Noise Reduction Analysis (Physics)

NOISE R EDUCTION A NALYSIS Uno Ingard P H Y S I C S S E R I E S NOISE REDUCTION ANALYSIS NOISE REDUCTIO...

Author: Uno Ingard

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NOISE R EDUCTION A NALYSIS Uno Ingard

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NOISE REDUCTION ANALYSIS

NOISE REDUCTION ANALYSIS Uno Ingard

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Jones and Bartlett’s books and products are available through most bookstores and online booksellers. To contact Jones and Bartlett Publishers directly, call 800-832-0034, fax 978-443-8000, or visit our website www.jbpub.com. Substantial discounts on bulk quantities of Jones and Bartlett’s publications are available to corporations, professional associations, and other qualified organizations. For details and specific discount information, contact the special sales department at Jones and Bartlett via the above contact information or send an email to [email protected]. Copyright © 2010 by Jones and Bartlett Publishers, LLC All rights reserved. No part of the material protected by this copyright may be reproduced or utilized in any form, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owner. The publisher recognizes and respects all marks used by companies, manufacturers, and developers as a means to distinguish their products. All brand names and product names mentioned in this book are trademarked or service marks of their respective companies. Any omission or misuse (of any kind) of service marks or trademarks, etc., is not an attempt to infringe on the property of others. ISBN: 978-1-934015-31-5 ISBN: 978-0-7637-8289-4 (e) Cover Design: Tyler Creative Library of Congress Cataloging-in-Publication Data Ingard, K. Uno. Noise reduction analysis / Uno Ingard. p. cm. ISBN 978-1-934015-31-5 (hardcover) 1. Noise control. I. Title. TD892.I56 2010 620.2’3–dc22 2009000229

6048 8875 Printed in the United States of America 13 12 11 10 09 10 9 8 7 6 5 4 3 2 1

Preface This book is based on notes resulting from my sporadic involvement in a variety of projects on noise and vibration and from lectures given in the department of Aeronautics and Astronautics at M. I. T. In principle, the most obvious and effective way of noise reduction is of course to eliminate the sources of the noise, providing that they can be identified and that their removal does not adversely affect the function of the machine or facility involved. The sources are of many different kinds and the noise reducing measures may be quite different. An example comes to mind from direct personal involvement. A power plant was forced to shut down because it emitted a very powerful howling tone which was deemed environmentally unacceptable. The source of the noise was traced to a number of solid bars which had been installed between opposing walls of the exaust stack to stabilize the stack. It so happened that the Aeolian tone from the wakes of the bars happened to coincide with the first acoustic cross mode of the duct and that led to a feedback instability and a very powerful excitation of this mode. Actually, the process was a little more complicated as the howling was periodic with a period of a few seconds. The cross mode became so intense that the related flow instability apparently increased the overall flow resistance of the duct, which in turn reduced the flow speed. Because of the flow dependence of the frequency, this led to a reduction of the frequency of the Aeolian tone from the rods and thus a removal of the coincidence with the acoustic cross mode frequency of the duct. This broke the feedback process and reduced the overall flow resistance of the rods resulting in an increase of the duct flow. In this manner the process was repeated periodically with a corresponding very intense periodic ‘howling.’ The problem to eliminate the noise in this case simply involved cutting down the rods. The stabilizing of the walls, if necessary, could be done externally. Often, as in this case, the mechanism of noise generation is unique to each particular facility. On the other hand, noise reduction by means of absorbers and attenuators is less dependent on the sources involved and lends itself to a treatment as given here. Discussions with students and colleagues at M. I. T. and engineers in industry have been very helpful, and they are gratefully acknowledged. Special thanks go to my former students at M. I. T., Dr. William Patrick, United Technologies, and Dr. George Maling, Du Pont, and to Terry Dear, Du Pont, for many helpful discussions and

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vi comments on a variety of interesting problems in noise control and for providing many of the references assembled in the lists at the end of the book. A grant from the Du Pont Company to the M. I. T. Aeronautics Department to support work in acoustics is also gratefully acknowledged. Uno Ingard, Professor Emeritus of Physics and of Aeronautics, M. I. T., October 2008.

