Formulas of Acoustics

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

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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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226 230 238 242 244 248 260 263 264 266 268 270 275 279 282

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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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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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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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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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1075 1078 1081 1082 1085 1086 1088 1091 1094 1094 1095 1096 1098 1098 1099

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 . . . . . . . . . . . . . . . . . . . . . .

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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 . . . . . . . . . . . . . . . . . . .

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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 = −grad p. 0 ∂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 + k02 p = −j k0 Z0 · q · ƒ(r − rq ). with monopole source at 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 ). ∂v 1 1 Impulse equation: + grad p1 − Œ v − Œ grad div v = 0. (8) ∂t 0 3 ∂T1 + (‰ − 1) T0 div v − T1 = 0. Heat balance: (9) ∂t ∂1 + 0 div v = 0. Conservation of mass: ∂t p1 1 T1 − − = 0. (10) Equation of state: p0 0 T0 Heat conduction inside a ∂Ti1 bordering medium (index i): − i Ti1 = 0. (11) ∂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

v = −grad ¥ + rot ¦ 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:

(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 ‰– – ≈j −1 ± 1 − 4j 2 , 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 ‰– − jk, j k, 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 4 ‰ 4 ‰ 2 4‰ 4 1 1 1

+ + 2 + + + 2 − 2 + k2 k02 3kŒ2 k0 k02 3kŒ2 k0 k0 ‰k02 3kŒ2 = . 4 2‰ 1 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 4 Pr 1 k0 − 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

Ÿ k2 ≈ −(‰ − 1) 20 , Ÿ k0 ¢ ≈ ‰ j

k2

≈‰j

–

jk 2 ¢ ≈ 0 –

‰

(27)

(28)

k02 , –

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

(29)

4‰ Pr ¢ = −0.3033. ≈1− ¢ 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 4 see Fig. 1) using measured data are given in the form f (T) = ai · Ti/2 for the material i=−4

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

Specific heat

cp

1.00710 3

J/kgK

P const.

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//

c20 =‰P0 /0

Interrelations are: Prandtl number:

Pr = Œ/.

Specific heat at constant volume:

cv = c p −

Isothermal compressibility:

ß2 T . K0 1 ∂V 1 ∂ K=− = . V ∂P T ∂P T

(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 K00 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; K0 = compressibility; < 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)| I = − · grad œ(r) + j · grad ln |p(r)| . 2 k0Z0 2

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 (3)

p1 · v2 · ds − p2 · v1 · ds = p1 (r2 ) · q2 − p2 (r1 ) · q1 . 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. The homogeneous wave equation is p(r) + k02 · p(r) = 0. (2) 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


∂g . lim −4R R→0 ∂R lim g(r|rq , –) =

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

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:

f (rq , t0) · g(r|rq , t − t0 ) dt0 ,

p(r, t) =

(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 . (15b) belongs to the time function 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, ¦n · ¦m dV = ƒnm = (17) 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 , –) =

An ¦ 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, ¦n · ¦m∗ dV = ƒnm = (21) 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)

−∞

thus

SF (x) =

−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 g(r, rq |–) = ƒm cos (m(œ − œq )) ‰ d‰. 82 m=0 ‰2 − k02 +∞

(37)

−∞

One gets after evaluation of the integral

⎧ ∞ ⎨Jm (k0 r) · H(2) m (k0 rq ) −j ƒm cos (m(œ − œq )) g(r, rq |–) = ⎩ 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 j k0 (n − m) ! 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 cos (m(œ − œq )) · Pm × ƒm 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: · r = The plane is at x=0; the source at xq , yq , zq ; the image source at –xq , yq , zq . Let n 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 Nm = Tm (r) dr + Tm (r) dr. (1) j k0 Z0 a Za A1

A2

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


B.8

B

21

Source Conditions

See > 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. k0 Z0 q The source condition can be written Z0 vr (rq +0)−vr (rq − 0) =! · ƒ(˜ −˜q ). (2) in the form 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 Otherwise: lim r = 0. (1) 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 ): Set (with G1, G2 selected so that Re{Ga }≥0): Gs is the symmetrical and Ga the antisymmetrical part of the boundary conditions. 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 . It follows immediately that Comparing this with the boundary conditions of the original task, one sees the correspondence:

v1

=

G1 · p1 ;

v2

=

G2 · p2 .

B

23

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

(1)

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)

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 ƒ L dt + ƒW dt = 0. (1) 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 > Sect. B.11. 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,

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 dr. f2(r) = f1 (r) · 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 ∂u2 ∂u2 ∂u2 2 = grad 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 = gki 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: contravariant 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 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 = gijaj ,

where the last notations use the “summation rule” (summation over multiple indices). i = gij R i = gij N j j. Thus: N ; R (6) 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 = ai ak = gik ai ak . i,k

i,k

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 + = Bim Bkn R Rs . 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}: # $ # $ j or •R k. = ikj R ikj = R 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 D ai ∂ai duk duk j = = ∇k ai − aj 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 as k + + ai s , ∇j aik = js js ∂uj s k k ∂a s i ais − + ask , ∇j aik = js ji ∂uj s ∂ai k s s as k − ∇j ai k = − ai s . ji jk ∂uj

j

j ik

aj . (27)

(28)

s

It holds that

∇j ai bk = ∇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) ;

i

g = det(gij) = H1 H2H3 .

ai∗ ei

a • b =

Scalar product:

(30)

;

ai∗ = a • ei =

ai = Hi a i . 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 = ∇

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

i

The rotation of a vector is a vector: ⎛ H1e1 1 × a = ⎝ ∂/∂u1 rot a = ∇ H1 H2 H3 a1 ⎛

H1 e1 1 ⎝ ∂/∂u1 = H1 H2 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 H1 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 • V 1 • grad V 2 = V 2 + V 2 • grad V 1 + V 1 × rot V 2 + V 2 × rot V 1, grad V = U · div V +V · grad U, div U · V 1 × V 2 = V 1 − V 1 • rot V 2, 2 • rot V div V = U · rotV + grad U × V, rot U · V 1 × V 2 = V 1 − V 1 • grad V 2 + V 1 div V 2 − V 2 div V 1, 2 • grad V rot V • ∇ ×V = div rot V = 0, ∇ × ∇U ∇ = rot grad U = 0, • ∇U ∇ = div grad U = U. Some co-ordinate systems (see > Sects. B.10, B.13 for more systems): a scalar function: U. A vector: V;

