotrope 36 conservation – of impulse 5 – of mass 5, 6 – of wave numbers 39 Corner conditions 40 cross impedance 24 D density variation 36–38 density wave 6, 7, 37 differential operator 31, 33, 34 Differential relations of acoustics 34 Dirac delta function 5, 12, 13, 16, 21 divergence 32 Doppler shift 38 E entropy variation 36–38 equation – of continuity 34 – of energy 35 – of state 5 F Field admittance 6, 11 field quantities 6, 36, 44

1216

Chapter Index

Field variables 34 Fundamental differential equations 5 G gas constant 9, 10, 36 gradient 12, 31 Green’s function 13–18 – in 2-dimensional polar coordinates 17 – in 3-dimensional infinite space 17 – in Cartesian co-ordinates 19 – in cylindrical co-ordinates 18 – in spherical harmonics 17 – of a set of plane waves 16 – of cylindrical wave 16 – of point source 13 Green’s functions and formalism 13 Green’s integral 12 Grid on porous layer – grid with finite thickness, narrow slits 53 – grid with finite thickness, wide slits 56, 57 – thin grid 50, 52, 53 H Hamilton’s principle 25 Hartree harmonics 44 heat balance 6 heat conduction 6, 35, 37 heat wave 6 Helmholtz’s wave equation 5 higher modes 48, 56 I impulse equation 6 Integral relations 12 L Lagrange density 25 Lagrange function 25 Lagrange multiplier 25 Laplacian – of a scalar 32 – of a vector 32 line source 21, 24

M Material constants – of air 8, 9 mirror source 19 mode in a duct 19, 20 modes – orthogonality of 19, 20 molecular mean free path length 35 monopole source 5 N nabla operator 31 Navier-Stokes equation 35 normal vectors on co-ordinate surface 27 O orthogonality integral 15 Orthogonality of modes 19 – in a bulk reacting duct 20 – in a locally reacting duct 19 orthogonality relation 17 orthonormal basis vectors 31 P particle velocity 5, 13, 23, 24, 37, 39 Periodic structures – admittance grid 44 – grooved wall, narrow grooves 46 – grooved wall, wide grooves 48 point source – above a locally reacting plane 19 – above hard or soft plane 18 potential – scalar 6, 7 – vector 6 potential function 13, 25 Prandtl number 8, 9 pressure variation 38 Principles of superposition – for unsymmetrical walls 22 – hard-soft superposition 24 – in a symmetrical space 23 R radiating spatial harmonics 46, 51 Rayleigh’s postulate 6

Chapter Index

reciprocity principle 12 reflection coefficient 52, 54, 55, 57, 58 reflection factor 46–50, 52 rotation of a vector 32 S scalar product 16, 19, 27, 29, 31 shear stress 5 – by viscosity 35 shear wave 6 shock front 39 solid-liquid interface 40 Sommerfeld’s condition 12, 21, 22 sound pressure coefficient 7 sound velocity 9, 10, 38 – in gas mixture 10 Source conditions 21 spatial harmonic 44–47, 49–52 specific heat 6, 9, 10, 35 stationary flow 5 Surface wave – along a locally reacting cylinder 42 – along a locally reacting plane 41 T Table with – material constants of air 9 – regression constants for data of air 11 tangent vectors – at co-ordinate lines 27 temperature coefficient 7

temperature variation 37 temperature wave 38 tensor – derivative of 30 thermal expansion coefficient 36 thermal pressure coefficient 36 thermal wave 7 thermodynamic relations 36 total time derivative 34 U unitary tensors 27, 30 V vector – angle between 29 – derivative along a curve 30 – derivatives of basis vectors 29 – length of 29 – scalar product 29 – triple product 29, 31 – vector (cross) product 29, 31 Vector algebra 28 viscous energy loss 35, 36 viscous shear 5, 6 W wave equation – adjoint 26 – homogeneous 5, 13, 14, 20, 21 – inhomogeneous 13 wave numbers – characteristic 20

C. Equivalent Networks ¢

D

¢ networks 66 ¢-circuit impedance 66

Distributed network elements 65

C Chain circuit 73 characteristic – impedance 66 – propagation constant 66

E electro-acoustic analogy 59 – UK-analogy 64 – Uv-analogy 65 electromagnetic quantities 59 Elements with constrictions 71 end correction 59, 69, 71, 75

1217

1218

Chapter Index

Equivalent Networks 59, 66, 71, 74, 75, 126 equivalent oscillating mass 71, 72 F four-pole equations 66 H Helmholtz’s source theorem 62 L lumped elements 65 M mesh theorem 62 N node theorem 62 O open-circuit source 62 R reciprocal invariant 63 Reciprocal networks 63, 64 rule for the construction of a reciprocal network 63

S short-circuit source 62 Sources 62, 63, 73 Superposition of multiple sources 73 T T-circuit impedances 66 T-networks 66 Table – with defining relations for passive mechanical components 61 – with electromagnetic quantities 59 – with passive electrical and mechanical circuit components 60 – with reciprocal electrical elements 63 – with UK-analogous elements 64 – with Uv-analogous elements 65 tube section – terminated with Helmholtz resonator 69 – with a layer of air 70 – with a layer of porous material 69 – with hard termination 67 – with open termination 68 – with perforated plate in the tube 70

D. Reflection of Sound A absorber – uneven 141 – with random admittance 142 – with variable admittance 141 absorber in baffle wall – circular absorber 140 – in constant height above baffle 142 – rectangular absorber 141 – strip in baffle wall 144 absorption coefficient 127, 128, 130, 133–136, 139 absorption cross section 138, 145, 147 absorption of finite-size absorbers 136, 153

acoustic corner effec 152 angular far field 139, 141, 147 B border scattering 153 bulk reacting layer 132, 136, 166 C characteristic values 130, 132, 150 – of Mathieu functions 150 cross impedance 163, 179 cross section – absorption 138 – extinction 138 – scattering 138

Chapter Index

D diffuse absorption coefficient 133 – in 2 dimensions 133 – in 3 dimensions 133 diffuse sound incidence 140 diffuse sound reflection – at bulk reacting plane 165 – at locally reacting plane 133 E effective power – absorbed 138, 153 – scattered 138 elliptic-hyperbolic co-ordinates 149, 151 extinction theorem 139 F Fresnel’s integral 144 G Green’s function 137, 138, 143 L line source above an absorbing plane 164 locally reacting layer 131

1219

pass way 155, 156, 159,160, 171 path of steepest descent 155, 170, 171 plane wave reflection – at a locally reacting plane 127 – at a multilayer absorber 132 – at a porous layer 130 point source above absorber – exact saddle point integration 170 – exact solution 148 pole contribution 155, 158, 159,171, 173, 174 pole crossing – condition for 159 principle of hard-soft superposition 163 Q quantitative corner effect 152 R radiation impedance 154 reflection of sound 127 refracted angle 130, 132 S saddle point 156, 158–160, 163–165, 167, 170, 171,174,182 saddle point integration 156 scattered wave 137, 143, 144 scattering at finite-size local absorber 136 scattering at the border of an absorbent half-plane 142

M Mathieu differential equations 150 Mathieu functions – azimuthal 150 – of the Bessel type 150 – of the Hankel type 150 – of the Neumann type 150 – radial 150 mirror source 138, 156, 161, 163, 167 mirror source approximation 161 mode norms 151

V variational principle 146

P pass integration 156, 159, 161, 162

W wall admittance 127, 142

U uniform pass integration 159

1220

Chapter Index

E. Scattering of Sound A absorbent dam – flat 214 – high 214 – semicircular 214, 227 absorption cross section 187 – for diffuse sound incidence 188 – of a cylinder 197 artificial medium of scatterers 198 B backscattering cross section 188, 263 boundary conditions 254 bubble oscillation 263 bulk reacting cylinder 185 C composite medium 230, 231, 233, 247– 249 D delta pulse 266 diffracted wave 266–268 Dirac delta function 266, 272 E effective – compressibility 247 – density 198, 247, 249 – wave impedance 244 – wave number 236 elliptic-hyperbolic co-ordinates 217 elliptical cylinder atop a screen 209 extinction cross section 187, 188, 191, 238 extinction theorem 188 G geometrical shadow limit 217 H high dam 215, 218, 219,221, 223 – hard ground 215, 219, 222 – line source 223 – on absorbent ground 221, 223

– on hard ground 218 L line source 199–201, 204–207,209, 210, 223, 229 locally reacting cylinder 198, 199, 201 M massivity 198, 238, 240,242, 244, 247, 248 Mathieu functions 209, 211, 214,218, 219, 285 mirror source 222, 266, 267 mixed monotype 244, 248 modal reflection factors 202, 210, 272 monotype scattering 231–233, 238, 244, 246, 248 multiple scattering – elastic scatterers 258 – porous scatterers 258 – rigid scatterer at rest 258 multiple scattering at cylinders and spheres 198 P polar mode numbers 275, 283 principle of superposition 218, 224 Q quality factor of shell resonance 262 R radiating mode in resonance 262 radiation impedance of shell modes 261 refracted angle 186 Reiche’s experiment 233–235 S scattered far field 187, 190, 234, 237, 244, 246, 249,262,266 scattered field 186, 195, 199, 200, 224, 225, 228, 229,231,234, 244,246,249, 260, 264, 274,283

Chapter Index

scatterer – bulk, movable 241 – hard 198, 258, 260 Scattering – at a cone 270, 282 – at a corner 201 – at a flat dam 223 – cross section 187, 191, 194 – in random media 230, 232, 244,248 – mixed monotype 248 – monotype scattering 244 – triple-type scattering 248 – of a plane wave – at a screen with mushroom-like hat 213 – of plane wave – at a liquid sphere 263 – at cylinder 185, 188 – at cylinder and sphere 188 – of spherical wave – at a soft cone 271 scattering – at a hard screen 208, 209, 268 – at a perfectly absorbing wedge 264 – at a screen with cylinder atop 209 – at a semicircular absorbent dam 214 – at elastic cylindrical shell 260

1221

– of impulsive spherical wave at a hard wedge 266 – of plane or cylindrical wave at a corner 201 – of Sound 185 shadow zone of a corner 203 shell resonances 261, 262 sound attenuation in a forest 242 source condition 202, 210, 271–273 surface impedance – of modes 186 symmetrical scatterers 236, 237, 246 T Table – with grad and rot components cylinder or sphere 252 – with strain velocity at cylinder or sphere 252 – with survey for scattering in random media 232 triple-type scattering 231, 232, 248 W wave impedance – in random media 244 wave impedancee 185

F. Radiation of Sound A array with small elementary radiators 331 B back orifice end correction 325 Bouwkamp’s integral 303, 306, 314 C complete elliptic integrals 304, 329 D directivity – of circular membrane or plate 334 – of circular piston in baffle 332

– of circular radiator with nodal lines 334 – of clamped rectangular plate in fundamental node 333 – of dense circular array of point sources 332 – of dense linear array of point sources 332 – of free rectangular plate in fundamental mode 333 – of radiator arrays 330 – rectangulat piston in baffle 305, 333 – two point sources 331 directivity coefficient 330 directivity factor 330, 331, 334, 335

1222

Chapter Index

directivity function 291, 303, 306,314 directivity index 330 directivity value 330 E end correction – of a slit in a slit array 326 End corrections 287, 309, 316, 318,346 F far field 290, 291, 303,306, 314, 330, 331, 334, 336, 340,345 field admittance 288 G Green’s function 289 H Hankel transform 291, 341, 343 Helmholtz resonator 297 higher modes in the neck 321, 322, 324 Huygens-Rayleigh integral 331 I interior end correction 316 – in a slit resonator array 322 – with only plane wave 324 interior orifice impedance 325, 326 – with slit in contact with porous material 325 – with slit in some distance to porous material 326 – with viscous and caloric losses 325 L long source 341 M modal impedances 292 mode norms 292 monopole 337 O oscillating mass 287, 288, 293,316, 317, 326, 328, 342

– of a fence in a hard tube 328 P piston radiator – circular 302, 330 – elliptic, in a baffle 303 – free circular disk 303 – in a hard tube 327 – on a sphere 297 – plane 291, 301, 309 – rectangular 305 – ring-shaped 330 – small circular 330 – strip-shaped 299 – with radial symmetry 341 plane radiator 309, 311, 313, 331, 339 point source – dipole 338 – lateral quadrupole 339 – linear quadrupole 339 – monopol 337 R Radiation and excitation efficiencies 344 radiation directivity 330 radiation factor 288 Radiation impedance 287, 289–291, 293, 296, 298,300,301, 304,306,310, 312, 314, 316,328,329, 340,342,345, 346 – definition 287, 314 radiation impedance – evaluation 301 – mechanical 287, 329 radiation loss 287 radiation of plates 344 radiation reactance 309, 320–322 radiator – breathing sphere 293 – cylindrical 295 – finite length cylinder 335 – in a baffle 339 – in spherical mode patterns 291 – line source 344 – monopole and multipole 337 – oscillating cylinder 317 – plane 309, 311, 313,331,339

Chapter Index

– – – – – – – – –

rectangular, wide, field excited 313 rigid sphere 293 strip-shaped 299 strip-shaped, narrow, field excited 309 strip-shaped, on a cylinder 299 strip-shaped, wide, field excited 311 with 1-dimensional pattern 343 with central symmetry 331 with nearly periodic pattern 343

1223

T Table – with end corrections of different radiators 318 – with oscillating masses of spherical and cylindrical radiators 317 U uniform end correction – of plane piston radiator 309

S Spherical radiator 288, 291

G. Porous Absorbers A absorber variable 349, 350, 353, 355, 358, 373, 394 B Biot wave 391 Biot’s theory 385 bulk compression modulus 388 bulk density 347, 351 C characteristic constants – from fitted theoretical models 399 – wave impedance 347, 371, 375 characteristic propagation constant 347, 371 closed cell model 371–374 compressibility 351, 353, 355, 358, 360, 371, 372, 375, 378 compressional waves 388–390 continued-fraction approximation 354 coupling coefficients 386 coupling density 386 D Delany & Bazley regression 394 density wave 352, 355, 358,371–373, 378, 381, 383, 384

E effective – air compressibility 353, 355, 358 – air density 353, 355, 358 – resistivity 351 effective wave multiple scattering – in transversal fibre bundle 379– 381 empirical relations for characteristic constants 394 equations of motion 386, 392 equivalent network of porous material 351 equivalent radius 387 F fibre bundle 356, 371, 373–376,379– 381, 383–385 fibrous material 347, 350, 356 – parallel fibres 350 – random arrangement 357 – regular fibre arrangement 369 – transversal fibres 373 fitted theory 400 – of flat capillaries 401 – of the quasi-homogeneous material 399 flow resisitvity – of random fibre orientation 350

1224

Chapter Index

flow resistivity 347, 349–351,353–357, 360, 369, 370, 373,379, 387, 394,399 – of parallel fibres 350, 356 – of transversal fibres 350 – with random fibre diameter and orientation 356 flow resitivity 349 flow velocity profile 353, 355 friction 350, 351 fundamental equations 386, 388, 389 I inertia of the air in pores 351 input impedance – of a porous layer 391, 399 – of a porous layer with adhesive front foil 392 M massivity 347, 350, 353, 357–359, 361, 369, 371, 373, 376,384 matrix material 347, 350, 352, 355, 388 matrix vibration 351 Mechel & Grundmann regression 398 mineral fibre material 347, 348, 352, 398 multiple scattering model 371, 375, 376, 380 N Navier-Stokes equation 361 O open-cell model 371, 373, 375 open-cellular foam 348 P Poisson distribution 357, 359, 370,394 Porous Absorbers 347, 385 Porous material – model with flat capillaries 355 porous material 347, 349, 351,352, 355, 370, 381, 385–387, 391, 399 – model with flat capillaries 354, 356 potential coupling factor 387 potentials 372, 373, 388,390 pressure force 367 propagation constant 347, 352, 355, 358, 371, 375, 381,394, 400

Q quasi-homogeneous material 347, 350, 352, 399 R randomised fibres 356, 359, 369 Rayleigh’s capillary model 352 regular fibre arrangement 356 relaxation 350–352, 358 relaxation frequency 352, 358 S Snell’s law 391 sound propagation – parallel to the fibres 358 – transversal to the fibres 370 strain-stress equations 386–388 structure factor 347, 349–351, 386 structure parameters 347 T Table – with ranges of porosity of materials 348 – with volume and surface porosities, and structure factors of different structures 349 Theory of the quasi-homogeneous material 350 tortuosity 386, 387, 390 transversal fibre bundle 373–376, 379–381, 383, 384 V velocity profile 353, 355, 357,384 velocity profile around a fibre 384 viscous and thermal losses 352, 355, 358 viscous force 367, 369, 387 volume porosity 347, 350, 351, 386 W wave equations 388, 389 wave impedance 347, 349, 351, 352, 355, 358, 371,375,381, 394 wave number of the shear wave 390 weak coupling 390