Contents Preface

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I Absorbers 1 Introduction 1.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Terminology and Notation . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Chapter Organization . . . . . . . . . . . . . . . . . . . . . 2 Sound Absorption Mechanisms 2.1 Brief Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Steady Flow Through a (Narrow) Channel . . . . . . . . . . . . . . 2.2.1 Flow Resistance . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Acoustic Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The Viscous Boundary Layer . . . . . . . . . . . . . . . . . 2.3.2 The Thermal Boundary Layer . . . . . . . . . . . . . . . . . 2.3.3 Power Dissipation, Visco-Thermal Boundary Layer . . . . . 2.4 Sound Propagation in a Narrow Channel . . . . . . . . . . . . . . . 2.4.1 Propagation Constant . . . . . . . . . . . . . . . . . . . . . 2.4.2 Velocity and Temperature Profiles . . . . . . . . . . . . . . 2.4.3 Effect of Internal Damping of Flexible Wall . . . . . . . . . 2.4.4 Relaxation Times and a Note on Complex Compressibility in a Channel . . . . . . . . . . . . . . . . . 2.5 Impedances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Impedance Per Unit Length . . . . . . . . . . . . . . . . . . 2.5.2 Complex Density and Wave Impedance . . . . . . . . . . . 2.5.3 Perforated Plate . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Wire Mesh Screen . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Perforated Plate-Screen Combination, Laminates . . . . . . 2.5.6 Effect of Acoustically Induced Motion . . . . . . . . . . . . 2.5.7 A Note on the Interpretation of Steady Flow Resistance Data 2.6 Visco-Thermal Admittance and Absorption Coefficient of a Rigid Wall 2.6.1 Equivalent Admittance . . . . . . . . . . . . . . . . . . . . 2.6.2 Absorption Coefficient . . . . . . . . . . . . . . . . . . . . . vii

3 3 4 6 7 7 9 9 10 10 12 13 14 15 18 19 19 21 22 23 24 28 29 30 30 31 31 33

viii 2.7

Mathematical Supplement . . . . . . . . . . . . . . . 2.7.1 Steady Flow Through a Narrow Channel . . . 2.7.2 Oscillatory Flow and Viscous Boundary Layer 2.7.3 The Thermal Boundary Layer . . . . . . . . . 2.7.4 Power Dissipation in the Boundary Layer . . 2.7.5 Sound Propagation in a Narrow Channel . . . 2.7.6 Impedances . . . . . . . . . . . . . . . . . .

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53 53 53 55 55 56 56 57 59 62 62 63 66 66 68 69 70 77 77 80 82 83 83 85 87 87 90 94 98 99

4 Resonators 4.1 Introduction and Summary . . . . . . . . . . . . . . 4.2 Absorption and Scattering . . . . . . . . . . . . . . . 4.2.1 Q-Value . . . . . . . . . . . . . . . . . . . . . 4.2.2 Helmholtz Resonator . . . . . . . . . . . . . 4.2.3 Resonator Absorber in a Diffuse Sound Field 4.2.4 Two-Dimensional Arrays of Resonators . . . .

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105 105 106 109 109 110 112

3 Sheet Absorbers 3.1 Introduction and Brief Summary . . . . . . . . . 3.1.1 Single Sheet Surface Absorber . . . . . . 3.1.2 Multisheet Absorber . . . . . . . . . . . . 3.1.3 Single Sheet as a ‘Volume’ Absorber . . . 3.2 Rigid Single Sheet with Cavity Backing . . . . . . 3.2.1 Flow Resistance and Impedances . . . . . 3.2.2 Resonances and Anti-Resonances . . . . . 3.2.3 Absorption Spectra . . . . . . . . . . . . . 3.2.4 Wire Screens . . . . . . . . . . . . . . . . 3.2.5 Effect of Honeycomb Cell Size . . . . . . 3.2.6 Examples and Comments . . . . . . . . . 3.3 Flexible Porous Sheet with Cavity Backing . . . . 3.3.1 The ‘Equivalent’ Impedance . . . . . . . 3.3.2 A Low Frequency Resonance . . . . . . . 3.3.3 Absorption Spectra . . . . . . . . . . . . . 3.3.4 Examples and Comments . . . . . . . . . 3.4 Lattice Absorbers . . . . . . . . . . . . . . . . . . 3.4.1 Periodic Lattice . . . . . . . . . . . . . . 3.4.2 Nonperiodic Lattice . . . . . . . . . . . . 3.5 ‘Volume’ Absorbers . . . . . . . . . . . . . . . . . 3.5.1 Reflection, Transmission, and Absorption . 3.5.2 Absorption Spectra, Infinite Sheet . . . . 3.5.3 Finite Sheet, Effect of Diffraction . . . . 3.6 Mathematical Supplement . . . . . . . . . . . . . 3.6.1 Rigid Single Sheet Cavity Absorber . . . . 3.6.2 Flexible Sheet Cavity Absorber . . . . . . 3.6.3 Uniform (Periodic) Lattice . . . . . . . . 3.6.4 Nonuniform Lattice . . . . . . . . . . . . 3.6.5 Sheet as a Volume Absorber . . . . . . . .