(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 Transformation: x = r · cos œ y = r · sin œ z=z

Line, surface, and volume elements:

r = x2 + y 2 œ = arctan (y/x) 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: 1 ∂U ∂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: ∂U 1 1 ∂U ∂U er + eœ + e˜ , ∂r r sin ˜ ∂œ r ∂˜ ∂Vœ 1 1 1 ∂ r 2 Vr ∂ (sin ˜ V˜ ) = + + , div V 2 r ∂r r sin ˜ ∂˜ r sin ˜ ∂œ ∂ sin ˜ Vœ 1 ∂V˜ = rot V − er r sin ˜ ∂˜ ∂œ 1 ∂(rV˜ ) ∂Vr ∂Vr 1 ∂(rVœ ) 1 − e˜ + − eœ , + r sin ˜ ∂œ r ∂r r ∂r ∂˜ 1 ∂U 1 ∂2U 1 ∂ ∂ 2 ∂U r + 2 sin ˜ + . U = 2 r ∂r ∂r r sin ˜ ∂˜ ∂˜ r2 sin2 ˜ ∂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 4 D v¯ = −grad (¯p + ¯ ¥ − (‹ + †) div v¯ ) ¯ Dt 3 −† rot (rot v¯ ),

35

(53) (54)

with ¥ = potential of an external force per unit mass (e.g. gravity); 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 v = v + vt with (57) div vt = 0

linearised and v0 =0; ¥ =0: Apply and compose: ‰ Cv Cp †

= = = = =

0

∂v ∂t

=

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. 5 0 c0 Cv ≈ †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: 1 dU 1 dE = = 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: dS dQ = 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 · p˜ + ·T S˜ = ideal gas ∂ p¯ T T0 ‰ T0 ‰ P0 ∂ T¯ p Thermodynamic relations: 1 ∂V isothermal compressibility; KT = − V ∂P T 1 ∂V isotrope compressibility; KS = − V ∂p S 1 ∂V thermal expansion coefficient; ß= V ∂T p ß ∂p = = thermal pressure coefficient; ∂T V KT

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


Cp Cv

‰=

B

adiabatic exponent;

KT = ‰ 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)

ß ß 0 c20 P0 = ; −−−−−−→ 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 – – – k2 ≈ 1 + v − j (‰ − 1) h ; c0 c0 c0 with ∂ p˜ 1 = j – · p˜ ; c20 = ˜p = −k2 p˜ ; ; ∂t 0 Ks – ‰−1 ˜ 1 + j h · p˜ ; temperature variation: T= ‰ c0 1 – density variation: ˜ = 2 1 − j (‰ − 1) h · p˜ ; c0 c0 entropy variation: longitudinal particle velocity:

Cp ‰ − 1 – h · p˜ ; T 0 ‰ 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 – ˜ 1 + j (‰ − 1) S˜ = 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 ∂ ∂t + U0 ∂ ∂x = c2i ∂ 2 ∂x2 (1) with general solutions –1 x – ¯ 1x + G1 – ¯ 1t + 1 = F1 –1 t − 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 + M1 1 − M1 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 = k1

c1 1 + M 1 . c2 1 + M 2

(10)

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

;

–2 = (c2 + U2) k2 ,

U2 1 + M1 –2 = 1+ –1 c2 1 + M 2

;

k 2 c1 1 + M 1 = . k 1 c2 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 v4 = u 3 =

−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) − œ(x, y) + j · which reads Z0 G = 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 ; ‚n2 = k02 sin ˜ + n Š0 T − 1 , (4) Thus: ßn = k0 sin ˜ + n Š0 T 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 gn = ; n = ±1, ±2, . . . ; g0 = . 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 cos ˜ − g0 A0 = 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) ; m = 1, 2, 3, . . . . (16) g0 = Gs ; gm = g−m = Gs 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 A0 · s(m)+ An ⎢ ⎣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 > Sect. B.19 the grooves are no longer narrow; higher modes may exist in them. 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 − (1) pk (x, zŒ ) = e + Cm · e Bm · e · cos m 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‹ j ‚ a m ⎦ An ⎣ (−1)‹ s−‹,n · s‹,m − ƒn,m 2T ‹=0 2 k0 1 + R‹ k0 n=−∞ (4) ⎡ ⎤ ∞ ƒ‹ ‰‹ 1 − R‹ a (−1)‹ s−‹,0 · s‹,m ⎦ = Pe · ⎣ƒ0,m · cos ˜ − 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) + (−1) . sm,n = e m/2 − ßn a/2 m/2 + ßn a/2

(5)

The amplitudes Bm follow with the An from

! (−1)m ƒm Pe · s−m,0 + Bm = An · s−m,n . 2 (1 + Rm ) n=0,±1,±2...

(6)

The reflection factor r of the arrangement follows with the An as in the previous > Sect. B.19.

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) = vIIIx (x, z) =

+∞ n=−∞

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 m=0: A0 an Zan cos ˜ + an Zan cos ˜ − + ; (6) An s(n) = Pe a k0 k0 a k0 n=0 —n ‚m —0 T —0 = −Pe · s(m) ; (7) m=0: A0 · s(m) + An s(m − n) − ƒm,n · an Zan j 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 ˜ an Zan = − 1⎠ ⎝ an Zan + 1⎠. 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 s(m − n) + ƒm,n · An · j 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) C sin (ß0 a/2) (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 S: = j 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/T a/T a/2) 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) B (1 + S) e+j k0 d =2 , 2 Pe ß0 a/2 (1 + S) e+j k0 d − (1 − S)2 e−j k0 d sin (ß0 a/2) C (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 j · s−m,Œ · sn,Œ + (−1) Xn k0 Œ=0,±1,... ‚Œ ƒm a 1 − e−2j ‰m d n≥0 2(−1)m e−j ‰m d 4T Pe · s−m,0 + · Y = m , a ƒm 1 − e−2j ‰m d (3) ! −2j ‰m d ƒ 4T Z 1 + e ‰n k0 k 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 ƒ0,Œ · Pe · cos ˜ − · sm,n , An = j (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