Chapter Index

1225

H. Compound Absorbers ¢ ¢-fourpole 445 A absorbed power 493–495 absorbent neck walls 430, 431 – narrow neck 431 – wide neck 431 Absorber of flat capillaries 404 absorption coefficient 405–407, 481– 483, 490–492, 495, 500, 501 admittance – average 403, 477 – homogenised 403 angular distribution 476, 478 array – of Š/4-resonator 474 – of circular necks 434 – of Helmholtz resonators 435, 456 – of parallel slits 407, 437, 449 average input impedance 456 average surface admittance 456, 459, 464 B backscattering 477 bending loss factor 485, 496 C capillaries 404, 405, 421, 422,431, 457, 463, 476, 497, 502 capillary wave modes 483 characteristic equation 422, 431, 460, 483, 484 coincidence frequency 467 complex bending modulus 467 Compound Absorbers 403 critical frequency 462, 467, 485,496 cylindrical shell 467, 469 D density wave 421, 483 directivity 477, 479

E effective bending modulus 497 – for perforated panel 497 effective plate partition impedance 498 effective surface mass density 457– 459, 462, 467 elastic panel 485, 496 elliptic-hyperbolic co-ordinates 486 end correction 410, 415, 417,418, 425, 434, 436, 437, 445–448, 452,457–459, 472, 497 equivalent network – for the perforated panel 498 – of a Helmholtz resonator 436 F foil – elastic 457, 461, 462, 465–467, 471 – elastic, with losses 467 – limp or elastic 403, 465 – porous and elastic 467 – porous and limp 466 – tight and elastic 466 – tight and limp 466 – tight or porous 403, 465 Foil resonator 469, 470 Free plate with an array of circular holes 429 friction force coupling 499, 500 H Helmholtz resonator 403, 419, 435, 436, 444, 449, 452, 456, 502 Helmholtz’s source superposition theorem 462 L loss factor 420, 426, 428, 467,485,496 lowest resonance 419, 420, 429,470 M mass reactance – of resonator orifice 417

1226

Chapter Index

matching – average sound pressures 409 Mathieu differential equations 487 Mathieu functions – azimuthal 487 – radial 487 micro structure – of perforation 497 modal partition impedances 488 modal reflection factors 440, 443, 449, 451 mode coupling coefficients 412, 454, 493 multilayer absorber 403 O orifice impedance – back side 409, 417, 433,436, 445– 447 – front side 409, 413, 417,419 oscillating mass 403, 419, 426,436, 445, 448, 452 P panel perforation 497 partition impedance – of membranes 469 perforation pattern 497 periodic structure 403, 438, 474, 480 plate vibration modes 488 Plate with narrow slits 407 Plate with wide slits 411 poro-elastic foil 457, 461, 462, 465,466, 471 porous foil 452 porous panel absorber – rigorous solution 496 pressure source 403 primitive root diffuser 475 propagation constant – in a flat capillary 405 pseudo-random variation 474 Q quadratic residue diffuser 475

R radiation impedance 403, 404, 414 reflection factor 404, 405, 440, 443, 449, 451, 495 resonance condition 419, 437 resonance formula 420 resonance frequency – of foil resonator 469 – of slit resonators 419 Ring resonator 471, 473 S scattered far field 474, 476–479 scattered field 491, 495 simply supported plate 485, 496 Slit array – with viscous and thermal losses 420 Slit resonator – dissipationless 415 – with viscous and thermal losses 426 slit resonator array – covered with a foil 462 – with porous layer on back orifice 449 – with porous layer on front orifice 452 – with subdivided neck plate 461 – with subdivided neck plate and floating foil 457, 461 spatial harmonics 451 spherical shell 468 surface impedance 456, 469, 493 surface porosity 404, 436 T thermal wave 421 Tight panel absorber – approximations 493 – rigorous solution 485 V viscous and thermal losses 404, 407, 420, 426, 430, 431, 434, 438, 476 viscous wave 421

Chapter Index

W wave equation of a membrane 469 wave impedance – in a flat capillary 405

1227

wide-angle absorber – near field and absorption 480 – scattered far field 474

I. Sound Transmission A absorber variable 504, 546 B bending loss factor 535, 536, 542, 552 bending wave equation 551, 552, 575 bending wave impedance 593 bending wave velocity 526 Berger’s law 534 bulk reacting lining 518 C chambered joint 520 characteristic equation 517, 522, 545, 553, 554, 576, 583 characteristic values 506, 518, 538, 540, 541, 553, 567,571 coincidence frequency 526, 571 continued fraction 518 corner impedance 527 coupling coefficient 522, 556, 558,563, 566, 569, 574, 583,585 – between duct modes and plate modes 556 critical frequency 542, 566 D diffracted angle 588 diffuse sound incidence 508, 513, 525, 534–536, 540,559, 567, 581,582 double sheet 594 double-shell – wall with absorber fill 538 – wall with thin air gap 540 double-shell resonance 540 – for an empty interspace 540 duct modes 555–561, 563, 570, 573, 574, 584

E elliptic-hyperbolic co-ordinates 565 emission room 577, 578, 581,584 empty and unsealed hole 513, 514 end correction 507, 521 equivalent chain network 520 equivalent network 515, 538, 541 equivalent network method 541 F field matching 523, 562, 579 filled interspace 538, 539 flat capillary 509, 540 foil 506, 511, 519 force impedance 527 free bending wave number 525, 529 G generalized bending stiffnesses 537 H hole transmission with equivalent network 515 hole, sealed, filled 515 I isotropic plate 537 L least attenuated mode 517, 518 longitudinal wave number 526, 529, 535 loss factor 529, 535, 536, 542,552,566, 590 M matching of fields 579 Mathieu functions 565–567

1228

Chapter Index

modal partition impedance 552, 563, 571, 576 mode coupling coefficients 566, 569, 574, 585 mode norms 522, 553–556,561,562, 566, 569, 571, 573,583 momentum impedance 527 N network elements 538, 541 niche effect 559 niche modes 560, 561, 563 noise barriers 503 noise sluice 503, 521, 523,525, 594 O office fences – with second principle of superposition 584 P partition impedance 527–529, 533, 535, 538, 542, 545,552, 563, 566,571, 575, 576 plate mode norms 562, 569 plate modes 552, 555, 557, 558, 560, 562, 565–567,570, 573 Plate with absorber layer behind 541 Plate with bending losses 535, 552 Plenum modes 574–576, 578 plug of porous absorber material 503 Poisson’s ratio 526, 589, 593 porous material 504, 506, 509,511, 519, 538, 541, 546, 571 propagation constant 503, 509, 511, 517, 544, 545, 575 R radiation impedance 507, 508, 512, 513, 516, 519, 520 receiving room 559, 577, 578,581 S Sandwich Panel 542 – with elastic core 589 – with porous board as core 589 – with porous board on back 547

– with porous board on front 544 sealing impedance 507 shear wave number 529, 535 simple plates 536, 538, 594 simply supported plate 552, 557, 567 slit – narrow, empty 509 – narrow, empty with sealing 509 – narrowe, filled with porous material 506 sound transmission – through a hole in a wall 511 – through a simple plate 532, 534 – through a slit in a wall 506 – through finite size double wall with porous absorber core 571 – through finite size plate with a front side absorber 567, 570 – through infinite plate between two differnt fluids 587 – through lined slits in a wall 517 – through office fences 503, 582 – through simply supported plates 565, 567 – through single plate in a wall niche 559 – through strip-shaped wall in infinite baffle wall 564 – through suspended ceilings 577 sound transmission factor 503 sound transmission through plates – with equivalent circuit 534, 588 Struve functions 508 suspended ceiling 503, 574, 577,580, 581, 594 T Table – elastic data of materials 531 – with characteristic values for clamped plates 553 – with characteristic wave speeds in plates 526 – with classical boundary conditions for plates 527 – with effective bending moduli of sandwich panels 543 – with elastic data of materials 529

Chapter Index

– with mode norms for clamped plates 554, 555 – with relations of elastic constants 531 – with types of sandwich panels 543 Timoshenko-Mindlin theory 529 trace wave number 528, 529, 535,590 transmission coefficient 504, 505, 508, 512, 513, 516, 520,523, 533–535,537, 539, 541, 547, 549,551, 557, 559,564, 567, 570, 571, 574,577–579,582,590

1229

transmission factor 503, 505, 537, 540, 587, 592, 593 transmission loss 504, 505, 512, 519, 523, 525, 527,536,537, 540,542,547, 549, 559, 564,570,577, 581,593 triple sheet 594 W Wall of multiple sheets with air interspaces 591 wave impedance 503, 509, 511, 517, 518, 575, 587

J. Duct Acoustics A adiabatic boundary condition 598 adjoint absorber 725–729, 731 admittance equation 610 admittance of annual absorbers – approximated with flat absorbers 643 algorithm – for optimisation of generalised Cremer admittance 729 Amplitude nonlinearity of fences 742 Annular ducts 647, 754, 762,764, 790 anti-symmetrical modes 606, 611, 613, 615, 619, 620, 627,707, 712, 715,718, 733, 776 axial propagation constant 595, 599, 602, 622, 671, 676,695, 706, 750 axial wave impedance 596, 700 B baffles 657, 680 Bent 752, 762, 781,783, 784, 788 – and straight ducts – with unsymmetrical linings 785 – flat ducts with locally reacting lining 762 bent – duct section 781, 783 Bessel differential equation 629, 703, 756 branch point of symmetrical modes 720

branch points 613–616, 618, 633–637, 640, 641, 643,720,736, 737 C Capillary – circular 598 – flat 595, 598 capillary – circular, with isothermal boundaries 598 – flat, with adiabatic boundaries 598 – flat, with isothermal boundaries 595 characteristic equation 596, 598–602, 606–608, 610, 611, 613, 614,616, 618, 620–623, 626–631, 645–647, 649,652, 654–657, 659, 660, 672, 678,695, 703, 706, 721, 732,733,736, 745,751,752, 754–756, 758, 760–762,764–766,768– 774, 776, 786 co-ordinate surface 601 concentrated absorber in an otherwise homogeneous lining 675 conical duct transition 702 – hard walls 702 – lined walls, stepping admittance approximation 712 – lined walls, stepping duct approximation 705 continued fraction 603, 604, 611, 612, 616, 623, 627,630,643, 647,652,751, 752, 754, 774

1230

Chapter Index

converging cone 707, 709 corner area 694 coupling coefficients 656, 659, 661, 674, 682, 690, 693,695, 699, 706,707, 713 cover foil – on bulk layer 623 Cremer’s admittance 720–722, 725 – with parallel resonators 725 cross-joints of ducts 697 D density wave 595, 596, 599 design point 722, 723, 725–728,730 diverging cone 708, 710 Duct section with feedback between sections without feedback 672

– with unsymmetrical local lining 750 flow resistance coefficient 743 flow-induced nonlinearity of perforated sheets 748 G graphical evaluation of attenuation 644 Grigoryan’s expansion of the characteristic equation 752 Grigoryan’s method 771 I Influence of flow on attenuation 731 Influence of temperature on attenuation 740

E effective compressibility 600 effective density 600 elliptic-hyperbolic co-ordinates 697 euquivalent annular and flat ducts 762 evaluation of sets of mode solutions 613, 633 expansion of characteristic equation – with Grigoryan’s method 752, 756 – with theorem of multiplication 758

L lateral mode profiles 681, 689, 706 least attenuated mode 602, 607, 608, 612, 623, 626–628, 644, 647,686, 688, 720, 721, 727,737,738, 750,754,776, 782, 784 – in rectangular duct 607 Lined duct corners and junctions 693, 788 Lined ducts 595, 601, 607, 613,650, 693, 697, 762,781,788

F feedback 671–673, 675 fictitious – adjoint absorber 725 – ducts 694 – volume source 676 field matching 681, 691, 699, 713 Flat duct – with a bulk reacting lining 619, 621, 623 – with an anisotropic lining 621 – with an unsymmetrical bulk lining 628 – with cross-layered lining 650, 651, 657 – with unsymmetrical bulk lining 628

M Mach number 732, 736, 744, 746,748, 789 matching of fields 656, 681, 691, 695, 699 Mathieu functions 697, 698, 790 modal angle 602, 624, 688, 716 modal reflection factors 695 mode approximation – by continued-fraction expansion 754 – by identical transformation 609 – by power series expansion 608, 649 – from the admittance equation 610 – in round ducts by iteration over resistance 646 mode charts 630, 633

Chapter Index

mode coupling coefficients 659, 661, 674, 693, 695, 706 mode coupling integrals 778, 780 mode eigenvalues 752, 754, 755,771, 774, 777, 781 mode excitation coefficients 718 mode jumping 618, 619 mode mixture 715 – with equal mode amplituted 717 – with equal mode energy densities 717 – with equal mode powers 717 mode norms 607, 656, 659, 661,676, 678, 681, 690, 695,706, 713, 716,718, 778, 780, 785, 786 mode orthogonality 661 mode profiles 629, 655, 681,689,692, 694, 706, 715, 762 mode range limit 616, 643 mode solution – by iteration through k0 h 624 – iteration through the modal angle 624 mode solutions 602, 613, 618, 619, 623, 633, 642, 734, 750,752, 754, 758,765, 767, 768, 771 mode solutions in bent ducts – both walls soft 770 – inner wall hard, outer wall soft 764 – with ideally reflecting walls 764 – with locally recating walls 764 modes – anti-symmetrical 611, 613, 619,620, 627, 707, 712, 715,718, 733, 776 – symmetrical 602, 606–608, 611,613, 618–620, 627,707, 712, 713,715,718, 733, 776, 785 Morse chart 607, 612, 621, 623 Muller’s procedure 603, 647, 752,754, 758, 761, 762, 766,767, 772 N nonlinearities – by amplitude and/or flow 742 – by flow along fibre absorber 746 – by flow over orifices 743 – by flow through an orifice 744

1231

– by flow through porous absorber 747 O optimised parameters 729 orifice input admittance 678 orthogonality – of modes in ducts with bulk lining 621 orthogonality relation 621, 655 P particle velocity profiles 597 partition impedance 620, 628, 645, 692, 742, 748,749 perforated sheets 742, 743, 748, 749 pine-tree baffles 657 pine-tree silencer 657 polyurethane foam 748 primary absorber 726, 728, 729,731 propagation constant 595, 598–600, 602, 622, 650,671,673, 676,695,706, 740, 747, 748,750 R radial mode 630, 633, 641,642, 647 radiation impedance 700, 701 radiation loss 699–702 ray formation 721, 722, 784, 789 reciprocity at duct joints 750 reflection factor 657, 695, 745, 746 resonator neck 676, 729 resonators in parallel 725 resonators in series 722, 723 ring-shaped duct 754 round duct – with a bulk reacting lining 645 – with a locally reacting lining 629, 634, 636, 642 S sectional admittances 712 Sections and cascades of silencers 671 secular equation 595, 598, 599, 601, 602, 606, 620,622,624, 629,659,660, 671, 732, 763

1232

Chapter Index

sets of mode solutions 618, 642 – in annular ducts via modes in flat ducts 762 – in annular ducts with unsymmetrical lining 754 – in flat ducts with bulk reacting lining 623 – in rectangular ducts 605 silencer modes 681, 682 silencer with rectangular turningvane splitters 787 sound radiation from a lined duct orifice 697 sound transmission coefficient 682 source contributions 676 spatial harmonics 653, 654, 680,683– 685, 689 Splitter type silencer – with bulk reacting splitters 688 – with locally reacting splitters 680, 683 stagnation enthalpy 744, 745 Stationary flow resistance of splitter silencers 741 step in duct – contracting, lateral-lateral 664, 665 – contracting, lateral-local 666 – contracting, local-lateral 668, 669 – contracting, local-local 662 – expanding, lateral-lateral 663, 665 – expanding, lateral-local 667, 668 – expanding, local-local 662 – expaning, local-lateral 666 Step of duct height and/or duct lining 658 substitutions induced by flow 732 surface admittance – of annular absorbers 643 – of annular lining 646 surface wave mode 614, 617–619, 641, 642, 719, 720 symmetrical modes 613, 619, 620,707, 712, 718, 776