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4.2.5 Three-Dimensional Lattice of Resonators . . . . . . . 4.2.6 Transient Response and Reverberation . . . . . . . . . Acoustic Nonlinearity . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Perforated Plate With (Porous) Cavity Backing . . . . . 4.3.2 Nonlinear Absorption Characteristics . . . . . . . . . . Effects of Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Flow Induced Acoustic Resistance . . . . . . . . . . . 4.4.2 Flow Excitation of Pipes and Orifices . . . . . . . . . . 4.4.3 Resonator in Free Field With Grazing Flow . . . . . . 4.4.4 Flow Excitation of a Side-Branch Resonator in a Duct . Mathematical Supplement . . . . . . . . . . . . . . . . . . . 4.5.1 Impedance of a Tube Resonator . . . . . . . . . . . . . 4.5.2 Absorption and Scattering Cross Sections . . . . . . . 4.5.3 Helmholtz Resonator . . . . . . . . . . . . . . . . . . 4.5.4 Three-Dimensional Array of Resonators . . . . . . . . 4.5.5 Acoustic Nonlinearity, Perforated Plate . . . . . . . . .

5 Rigid Porous Materials 5.1 Introduction and Summary . . . . . . . . . . . . . . 5.2 The Slot Absorber . . . . . . . . . . . . . . . . . . . 5.2.1 Input Impedance, Absorption Spectra . . . . 5.3 Isotropic Porous Layer, Physical Parameters . . . . . 5.3.1 Porosity . . . . . . . . . . . . . . . . . . . . . 5.3.2 Flow Resistance and Impedance . . . . . . . 5.3.3 Structure Factor . . . . . . . . . . . . . . . . 5.3.4 Mass Density of a Porous Material . . . . . . 5.3.5 Compressibility . . . . . . . . . . . . . . . . . 5.3.6 Discussion . . . . . . . . . . . . . . . . . . . 5.4 Wave Motion . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Propagation Constant . . . . . . . . . . . . . 5.4.2 Penetration Depth . . . . . . . . . . . . . . . 5.5 Absorption Spectra . . . . . . . . . . . . . . . . . . . 5.5.1 Infinite Layer . . . . . . . . . . . . . . . . . . 5.5.2 Finite Layer . . . . . . . . . . . . . . . . . . 5.5.3 Examples and Comments . . . . . . . . . . . 5.5.4 Effect of a Perforated Facing, Its Nonlinearity and Induced Motion . . . . . . . . . . . . . . 5.5.5 Effect of a Screen on a Porous Layer . . . . . 5.5.6 Nonuniform Porous Absorbers . . . . . . . . 5.5.7 Sheet Absorbers vs Uniform Porous Layer . . 5.6 Effect of Refraction in Grazing Flow . . . . . . . . . 5.6.1 View Angle vs Emission Angle . . . . . . . . . 5.6.2 The Boundary Layer . . . . . . . . . . . . . . 5.6.3 Effect on Absorption . . . . . . . . . . . . . . 5.6.4 Region of Total Reflection . . . . . . . . . . . 5.7 Mathematical Supplement . . . . . . . . . . . . . . .

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113 115 116 117 120 122 122 123 127 128 132 132 133 136 137 139

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166 167 169 171 173 174 175 178 179 181