Definition U I s I dt C= U

R

R=

Capacitor Capacity

C

Coil

L

L=

Complex Impedance

Z

Z=

Complex Admittance

G

Inductivity

U I I 1 G= = U Z U = 0; I = 0

Connection Transformer

I1

I

U1

U2

2

I I

1

U

U1

Element

Quantity

u=

w2 U2 I1 u= = w1 U1 I2

2

2

Symbol

Letter

Resistor

Friction

R

Spring

Compliance

C

Mass

Inertance

M

Complex Impedance

Z

Complex Admittance

G

Definition 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

U dI=dt

l2 u=

l1 F1 F1 F2 l2 l1

F2

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

−F

F1 x

x

10

Mass

F = F1 =

1

x x

1

2

1 v j–C

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 2C

Z

r 2/ Z

G

r G

2

Capacity Inductivity Admittance

2

Impedance

U/ r

Voltage source

U

Current source

Reciprocal resistance

~

Current source

Zi

2

r / Zi

I

Voltage source

Z

i

r I

~

2 r / Zi

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. ∗)

See Preface to the 2nd edition.

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Below is an example illustrating the application of this rule.

R

U

L

1

L

1

~

2

C

R

2

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

~

Zi

Velocity source

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 1 0 c20 = , 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 1 0 c20 Za ≈ . 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 Z0 k0t · (a /k0)2 ; Z −−−−→ a Za t ≈ tŠa ‰

¡t −−−→ E1

;

j – 0 t −−−→ E1

(9)

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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=

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

p Sa 1 Sa Zs = 0 c20 . ≈ · Z −−−−→ 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

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Equivalent Networks

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.

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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 Gv = · 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 = B0 e−j k0 x + j Cm e+‰m x J0 (—m r) 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) 0 m = 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 n 0 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)

a2 J1 (†na) ; −−−−−−→ 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 ! Dn sinh ‚n t J0 (†n r) = −j ‰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)

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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 B0 S0,n + j (22) Cm Sm,n . Dn =

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 ‚n −‚n x j @p 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 Dn e J0 (†n r) = −j 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 eix ] J0 (—i r) = i Zi @x Zi


pII (x; r) =

n0

Z0 vIIx (x; r) =

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

eff 1 C = ; 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 Z0 vIx (x; r) = = B0 e−j k0 x − C0 ej k0 x k0 @x


pII (x; r) =

n0

Z0 vIIx (x; r) =

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


;

J1 (†n a) a2 ; −−−−! 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 =j Dn sinh (‚n (x − t)) J0 (†n r) 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) =

n0

Dn cosh (‚n (x − t)) J0 (†n r) ;

‚n2 = †2n + a2 Z0 vIIx (x; r)

=

−1 @p a Zi @x

=

‚n −1 Dn sinh (‚n (x − t)) J0 (†n r) i Zi 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 i i 0;n ⎦ 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 −−−! †n a 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)) Z 2 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 Fn sinh (‰n (x − t)) J0 (†n r) =j k0 @x k0

n0

Z0 vIIIx (x; r) =

pIII (x; r) =

‚n2 = †2n + a2

−1 @p 1 ‚n

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


Dn e−‚n x + En e+‚n x J0 (†n r) ; pII (x; 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) D0 + E0 hpII (0; r)ib

= 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 −−−! †n a 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

‰n −1 @p −1 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

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

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 (‰n s)

‚n i Zi − j ‰n tanh (‰n s) 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

j k0 S0;n ‚n i Zi + j ‰n tanh (‰n s)

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

n0

⎤ −1 k0 S20;n ‚n i Zi + j ‰n tanh (‰n s) 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 (‰n s) ⎡

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) −−−! †n a 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 ⎪ ⎪ ⎨ 0 ; n 6= n (y) (y) (II) (z) Nn;Œ = b2 Nn NŒ ; Nn = 1 ; n = n0 6= 0 ; ⎪ ⎪ ⎪ ⎩

Mode norms in (II)

2 ; n = n0 = 0

⎧ ⎪ 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

cos(†n y) cos(†Œ z) ds ! J1 a †2n + †2Œ 2 −−−−−−−! a2 = 2 a †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 ‚n;Œ −‚n;Œ x 1 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(†Œ 0 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 .




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

cos(†n y) cos(†Œ z)

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

N(II) n;Œ

Sn;Œ (B0 − C0 )

(‚n;Œ k0 ) sinh ‚n;Œ t N(II) n;Œ

+b +b = cos(†n y) cos(†Œ z) cos(†n0 y) cos(†Œ 0 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 .




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 Zi 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 )


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

Mode norms in (II)


in s= a2

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 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

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 Zi 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 a (‚n;Œ k0 ) N(II) 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) =




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 @p 1 ‚n;Œ

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) =


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

= Zi = Zi ZK = 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) =


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

⎝B0 S0;Œ 0;n (‚n;Œ k0 ) N(II) n;Œ

+j

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 )) ⎪ ⎪ a 1 + ; —m = †n 6= 0 ⎪ ⎪ 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 k (‚ ) N 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

Dn;Œ =


N(II) z;Œ =

SŒn

⎧ ⎨ ⎩

b ; n 6= 0

;

2b ; n = 0

2c ; Œ = 0 +a +a

=

cos(†n y) cos(”Œ z) dy dz ⎧ ⎨

Mode coupling (I)–(II) =

⎩

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

ZMi = Zsh − ZK Zsh

⎩

c ; Œ 6= 0

−a −a


⎧ ⎨

;

⎫ ⎬ ⎭

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 Z0 vIx (x; y; z) = = B0 e−j k0 x − C0 e+j k0 x 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) =

‚n;Œ −‚n;Œ x 1 Dn;Œ e i Zi 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 ⎡ (SŒn )2 i Zi ⎣ ⎦ 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 4a (‚n;Œ k0 ) N(II) n;Œ n;Œ0 ⎡