T T-joints of ducts 697 T-shaped Helmholtz resonator 678, 679 Table – with branch points of characteristic equation in round ducts 634 – with coefficients of polynomial approximation to circular modes 631 – with relations in flat duct with anisotropic bulk reacting layer 622 temperature wave 595, 599 temperature-dependent input parameters 740 test of mode approximations 612 three-dimensional mode charts 607, 633 time derivative – partial 731 – substantial 731 transmission coefficient 682, 692, 699 transmission loss 602, 603, 684, 687, 688, 696, 781–784, 788 triple Helmholtz resonator 723 turning-vane splitter silencer 775, 781, 788, 789 – transmission loss 781 U unsymmetrical duct 627 V viscosity wave 595, 599, 747 viscous and thermal losses 595 W wave equation 595, 601, 605, 606, 620, 622, 652, 676,690,703, 715,732 wave impedance 596, 598, 599, 700, 740, 747, 748,755

Chapter Index

1233

K. Muffler Acoustics A absorber variable 837 Acoustic power 793–797, 801 Acoustically Lined Circular Duct 837 anechoic termination 797, 801 Annular Airgap Lined Duct 834, 840 B baffle silencers 832 Bellow – flexible-wall 832 bellow – multi-step 832 bulk-reacting lining 839 C Capillary Tube Monolith 829 cascaded one-dimensional systems 796 catalyst pellets 828 Catalytic converter 828 common perforated section 819, 820, 822, 824, 825, 836 Concentric-Tube Resonator 806, 808 conical duct transitions 832 Conical tube 803, 804 convected quantities 793, 794 convective radiation flow impedance 795 convergent horn 804 cross-flow 806, 809, 816,818,821, 823 cross-flow contraction element 809 cross-flow expansion element 809 cross-flow, closed-end, extented perforation element 823 cross-flow, open-end, extended perforation element 821 cross-flow, three-duct, closed-end element 816 cross-flow, three-duct, open-end element 818 D divergent horn 804 downsound cross sections 793

downstream end 797 dynamic head 800, 802 E effect of meanflow 795, 833 eigenmatrix 807, 815 end correction 795, 800, 827 Exponential horn 804 Extended Inlet/Outlet 801 extended outlet element 828 extended perforated pipes 822, 823, 825 Extended-Tube Three-Pass Perforated Element 828, 840 F flow impedance 793–795, 797, 802, 805, 818–820, 830, 831, 833,836, 837 flow resistivity 837 flush-tube three-pass perforated element 827 four-pole parameters 796, 797 G granular pellets 829 grazing flow 808, 818, 834, 841 grazing-flow impedance 818, 836, 837 H Helmholtz resonators 830 Hose 801, 804–806, 820, 822, 832,835, 840 hose-wall impedance 805 I impedance jump 801 In-line Cavity 831 insertion loss 796, 797 J junction of pipe and chamber 800 L level difference 796, 798

1234

Chapter Index

M Mach number 794, 795, 798–802,808, 820, 833, 834, 837,838, 840 matrix equation 793, 827 mean flow 793, 799, 800,802,832, 840, 841 micro-perforated Helmholtz panel parallel baffle muffler 836 modal matrix 807, 815, 826 Muffler Acoustics 793 Muffler performance parameters 796 N noise reduction 796, 798 O open-end elements 818, 821 P Parallel Baffle Muffler 832, 836, 839 partition impedance 807, 837, 838 Pellet Block Element 829 Perforated Extended Inlet 814 Perforated Extended Outlet 813 perforates in stationary media 808 – with cross flow 807 – with grazing flow 808 Pod Silencer 832 porous blocks 828 principle of impedance mismatch 801 Q Quincke tube 833, 841 R Radiation – flow resistance 795 – impedance 793, 795, 797,805 radiation – from the open end of a flow duct 795 reflection factor 794, 795, 801 reversal contraction, two-duct, openend perforated element 812 reversal expansion element 812, 827, 828

reversal-contraction 820 reversal-expansion 820 reversal-expansion, two-duct, openend perforated element 811 Reverse-Flow 810, 817, 819,824, 840 Reverse-Flow Contraction Element 810, 819 Reverse-Flow Expansion Element 810, 819 Reverse-Flow, open-end, extended perforated element 824 Reverse-Flow, open-end, three-duct element 819 Reverse-Flow, three-duct, closed-end element 817 reversed-flow elements 818 S stagnation pressure 799–802 stationary medium 794, 795, 799,801, 803, 806 Sudden area changes 799, 801, 829 sudden contraction element 827 T Three-Duct Perforated Element 814, 816, 820, 840 Three-duct perforated elements with extended perforations 820 three-duct perforated section – common 814, 820 – particular 816 Three-Pass perforated elements 825, 827 transfer matrix 796, 797, 799–804, 806–808, 812, 814, 816, 818–833, 835– 840 transfer matrix representation 796 transformation matrix 793, 796 transmission loss 796, 797, 801, 832, 839 two-duct elements 808, 825 Two-Duct Perforated Elements 806 U Uniform tube – with flow and viscous losses 798

Chapter Index

upsound cross sections 793 upstream end 796

1235

V volume flow impedance 793, 805, 819, 820 volume flow velocity 793

L. Capsules and Cabins A absorbent sound source in a capsule 847 B bending loss factor 845, 848, 862,866 C Cabin – semicylindrical model 861 – with plane walls 865 – with rectangular cross section 869 capsule efficiency 843 capsules – energetic approximation 843, 844, 847, 852 D diffuse sound incidence 846, 847, 852, 861, 869 E equivalent circuit elements 866 equivalent network 865, 867 H Helmholtz’ source theorem 847 Hemispherical source and capsule 857 I insertion coefficient 846 insertion loss 843, 846, 851,852, 856, 857, 859, 860

M modal partition impedance 862 modal radiation impedance 855, 859 multi-modal excitation 856, 857, 859, 860 N narrow capsule 850 O oncoherent sound incidence 868 P partition impedance 844, 845, 848, 854, 858, 862,871 R radiated power 844 S Semicylindrical source and capsule 853 shadow field 867 sound protection measure 861, 864, 867, 868 sound transmission factor 844 sound transmission loss 846 source impedance 853 T the source is a pressure source 849 the source is a velocity source 850

1236

Chapter Index

M. Room Acoustics A addition theorem 908 alogorithms for mirror source construction 939 anti-symmetrical mode 874 aperture 889, 897–899 average absorption coefficient 879, 936, 942 averaging over time intervals 922 B beams 937 break-off criteria for scouting 932 C centre time 941 characteristic equations 875 clarity 941, 943 collection in a corner source 907 Combination of corner fields 906 concave corners 889, 897, 903, 913, 917, 930 concave model room 891 condition of visibility 899 conditions for statistical methods 876 cone approach 937 continued-fraction expansion 875 convex corners 884–887, 889, 890, 896, 897, 899, 900, 913,917, 930 corner – real 890, 905 – virtual 890, 905 corner field 906 corner set of mirror sources 905–907 corner source 907, 909, 910, 912, 921 D daughter source 882–884, 887, 889, 896, 897, 903, 904,922, 923, 932 decay curve 923, 937, 939 decay of the average energy density 923 decay rate 878, 881 definition 883, 884, 920,922, 923, 940, 941, 943

Density of eigenfrequencies 876 diffuse sound field 879, 880, 936,937 directivity factor 883 distance of limit between direct and reverberant fields 879 E early and late energy densities 942 early decay time 939, 940 edge condition 909 efficiency checks 899, 905, 909, 912 eigenfrequencies 875, 876 Eigenfunctions in parallelepipeds 873 energy balance 879 energy decay 936, 937 energy density 879, 880, 923,939 energy integral 937, 942 equivalent absorption area 880 equivalent source for the corner field 907 equivalent sources 922 F field angle 884, 885, 887,888,893, 896, 898, 905, 907,909,932, 934 field angle criterion 884 field angle of a mirror source 884 foundation of the mirror source approximation 882 fundamental solutions 873 G General criteria for mirror sources 883 Geometrical room acoustics in parallelepipeds 877 geometrical subtasks 889, 926 Global application of the principle of superposition 921 goal of scouting 931 grouping of mirror sources 903, 905 I illegal mirror source 883, 884, 886, 887, 896, 909,912

Chapter Index

impulse response 938–941 incoherent sources 923 ineffective mirror source 884 Inside checks 888, 889, 928 inside criterion 883, 885, 888,904, 916, 918, 930 intensity of a reflection 879 intensity of reverberant field 878 interaural cross-correlation coefficient 942 interaural cross-correlation function 941 interrupt checks 887, 899, 905, 909, 912, 931 L Lambert scattering 937 Lambert’s law 935, 939 late lateral sound level 942 lateral energy fraction 941 level decay 878, 881 looping in mirror source scouting 932, 933 M mean free path length 877 – energetic average 877 – geometrical average 878 mean reflection rate 879, 936 mirror source 877, 882–884, 904–906, 909, 914, 918, 924,930–932,938,939 Mirror source model 882 mirror sources of parallel walls 907, 910 mirror-receiver method 910 mode overlap 876 model room with convex corners 899 modes 873, 875, 876,935 Monte Carlo method 936, 937 mother source 883, 884, 888–890, 893, 899, 910, 915, 932 multiple covering of mirror sources 885 N nested loops 890, 932 number of mirror sources 877, 882 number of reflections 877, 879, 880

1237

O orchestra pit 897–900 original source 882, 883, 887,889,890, 907, 911, 912,932 P parallel walls 891, 907, 910, 911 principle of superposition for unsymmetrical absorption 920 probability density 877, 936, 939 R random scattering 935 random-incidence absorption 936 ray energy 935–937 ray path 938 Ray sources 938 ray tracing 935, 937 reciprocity in the mirror source method 910 reflection factor of a plane wave 883 reflection rate 879, 936, 942 reverberation curve 923–926 reverberation time 876, 878, 881,922, 923, 925, 926,940,942 – definition 922 – Eyring 881 – Millington-Sette 881 – Pujolle 881 – Sabine 881 reverberation time with mirror source results 922 room acoustical parameters 940, 942 room impulse responses 939 room transfer function 877 room with concave edges 924 S scaled co-ordinates 891 scattered field 914, 917–919 scouting of mirror sources 930–932 secular equation 874, 875 shading 889, 896 sound strength 941, 942 source factor 883, 886–888, 895, 902, 904, 910, 911,915,918, 922,932 source list 889, 890, 899,932–934

1238

Chapter Index

sources as a network 922 specular reflection 935, 937 speech intelligibility 941 statistical methods 876 Statistical room acoustics 879 subjective transparency 941 symmetrical mode 874, 875

temporal density of reflections 877 total number of reflections 877 V visibility check 899

T Table – with numbers of legal and effective mirror sources 901

N. Flow Acoustics A Acoustic analogy – with effects of solid boundaries 984 – with mean flow effects 970 acoustic efficiency 1003 acoustic power 1003, 1004 Acoustics of moving sources 993 adiabatic flow 946 aerodynamic sound 949, 964, 1000 – Doak’s theory of aerodynamic sound 981 – fluctuations of flames 991 – turbulent flames 991 Aerodynamic sound sources 1000, 1012 arbitrarily moving surface 999 autocorrelation 953, 1004 average – ensemble 950 – mass-weighted 950 – phase 950 – spatial 949 – time 949 averaging 948–952, 959 B Basic equations of fluid motion 954 Biot-Savart law 979 blade thickness noise 1007 boundary layer 948, 949 bulk modulus – adiabatic 946

– isothermal 946 C coefficient of expansion 946, 956 compressibility 946, 959, 968, 990 compressible flow 946, 952, 968,969, 979 Concepts and notations in fluid mechanics 945 conservation – of mass 960 – of momentum 960 continuity equation 951, 952, 954, 957, 959–961, 963, 971, 973, 976,977, 981, 984, 992 convected scalar wave equation 982 convected wave operators 980 – Doak’s form 981 – Howe’s form 981 – Möhring’s form 981 convective operator 972 convective wave equation 952, 962, 972, 974, 998 Correlations 953, 1017 Curle’s equation 988 cylindrical co-ordinates 962 D Decomposition 950–952, 958, 959, 961, 965, 968, 972,975 – of field qualities 965 dipole point source 995, 996

Chapter Index

dipole source 965, 976, 985, 989, 990, 995, 996, 1012 dissipation rate 954, 992 double subscript summation rule 945 E energy density 955 – fluctuating part 958 – stationary part 958 energy equation 954, 955, 957,981 enthalpy – specific 946, 955, 972, 977 – stagnation 947, 955, 977 – total 955, 977–982, 1015 entropy 946, 947, 956, 970,973, 978, 979, 983, 988, 990 equation of continuity 960, 963, 964 equation of motion 960, 963, 964 equation of state 951, 956, 961 Equations of linear acoustics 960, 961 Euler equations 954, 956, 958 external source flux 964 F Ffowcs Williams-Hawkings equation 1015, 1016 fluid – density 955 – energy 955 – ideal 945, 964 – Newtonian 945 – non-Newtonian 945 – pressure 946 – viscosity 970 free-space solution 965 G gas constant 945, 956 generalised wave equation 987 H Heaviside function 984 Helmholtz vorticity equation 979, 980 homentropic flow 947, 978, 982

1239

I ideal flow 946 inhomogeneous wave equation – convective 970, 971 – Dowling’s form 988 – Ffowcs Williams-Hawking’s form 984 – Goldstein’s third-order form 973 – Goldstein-Howe’s form 1005 – Howe’s form 978 – including the stream function 975 – Lighthill’s form 963, 988 – Lilley’s convective inhomogeneous form 971 – Lilley’s form 971 – Meecham’s form 969 – Phillip’s convective imhomogeneous form 970 – Phillip’s convective inhomogeneous form 973 – Ribner’s form 968 integral length scale 953, 1005, 1006 integral time scale 953 internal energy 946, 955, 977,981 inviscid flow 946, 961 irrotational flow 947, 961 isentropic condition 961 isentropic flow 946, 964, 979 isothermal flow 947 J jet noise 989, 990, 1000, 1003, 1016, 1017 K Kirchhoff source term 987 L Lamb vector 977, 979 laminar flow 947 Laplace operator 962 – cylindrical 962 – spherical 963 Lighthill stress tensor 964, 972, 997, 1010 Lighthill’s equation 963, 970, 973, 974 loading noise 1008, 1009

1240

Chapter Index

M Mach number 962, 964, 969, 970, 976, 978–980, 987,991, 994, 997,999,1002, 1007, 1008,1010, 1011,1013, 1014, 1016 models of fluid flows 946 – Adiabatic flow 946 – boundary layer flow 948 – compressible flow 946 – homentropic flow 947 – ideal flow 946 – incompressible flow 946 – inviscid flow 946 – irrotational flow 947 – isentropic flow 946 – isothermal flow 947 – laminar flow 947 – non-uniform flow 947 – Real flow 946 – rotational flow 947 – stationary flow 947 – steady flow 947 – turbulent flow 947 – uniform flow 947 – unsteady flow 947 – viscious flow 946 momentum equation 947, 949, 952, 954, 955, 957, 959–961, 963,968,971, 973, 976–978,981, 984, 985,993 momentum flux 949, 989, 995–997 monopole source 965, 985, 989,990, 994, 995, 1012 moving Kirchhoff surfaces 998 moving source 947, 970, 984, 986, 993, 994, 998, 999, 1002, 1007,1014–1016 N Navier-Stokes equation 952, 960 near-field and far-field solution 965 Newton’s law 960 non-linear disturbance 956, 957 non-uniform flow 947, 971 O octupole source 964, 967, 989,996

P perturbation quantity 956 Poisson’s equation 968, 969 power law 1002, 1012–1014 – of compact aerodynamic multipoles 1012 – of moving aerodynamic sources 1014 – of noncompact aerodynamic multipoles 1012 power law for the acoustic intensity 1002 power law for the aerodynamic sound sources 1012 power spectral density 954, 1000, 1003 premises of linear acoustics 961 Pressure-source 968, 969, 1016 pressure-source theory – by Meecham 969 pseudo-sound 968 Q quadrupole noise 1009, 1016 quadrupole source 964, 965, 976,985, 990, 995–997, 1007, 1012 R random fluctuations 951, 953 ratio of the specific heats 956 real flow 946 retardation time 965 retarded source strength 965 Reynolds averaging 950, 952 Reynolds stress 949, 953, 964,993 Ribner’s source term 974 rigid boundaries 966 rotational flow 947, 948 rotor – dipole sound 1011, 1012 – monopole sound 1011 Rotor noise 1007, 1010, 1017 S Scales 948, 953 self-noise 960, 967, 975 sexdecupole source 997 shear noise 960, 967, 971, 975