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Slot Absorber . . . . . . . . . . . . . . . . . . . . . . . . Isotropic Porous Layer . . . . . . . . . . . . . . . . . . . Interaction Impedance, Impedance Per Unit Length, and Complex Density . . . . . . . . . . . . . . . . . . . 5.7.4 Propagation Constant and Wave Impedance . . . . . . . 5.7.5 Angle of Refraction . . . . . . . . . . . . . . . . . . . . 5.7.6 Input Impedance and Admittance, Absorption Coefficient . . . . . . . . . . . . . . . . . . . 5.7.7 Perforated Facing, Its Nonlinearity and Induced Motion 5.7.8 Anisotropic Layer . . . . . . . . . . . . . . . . . . . . . 5.7.9 Effect of Grazing Flow . . . . . . . . . . . . . . . . . . 5.7.10 Computational Considerations . . . . . . . . . . . . . . 6 Flexible Porous Materials 6.1 Introduction and Summary . . . . . . . . . . . 6.2 Coupled Waves . . . . . . . . . . . . . . . . . . 6.3 Dispersion Relation . . . . . . . . . . . . . . . 6.4 Field Distributions . . . . . . . . . . . . . . . . 6.4.1 Pressure and Velocity Fields . . . . . . . 6.4.2 Dissipation Function . . . . . . . . . . . 6.4.3 Examples and Comments . . . . . . . . 6.5 Absorption Spectra . . . . . . . . . . . . . . . . 6.5.1 General Comments . . . . . . . . . . . 6.5.2 Absorption Peaks But Not at Resonances 6.5.3 Effect of a Bonded Perforated Facing . 6.5.4 Examples and Comments . . . . . . . . 6.5.5 Porous Material with Closed Cells . . . 6.6 Nonlinear Effects and Shock Wave Reflection . 6.6.1 Apparatus . . . . . . . . . . . . . . . . . 6.6.2 Amplitude Dependence of Wave Speed 6.6.3 Reflection From a Flexible Porous layer 6.7 Measurement of Complex Elastic Modulus . . . 6.7.1 Apparatus . . . . . . . . . . . . . . . . . 6.7.2 Data Analysis . . . . . . . . . . . . . . . 6.8 Mathematical Supplement . . . . . . . . . . . . 6.8.1 Limp Material . . . . . . . . . . . . . . 6.8.2 Equations for Coupled Waves . . . . . . 6.8.3 Pressure and Velocity Fields . . . . . . . 6.8.4 Absorption Coefficients . . . . . . . . .

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197 197 198 199 201 201 203 204 204 204 205 206 208 210 211 212 214 214 217 217 218 219 219 221 225 228

II Duct Attenuators 7 Duct Acoustics 235 7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 7.2 Wave Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

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7.4 7.5 7.6 7.7

7.2.1 Simple Illustration . . . . . . . . . . . . . . . . Measures of Silencer Performance . . . . . . . . . . . 7.3.1 Attenuation . . . . . . . . . . . . . . . . . . . . 7.3.2 Transmission Loss, TL and TL0 . . . . . . . . . 7.3.3 Insertion Loss, IL . . . . . . . . . . . . . . . . 7.3.4 Noise Reduction, NR . . . . . . . . . . . . . . 7.3.5 Numerical Examples . . . . . . . . . . . . . . . 7.3.6 Pressure Drop and Flow Noise (Self-Noise, SN) Lined Ducts . . . . . . . . . . . . . . . . . . . . . . . . ‘Reactive’ Silencers . . . . . . . . . . . . . . . . . . . . Acoustically Equivalent Silencers . . . . . . . . . . . . Additional Comments on Silencer Testing . . . . . . .

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8 Lined Ducts 8.1 Attenuation Mechanisms . . . . . . . . . . . . . . . . . . 8.1.1 Dissipation in Duct Liners . . . . . . . . . . . . . 8.1.2 Interference . . . . . . . . . . . . . . . . . . . . 8.2 Rectangular Ducts . . . . . . . . . . . . . . . . . . . . . 8.2.1 Locally Reacting Liner . . . . . . . . . . . . . . . 8.2.2 Nonlocally Reacting Liner . . . . . . . . . . . . . 8.2.3 Locally vs Nonlocally Reacting Liner, An Example 8.2.4 Attenuation vs Flow Resistance of Liner . . . . . 8.2.5 Example: A Proposed Air Intake Silencer for an Automobile . . . . . . . . . . . . . . . . . 8.3 Additional Duct Shapes . . . . . . . . . . . . . . . . . . 8.3.1 Rectangular Duct With All Sides Lined . . . . . . 8.3.2 Circular Duct . . . . . . . . . . . . . . . . . . . . 8.3.3 A Comparison, Circular vs Square Lined Duct . . 8.3.4 Annular Duct . . . . . . . . . . . . . . . . . . . . 8.4 Ducts in Series and in Parallel . . . . . . . . . . . . . . . 8.4.1 Ducts in Series . . . . . . . . . . . . . . . . . . . 8.4.2 Parallel Ducts, Interference Filter . . . . . . . . 8.5 Duct Liner Configurations . . . . . . . . . . . . . . . . . 8.5.1 Effects of a Perforated Facing . . . . . . . . . . . 8.5.2 Effect of Duct Liner Flexibility . . . . . . . . . . 8.5.3 Multilayer Liners . . . . . . . . . . . . . . . . . . 8.5.4 Slotted Liner . . . . . . . . . . . . . . . . . . . . 8.5.5 Effect of Partition Spacing . . . . . . . . . . . . . 8.6 Effects of Higher Modes and Flow . . . . . . . . . . . . 8.6.1 Higher Modes . . . . . . . . . . . . . . . . . . . 8.6.2 Convection . . . . . . . . . . . . . . . . . . . . . 8.6.3 Refraction . . . . . . . . . . . . . . . . . . . . . 8.6.4 Scaling Laws . . . . . . . . . . . . . . . . . . . . 8.6.5 Static Pressure Drop in Ducts . . . . . . . . . . . 8.7 Liquid Pipe lines, Elementary Aspects . . . . . . . . . . 8.7.1 Liquid Pipe Line with Slightly Compliant Walls .