⎤ −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) =

n;Œ0

Z0 vIIx (x; y; z) =




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 4a (‚n;Œ k0 ) N(II) n;Œ n;Œ0 ⎡

⎤ −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(a t)

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

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

= Zi = Zi ZK = 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 Z0 vIx (x) = = B0 e−j k0 x − C0 e+j k0 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 ) †n b = 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 ) †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) =

C


Table 26 continued



Mode coupling (I)–(II) in s = 4a2 = −a −a

2b ; n = 0

b ; n 6= 0

⎩

⎧ ⎨

⎩

a ; n 6= 0

⎭

⎫ ⎬ ⎩

⎧ ⎨

a ; Œ 6= 0

2a ; Œ = 0

2c ; Œ = 0

c ; Œ 6= 0

2a ; n = 0

; N(II) z;Œ =

⎭

⎫ ⎬

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

⎩

⎧ ⎨

cos(†n y) cos(”Œ z) dy dz =

ZMi = Zsh − ZK

SŒn


Mode norms in (II) +a +a

Fn;Œ = ⎧ ⎨

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

e (II) ‚n;Œ Nn;Œ 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

Dn;Œ e−‚n;Œ (t−s) + En;Œ e+‚n;Œ (t−s)

cosh ‰n;Œ s

En;Œ = (B0 − C0 )

Equations for Fn;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:

j ∂p(0, y)/∂x vx (0, y) = . 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 g2 ) 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 ;

m = 1, 2, . . . , M

Characteristic values:

am , Zam ;

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 :

g2 − g2 g 1 + 2 g 3−dif = 8 g 1 + · arctan − g · ln 1 + g g + g2 + g2 g2 + g2

g2 1 + 2 g (3) 1 + → 8 g − g · ln 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 z 2 (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 ‡: (‡) = 1 − |R(‡)|2 = R(‡) =

4g1 cos ‡ , (cos ‡ + g1)2 + g22

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

Ÿ (‡) cos ‡ sin ‡ d‡ = 0

8g1 cos ‡ → t −−−−−−−−−−−−−−−→= 2 − sin ‡ d‡ = dt sin Ÿ

1 cos Ÿ

8g1 sin2 Ÿ

Ÿ 0

cos2 ‡ sin ‡ d‡ (cos ‡ + g1 )2 + g22

8g1 g2 − g22 t2 dt = 1 − cos Ÿ + 1 2 2 2 g2 (t + g1 ) + g2 sin Ÿ

1 + g1 g1 + cos Ÿ arctan − arctan g2 g2

g2 + g22 + 2g1 cos Ÿ + cos2 Ÿ + g1 ln 1 1 + g12 + g22 + 2g1

g12 − g22 1 + g1 g1 g12 + g22 arctan +g1 ln −arctan −−−−−−−−→ 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:

∂ ∂ G (s|s0 ) · p(s0 ) − p(s0 ) · G (s|s0 ) ds0 , ps (s) = ∂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 dS. = Re · Qs = Ii P∗i Pi /Z0 ∗ ¢a vn p Qa = dS. = − Re · Ii P∗i Pi /Z0

Scattering cross section of A: Absorption cross section of A: Extinction cross section of A:

Qe = Qa + 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 The far field ps can be separated . (10) ps (s) −−−−→ Pi · ¥s ˜i , œi |˜ , œ · s→∞ s into an angular and a radial function: 4 The extinction theorem: Qe = − (11) · 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 Qa = − · Im{¥s ˜i , œi |˜r , œr } 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 ¥s = (G − F0) e−j (‹x x0 +‹y y0 ) dx0 dy0 , 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)

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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

−(d/d… )2 /2

K… (d) = … A · e

/2

,

(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 p(x, 0) = Pi e−jkx·sin Ÿi 2 − 2 − U(−Ÿi , kx) + jV(−Ÿi , kx) , 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

x0

P(x,z)

s R

x +a/2

See > Sect. D.9 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 ps (s) = −Pi · ¥s (Ÿi |˜ ) · √ , (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

2 Qa = 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)

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D

Reflection of Sound

The resulting Qa is +/2 2ka G cos Ÿi 2 G cos Ÿi Qa = 4Re − si 2 (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 − k 2 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) · e−jkx·sin Ÿi p(x, 0) = 2Pi 1 − 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

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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 + + ez · , grad u = ∂ ∂˜ ∂z c sinh2 + sin2 ˜ c sinh2 + sin2 ˜ and therefore the particle velocity v v in the direction of the hyperbolic -lines:

=

j j ∂p/∂ . 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

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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: G 2 ¢ 1 · 1 + CE(˜ ) = = ¢∞ 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 (kr) = e−j kr·sin Ÿ dŸ ; Re{kr} > 0 . H(2) 0

(4)

C(Ÿ)

whith path C(Ÿ): −j∞ → 0 → → + j∞. Thus the second term yields 1 pr = R(Ÿ − ˜ ) · e−j kr·(sin(Ÿ−2˜ )+2h/r·cos(Ÿ−˜ )) dŸ,

(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

2

2

1 + G − cos ‡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 1 · 3 · 5 · . . . (2n − 1) (2n) 1 −j kr ¥ (0) + e ¥ (0) + . . . + ¥ (0) , kr 4 kr (2n)!(2 kr)n

(8)

(primes at ¥ indicate derivatives with respect to s), with: 2j 2j cos(œ(s) + ‡s ) − G · ¥ (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 =

j 9 75 j 3675 2j −j kr · R(‡s ) e · 1 + − − + kr 8 kr 128(kr)2 1024(kr)3 32768(kr)4 (2) 5 R (‡s ) 259 j 3229 j − · − + + 2 16 kr 768(kr)2 6144(kr)3 kr (4) 35 j R (‡s ) 329 1 + · − − 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 pr (r , 0) = e · 1+ − − + kr 8 kr 128 (kr)2 1024 (kr)3 32768 (kr)4 5 2G 259 j 3229 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)

158

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 5 4 259 j 3229 j − − · 2+ − + (14) 2 16 kr 768 (kr)2 6144 (kr )3 kr G 35 j 7 48 − 16 G2 329 j 1 + − · − + + 8 192 kr 1024 (kr)2 (kr )2G4 48 128 kr , 1 2 40320(1 − G2 ) + 8064 G4 − 128 G6 2(720 − 480 G2 + 32 G4 ) . · − × (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 ) I=e ·T ; Im{b} ≷ 0 , · ±ja · W ±b x + x

√ 2 Im{b} = · T − ja · e−x·b ; ex·f (œs ) · ja · W b x + Re{b} = x · T ; Im{b} = 0 , ex·f (œs ) · x

0 0

,

(19)

160

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 > Sect. B.10) gives pa (r) =

1 · ph (r) + G · X(s) · ps (r) , 1 + G · X(s)

(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 ) .