Chapter Index

shear stress 945, 946, 949 solid boundaries in flow 966 solution of inhomogeneous wave equation 991 sound field of moving point sources 994 sound from free turbulence 966 sound generation – by flow interaction 984 – by fluctuating heat sources 988 – by interaction of turbulence and sound 960 – by turbulent interaction 960 – by turbulent two-phase flow 992, 1015 sound propagation – in a shear flow 972 – in non-uniform flow 971 source terms 964, 967, 970–972,976, 983, 987 specific – enthalpy 946, 955 – entropy 946 – heat 946, 956 – heat ratio 946 – internal energy 946 speed of sound 946, 956, 963, 970, 1013 spherical co-ordinates 963 stationary flow 947 steady flow 947, 989 subsonically moving surface 999 substantial derivative 969, 970 supersonic jet flow 1003 supersonically moving surface 999

1241

T thermal conductivity 945, 946, 970, 971, 981 Thermodynamic relationships 955 thickness and loading noise 1009 Tools in fluid mechanics 949 turbulence 947, 948, 951, 954, 960,966, 984, 992, 1001,1002, 1005–1007, 1013, 1015, 1016 turbulence level 948 turbulent flow 947–949, 959, 960, 986, 1015 U uniform flow 947, 962 V vector Green’s function 980 velocity potential 962 viscous flow 946, 948, 1017 viscous stresses 963, 967, 970,986 vortex sound 976, 1016, 1017 vorticity 947, 976–980, 1016 W wave equation – convected 972 – in uniform flow 962 – inhomogeneous 949, 963–965, 968– 971, 973, 975,976,978, 987–989, 991, 993, 1000 – linear, homogeneous 962

O. Analytical and Numerical Methods in Acoustics A absorbing boundary condition operators 1080 absorption coefficient 1019–1021, 1025, 1027 acoustic equilibrium equation 1075 acoustic-structural analogy 1075 average admittance 1026 averaged square error 1020 axial propagation constant 1029

B backward propagating modes 1029 boundary element method 1019, 1040, 1046, 1049, 1051, 1059,1071 boundary equations 1041, 1042, 1061 boundary error 1040, 1042, 1053,1056, 1059 boundary integral equation 1019, 1059, 1061, 1063, 1065, 1068,1072, 1073, 1103–1105

1242

Chapter Index

boundary integral methods 1061 boundary operator 1035, 1044 branch points 1035 bulk modulus 1074 Burton and Miller method 1068 C capillaries 1028 cascades of ducts 1033 Cat’s Eye model 1081, 1082, 1107 circular piston in an infinite rigid baffle 1049 collocation method 1063 Combined integral equation formulation 1066, 1106 complex source point 1049, 1051, 1071, 1103,1104 Computational optimisation of sound absorbers 1019 Computing with mixed numericsymbolic expressions 1028 condition number 1055, 1069, 1105 contracting duct 1030, 1031 coupled fluid-structure equation 1080 Critical frequencies 1046, 1047, 1052, 1060, 1066,1069, 1071 D damping factor 1072 differentiated integral equation 1068 diffuse sound incidence 1020, 1023, 1026, 1027 dipole terms 1063 directivity pattern 1057, 1058 Dirichlet condition 1036 discretization of the boundary integral equation 1063 discretized functional 1076, 1077 discretized global functional 1078 double-layer potential 1061, 1062, 1068, 1069 E effective sound power 1035, 1056 eigenfrequencies 1037, 1075,1078 eigenfunctions 1072 eigenmodes 1037, 1072,1075

eigenvalue problem 1037, 1078 equilibrium of stresses 1074 equivalent – radiant problem 1036 – source system 1042 error field 1046 Euler-Lagrange equation 1076 expanding duct 1030, 1031 exterior Dirichlet problem 1061, 1062, 1105 F field admittance 1035, 1036 field matching 1086, 1096 finite element method (FEM) 1019, 1041, 1059, 1071, 1074 finite element models (FEM) 1064 fixed parameters 1021 fluid-elastic structure interaction problem 1037 fluid-structure coupling 1078 forward propagating modes 1029 frequency response curve 1020 Frobenius norm 1055 full-field equations – of the first kind 1046, 1048,1052 – of the second kind 1047, 1048, 1052, 1054 G Galerkin method 1045 Gaussian elimination 1064 general impedance boundary 1073 general termination 1033 generalised – eigenvalue problem 1078 generalised full-field equations, 1052 global shape functions 1077 Gram-Schmidt technique 1048 Green’s function 1069, 1070, 1072 Green’s function for a rectangular enclosure, 1071 H Hamilton’s principle 1076 Helmholtz equation 1034, 1036–1040, 1042, 1049, 1059, 1070, 1075,1076, 1103–1105

Chapter Index

Helmholtz formula 1060, 1072, 1080 Helmholtz integral equation 1040, 1046, 1059–1063,1065–1068,1070, 1071, 1103–1106 hybrid evaluation 1033 I impedance boundary condition 1036 incident wave 1032, 1036, 1037, 1057 integral operators 1063, 1065, 1068, 1069, 1105 interior – Dirichlet problem 1046, 1061, 1066, 1067, 1069 – Helmholtz integral formula 1070 – Neumann problem 1061 Irregular shaped cavities 1075, 1078, 1107 iterative – linear system of equations 1028 – solvers 1065 – systems, 1033 J Jacobi iteration 1065 L Lagrange function 1076 Lagrange multiplier 1067 Least squares minimization technique 1047 linearly elastic structure 1037, 1038 M matching – for the sound pressure 1087 matching conditions 1086 matrix norm 1055 mechanical analogy 1074 method of mirror sources 1071 method of weighted residuals 1040, 1042 minimization process 1042 minimum search 1021 mixed boundary conditions 1048 mixed numerical symbolic evaluation 1033

1243

mode coupling integrals 1088, 1096– 1098 Mode norms 1029, 1030,1096, 1098 mode orthogonality 1085, 1086 mode profiles 1029 modified Green’s functions 1069 monopole terms 1063, 1070 multi-layer absorber 1019–1021, 1026 multi-point multipole method 1051, 1056 multigrid method 1060, 1066, 1104, 1105 multipoles 1044, 1051,1057 N Neumann boundary value problem 1034 Neumann symbol 1072 nodal variables 1076, 1078 nodes 1064, 1067, 1076–1078 normalizing factors 1043 null-field equations 1040, 1045,1046, 1048, 1051, 1052, 1055, 1057,1058, 1067, 1068 numerical sound intensity method 1035 O one-point multipole method 1051, 1056 Orange model 1094 Orthogonality of modes 1095 orthogonalization method 1048 orthonormal 1043, 1048, 1067 P parameters – variable 1022, 1023, 1026,1027 particle velocity source 1032 Picard iteration 1065, 1066 potential-layer approach 1061 procedure for optimisation 1019, 1021 R radiation condition 1035–1037, 1039, 1040, 1060, 1070, 1080, 1081 radiation efficiency 1034, 1035

1244

Chapter Index

strain tensor 1038 stress tensor 1038 structural impedance matrix 1079 successive overrelaxation method 1065 superposition method 1040, 1051, 1103 surface velocity distribution 1073 surface velocity error 1047, 1053,1056 symmetry relation 1042–1047, 1052

Radiation problem 1034–1036, 1040, 1042, 1046,1063, 1065,1066, 1070, 1072, 1081,1082, 1092,1102–1105 Reduction of the system of equations 1091 S scattering problem 1019, 1036, 1037, 1040, 1056,1059, 1072,1079, 1081, 1082, 1103 secular equation 1029 self-adjoint formulation 1065 shape functions 1064, 1076–1078 single-layer potential 1061, 1062, 1066 six elements per wavelength rule 1064 Sommerfeld’s radiation condition 1035, 1039 Sound field in interior spaces 1037 Sound fields in rooms and halfspaces 1070 sound pressure source 1031 sound radiation into a half-space 1071 source condition 1031–1033 source function 1040–1042, 1044, 1048, 1051,1054, 1056 source simulation technique (SST) 1040 source strengths 1055 source superposition 1055 spherical harmonics 1043, 1093 spherical wave functions 1040, 1042– 1045, 1047,1048, 1051,1054, 1067, 1080 stability and the condition number 1055

T target quantity 1021, 1028 target strength 1037, 1057, 1058 termination condition 1031, 1033 transmission condition 1039, 1073 transmission problem 1019, 1039, 1072, 1073 Treatment of singularities 1070 triangularization 1076 two-grid method 1066 V variable absorber parameters 1021, 1023 variational principle 1046, 1075, 1079 velocity potential 1038 viscoelastic material 1038 viscous and thermal losses 1028 W weight function 1021, 1024 weighted residual equations 1042, 1044–1047 weighted residuals 1040, 1042, 1044, 1057 weighting functions 1040

P. Variational Principles in Acoustics A acoustics of porous media 1112 anisotropic absorbent 1118 anisotropic and inhomogeneous lining 1121

B boundary conditions – Diriclet 1110 – Neumann 1110 boundary layer thickness 1115

Chapter Index

C circular annuli 1122 circular tube 1114, 1131 coupled eigenproblem 1128 cut-on frequencies 1110, 1112 D dispersion relation 1117, 1120, 1123 duct with – flexible walls, no mean flow and no lining 1126 – rigide walls and mean gas flow 1126 – two cross-sectional lines of symmetry 1121 E Eigenfrequencies of a rigid-walled cavity 1110 eigenfrequency 1111 eigenmodes 1110, 1127 eigenvalue problems 1110, 1130 energy equation 1113 equivalent fluid 1117, 1118, 1121 Euler equations 1110, 1112, 1113, 1117, 1118,1121, 1122,1125, 1128 F flat-oval duct 1110, 1111, 1130 forced boundary condition 1114, 1117 functional 1109, 1112, 1113,1117, 1118, 1121, 1122,1124, 1125,1128, 1129 G governing differential equations 1110, 1124

1245

H heat-conducting walls 1113 Helmholtz equations 1117, 1127 L Lagrange density 1109 Lagrange equations 1109 Lagrange-Euler equation 1109 Lagrangian 1109 low frequency approximation 1115, 1117, 1125 M Mach number 1118 mean flow 1123, 1126 mode functions 1111 mode shape 1120, 1123 N Navier-Stokes equation 1113 S stationary value 1109 T thermal boundary layer 1114, 1115 trial functions 1109–1112, 1114,1116, 1119, 1122, 1128–1130 V variational functional 1118, 1121,1122 variational methods 1109, 1110 variational principle 1109, 1110, 1113, 1117 variational statement 1118, 1125, 1128 variational techniques 1110 velocity potential 1109 viscous and thermal boundary layer 1114

Q. Elasto-Acoustics A acoustic Poynting vector 1143 Anderson localisation 1146 angle of sound incidence 1175

Anisotropic media 1133, 1140, 1151, 1195, 1196 anisotropy – cubic 1137

1246

Chapter Index

– orthotropic 1136 Anisotropy and isotropy 1135

divergence relations for intensity 1144

B balance of forces 1135, 1139 bar – bending modulus 1168 – bending wave velocity 1168 – longitudinal wave velocity 1167 Bending mode 1157, 1158 bending stiffness 1158, 1160, 1161, 1165, 1175,1176, 1179,1180, 1182, 1183, 1190 bending waves 1160, 1162, 1164, 1165 – wave equation 1175 – wave number 1170 Bernoulli-Euler model 1164, 1165 Bloch wave 1145–1147, 1195 bulk modulus 1137, 1138

E Effective moduli 1148–1150 eigenfrequencies – of spherical shells 1191 eigenvalues 1152, 1171, 1187–1189 elastic mean free path 1145 elastic-viscoelastic correspondence 1142 elementary deformations 1138 energy balance 1143 energy density 1133, 1144, 1147, 1152, 1156 energy velocity 1145 equation of motion 1134, 1135,1147, 1151

C Christoffel’s equation 1151, 1153 Circular cylindrical shell 1177 coincidence frequency 1163, 1169, 1175, 1192–1194 Cole-Cole equation 1142 complex modulus 1141, 1175 compression 1133, 1134, 1137–1139, 1149, 1150,1152, 1167,1184 compression modulus 1138, 1150, 1167 continuity of displacements 1139 creep compliance 1141 critical frequency 1162, 1169, 1175 cylindrical shell 1177, 1186 – longitudinal vibration 1188 – radial vibration 1188 – ring-shaped vibration 1189 – torsion 1188 D D’Alembert’s principle 1135 Density of eigenfrequencies in plates 1178 diffusion coefficient 1146 Dilatation modulus 1167 dispersion relation 1163, 1165

F fading memory 1141 far field of a plate – excited by a point force 1193 Fresnel equations 1140 Fundamental equations of motion 1133 G group velocity 1133, 1145, 1152, 1153, 1155, 1156, 1158, 1162, 1184,1195 H half-value bandwidths 1143 Halpin-Tsai equations 1150 Hamilton’s principle 1134 Hashin-Shtrikman bounds 1149 hollow circular cylinder 1164, 1165 Homogeneous isotropic plate 1160, 1161, 1163 Homogenisation 1145, 1147, 1148, 1151 Hooke’s law 1134, 1135 I intensity 1133, 1143, 1144, 1147, 1152– 1156, 1160, 1162, 1163, 1195 Interface conditions 1139 isotropic fibre materials 1150

Chapter Index

isotropic media 1133–1135, 1140, 1151, 1153,1195 K Kelvin-Voigt model 1141 Kirchhoff vector 1143, 1144 Kramers-Kronig relations 1141, 1195 L Lagrange-Euler equations 1134 Lagrangian density 1133, 1134 Lamé constants 1166 Lamb waves 1154, 1156 lateral contraction 1139, 1164, 1167 line force 1179, 1191,1193 linearised theory 1134 logarithmic decrement 1141 longitudinal polarisation 1153 longitudinal waves 1139, 1147,1153, 1163 loss factor 1141, 1143, 1175,1192, 1194 loss tangent 1141 M material – as composite sphere assembly 1150 – with cubic grains 1149 – with periodically spaced parallel fibres 1150 – with polycrystals 1149 – with sperical inclusions 1149 – with stack of layers 1150 Material damping 1140 Maxwell model 1141 measurement of the intensity 1162 Modes of rectangular plates 1170 moduli of isotropic materials 1165 O one-dimensional periodicity 1147 P Partition impedance of plates 1174 Partition impedance of shells 1176 Periodic media 1146, 1151 Plane waves in unbounded homogeneous media 1151

1247

plate – free 1168, 1169, 1192,1194, 1195 – simply supported 1174 plate bending modulus – wave equation 1171 – wave velocity 1168 plate dilatational stiffness 1176 Plate waves 1154, 1195 point force 1179, 1183, 1192,1193 Poisson numbers 1136 Poisson ratio 1162, 1167, 1173, 1180, 1182, 1186, 1190, 1192 polarisation 1151–1153 pure mode 1152 Q quality factor 1141 quasi-longitudinal mode 1156, 1158 quasi-longitudinal waves 1160, 1163 R radiation 1191–1193, 1195 radiation of plate – excited by a line force 1191 – excited by a point force 1191 – finite plate excited by a point force 1192 Random media 1145, 1196 Rayleigh velocity 1163, 1165 Rayleigh waves 1159, 1195 Rayleigh’s principle 1144, 1145, 1155 Rayleigh-Lamb modes 1155, 1195 reciprocal lattice 1146, 1147 reflected waves 1140 refracted waves 1140 relaxation modulus 1141 relaxation time 1142 ring-shaped bar 1189 rotatory inertia 1163, 1165 S secular equation 1176, 1177 shear moduli 1136, 1137, 1149 shear motion 1187 shear stiffness 1176, 1181 shear waves 1138, 1154, 1156 Similarity relations for spherical shells 1190

1248

Chapter Index

simply supported plate 1174 slip assumption 1139 slowness vector 1152 Snell’s law 1140 sound radiation from plates 1191 Spherical shell 1177, 1190,1191 strain tensor 1134 stress tensor 1134, 1135 summation rule 1133 surface intensity 1143, 1195 T Table – with bending moduli of sandwich panels 1170 – with foot point admittances 1180 – with relations between istropy parameters 1138 – with transmission coefficients at steps 1185 – with types of sandwich panels 1169 Time-harmonic wavefields 1144

Timoshenko bar 1181 Timoshenko mode 1165 Timoshenko-Mindlin model 1163 torsional stiffness 1164 Torsional waves 1138, 1164 transmission coefficient 1184–1186 Transmission loss at steps 1184 transversal plane wave 1153 transversal polarisation 1152, 1153 triclinic anisotropy 1135 V velocity of a plate when excited by a diffuse sound field 1193 viscoelastic models 1140 Voigt and Reuss averages 1148, 1149 Voigt constants 1161 W Waves in bounded media 1154 Waves in thin beams 1163 Waves in thin plates 1160

R. Ultrasound Absorption in Solids A

F

Absoprtion in amorphous solids and glasses 1210 absorption 1197, 1204–1212 Absorption and dispersion in solids by dislocations 1204 amorphous solids 1210, 1211

Fermi surface 1208, 1209 Fermi velocity 1197, 1208 G Generation of ultrasound 1198 geometrical losses 1201

B Burger’s vector 1204 C collision time 1209 D damping coefficient 1200 de Haas van Alphen effect 1209 diffraction losses 1201 dispersion 1204, 1207, 1210,1211

I Interaction of ultrasound with electrons in metals 1208 K Kramers-Kronig relation 1211 L Lamé constants 1202 logarithmic decrement 1200

Chapter Index

M magnetic field dependence of absorption 1209 P Phonon interactions 1207 piezoelectric disc transducer 1199 piezoelectric equations 1198 piezoelectric transducers 1198 polycrystals 1202, 1207 Q Q-value 1197 quantum oscillations 1209 R resonance phenomena of electrons 1209 S scattering – at elastic spheres 1202

– – – –

1249

cross-section 1201 in polycristalline materials 1202 losses 1201 parameter for longitudinal waves 1202 – parameter for transversal waves 1202 stress field in a piezoelectric solids 1210 superconducting metal 1209 T Thermoelastic effect 1206–1208 thin film transducer 1199 U Ultrasonic attenuation 1199 ultrasonic backscattering 1204 W wave equation 1198 wave propagation in piezoelectric semiconducting solids 1210

General Index Capitalised initials of entries (if not names) indicate Sections. ¢ ¢ networks 66 ¢-circuit impedance 66 ¢-fourpole 445 A Absoprtion in amorphous solids and glasses 1210 absorbed power 493–495 absorbent dam – flat 214 – high 214 – semicircular 214, 227 absorbent neck walls 430, 431 – narrow neck 431 – wide neck 431 absorbent sound source in a capsule 847 absorber – uneven 141 – with random admittance 142 – with variable admittance 141 absorber in baffle wall – circular absorber 140 – in constant height above baffle 142 – rectangular absorber 141 – strip in baffle wall 144 Absorber of flat capillaries 404 absorber variable 349, 350, 353,355, 358, 373, 394, 504, 546,837 absorbing boundary condition operators 1080 absorption 1197, 1204–1212 Absorption and dispersion in solids by dislocations 1204 absorption coefficient 46, 127, 128, 130, 133–136, 139, 405–407,481–

483, 490–492, 495, 500, 501, 1019– 1021, 1025, 1027 absorption cross section 138, 145, 147, 187 – for diffuse sound incidence 188 – of a cylinder 197 absorption of finite-size absorbers 136, 153 Acoustic analogy – with effects of solid boundaries 984 – with mean flow effects 970 acoustic corner effec 152 acoustic efficiency 1003 acoustic equilibrium equation 1075 Acoustic power 793–797, 801 acoustic power 1003, 1004 acoustic Poynting vector 1143 acoustic-structural analogy 1075 Acoustically Lined Circular Duct 837 addition theorem 908 adiabatic boundary condition 598 adiabatic exponent 6, 9, 10, 35, 37 adiabatic flow 946 adjoint absorber 725–729, 731 admittance – average 403, 477 – homogenised 403 admittance equation 610 admittance of annual absorbers – approximated with flat absorbers 643 admittance relation 11, 42 admittance rule 11 aerodynamic sound 949, 964, 1000 – Doak’s theory of aerodynamic sound 981 – fluctuations of flames 991 – turbulent flames 991

1252

General Index

Aerodynamic sound sources 1000, 1012 algorithm – for optimisation of generalised Cremer admittance 729 alogorithms for mirror source construction 939 amorphous solids 1210, 1211 Amplitude nonlinearity of fences 742 Anderson localisation 1146 anechoic termination 797, 801 angle of sound incidence 1175 angular distribution 476, 478 angular far field 139, 141, 147 anisotropic absorbent 1118 anisotropic and inhomogeneous lining 1121 Anisotropic media 1133, 1140, 1151, 1195, 1196 anisotropy – cubic 1137 – orthotropic 1136 Anisotropy and isotropy 1135 Annular Airgap Lined Duct 834, 840 Annular ducts 647, 754, 762,764, 790 anti-symmetrical modes 606, 611, 613, 615, 619, 620, 627,707,712, 715, 718, 733, 776, 874 aperture 889, 897–899 arbitrarily moving surface 999 array – of Š/4-resonator 474 – of circular necks 434 – of Helmholtz resonators 435, 456 – of parallel slits 407, 437, 449 – with small elementary radiators 331 artificial medium of scatterers 198 autocorrelation 953, 1004 average – ensemble 950 – mass-weighted 950 – phase 950 – spatial 949 – time 949 average absorption coefficient 879, 936, 942 average admittance 1026

average input impedance 456 average surface admittance 456, 459, 464 averaged intensity 11 averaged square error 1020 averaging 948–952, 959 averaging over time intervals 922 axial propagation constant 595, 599, 602, 622, 671, 676, 695, 706, 750,1029 axial wave impedance 596, 700 B back orifice end correction 325 backscattering 477 backscattering cross section 188, 263 backward propagating modes 1029 baffle silencers 832 baffles 657, 680 balance – of entropy 36 – of heat 36 – of internal energy 36 balance of forces 1135, 1139 bar – bending modulus 1168 – bending wave velocity 1168 – longitudinal wave velocity 1167 Basic equations of fluid motion 954 beams 937 Bellow – flexible-wall 832 bellow – multi-step 832 bending loss factor 485, 496, 535, 536, 542, 552, 845, 848, 862, 866 Bending mode 1157, 1158 bending stiffness 1158, 1160, 1161, 1165, 1175, 1176, 1179,1180, 1182, 1183, 1190 bending wave equation 551, 552, 575 bending wave impedance 593 bending wave velocity 526 bending waves 1160, 1162, 1164,1165 – wave equation 1175 – wave number 1170 Bent 752, 762, 781,783, 784, 788 – and straight ducts – with unsymmetrical linings 785

General Index

– flat ducts with locally reacting lining 762 bent – duct section 781, 783 Berger’s law 534 Bernoulli-Euler model 1164, 1165 Bessel differential equation 629, 703, 756 bilinear concomitant 26 Biot wave 391 Biot’s theory 385 Biot-Savart law 979 blade thickness noise 1007 Bloch waves 44, 1145–1147, 1195 border scattering 153 Boundary Condition – at a moving boundary 38 – at liquids and solids 39 boundary conditions 254 – adiabatic 8 – Diriclet 1110 – for particle velocity 5, 13, 23, 24, 39 – for sound pressure 5 – isothermal 8 – Neumann 1110 boundary element method 1019, 1040, 1046,1049, 1051, 1059, 1071 boundary equations 1041, 1042,1061 boundary error 1040, 1042, 1053,1056, 1059 boundary integral equation 1019, 1059, 1061,1063, 1065, 1068, 1072, 1073, 1103–1105 boundary integral methods 1061 boundary layer 948, 949 boundary layer thickness 1115 boundary operator 1035, 1044 Bouwkamp’s integral 303, 306, 314 branch point of symmetrical modes 720 branch points 613–616, 618, 633–637, 640, 641, 643, 720, 736,737, 1035 break-off criteria for scouting 932 bubble oscillation 263 bulk compression modulus 388 bulk density 347, 351 bulk modulus 1074, 1137,1138 – adiabatic 946

1253

– isothermal 946 bulk reacting cylinder 185 bulk reacting layer 132, 136, 166 bulk reacting lining 518, 839 Burger’s vector 1204 Burton and Miller method 1068 C Cabin – semicylindrical model 861 – with plane walls 865 – with rectangular cross section 869 Capillary – circular 598 – flat 595, 598 capillaries 404, 405, 421,422,431, 457, 463, 476, 497, 502,1028 – circular, with isothermal boundaries 598 – flat, with adiabatic boundaries 598 – flat, with isothermal boundaries 595 Capillary Tube Monolith 829 capillary wave modes 483 capsule efficiency 843 capsules – energetic approximation 843, 844, 847, 852 cascaded one-dimensional systems 796 cascades of ducts 1033 Cat’s Eye model 1081, 1082, 1107 catalyst pellets 828 Catalytic converter 828 centre time 941 Chain circuit 73 chambered joint 520 characteristic – impedance 66 – propagation constant 66 characteristic constants – from fitted theoretical models 399 – wave impedance 347, 371, 375 characteristic equation 422, 431, 460, 483, 484, 517, 522, 545, 553, 554,576, 583, 596, 598–602, 606–608,610, 611, 613, 614, 616, 618, 620–623,626–631, 645–647, 649, 652, 654–657, 659,

1254

General Index

660, 672, 678, 695,703, 706, 721, 732, 733, 736, 745, 751,752, 754–756, 758, 760–762, 764–766, 768–774, 776, 786, 875 characteristic propagation constant 347, 371 characteristic values 130, 132, 150, 506, 518, 538, 540, 541, 553, 567, 571 – of Mathieu functions 150 Christoffel symbols – of first kind 30 – of second kind 29 Christoffel’s equation 1151, 1153 circular annuli 1122 Circular cylindrical shell 1177 circular piston in an infinite rigid baffle 1049 circular tube 1114, 1131 clarity 941, 943 closed cell model 371–374 co-ordinate surface 601 co-ordinate system 16, 20, 21, 26,30,32 – Cartesian 33 – curvilinear 26 – cylindrical 21 – reciprocal 27 – spherical 34 co-ordinates – contravariant 27 – covariant 27 – mixed 27 – transformation 28 coefficient of expansion 946, 956 coincidence frequency 467, 526, 571, 1163, 1169, 1175, 1192–1194 Cole-Cole equation 1142 collection in a corner source 907 collision time 1209 collocation method 1063 Combination of corner fields 906 Combined integral equation formulation 1066, 1106 common perforated section 819, 820, 822, 824, 825, 836 complete elliptic integrals 304, 329 complex bending modulus 467 complex modulus 1141, 1175

complex source point 1049, 1051, 1071, 1103, 1104 composite medium 230, 231, 233, 247– 249 Compound Absorbers 403 compressibility 351, 353, 355, 358,360, 371, 372, 375, 378, 946, 959, 968,990 – isothermal 9, 36 – isotrope 36 compressible flow 946, 952, 968,969, 979 compression 1133, 1134, 1137–1139, 1149, 1150, 1152, 1167, 1184 compression modulus 1138, 1150, 1167 compressional waves 388–390 Computational optimisation of sound absorbers 1019 Computing with mixed numericsymbolic expressions 1028 concave corners 889, 897, 903, 913, 917, 930 concave model room 891 concentrated absorber in an otherwise homogeneous lining 675 Concentric-Tube Resonator 806, 808 Concepts and notations in fluid mechanics 945 condition number 1055, 1069, 1105 condition of visibility 899 conditions for statistical methods 876 cone approach 937 conical duct transition 702, 832 – hard walls 702 – lined walls, stepping admittance approximation 712 – lined walls, stepping duct approximation 705 Conical tube 803, 804 conservation – of impulse 5 – of mass 5, 6, 960 – of momentum 960 – of wave numbers 39 continued fraction 518, 603, 604, 611, 612, 616, 623, 627, 630, 643, 647,652, 751, 752, 754, 774

General Index

continued-fraction approximation 354 continued-fraction expansion 875 continuity equation 951, 952, 954,957, 959–961, 963, 971, 973, 976, 977,981, 984, 992 continuity of displacements 1139 contracting duct 1030, 1031 convected quantities 793, 794 convected scalar wave equation 982 convected wave operators 980 – Doak’s form 981 – Howe’s form 981 – Möhring’s form 981 convective operator 972 convective radiation flow impedance 795 convective wave equation 952, 962, 972, 974, 998 convergent horn 804 converging cone 707, 709 convex corners 884–887, 889, 890, 896, 897, 899, 900, 913,917,930 corner – real 890, 905 – virtual 890, 905 corner area 694 Corner conditions 40 corner field 906 corner impedance 527 corner set of mirror sources 905–907 corner source 907, 909, 910, 912, 921 Correlations 953, 1017 coupled eigenproblem 1128 coupled fluid-structure equation 1080 coupling coefficient 386, 522, 556,558, 563, 566, 569, 574, 583, 585, 656, 659, 661, 674, 682, 690, 693, 695, 699, 706, 707, 713 – between duct modes and plate modes 556 coupling density 386 cover foil – on bulk layer 623 creep compliance 1141 Cremer’s admittance 720–722, 725 – with parallel resonators 725

1255

Critical frequencies 1046, 1047,1052, 1060, 1066, 1069, 1071 critical frequency 462, 467, 485, 496, 542, 566, 1162, 1169, 1175 cross impedance 24, 163, 179 cross section – absorption 138 – extinction 138 – scattering 138 cross-flow 806, 809, 816,818, 821, 823 cross-flow contraction element 809 cross-flow expansion element 809 cross-flow, closed-end, extented perforation element 823 cross-flow, open-end, extended perforation element 821 cross-flow, three-duct, closed-end element 816 cross-flow, three-duct, open-end element 818 cross-joints of ducts 697 Curle’s equation 988 cut-on frequencies 1110, 1112 cylindrical co-ordinates 962 cylindrical shell 467, 469, 1177, 1186 – longitudinal vibration 1188 – radial vibration 1188 – ring-shaped vibration 1189 – torsion 1188 D D’Alembert’s principle 1135 damping coefficient 1200 damping factor 1072 daughter source 882–884, 887, 889, 896, 897, 903, 904, 922, 923, 932 de Haas van Alphen effect 1209 decay curve 923, 937, 939 decay of the average energy density 923 decay rate 878, 881 Decomposition 950–952, 958, 959, 961, 965, 968, 972, 975 – of field qualities 965 definition 883, 884, 920,922, 923,940, 941, 943 Delany & Bazley regression 394 delta pulse 266

1256

General Index

Density of eigenfrequencies 876 – in plates 1178 density variation 36–38 density wave 6, 7, 37, 352, 355,358, 371–373, 378, 381, 383, 384, 421,483, 595, 596, 599 design point 722, 723, 725–728,730 differential operator 31, 33, 34 Differential relations of acoustics 34 differentiated integral equation 1068 diffracted angle 588 diffracted wave 266–268 diffraction losses 1201 diffuse absorption coefficient 133 – in 2 dimensions 133 – in 3 dimensions 133 diffuse sound field 879, 880, 936, 937 diffuse sound incidence 140, 508, 513, 525, 534–536, 540, 559, 567, 581,582, 846, 847, 852, 861, 869,1020, 1023, 1026, 1027 diffuse sound reflection – at bulk reacting plane 165 – at locally reacting plane 133 diffusion coefficient 1146 Dilatation modulus 1167 dipole point source 995, 996 dipole source 965, 976, 985, 989, 990, 995, 996, 1012 dipole terms 1063 Dirac delta function 5, 12, 13, 16,21, 266, 272 directivity 477, 479 – of circular membrane or plate 334 – of circular piston in baffle 332 – of circular radiator with nodal lines 334 – of clamped rectangular plate in fundamental node 333 – of dense circular array of point sources 332 – of dense linear array of point sources 332 – of free rectangular plate in fundamental mode 333 – of radiator arrays 330 – rectangulat piston in baffle 305, 333

– two point sources 331 directivity coefficient 330 directivity factor 330, 331, 334, 335, 883 directivity function 291, 303, 306,314 directivity index 330 directivity pattern 1057, 1058 directivity value 330 Dirichlet condition 1036 discretization of the boundary integral equation 1063 discretized functional 1076, 1077 discretized global functional 1078 dispersion 1204, 1207, 1210,1211 dispersion relation 1117, 1120, 1123, 1163, 1165 dissipation rate 954, 992 distance of limit between direct and reverberant fields 879 Distributed network elements 65 divergence 32 divergence relations for intensity 1144 divergent horn 804 diverging cone 708, 710 Doppler shift 38 double sheet 594 double subscript summation rule 945 double-layer potential 1061, 1062, 1068, 1069 double-shell – wall with absorber fill 538 – wall with thin air gap 540 double-shell resonance 540 – for an empty interspace 540 downsound cross sections 793 downstream end 797 duct modes 555–561, 563, 570,573, 574, 584 Duct section with feedback between sections without feedback 672 duct with – flexible walls, no mean flow and no lining 1126 – rigide walls and mean gas flow 1126 – two cross-sectional lines of symmetry 1121