One gets for the cross impedance X(y) 1 0 k 0 rq −Z0 vQsx H(2) (k r ) 1 0 = · 1 + 2(2) = −j · X(y) pQh 2 H0 (k0r ) 0 1 H(2) (k0 y)2 + (k0 rq )2 k 0 rq 2 · 1 + (2) = −j · . 2 H0 (k0 y)2 + (k0 rq )2

(4)

(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)

1/X(y) − G · pSh (x, y) . = pQ (x, y) + 1/X(y) + G Numerical comparison with saddle point integration (see > Sect. D.14):

(6)

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 ∞ p1 (r, z) y · J0 (y k1r) · e−k1 z y −1 = (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 pr i = P0 2 k1 i p2 = 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 k 1 r1 k 1 r2 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 1 ∂ 2 ei k2 r1 ei k1 r2 p1 (r, z) ei k1 r1 ei k1 r2 = + − · r0 dr0 dz dœ , P0 k 1 r1 k 1 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

168

Reflection of Sound

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 pr i 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

D

169

With a field composition p(r, z; h) ei k0 r1 ei k0 r2 p1 (r, z + h; 0) = − + P0 k 0 r1 k 0 r2 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 1+Z 2 (n − m) − 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 k 0 r1 k 0 r2 P0 √ √ 2 2 2 2 p(r, z; h) ei 2 (r/Š0 ) +(z/Š0−h/Š0 ) ei 2 (r/Š0 ) +(z/Š0+h/Š0 ) = + 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 > Sect. D.17, after multiplication and division of the integrand by exp(±ik0r · sin ˜ ): Q G

h

z

r1

h Θ0

i pr = P0 2

Q' C

P

Θ0

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

sin(˜Pw − Ÿ0 ) = − tanh ˜Pw ,

(4)

D

172

Reflection of Sound

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 · sinh ˜Pw + i −−−−−−→ −(˜Pw ) +i. f (˜Pw ) = − tanh ˜Pw

(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 the general integral I is transformed into With the transition ˜ → ˜Pw ⎧ g(˜Pw ) = i − tanh ˜Pw · sinh ˜Pw , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −∞ d˜ ⎨ G(˜ ) = F ˜ (˜ ) a·g(˜Pw ) Pw Pw , I= e · G(˜Pw ) d˜Pw ; d˜Pw ⎪ ⎪ ⎪ +∞ ⎪ dg(˜Pw ⎪ )/d˜Pw ⎪ ⎩ . = F ˜ (˜Pw ) df (˜ )/d˜

(9)

The last fraction becomes sinh ˜Pw dg(˜Pw 2 − tanh2 ˜Pw )/d˜Pw · (2 − tanh 2 ˜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

D

173

In the special case h = z = 0, i.e. Ÿ0 = /2, one gets cos ˜Pw

= sinh ˜Pw − i · tanh ˜Pw = − cos −˜Pw

, sin ˜Pw

= 1 + i · tanh ˜Pw · sinh ˜Pw = sin −˜Pw

,

Z · sinh ˜Pw − i · tanh ˜Pw −1 = 1/R −˜Pw

, 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 pr (r, z) ei k0 r2 i R(Ÿ0 ) − . = R (Ÿ0 ) cot Ÿ0 + R (Ÿ0 ) + P0 k 0 r2 2 k 0 r2 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 ei k0 r2 pr (r, z) =− P0 k 0 r2

2 3 prp 1 1 , + i · F(Ÿ0) + F (Ÿ0 ) + 8 k 0 r2 2 k 0 r2 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 a·f (˜s )

∞

I=e

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 + (1 − i) H(1) · sin Ÿ0 0 (k0 r sin Ÿ0 ) · e cosŸ0 + G ∞ 2 2 2 e−x dx . W(u) = e−u · erfc(−iu) ; erfc(z) = √

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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Reflection of Sound

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 √ 2 · u · eu · erfc (−u) k0 r2 G + cos Ÿ0 ; u = (1 − i) 2 sin Ÿ0

(13)

F(u) = 1 +

with

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: ei k0 r2 ps pp pr = + + , P0 k 0 r2 P0 P0 with

(17)

6 1 1 + G cos Ÿ0 −1 + √ 1 + √ + 2 sin Ÿ0 1 − G2 i sin Ÿ0 √ + 1 + G cos Ÿ − 1 − G2 √ 0 k0 r2 (G + cos Ÿ0 )2 8 2 ,7 1 + G cos Ÿ0 1 + G cos Ÿ0 3/2 3+ ; · 1+ √ √ sin Ÿ0 1 − G2 sin Ÿ0 1 − G2

2G ei k0 r2 ps = 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

D

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 √ ei k0 r2 pr −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) (2m) ! pr ei k0 r2 1 − 2 k 0 r2 G = · Im 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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Reflection of Sound

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 = · e−Š · erfc (−i Š) , a0 = 1 ; an = 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 k 0 r2

;

n−1 · En−2 ; 2 k 0 r2

Kn = −B · Kn−1 − i

n−1 · Kn−2 . 2 k 0 r2

(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 −Z0 vsz (r, z = 0) i ∂ pQ (r, 0) − pQ (r, 0) /∂z 1 = = , (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 k h i + k h 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

Reflection of Sound

The approximation by Delany / Bazley starts from Van der Pol’s exact form; it is ei k0 r2 pr = +2ik P0 k 0 r2

∞ 0

ei (k1 r3 +ky) dy ; k 1 r3

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 k 2 −1 + · e + · e , (k1 r)2 (k1 r)2

kz kZ ; C1 (z) = −2 i (1 + kZ) +√ 2 1−k 1 − k2 6 k Z2 − 1 ; ‚ . C2 = −2 i (1 + kZ) /k = k 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

F2 (H) 1+ƒ F1 (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 + k1H · ƒ · 1 − k 2

(6)

Reflection of Sound

F1 (H) =

D

181

√ 1 . 4 − 3(k1 H)2 + k 2 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 k 4(k1 H)2 2

4

2

2

+ 48 − 24(k1H) + 5(k1H) + k 72 − 12(k1H) − 5(k1H)

4

(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 − k 2 − 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 k 2ƒ4 .