General Index

dynamic head 800, 802 E early and late energy densities 942 early decay time 939, 940 edge condition 909 effect of meanflow 795, 833 effective – air compressibility 353, 355, 358 – air density 353, 355, 358 – compressibility 247 – density 198, 247, 249 – resistivity 351 – wave impedance 244 – wave number 236 effective bending modulus 497 – for perforated panel 497 effective compressibility 600 effective density 600 Effective moduli 1148–1150 effective plate partition impedance 498 effective power – absorbed 138, 153 – scattered 138 effective sound power 1035, 1056 effective surface mass density 457– 459, 462, 467 effective wave multiple scattering – in transversal fibre bundle 379– 381 efficiency checks 899, 905, 909, 912 eigenfrequencies 875, 876, 1037,1075, 1078, 1111 – of spherical shells 1191 Eigenfrequencies of a rigid-walled cavity 1110 eigenfunctions 1072 Eigenfunctions in parallelepipeds 873 eigenmatrix 807, 815 eigenmodes 1037, 1072,1075, 1110, 1127 eigenvalue problems 1037, 1078, 1110, 1130 eigenvalues 1152, 1171,1187–1189 elastic mean free path 1145 elastic panel 485, 496

1257

elastic-viscoelastic correspondence 1142 electro-acoustic analogy 59 – UK-analogy 64 – Uv-analogy 65 electromagnetic quantities 59 elementary deformations 1138 Elements with constrictions 71 elliptic-hyperbolic co-ordinates 149, 151, 217, 486, 565, 697 elliptical cylinder atop a screen 209 emission room 577, 578, 581,584 empirical relations for characteristic constants 394 empty and unsealed hole 513, 514 end correction 59, 69, 71, 75,410, 415, 417, 418, 425,434,436, 437,445–448, 452, 457–459, 472, 497, 507,521,795, 800, 827 – of a slit in a slit array 326 End corrections 287, 309, 316,318,346 energy balance 879, 1143 energy decay 936, 937 energy density 879, 880, 923,939,955, 1133, 1144, 1147, 1152,1156 – fluctuating part 958 – stationary part 958 energy equation 954, 955, 957, 981, 1113 energy integral 937, 942 energy velocity 1145 enthalpy – specific 946, 955, 972,977 – stagnation 947, 955, 977 – total 955, 977–982, 1015 entropy 946, 947, 956,970,973, 978, 979, 983, 988, 990 entropy variation 36–38 equations – of continuity 34, 960, 963,964 – of energy 35 – of motion , 960, 963, 964, 1134,1135, 1147, 1151 – of state 5, 951, 956,961 Equations of linear acoustics 960, 961 equations of motion equilibrium of stresses 1074

1258

General Index

equivalent – radiant problem 1036 – source system 1042 equivalent absorption area 880 equivalent chain network 520 equivalent circuit elements 866 equivalent fluid 1117, 1118, 1121 equivalent network 515, 538, 541,865, 867 – for the perforated panel 498 – of a Helmholtz resonator 436 equivalent network method 541 equivalent network of porous material 351 Equivalent Networks 59, 66, 71, 74, 75, 126 equivalent oscillating mass 71, 72 equivalent radius 387 equivalent source for the corner field 907 equivalent sources 922 error field 1046 Euler equations 954, 956, 958, 1110, 1112, 1113, 1117, 1118,1121, 1122, 1125, 1128 Euler-Lagrange equation 1076 euquivalent annular and flat ducts 762 evaluation of sets of mode solutions 613, 633 expanding duct 1030, 1031 expansion of characteristic equation – with Grigoryan’s method 752, 756 – with theorem of multiplication 758 Exponential horn 804 Extended Inlet/Outlet 801 extended outlet element 828 extended perforated pipes 822, 823, 825 Extended-Tube Three-Pass Perforated Element 828, 840 exterior Dirichlet problem 1061, 1062, 1105 external source flux 964 extinction cross section 187, 188, 191, 238 extinction theorem 139, 188

F fading memory 1141 far field 290, 291, 303, 306, 314,330, 331, 334, 336, 340, 345 far field of a plate – excited by a point force 1193 feedback 671–673, 675 Fermi surface 1208, 1209 Fermi velocity 1197, 1208 Ffowcs Williams-Hawkings equation 1015, 1016 fibre bundle 356, 371, 373–376,379– 381, 383–385 fibrous material 347, 350, 356 – parallel fibres 350 – random arrangement 357 – regular fibre arrangement 369 – transversal fibres 373 fictitious – adjoint absorber 725 – ducts 694 – volume source 676 Field admittance 6, 11 field admittance 288, 1035, 1036 field angle 884, 885, 887,888,893, 896, 898, 905, 907, 909,932, 934 field angle criterion 884 field angle of a mirror source 884 field matching 523, 562, 579, 681,691, 699, 713, 1086, 1096 field quantities 6, 36, 44 Field variables 34 filled interspace 538, 539 finite element method (FEM) 1019, 1041, 1059, 1071, 1074 finite element models (FEM) 1064 fitted theory 400 – of flat capillaries 401 – of the quasi-homogeneous material 399 fixed parameters 1021 flat capillary 509, 540 Flat duct – with a bulk reacting lining 619, 621, 623 – with an anisotropic lining 621

General Index

– with an unsymmetrical bulk lining 628 – with cross-layered lining 650, 651, 657 – with unsymmetrical bulk lining 628 – with unsymmetrical local lining 750 flat-oval duct 1110, 1111, 1130 flow impedance 793–795, 797, 802, 805, 818–820, 830, 831, 833, 836,837 flow resisitvity – of random fibre orientation 350 flow resistance coefficient 743 flow resistivity 347, 349–351,353–357, 360, 369, 370, 373,379,387, 394, 399, 837 – of parallel fibres 350, 356 – of transversal fibres 350 – with random fibre diameter and orientation 356 flow resitivity 349 flow velocity profile 353, 355 flow-induced nonlinearity of perforated sheets 748 fluid – density 955 – energy 955 – ideal 945, 964 – Newtonian 945 – non-Newtonian 945 – pressure 946 – viscosity 970 fluid-elastic structure interaction problem 1037 fluid-structure coupling 1078 flush-tube three-pass perforated element 827 foil 506, 511, 519 – elastic 457, 461, 462, 465–467, 471 – elastic, with losses 467 – limp or elastic 403, 465 – porous and elastic 467 – porous and limp 466 – tight and elastic 466 – tight and limp 466 – tight or porous 403, 465 Foil resonator 469, 470

1259

force impedance 527 forced boundary condition 1114, 1117 forward propagating modes 1029 foundation of the mirror source approximation 882 four-pole equations 66 four-pole parameters 796, 797 free bending wave number 525, 529 Free plate with an array of circular holes 429 free-space solution 965 frequency response curve 1020 Fresnel equations 1140 Fresnel’s integral 144 friction 350, 351 friction force coupling 499, 500 Frobenius norm 1055 full-field equations – of the first kind 1046, 1048,1052 – of the second kind 1047, 1048, 1052, 1054 functional 1109, 1112, 1113,1117, 1118, 1121, 1122, 1124, 1125, 1128,1129 Fundamental differential equations 5 fundamental equations 386, 388, 389 Fundamental equations of motion 1133 fundamental solutions 873 G Galerkin method 1045 gas constant 9, 10, 36, 945, 956 Gaussian elimination 1064 General criteria for mirror sources 883 general impedance boundary 1073 general termination 1033 generalised eigenvalue problem 1078 generalised full-field equations, 1052 generalised wave equation 987 generalized bending stiffnesses 537 Generation of ultrasound 1198 geometrical losses 1201 Geometrical room acoustics in parallelepipeds 877 geometrical shadow limit 217 geometrical subtasks 889, 926

1260

General Index

Global application of the principle of superposition 921 global shape functions 1077 goal of scouting 931 governing differential equations 1110, 1124 gradient 12, 31 Gram-Schmidt technique 1048 granular pellets 829 graphical evaluation of attenuation 644 grazing flow 808, 818, 834, 841 grazing-flow impedance 818, 836, 837 Green’s function 13–18, 137, 138,143, 289, 1069, 1070, 1072 – for a rectangular enclosure, 1071 – in 2-dimensional polar coordinates 17 – in 3-dimensional infinite space 17 – in Cartesian co-ordinates 19 – in cylindrical co-ordinates 18 – in spherical harmonics 17 – of a set of plane waves 16 – of cylindrical wave 16 – of point source 13 Green’s functions and formalism 13 Green’s integral 12 Grid on porous layer – grid with finite thickness, narrow slits 53 – grid with finite thickness, wide slits 56, 57 – thin grid 50, 52, 53 Grigoryan’s expansion of the characteristic equation 752 Grigoryan’s method 771 group velocity 1133, 1145, 1152,1153, 1155, 1156, 1158, 1162,1184, 1195 grouping of mirror sources 903, 905 H half-value bandwidths 1143 Halpin-Tsai equations 1150 Hamilton’s principle 25, 1076, 1134 Hankel transform 291, 341, 343 Hartree harmonics 44 Hashin-Shtrikman bounds 1149 heat balance 6

heat conduction 6, 35, 37 heat wave 6 heat-conducting walls 1113 Heaviside function 984 Helmholtz equation 1034, 1036–1040, 1042, 1049, 1059, 1070, 1075, 1076, 1103–1105, 1117,1127 Helmholtz formula 1060, 1072, 1080 Helmholtz integral equation 1040, 1046, 1059–1063, 1065–1068, 1070, 1071, 1103–1106 Helmholtz resonator 297, 403, 419, 435, 436, 444, 449, 452, 456, 502,830 Helmholtz vorticity equation 979, 980 Helmholtz’ source theorem 62, 847 Helmholtz’ source superposition theorem 462 Helmholtz’ wave equation 5 Hemispherical source and capsule 857 high dam 215, 218, 219, 221, 223 – hard ground 215, 219, 222 – line source 223 – on absorbent ground 221, 223 – on hard ground 218 higher modes 48, 56 higher modes in the neck 321, 322, 324 hole transmission with equivalent network 515 hole, sealed, filled 515 hollow circular cylinder 1164, 1165 homentropic flow 947, 978, 982 Homogeneous isotropic plate 1160, 1161, 1163 Homogenisation 1145, 1147, 1148, 1151 Hooke’s law 1134, 1135 Hose 801, 804–806, 820, 822, 832,835, 840 hose-wall impedance 805 Huygens-Rayleigh integral 331 hybrid evaluation 1033 I ideal flow 946 illegal mirror source 883, 884, 886, 887, 896, 909, 912 impedance boundary condition 1036

General Index

impedance jump 801 impulse equation 6 impulse response 938–941 In-line Cavity 831 incident wave 1032, 1036, 1037, 1057 incoherent sources 923 ineffective mirror source 884 inertia of the air in pores 351 Influence of flow on attenuation 731 Influence of temperature on attenuation 740 inhomogeneous wave equation – convective 970, 971 – Dowling’s form 988 – Ffowcs Williams-Hawking’s form 984 – Goldstein’s third-order form 973 – Goldstein-Howe’s form 1005 – Howe’s form 978 – including the stream function 975 – Lighthill’s form 963, 988 – Lilley’s convective inhomogeneous form 971 – Lilley’s form 971 – Meecham’s form 969 – Phillip’s convective inhomogeneous form 970, 973 – Ribner’s form 968 input impedance – of a porous layer 391, 399 – of a porous layer with adhesive front foil 392 insertion coefficient 846 insertion loss 796, 797, 843,846, 851, 852, 856, 857, 859,860 Inside checks 888, 889, 928 inside criterion 883, 885, 888,904, 916, 918, 930 integral length scale 953, 1005, 1006 integral operators 1063, 1065, 1068, 1069, 1105 Integral relations 12 integral time scale 953 intensity 1133, 1143,1144, 1147,1152– 1156, 1160, 1162, 1163,1195 – of a reflection 879 – of reverberant field 878

1261

Interaction of ultrasound with electrons in metals 1208 interaural cross-correlation coefficient 942 interaural cross-correlation function 941 Interface conditions 1139 interior – Dirichlet problem 1046, 1061, 1066, 1067, 1069 – Helmholtz integral formula 1070 – Neumann problem 1061 interior end correction 316 – in a slit resonator array 322 – with only plane wave 324 interior orifice impedance 325, 326 – with slit in contact with porous material 325 – with slit in some distance to porous material 326 – with viscous and caloric losses 325 internal energy 946, 955, 977,981 interrupt checks 887, 899, 905,909, 912, 931 inviscid flow 946, 961 Irregular shaped cavities 1075, 1078, 1107 irrotational flow 947, 961 isentropic condition 961 isentropic flow 946, 964, 979 isothermal flow 947 isotropic fibre materials 1150 isotropic media 1133–1135, 1140, 1151, 1153, 1195 isotropic plate 537 iterative – linear system of equations 1028 – solvers 1065 – systems 1033 J Jacobi iteration 1065 jet noise 989, 990, 1000, 1003, 1016, 1017 junction of pipe and chamber 800

1262

General Index

K Kelvin-Voigt model 1141 Kirchhoff source term 987 Kirchhoff vector 1143, 1144 Kramers-Kronig relations 1141, 1195, 1211 L Lagrange density 25, 1109 Lagrange equations 1109 Lagrange function 25, 1076 Lagrange multiplier 25, 1067 Lagrange-Euler equation 1109, 1134 Lagrangian 1109 Lagrangian density 1133, 1134 Lamé constants 1166, 1202 Lamb vector 977, 979 Lamb waves 1154, 1156 Lambert scattering 937 Lambert’s law 935, 939 laminar flow 947 Laplace operator 962 – cylindrical 962 – spherical 963 Laplacian – of a scalar 32 – of a vector 32 late lateral sound level 942 lateral contraction 1139, 1164, 1167 lateral energy fraction 941 lateral mode profiles 681, 689, 706 least attenuated mode 517, 518, 602, 607, 608, 612, 623, 626–628, 644,647, 686, 688, 720, 721, 727, 737, 738, 750, 754, 776, 782, 784 – in rectangular duct 607 Least squares minimization technique 1047 level decay 878, 881 level difference 796, 798 Lighthill stress tensor 964, 972, 997, 1010 Lighthill’s equation 963, 970, 973,974 line force 1179, 1191,1193 line source 21, 24, 199–201,204–207, 209, 210, 223, 229 – above an absorbing plane 164

linearised theory 1134 linearly elastic structure 1037, 1038 Lined duct corners and junctions 693, 788 Lined ducts 595, 601, 607, 613,650, 693, 697, 762, 781, 788 loading noise 1008, 1009 locally reacting cylinder 198, 199, 201 locally reacting layer 131 logarithmic decrement 1141, 1200 long source 341 longitudinal polarisation 1153 longitudinal wave number 526, 529, 535 longitudinal waves 1139, 1147, 1153, 1163 looping in mirror source scouting 932, 933 loss factor 420, 426, 428, 467,485,496, 529, 535, 536, 542,552, 566, 590, 1141, 1143, 1175, 1192, 1194 loss tangent 1141 low frequency approximation 1115, 1117, 1125 lowest resonance 419, 420, 429,470 lumped elements 65 M Mach number 732, 736, 744, 746,748, 789, 794, 795, 798–802, 808,820,833, 834, 837, 838, 840,962, 964, 969,970, 976, 978–980, 987, 991, 994,997,999, 1002, 1007, 1008, 1010,1011, 1013, 1014, 1016, 1118 magnetic field dependence of absorption 1209 mass reactance – of resonator orifice 417 massivity 198, 238, 240,242, 244, 247, 248, 347, 350, 353, 357–359,361,369, 371, 373, 376, 384 matching – average sound pressures 409 – for the sound pressure 1087 – of fields 579, 656, 681, 691, 695,699 matching conditions 1086

General Index

material – as composite sphere assembly 1150 – with cubic grains 1149 – with periodically spaced parallel fibres 1150 – with polycrystals 1149 – with sperical inclusions 1149 – with stack of layers 1150 Material constants – of air 8, 9 Material damping 1140 Mathieu differential equations 150, 487 Mathieu functions 209, 211, 214, 218, 219, 285, 565–567, 697, 698, 790 – azimuthal 150, 487 – of the Bessel type 150 – of the Hankel type 150 – of the Neumann type 150 – radial 150, 487 matrix equation 793, 827 matrix material 347, 350, 352, 355, 388 matrix norm 1055 matrix vibration 351 Maxwell model 1141 mean flow 793, 799, 800,802,832, 840, 841, 1123, 1126 mean free path length 877 – energetic average 877 – geometrical average 878 mean reflection rate 879, 936 measurement of the intensity 1162 mechanical analogy 1074 Mechel & Grundmann regression 398 mesh theorem 62 method of mirror sources 1071 method of weighted residuals 1040, 1042 micro structure – of perforation 497 micro-perforated Helmholtz panel parallel baffle muffler 836 mineral fibre material 347, 348, 352, 398 minimization process 1042 minimum search 1021

1263

mirror source 19, 138, 156, 161, 163, 167, 222, 266, 267, 877, 882–884,904– 906, 909, 914, 918, 924, 930–932,938, 939 mirror source approximation 161 Mirror source model 882 mirror sources of parallel walls 907, 910 mirror-receiver method 910 mixed boundary conditions 1048 mixed monotype 244, 248 mixed numerical symbolic evaluation 1033 modal angle 602, 624, 688, 716 modal impedances 292 modal matrix 807, 815, 826 modal partition impedance 552, 563, 571, 576, 862 modal partition impedances 488 modal radiation impedance 855, 859 modal reflection factors 202, 210, 272, 440, 443, 449, 451, 695 mode approximation – by continued-fraction expansion 754 – by identical transformation 609 – by power series expansion 608, 649 – from the admittance equation 610 – in round ducts by iteration over resistance 646 mode charts 630, 633 mode coupling coefficients 412, 454, 493, 566, 569, 574, 585, 659, 661,674, 693, 695, 706 mode coupling integrals 778, 780, 1088, 1096–1098 mode eigenvalues 752, 754, 755, 771, 774, 777, 781 mode excitation coefficients 718 mode functions 1111 mode in a duct 19, 20 mode jumping 618, 619 mode mixture 715 – with equal mode amplituted 717 – with equal mode energy densities 717 – with equal mode powers 717 Mode norms 1029, 1030,1096, 1098

1264

General Index

mode norms 151, 292, 522, 553–556, 561, 562, 566, 569, 571, 573, 583, 607, 656, 659, 661, 676, 678, 681, 690, 695, 706, 713, 716, 718,778, 780, 785, 786 mode orthogonality 661, 1085, 1086 mode overlap 876 mode profiles 629, 655, 681,689,692, 694, 706, 715, 762, 1029 mode range limit 616, 643 mode shape 1120, 1123 mode solutions 602, 613, 618, 619, 623, 633, 642, 734, 750, 752,754, 758, 765, 767, 768, 771 – by iteration through k0 h 624 – iteration through the modal angle 624 mode solutions in bent ducts – both walls soft 770 – inner wall hard, outer wall soft 764 – with ideally reflecting walls 764 – with locally recating walls 764 model room with convex corners 899 models of fluid flows 946 – Adiabatic flow 946 – boundary layer flow 948 – compressible flow 946 – homentropic flow 947 – ideal flow 946 – incompressible flow 946 – inviscid flow 946 – irrotational flow 947 – isentropic flow 946 – isothermal flow 947 – laminar flow 947 – non-uniform flow 947 – Real flow 946 – rotational flow 947 – stationary flow 947 – steady flow 947 – turbulent flow 947 – uniform flow 947 – unsteady flow 947 – viscious flow 946 modes 873, 875, 876,935 – anti-symmetrical 611, 613, 619,620, 627, 707, 712, 715, 718,733, 776 – orthogonality of 19, 20

– symmetrical 602, 606–608,611,613, 618–620, 627, 707, 712, 713,715,718, 733, 776, 785 Modes of rectangular plates 1170 modified Green’s functions 1069 moduli of isotropic materials 1165 molecular mean free path length 35 momentum equation 947, 949, 952, 954, 955, 957, 959–961, 963,968,971, 973, 976–978, 981, 984, 985, 993 momentum flux 949, 989, 995–997 momentum impedance 527 monopole 337 monopole source 5, 965, 985, 989,990, 994, 995, 1012 monopole terms 1063, 1070 monotype scattering 231–233, 238, 244, 246, 248 Monte Carlo method 936, 937 Morse chart 607, 612, 621, 623 mother source 883, 884, 888–890, 893, 899, 910, 915, 932 moving Kirchhoff surfaces 998 moving source 947, 970, 984, 986,993, 994, 998, 999, 1002, 1007,1014–1016 Muffler Acoustics 793 Muffler performance parameters 796 Muller’s procedure 603, 647, 752,754, 758, 761, 762, 766,767, 772 multi-layer absorber 403, 1019–1021, 1026 multi-modal excitation 856, 857, 859, 860 multi-point multipole method 1051, 1056 multigrid method 1060, 1066, 1104, 1105 multiple covering of mirror sources 885 multiple scattering – at cylinders and spheres 198 – elastic scatterers 258 – porous scatterers 258 – rigid scatterer at rest 258 multiple scattering model 371, 375, 376, 380 multipoles 1044, 1051,1057

General Index

N nabla operator 31 narrow capsule 850 Navier-Stokes equation 35, 361, 952, 960, 1113 near-field and far-field solution 965 nested loops 890, 932 network elements 538, 541 Neumann boundary value problem 1034 Neumann symbol 1072 Newton’s law 960 niche effect 559 niche modes 560, 561, 563 nodal variables 1076, 1078 node theorem 62 nodes 1064, 1067,1076–1078 noise barriers 503 noise reduction 796, 798 noise sluice 503, 521, 523,525, 594 non-linear disturbance 956, 957 non-uniform flow 947, 971 nonlinearities – by amplitude and/or flow 742 – by flow along fibre absorber 746 – by flow over orifices 743 – by flow through an orifice 744 – by flow through porous absorber 747 normal vectors on co-ordinate surface 27 normalizing factors 1043 null-field equations 1040, 1045, 1046, 1048, 1051, 1052, 1055,1057, 1058, 1067, 1068 number of mirror sources 877, 882 number of reflections 877, 879, 880 numerical sound intensity method 1035 O octupole source 964, 967, 989,996 office fences – with second principle of superposition 584 oncoherent sound incidence 868 one-dimensional periodicity 1147

1265

one-point multipole method 1051, 1056 open-cell model 371, 373, 375 open-cellular foam 348 open-circuit source 62 open-end elements 818, 821 optimised parameters 729 Orange model 1094 orchestra pit 897–900 orifice impedance – back side 409, 417, 433, 436, 445– 447 – front side 409, 413, 417,419 orifice input admittance 678 original source 882, 883, 887,889,890, 907, 911, 912, 932 orthogonality – of modes in ducts with bulk lining 621 orthogonality integral 15 Orthogonality of modes 19, 1095 – in a bulk reacting duct 20 – in a locally reacting duct 19 orthogonality relation 17, 621, 655 orthogonalization method 1048 orthonormal 1043, 1048, 1067 orthonormal basis vectors 31 oscillating mass 287, 288, 293,316, 317, 326, 328, 342, 403, 419, 426,436, 445, 448, 452 – of a fence in a hard tube 328 P panel perforation 497 Parallel Baffle Muffler 832, 836, 839 parallel walls 891, 907, 910, 911 parameters – variable 1022, 1023, 1026,1027 particle velocity 5, 13, 23, 24, 37, 39 particle velocity profiles 597 particle velocity source 1032 partition impedance 527–529, 533, 535, 538, 542, 545, 552, 563, 566,571, 575, 576, 620,628, 645, 692, 742,748, 749, 807, 837,838, 844, 845, 848,854, 858, 862, 871 – of membranes 469 – of plates 1174

1266

General Index

– of shells 1176 pass integration 156, 159, 161, 162 pass way 155, 156, 159,160, 171 path of steepest descent 155, 170, 171 Pellet Block Element 829 Perforated Extended Inlet 814 Perforated Extended Outlet 813 perforated sheets 742, 743, 748, 749 perforates in stationary media 808 – with cross flow 807 – with grazing flow 808 perforation pattern 497 Periodic media 1146, 1151 periodic structure 403, 438, 474, 480 Periodic structures – admittance grid 44 – grooved wall, narrow grooves 46 – grooved wall, wide grooves 48 perturbation quantity 956 Phonon interactions 1207 Picard iteration 1065, 1066 piezoelectric disc transducer 1199 piezoelectric equations 1198 piezoelectric transducers 1198 pine-tree baffles 657 pine-tree silencer 657 piston radiator – circular 302, 330 – elliptic, in a baffle 303 – free circular disk 303 – in a hard tube 327 – on a sphere 297 – plane 291, 301, 309 – rectangular 305 – ring-shaped 330 – small circular 330 – strip-shaped 299 – with radial symmetry 341 plane radiator 309, 311, 313,331, 339 plane wave reflection – at a locally reacting plane 127 – at a multilayer absorber 132 – at a porous layer 130 Plane waves in unbounded homogeneous media 1151 plate – free 1168, 1169, 1192, 1194, 1195 – simply supported 1174

plate bending modulus – wave equation 1171 – wave velocity 1168 plate dilatational stiffness 1176 plate mode norms 562, 569 plate modes 552, 555, 557, 558,560, 562, 565–567, 570, 573 plate vibration modes 488 Plate waves 1154, 1195 Plate with absorber layer behind 541 Plate with bending losses 535, 552 Plate with narrow slits 407 Plate with wide slits 411 Plenum modes 574–576, 578 plug of porous absorber material 503 Pod Silencer 832 point force 1179, 1183, 1192,1193 point source – above a locally reacting plane 19 – above hard or soft plane 18 – dipole 338 – lateral quadrupole 339 – linear quadrupole 339 – monopol 337 point source above absorber – exact saddle point integration 170 – exact solution 148 Poisson distribution 357, 359, 370,394 Poisson numbers 1136 Poisson ratio 1162, 1167, 1173, 1180, 1182, 1186, 1190, 1192 Poisson’s equation 968, 969 Poisson’s ratio 526, 589, 593 polar mode numbers 275, 283 polarisation 1151–1153 pole contribution 155, 158, 159,171, 173, 174 pole crossing – condition for 159 polycrystals 1202, 1207 polyurethane foam 748 poro-elastic foil 457, 461, 462,465,466, 471 Porous Absorbers 347, 385 porous blocks 828 porous foil 452 Porous material – model with flat capillaries 355

General Index

porous material 347, 349, 351,352, 355, 370, 381, 385–387, 391, 399, 504,506, 509, 511, 519, 538, 541, 546, 571 – model with flat capillaries 354, 356 porous panel absorber – rigorous solution 496 potential – scalar 6, 7 – vector 6 potential coupling factor 387 potential function 13, 25 potential-layer approach 1061 potentials 372, 373, 388,390 power law 1002, 1012–1014 – for the acoustic intensity 1002 – for the aerodynamic sound sources 1012 – of compact aerodynamic multipoles 1012 – of moving aerodynamic sources 1014 – of noncompact aerodynamic multipoles 1012 power spectral density 954, 1000, 1003 Prandtl number 8, 9 premises of linear acoustics 961 pressure force 367 pressure source 403 pressure variation 38 Pressure-source 968, 969, 1016 pressure-source theory – by Meecham 969 primary absorber 726, 728, 729,731 primitive root diffuser 475 principle of hard-soft superposition 163 principle of impedance mismatch 801 principle of superposition 218, 224 – for unsymmetrical absorption 920 Principles of superposition – for unsymmetrical walls 22 – hard-soft superposition 24 – in a symmetrical space 23 probability density 877, 936, 939 procedure for optimisation 1019, 1021 propagation constant 347, 352, 355, 358, 371, 375, 381, 394, 400, 503, 509,

1267

511, 517, 544,545, 575, 595, 598–600, 602, 622, 650,671,673, 676, 695, 706, 740, 747, 748,750 – in a flat capillary 405 pseudo-random variation 474 pseudo-sound 968 pure mode 1152 Q Q-value 1197 quadratic residue diffuser 475 quadrupole noise 1009, 1016 quadrupole source 964, 965, 976,985, 990, 995–997, 1007, 1012 quality factor 1141 quality factor of shell resonance 262 quantitative corner effect 152 quantum oscillations 1209 quasi-homogeneous material 347, 350, 352, 399 quasi-longitudinal mode 1156, 1158 quasi-longitudinal waves 1160, 1163 Quincke tube 833, 841 R radial mode 630, 633, 641,642, 647 radiated power 844 radiating mode in resonance 262 radiating spatial harmonics 46, 51 Radiation – flow resistance 795 – impedance 793, 795, 797,805 radiation 1191–1193, 1195 – from the open end of a flow duct 795 Radiation and excitation efficiencies 344 radiation condition 1035–1037, 1039, 1040, 1060, 1070, 1080, 1081 radiation directivity 330 radiation efficiency 1034, 1035 radiation factor 288 Radiation impedance 287, 289–291, 293, 296, 298, 300, 301, 304, 306,310, 312, 314, 316, 328, 329, 340, 342,345, 346 – definition 287, 314

1268

General Index

radiation impedance 154, 403, 404, 414, 507, 508, 512, 513, 516, 519, 520, 700, 701 – evaluation 301 – mechanical 287, 329 – of shell modes 261 radiation loss 287, 699–702 radiation of plate 344 – excited by a line force 1191 – excited by a point force 1191 – finite plate excited by a point force 1192 Radiation problem 1034–1036, 1040, 1042, 1046, 1063, 1065,1066, 1070, 1072, 1081,1082, 1092, 1102–1105 radiation reactance 309, 320–322 radiator – breathing sphere 293 – cylindrical 295 – finite length cylinder 335 – in a baffle 339 – in spherical mode patterns 291 – line source 344 – monopole and multipole 337 – oscillating cylinder 317 – plane 309, 311, 313, 331,339 – rectangular, wide, field excited 313 – rigid sphere 293 – strip-shaped 299 – strip-shaped, narrow, field excited 309 – strip-shaped, on a cylinder 299 – strip-shaped, wide, field excited 311 – with 1-dimensional pattern 343 – with central symmetry 331 – with nearly periodic pattern 343 random fluctuations 951, 953 Random media 1145, 1196 random scattering 935 random-incidence absorption 936 randomised fibres 356, 359, 369 ratio of the specific heats 956 ray energy 935–937 ray formation 721, 722, 784,789 ray path 938 Ray sources 938 ray tracing 935, 937

Rayleigh velocity 1163, 1165 Rayleigh waves 1159, 1195 Rayleigh’s capillary model 352 Rayleigh’s postulate 6 Rayleigh’s principle 1144, 1145, 1155 Rayleigh-Lamb modes 1155, 1195 real flow 946 receiving room 559, 577, 578,581 reciprocal invariant 63 reciprocal lattice 1146, 1147 Reciprocal networks 63, 64 reciprocity at duct joints 750 reciprocity in the mirror source method 910 reciprocity principle 12 Reduction of the system of equations 1091 reflected waves 1140 reflection coefficient 52, 54, 55, 57, 58 reflection factor 46–50, 52, 404,405, 440, 443, 449, 451, 495, 657, 695,745, 746, 794, 795, 801 reflection factor of a plane wave 883 reflection of sound 127 reflection rate 879, 936, 942 refracted angle 130, 132, 186 refracted waves 1140 regular fibre arrangement 356 Reiche’s experiment 233–235 relaxation 350–352, 358 relaxation frequency 352, 358 relaxation modulus 1141 relaxation time 1142 resonance condition 419, 437 resonance formula 420 resonance frequency – of foil resonator 469 – of slit resonators 419 resonance phenomena of electrons 1209 resonator neck 676, 729 resonators in parallel 725 resonators in series 722, 723 retardation time 965 retarded source strength 965 reverberation curve 923–926 reverberation time 876, 878, 881,922, 923, 925, 926, 940,942