G1 (H) = −

An approximation given by Attenborough/Hayek/Lawther is

cos Ÿ0 iF ei k1 r2 pr −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 √ sin2 Ÿ0 cos Ÿ0 + kZ k 2 − sin 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 k 2 − 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 k 1 r2 2 k 1 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

ei k1 r2 i pr R(Ÿ0 ) − = R (Ÿ0 ) · cot Ÿ0 + R (Ÿ0 ) . P0 k 1 r2 2 k 1 r2

(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

E

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 1 J1 (j k0 a · an ) cos Ÿ . → =− 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 dS −−−→ Re{ps · Z0 vrs∗ } dS, · rs Qs = Re p0 =1 p0 p0 /Z0 2 Qs = 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, −2 Qe = 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 =

− Hm+1

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

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−

−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

1 + 2(k0 a)2 1 + (k0 a)2

X1 =

−k0 a

Xm =

(k0 a)2 k0 a 4(m − 1) − m (m − 2)(k0 a)2 1+ 4m (m − 1)

−

Qs at lowest resonance; R = 0; X = X0

Qs =S =

2 k0 a

4 (k0 a)2

Frequency of lowest backscatter minimum

k0 a =

2 jXj 2 + 3 jXj2 p jXjmax = 2=3 p (k0 a)max = 1= 6

6 jXj 3 + 5 jXj2

−k0 a

1+

k0 a m

1 (k0 a)2 1− 2m 2

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 4(k0 a)4 an 2 = 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

(1 − jC0 j2 ) 4 k0 a 1 + j an Zan an = − + 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

) −j an an Zan − j 3 (k0 a) − 1 + j +2 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/(0c20 ). (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) ps (r, ˜ ) = −P0 · ƒm · cm · H(2) 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

k0y

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: p(r, ˜ , z) = Z(kz z) R† (kr) · T(†˜ ).

(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) pa (r, ˜ ) = An · H(2) a ≤ r ≤ rq †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) k 0 rq cos (†n ˜q ) = cos (†n ˜q ). 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 · z3 + 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 p(r, ˜ ) = pQ (0) 2kr cos 2 2 2 √ ˜ − ˜q 2kr cos −jS 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 rq2 = xq2 + yq2 = c2 · cosh2 q · sin2 œq + sinh2 q · cos2 œq . (4) 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).

The term amplitudes am follow from the source condition; they are Z0 q · k0 c am = cen/2(œq ) · Icm,n (q ) 2 n≥0

(6)

(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 Ii (0) = (15) 2 + ; 2i = even; ⎪ ⎪ 2i − 1 2i + 1 ⎪ ⎪ ⎪ ⎪ ⎩ 0 ; 2i = odd = 1.

212

E

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 Icm,n () = (−1)s+ C2s (‹) · C2 (Œ) · I|(‹ −Œ )/2+s−| (). 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 Icm,n () → (−1)(‹+Œ)/2−1 cosh · B2s+1 (‹) · B2+1(Œ), 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.: j

⎧ ⎪ ⎨ 0

∂p1 = ⎪ ∂ 2 ⎩ −G · p k0 c sinh + cos2 œ c 1

; = c

; /2 ≤ œ ≤ 3/2, (26)

; =0

; else.

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 )

3/2 1 · − cem/2 (œ) · cen/2(œ) dœ 2 /2 1 (0) | cos œ| · cem/2(œ) · cen/2 (œ) dœ − j k0c · Gc · Hc(2) 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: 2 + j˜ = area cosh( + j†) = ln + jn ± ( + j†) − 1 ,

(2)

with = x/c, † = y/c. Geometrical parameters are with the elliptical radius c on the object: # h c = cosh c h ; b h = tanh c ; c = . (3) 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) pback (˜ > 0) = pe (˜ ) + p (−˜ ) − p(w) 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

= 8Pe pe + p(w) 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

= Pe p(w) 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) −Z0 ve + vr + vs = G · pe + p(ß) (17) 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) ·

* u 2 + sin2 t dt

;

(20) s ≥ 0.

s≥0

0

j (2) (2) Hs2r+1 (c ) + G · Hs2r+1 (c ) · Is2r+1,2s+1 = −8 (−j)2r se2r+1 (+ ) Sr · ƒr,s 2 k0 c r≥0 r≥0

j Js (c ) + G · Js2r+1 (c ) · Ic2r+1,2s+1 ; · ƒr,s 2 k0c 2r+1

(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˜ ) u 2 + 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˜ ) u 2 + 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 ): pe + p(h) (−j)m cem (+ )(1 + r · (−1)m ) · cem (˜ ) · Jcm (), 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: p(h) Cm · cem (˜ ) · Hc(2) 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 Jcm (c ) + Z0 G · Jcm (c ) · Icm,‹ ; (29) (−j) cem ( ) (1 + r · (−1) ) · ƒm,‹ = −4 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 Jsm (c )+Z0 G · Jsm (c ) · Ism,‹ .; . (30) (−j)m sem (+ ) (1 − r·(−1)m ) · ƒm,‹ =− 4 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

= Pe p(w) 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) Hs ƒ 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 (c ) + G · Hc2r+1 (c ) · Ic2r+1,2s+1 Cr ƒr,s 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) Hs (c ) + G · Hs2r+1 (c ) · Is2r+1,2s+1 ƒr,s 2k0c 2r+1 r≥0 = −4 (1 − r) (−j)2r+1 se2r+1 (+ )

S(w) r

r≥0

ƒr,s

(10)

j Js (c ) + G · Js2r+1 (c ) · Ic2r+1,2s+1 ; 2k0c 2r+1

s≥0

The integrals Icm,n , Ism,n are described in > Sect. E.8. The sound field is known after (ß) solving for the C(ß) r , Sr .