General Index

– definition 922 – Eyring 881 – Millington-Sette 881 – Pujolle 881 – Sabine 881 – with mirror source results 922 reversal contraction 820 reversal contraction, two-duct, openend perforated element 812 reversal expansion 820 reversal expansion, two-duct, openend perforated element 811 reversal expansion element 812, 827, 828 Reverse-Flow 810, 817, 819,824, 840 – Contraction Element 810, 819 – Expansion Element 810, 819 – open-end, extended perforated element 824 – open-end, three-duct element 819 – three-duct, closed-end element 817 reversed-flow elements 818 Reynolds averaging 950, 952 Reynolds stress 949, 953, 964, 993 Ribner’s source term 974 rigid boundaries 966 Ring resonator 471, 473 ring-shaped bar 1189 ring-shaped duct 754 room acoustical parameters 940, 942 room impulse responses 939 room transfer function 877 room with concave edges 924 rotation of a vector 32 rotational flow 947, 948 rotatory inertia 1163, 1165 rotor – dipole sound 1011, 1012 – monopole sound 1011 Rotor noise 1007, 1010, 1017 round duct – with a bulk reacting lining 645 – with a locally reacting lining 629, 634, 636, 642 rule for the construction of a reciprocal network 63

1269

S saddle point 156, 158–160, 163–165, 167, 170, 171, 174,182 saddle point integration 156 Sandwich Panel 542 – with elastic core 589 – with porous board as core 589 – with porous board on back 547 – with porous board on front 544 scalar product 16, 19, 27, 29,31 scaled co-ordinates 891 Scales 948, 953 scattered far field 187, 190, 234, 237, 244, 246, 249, 262, 266, 474, 476–479 scattered field 186, 195, 199, 200, 224, 225, 228, 229, 231,234, 244, 246,249, 260, 264, 274, 283, 491, 495, 914,917– 919 scattered wave 137, 143, 144 scatterer – bulk, movable 241 – hard 198, 258, 260 Scattering – at a cone 270, 282 – at a corner 201 – at a flat dam 223 – cross section 187, 191, 194 – in random media 230, 232, 244,248 – mixed monotype 248 – monotype scattering 244 – triple-type scattering 248 – of plane wave – at liquid sphere 263 – at cylinder 185, 188 – at cylinder and sphere 188 – at screen with mushroom-like hat 213 – of spherical wave – at a soft cone 271 scattering – at a hard screen 208, 209, 268 – at a perfectly absorbing wedge 264 – at a screen with cylinder atop 209 – at a semicircular absorbent dam 214 – at elastic cylindrical shell 260 – at elastic spheres 1202

1270

General Index

– at finite-size local absorber 136 – at the border of an absorbent halfplane 142 – cross-section 1201 – in polycristalline materials 1202 – losses 1201 – of impulsive spherical wave at a hard wedge 266 – of plane or cylindrical wave at a corner 201 – of Sound 185 – parameter for longitudinal waves 1202 – parameter for transversal waves 1202 scattering problem 1019, 1036, 1037, 1040, 1056, 1059, 1072,1079, 1081, 1082, 1103 scouting of mirror sources 930–932 sealing impedance 507 sectional admittances 712 Sections and cascades of silencers 671 secular equation 595, 598, 599, 601, 602, 606, 620, 622, 624,629, 659, 660, 671, 732, 763, 874, 875, 1029, 1176, 1177 self-adjoint formulation 1065 self-noise 960, 967, 975 Semicylindrical source and capsule 853 sets of mode solutions 618, 642 – in annular ducts via modes in flat ducts 762 – in annular ducts with unsymmetrical lining 754 – in flat ducts with bulk reacting lining 623 – in rectangular ducts 605 sexdecupole source 997 shading 889, 896 shadow field 867 shadow zone of a corner 203 shape functions 1064, 1076–1078 shear moduli 1136, 1137, 1149 shear motion 1187 shear noise 960, 967, 971,975 shear stiffness 1176, 1181

shear stress 5, 945, 946,949 – by viscosity 35 shear wave number 529, 535 shear waves 6, 1138, 1154, 1156 shell resonances 261, 262 shock front 39 short-circuit source 62 silencer modes 681, 682 silencer with rectangular turningvane splitters 787 Similarity relations for spherical shells 1190 simple plates 536, 538, 594 simply supported plate 485, 496, 552, 557, 567, 1174 single-layer potential 1061, 1062, 1066 six elements per wavelength rule 1064 slip assumption 1139 slit – narrow, empty 509 – narrow, empty with sealing 509 – narrow, filled with porous material 506 Slit array – with viscous and thermal losses 420 Slit resonator – dissipationless 415 – with viscous and thermal losses 426 slit resonator array – covered with a foil 462 – with porous layer on back orifice 449 – with porous layer on front orifice 452 – with subdivided neck plate 461 – with subdivided neck plate and floating foil 457, 461 slowness vector 1152 Snell’s law 391, 1140 solid boundaries in flow 966 solid-liquid interface 40 solution of inhomogeneous wave equation 991 Sommerfeld’s condition 12, 21, 22

General Index

Sommerfeld’s radiation condition 1035, 1039 sound attenuation in a forest 242 Sound field in interior spaces 1037 sound field of moving point sources 994 Sound fields in rooms and halfspaces 1070 sound from free turbulence 966 sound generation – by flow interaction 984 – by fluctuating heat sources 988 – by interaction of turbulence and sound 960 – by turbulent interaction 960 – by turbulent two-phase flow 992, 1015 sound pressure coefficient 7 sound pressure source 1031 sound propagation – in a shear flow 972 – in non-uniform flow 971 – parallel to the fibres 358 – transversal to the fibres 370 sound protection measure 861, 864, 867, 868 sound radiation from a lined duct orifice 697 sound radiation from plates 1191 sound radiation into a half-space 1071 sound strength 941, 942 sound transmission – through a hole in a wall 511 – through a simple plate 532, 534 – through a slit in a wall 506 – through finite size double wall with porous absorber core 571 – through finite size plate with a front side absorber 567, 570 – through infinite plate between two differnt fluids 587 – through lined slits in a wall 517 – through office fences 503, 582 – through plates with equivalent circuit 534, 588 – through simply supported plates 565, 567

1271

– through single plate in a wall niche 559 – through strip-shaped wall in infinite baffle wall 564 – through suspended ceilings 577 sound transmission coefficient 682 sound transmission factor 503, 844 sound transmission loss 846 sound velocity 9, 10, 38 – in gas mixture 10 source condition 202, 210, 271–273, 1031–1033 Source conditions 21 source contributions 676 source factor 883, 886–888, 895, 902, 904, 910, 911, 915,918, 922, 932 source function 1040–1042, 1044, 1048, 1051, 1054, 1056 source impedance 853 source list 889, 890, 899,932–934 source simulation technique (SST) 1040 source strengths 1055 source superposition 1055 source terms 964, 967, 970–972,976, 983, 987 Sources 62, 63, 73 sources as a network 922 spatial harmonics 44–47, 49–52, 451, 653, 654, 680,683–685, 689 specific – enthalpy 946, 955 – entropy 946 – heat 6, 9, 10, 35,946,956 – heat ratio 946 – internal energy 946 specular reflection 935, 937 speech intelligibility 941 speed of sound 946, 956, 963, 970,1013 spherical co-ordinates 963 spherical harmonics 1043, 1093 Spherical radiator 288, 291 Spherical shell 1177, 1190, 1191 spherical shell 468 spherical wave functions 1040, 1042– 1045, 1047, 1048, 1051,1054, 1067, 1080

1272

General Index

Splitter type silencer – with bulk reacting splitters 688 – with locally reacting splitters 680, 683 stability and the condition number 1055 stagnation enthalpy 744, 745 stagnation pressure 799–802 stationary flow 5, 947 Stationary flow resistance of splitter silencers 741 stationary medium 794, 795, 799,801, 803, 806 stationary value 1109 statistical methods 876 Statistical room acoustics 879 steady flow 947, 989 step in duct – contracting, lateral-lateral 664, 665 – contracting, lateral-local 666 – contracting, local-lateral 668, 669 – contracting, local-local 662 – expanding, lateral-lateral 663, 665 – expanding, lateral-local 667, 668 – expanding, local-local 662 – expaning, local-lateral 666 Step of duct height and/or duct lining 658 strain tensor 1038, 1134 strain-stress equations 386–388 stress field in a piezoelectric solids 1210 stress tensor 1038, 1134, 1135 structural impedance matrix 1079 structure factor 347, 349–351, 386 structure parameters 347 Struve functions 508 subjective transparency 941 subsonically moving surface 999 substantial derivative 969, 970 substitutions induced by flow 732 successive overrelaxation method 1065 Sudden area changes 799, 801, 829 sudden contraction element 827 summation rule 1133 superconducting metal 1209

superposition method 1040, 1051, 1103 Superposition of multiple sources 73 supersonic jet flow 1003 supersonically moving surface 999 surface admittance – of annular absorbers 643 – of annular lining 646 surface impedance 456, 469, 493 – of modes 186 surface intensity 1143, 1195 surface porosity 404, 436 surface velocity distribution 1073 surface velocity error 1047, 1053,1056 Surface wave – along a locally reacting cylinder 42 – along a locally reacting plane 41 surface wave mode 614, 617–619, 641, 642, 719, 720 suspended ceiling 503, 574, 577,580, 581, 594 symmetrical modes 613, 619, 620,707, 712, 718, 776, 874, 875 symmetrical scatterers 236, 237, 246 symmetry relation 1042–1047, 1052 T T-circuit impedances 66 T-joints of ducts 697 T-networks 66 T-shaped Helmholtz resonator 678, 679 Table – with bending moduli of sandwich panels 1170 – with branch points of characteristic equation in round ducts 634 – with characteristic values for clamped plates 553 – with characteristic wave speeds in plates 526 – with classical boundary conditions for plates 527 – with coefficients of polynomial approximation to circular modes 631

General Index

– with defining relations for passive mechanical components 61 – with effective bending moduli of sandwich panels 543 – with elastic data of materials 529, 531 – with electromagnetic quantities 59 – with end corrections of different radiators 318 – with foot point admittances 1180 – with grad and rot components cylinder or sphere 252 – with material constants of air 9 – with mode norms for clamped plates 554, 555 – with numbers of legal and effective mirror sources 901 – with oscillating masses of spherical and cylindrical radiators 317 – with passive electrical and mechanical circuit components 60 – with ranges of porosity of materials 348 – with reciprocal electrical elements 63 – with regression constants for data of air 11 – with relations between istropy parameters 1138 – with relations in flat duct with anisotropic bulk reacting layer 622 – with relations of elastic constants 531 – with strain velocity at cylinder or sphere 252 – with survey for scattering in random media 232 – with transmission coefficients at steps 1185 – with types of sandwich panels 543, 1169 – with UK-analogous elements 64 – with Uv-analogous elements 65 – with volume and surface porosities, and structure factors of different structures 349 tangent vectors – at co-ordinate lines 27

1273

target quantity 1021, 1028 target strength 1037, 1057, 1058 temperature coefficient 7 temperature variation 37 temperature wave 38, 595, 599 temperature-dependent input parameters 740 temporal density of reflections 877 tensor – derivative of 30 termination condition 1031, 1033 test of mode approximations 612 Theory of the quasi-homogeneous material 350 thermal boundary layer 1114, 1115 thermal conductivity 945, 946, 970, 971, 981 thermal expansion coefficient 36 thermal pressure coefficient 36 thermal wave 7, 421 thermodynamic relations 36 Thermodynamic relationships 955 Thermoelastic effect 1206–1208 thickness and loading noise 1009 thin film transducer 1199 three-dimensional mode charts 607, 633 Three-Duct Perforated Element 814, 816, 820, 840 Three-duct perforated elements with extended perforations 820 three-duct perforated section – common 814, 820 – particular 816 Three-Pass perforated elements 825, 827 Tight panel absorber – approximations 493 – rigorous solution 485 time derivative – partial 731 – substantial 731 Time-harmonic wavefields 1144 Timoshenko bar 1181 Timoshenko mode 1165 Timoshenko-Mindlin model 1163 Timoshenko-Mindlin theory 529 Tools in fluid mechanics 949

1274

General Index

torsional stiffness 1164 Torsional waves 1138, 1164 tortuosity 386, 387, 390 total number of reflections 877 total time derivative 34 trace wave number 528, 529, 535,590 transfer matrix 796, 797, 799–804, 806–808, 812, 814, 816, 818–833, 835–840 transfer matrix representation 796 transformation matrix 793, 796 transmission coefficient 504, 505, 508, 512, 513, 516, 520, 523,533–535,537, 539, 541, 547, 549, 551, 557, 559, 564, 567, 570, 571, 574, 577–579, 582, 590, 682, 692, 699, 1184–1186 transmission condition 1039, 1073 transmission factor 503, 505, 537, 540, 587, 592, 593 transmission loss 504, 505, 512,519, 523, 525, 527, 536, 537,540, 542, 547, 549, 559, 564, 570, 577, 581, 593, 602, 603, 684, 687, 688, 696, 781–784, 788, 796, 797, 801, 832,839 Transmission loss at steps 1184 transmission problem 1019, 1039, 1072, 1073 transversal fibre bundle 373–376, 379–381, 383, 384 transversal plane wave 1153 transversal polarisation 1152, 1153 Treatment of singularities 1070 trial functions 1109–1112, 1114,1116, 1119, 1122, 1128–1130 triangularization 1076 triclinic anisotropy 1135 triple Helmholtz resonator 723 triple sheet 594 triple-type scattering 231, 232, 248 tube section – terminated with Helmholtz resonator 69 – with hard termination 67 – with layer of air 70 – with layer of porous material 69 – with open termination 68 – with perforated plate in the tube 70

turbulence 947, 948, 951, 954, 960,966, 984, 992, 1001, 1002, 1005–1007, 1013, 1015, 1016 turbulence level 948 turbulent flow 947–949, 959, 960, 986, 1015 turning-vane splitter silencer 775, 781, 788, 789 – transmission loss 781 two-duct elements 808, 825 Two-Duct Perforated Elements 806 two-grid method 1066 U Ultrasonic attenuation 1199 ultrasonic backscattering 1204 uniform end correction – of plane piston radiator 309 uniform flow 947, 962 uniform pass integration 159 Uniform tube – with flow and viscous losses 798 unitary tensors 27, 30 unsymmetrical duct 627 upsound cross sections 793 upstream end 796 V variable absorber parameters 1021, 1023 variational functional 1118, 1121,1122 variational methods 1109, 1110 variational principle 146, 1046, 1075, 1079, 1109, 1110, 1113,1117 variational statement 1118, 1125, 1128 variational techniques 1110 vector – angle between 29 – derivative along a curve 30 – derivatives of basis vectors 29 – length of 29 – scalar product 29 – triple product 29, 31 – vector (cross) product 29, 31 Vector algebra 28 vector Green’s function 980 velocity of a plate when excited by a diffuse sound field 1193

General Index

velocity potential 962, 1038, 1109 velocity profile 353, 355, 357,384 velocity profile around a fibre 384 viscoelastic material 1038 viscoelastic models 1140 viscosity wave 595, 599, 747 viscous and thermal boundary layer 1114 viscous and thermal losses 352, 355, 358, 404, 407, 420, 426, 430, 431, 434, 438, 476, 595, 1028 viscous energy loss 35, 36 viscous flow 946, 948, 1017 viscous force 367, 369, 387 viscous shear 5, 6 viscous stresses 963, 967, 970,986 viscous wave 421 visibility check 899 Voigt and Reuss averages 1148, 1149 Voigt constants 1161 volume flow impedance 793, 805, 819, 820 volume flow velocity 793 volume porosity 347, 350, 351,386 vortex sound 976, 1016, 1017 vorticity 947, 976–980, 1016 W wall admittance 127, 142 Wall of multiple sheets with air interspaces 591 wave equation 388, 389, 595, 601, 605, 606, 620, 622, 652, 676, 690, 703, 715, 732, 1198

– – – – –

1275

adjoint 26 convected 972 homogeneous 5, 13, 14, 20, 21 in uniform flow 962 inhomogeneous 13, 949, 963–965, 968–971, 973, 975, 976, 978,987–989, 991, 993, 1000 – linear, homogeneous 962 – of a membrane 469 wave impedance 185, 347, 349, 351, 352, 355, 358,371, 375, 381, 394,503, 509, 511, 517,518,575, 587, 596, 598, 599, 700, 740,747,748, 755 – in a flat capillary 405 – in random media 244 wave number – characteristic 20 – of the shear wave 390 wave propagation in piezoelectric semiconducting solids 1210 Waves – in bounded media 1154 – in thin beams 1163 – in thin plates 1160 weak coupling 390 weight function 1021, 1024 weighted residual equations 1042, 1044–1047 weighted residuals 1040, 1042, 1044, 1057 weighting functions 1040 wide-angle absorber – near field and absorption 480 – scattered far field 474

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