E.10 Scattering at a Semicircular Absorbing Dam on Absorbing Ground

See also: Mechel, Vol. III, Ch. 22 (1998)

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 : ps = Pe ƒm (−j)m · Dm · [cos(m(˜ + Ÿ0 )) + r · cos(m(˜ − Ÿ0 ))] · H(2) 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 (k0h) − 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) pe = Pe H(2) 0 (k0 R1 ) + r · H0 (k0 R2 ) = 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 r e , 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

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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

;

re = −ra ·

1 − e−2y . 1 − r2a · 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): ¥ (r) = ¥e (r) + U(r) ; U(r) = us (r − rs ),

(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 sums over Hankel functions of the second kind H(2) cylindrical scatterers or 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< ). C0 k0

(26)

(27)

(28)

238

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Scattering of Sound

For low scatterer number density N → 0 will be |jC · g/k0| 1, and therefore kE2 C ≈ 1 + 2j · 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)! Hn (z) ! ; n>0 (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) −−−−−−!

e−jz jn 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

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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 D0 → D0 · . (5) ; 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 (6) · L f , a0, ‹0, D0 . L(f , a, ‹, D) = 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

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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 jk −jk +jk A e−jkE x = e−jk0 x · ⎣1 + e 0 · (A · S+ e E + B · R+ e E ) 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 > Sect. E.12. Integration yields the following system of equations: R− S+ − = −j kE − k0 kE + k0 B · R− A · S+ − = −j kE − k0 kE + k0

;

S− R+ − = −j kE − k0 kE + k0

;

B · S− · e+jkE D A · R− · e−jkE D − =0 kE − k0 kE + k0

(5)

and the solutions of the last two equations: A = (1 − Q) · F

;

B = (1 − Q ) Q e−2jkE D · F,

(6)

where (as in > 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

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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 > Sect. E.12, and the radial functions ⎧ ⎪ ⎪ ⎨ Jn (k r) ; Cylinder, Rn (r) = (16) ⎪ ⎪ ⎩ j (k r) ; Sphere. n

Scattering of Sound

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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 / / ZE kE eff Ceff eff Ceff = · ; = / (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 pii = p + 2† · ; i = j. (10) ; pij = † · + ∂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

1 @vr @v˜ 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 @ ¦z 1 @¦z 1 @2 ¦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 T0 T = T0 T0

=

0 =0 ;0

An [nHn (k a) − k aHn+1 (k a)] − Avn nHn (kv a)

=;

An nHn (k a) − Avn [nHn (kv a) − kv aHn+1 (kv a)]

A0 n nJn (kß0 a) − Av 0 n [nJn (kv 0 a) − kv 0 aJn+1 (kv 0 a)] =;

An Ÿ 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) − 2k aHn+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)] − Av 0 n nJn (kv 0 a)

Ÿe Jn (ke a) +

An

= †0 AŒ 0 n 2n[(n − 1)Jn (kv 0 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) − k aHn+1 (k a)]

− Avn [(2n(n − 1) − (kv a)2 )Hn (kv a) + 2kv aHn+1 (kv a)] A0 n [(n − 1)Jn (k0 a) − k0 aJn+1 (kß0 a)] = †0 2n 0 =0 ;0

− Av 0 n [(2n(n − 1) − (kv 0 a)2 )Jn (kv 0 a) + 2kv 0 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)

An hn (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 (kv 0 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) † p0 a2 An + ¢ + 2(n − n2 + (k a)2 ) hn (k a) − 4k a hn+1 (k a) † =; − 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) − kv 0 ajn+1 (kv 0 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) − k a hn+1 (k a)]

+ Avn [(2(n2 − 1) − (kv a)2 )hn (kv a) + 2kv a hn+1 (kv a)] A0 n [(n − 1)jn (k0 a) − k0 a jn+1 (k0 a)] = †0 2 0 =0 ;0

+ Av 0 n [(2(n2 − 1) − (kv 0 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: ‰ kE2 ‰(‰ − 1) kE2 ‰−1 ≈ , 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 ! Kx = (prr cos ˜ − pr˜ sin ˜ ) · dA = j–M · vx = j–M · jƒ1 , (24) 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)] − Avn nHn (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) − k a hn+1 (k a)] + Avn n(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) †

+ Av1 2(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 · ¥s (§0 , œ, ‡) , ps ≈ −2 Z0 ƒn · cos (nœ) ¥s (§0 , œ, ‡) = . ¥n = 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 ‡ 4 §0 cos ‡ 2n+1 . 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): h Im{Zsn } + h (n !)2 2 2n = n . 1+ qn = 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: 8 Re{Zsn } 1 §0 2n+1 = . 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: ¥nres 2 . = 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: 3 12 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 y

ρ

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 − t 2 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 k0 2 + q2 + (z − zq )2 p ≈ h(1) 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: ei (k0 R1 +/4) 1 1 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 − (r2 + 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 ## −q 0 1 1 + j j –‘0 1

pd1 (–) = * · √ ·e . 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 1 3 ‘1,2 1 −j –‘1,2 ·e · + erfc −j –‘1,2 − * + √ ‘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: 1 −q 0 ej –‘0 1 (−2t+ )n (n + ) · U(n + , n + 1 ; −2j –t+ ) pd6 (–) = 2 * 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: Œ(1 + Œ) d2 RŒ 2 dRŒ 2 (2) + k RŒ = 0 ⇒ RŒ (r) = A · h(1) + − 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

(x) ≡ 0, the value Œ = 0 is the trivial solution for the hard Because of P00 (x) = 1 ; Pm>0 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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Scattering of Sound

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) ; C → ; CŒ −−−−−→ 0. − − − − − − 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) p1 (r, ˜ , œ) = 0 hŒ (k0rq ) jŒ (k0r) − CŒ · h(2) 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 − x2 )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 0 →Œ 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)

‹

1 x 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 > Sect. J.4,“Lined ducts, general”). 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=0

k! (b1)k

zk

(6a)

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 . (7) Œ≈− 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 |vn (s)|2 dA. (2) ¢: = Zr · 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 . p(s)A . vn (s) = const(s) in magnitude and phase: Zr = vn 1 1 vn∗ (s) dA = p · q∗ , p(s) = const(s): ¢= p 2 2

Special case: Special case:

(7) (8)

A

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: Z 1 =¢ |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: k0Z0 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): k0 Z0 p(x, z) = 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 k0Z0 |V(kr )|2 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+j∞

2

|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

(8)

1 Vn = n + v(˜ ) · Pn (cos ˜ ) · sin ˜ d˜ , 2 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 ·

0 · 4a3 Zr0 = −−−−−→ 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 km > k02 .

A slow mode in the axial direction:

The modal radiation impedances then are:

⎤ ⎡ 2 − k2 a k K n+1 0 m n k0 ⎦ ⎣

− Zm,n (a) = j 0 c0 2 − k2 2 − k2 2 − k2 km a k a K k m 0 0 n 0 m

(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: v ; 0 ≤ ˜ < ˜0 v(˜ ) = 0 0 ; ˜0 < ˜ ≤ . Modal velocity amplitudes at r = a: 1 Vn = (n + 1/2) · v0

Pn (x) dx = cos ˜0

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

ƒm sin (nœ0 ) 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: S1 (2k0a) J1 (2k0a) Zr +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 x4 x6 x2 − + − +...; = 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 ): Zr 2 =1− Z0 x

2 · sin (x − /4) x

;

Zr 4 2 1− · sin (x + /4) . = 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: ∞ −8j k0 a Zr 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: ∞ (k0 a)2m Zr · 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

Zr 2 = 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 −(k a) 0 c1 = (k0a)2 /2 ; cn = · c ; 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 J1 (2k0a) Zr − (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 sin k0 b cos k0b − 1 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 Ci(k0a x2 + 1/2) + 2 Ci(k0b x2 + 2) · (1 − x) dx, I1 (k0 a , ) = 0

1 1 Si(k0a x2 + 1/2) + 2 Si(k0b x2 + 2) · (1 − x) dx. I2 (k0 a , ) =

(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) + √ √ 2 Zr · 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: 2 Zr 2 cos (k0a − /4) cos (k0b − /4) − =1− + [1 − cos k0a − cos k0b] 3/2 3/2 Z0 (k0a) (k0b) k02 ab (27) 2 2 3/2 √ 9 2 sin (k0 a − /4) sin (k0b − /4) 2 (a + b ) − + + 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) √ 9 2 cos (k0a − /4) cos (k0b − /4) 2 (a2 + b2 )3/2 − + + 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

Radiation of Sound

According to > Sect. F.1, because of

|v| = const,

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

;

k 2 = 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):

(ka)2 (ka)4 (ka)2 (ka)4 Zr + 1− + = k0 a 1 − Z0 6 64 3 45 2 4 4 (ka) (ka) 1 (ka) − + 1 − (ka)2 + , − 2 16 192 9

(8)

Zr (ka)2 (ka)4 (ka)2 (ka)4 ka 2 k0 a 1− + − 1− + ln +c = Z0 9 225 3 45 2

(ka)2 (ka)4 (ka)4 1 3(ka)2 + + (ka)2 − + 1 − (ka)2 + · 1− 4 64 4 128 9

2 4

2 4 (ka) 1 5(ka) 10(ka) ka 1 (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: C Zr = 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

sin2n Ÿ · b2n+1 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 B= · ln 2 · (−1) 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): k0 a · k0 b Zr = 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 2j Zr = 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 2j Zr ⎥ ⎢ = 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 = k0 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

−−−−−−! k0 a1

3 2

0 V

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 `

Object

a 0:785 = =4 < `=a 8=3 = 0:85

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 4 + 2 16 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=)

a = small b = large half axis = a=b < 1

K(1 − 2 )

=2 4 sin(x cos ) sin2 d ”0 (x) =

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

l 2 1 − (1 + 2 )3=2 = + a 3 2 p 2 1 ln + 1 + 2 2b +

1 p 1 + 1 + 2 + ln

` 1 1 sin n˜ 2 ln(k0 a) − = 2a˜ n n˜ 2

2a

Slit on a cylinder

n1

2aø

2ϑ

1 1 1 + cot ln tg 2 4 2 4 1 ; 0 Š 2 n a/L 0 n −1 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 a π

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 a k0 a/L Zsh + 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: " " # # Zb Zb0 Zb Zb0 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: a Za b =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 dA dA1 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 Zs 8 k 0 r0 2 2 2 (k0r0 ) + j (1 + )(1 − E()) + (1 − )K() ≈ A0 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: 1 J1 (k0 r1 sin ˜ ) r1 2 J1 (k0 r0 sin ˜ ) −2 · e−j k0 r . p(r, ˜ ) ≈ − a Z0 k0 r1 2 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) or = (1) Directivity factor: D0 (˜ , œ) = 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=

1 j Œk0 d sin ‚ sin (n) 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, . . .

with values at the maxima:

(6) DŒ =

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 3 D= 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: V =

1 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 = V0 + V1 + V2 + . . . + 2 3 n+1 (20) J1 () J2 () Jn+1 () −1 n+1 + 2 · 1 ! · V1 2 + . . . + 2 · n ! · Vn n+1 · 2V0 ; = k0a sin ‚. For a velocity distribution

V() = V0 · J0 (kB )

(21)

with the bending wave number kB on the radiator: D=

1 kB a 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

j p(r) . 1− v = vr = Z0 k0 r

1 0 2 2 . |q| (k0 r) + w= (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 1− · cos ˜ . p(r) = j k0Z0 · q · g(r| d) − g(r| − d) = −k0 Z0 D 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 |D| . (17) 12 0 Reactive energy outside the radius r: E = |D|2. (18) 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 Maximum velocity: Ud = 1 + j k0 a − (k0a) . (19) e 2a3 2 j k0Z0 · D −j k0 a Driving force: Fd = e 1 − j k0 a . (20) 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 e−j k0 r x 2 3x2 − r2 j + . − 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): +∞

k0Z0 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