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238 240 240 241 244 246 246 248 249 251 251 252

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Liquid Pipe Line With Air Layer Wall Treatment . . . . . . 295

9 Reactive Duct Elements 9.1 Uniform Duct Section . . . . . . . . . . . . . . . . . . . 9.1.1 Role of Source Impedance . . . . . . . . . . . . 9.2 Expansion Chamber . . . . . . . . . . . . . . . . . . . . 9.2.1 Transmission Loss . . . . . . . . . . . . . . . . . 9.2.2 Insertion Loss . . . . . . . . . . . . . . . . . . . 9.3 ‘Contraction’ Chamber . . . . . . . . . . . . . . . . . . . 9.3.1 Transmission Loss . . . . . . . . . . . . . . . . . 9.3.2 Insertion Loss . . . . . . . . . . . . . . . . . . . 9.4 Side-Branch Resonator in a Duct . . . . . . . . . . . . . 9.4.1 Transmission Loss . . . . . . . . . . . . . . . . . 9.4.2 Insertion Loss . . . . . . . . . . . . . . . . . . . 9.5 Perforated Plate . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Effect of Mean Flow On the Acoustic Resistance 9.5.2 Shock Wave Interaction With an Orifice Plate . . 9.6 Attenuation in Turbulent Flow in Ducts . . . . . . . . . . 9.6.1 Static Pressure Drop . . . . . . . . . . . . . . . . 9.6.2 Sound Attenuation . . . . . . . . . . . . . . . . . 9.6.3 A Proposed AeroAcoustic Instability . . . . . . . 9.7 Nonlinear Attenuation . . . . . . . . . . . . . . . . . . . 9.8 On Air Induction Acoustics . . . . . . . . . . . . . . . . 9.8.1 Sound Pressure and Radiated Power . . . . . . . 9.8.2 Pipe Impedance . . . . . . . . . . . . . . . . . . 9.8.3 Radiated Power . . . . . . . . . . . . . . . . . . 9.8.4 Acoustic ‘Supercharge’ . . . . . . . . . . . . . . . 9.8.5 Numerical Example . . . . . . . . . . . . . . . .

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10 Mathematical Supplements and Comments 10.1 Supplement to Section 8.1 . . . . . . . . . . . . . . . . . . . . . 10.1.1 High Frequency Attenuation of Fundamental Mode in Lined Duct, Average Compressibility . . . . . . . . . 10.2 Supplement to Section 8.2 . . . . . . . . . . . . . . . . . . . . . 10.2.1 Locally Reacting Liners . . . . . . . . . . . . . . . . . . 10.2.2 Nonlocally Reacting Liner . . . . . . . . . . . . . . . . . 10.3 Supplement to Section 8.3, Other Duct Types . . . . . . . . . . 10.3.1 Rectangular Duct Lined On All Sides . . . . . . . . . . . 10.3.2 Circular Duct . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Annular Duct . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Supplement to Section 8.6, Higher Modes and Flow . . . . . . . 10.5 Supplement to Section 8.7, Liquid Pipe Lines . . . . . . . . . . 10.5.1 Liquid Pipe Line With Slightly Compliant Walls . . . . . 10.5.2 Liquid Pipe Line With Air Layer Wall Treatment . . . . 10.6 Supplement to Section 9.1, Uniform Duct . . . . . . . . . . . . 10.7 Supplement to Section 9.6, Attenuation in Turbulent Duct Flow

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299 299 300 303 304 304 305 305 306 307 308 309 312 312 316 317 318 318 319 319 320 321 327 328 328 329

331 . . 331 . . . . . . . . . . . . . .

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331 332 332 337 341 341 342 344 346 349 349 352 356 357

xiii 10.7.1 Friction Factor in Turbulent Duct Flow . . . . . . . . . . . 357 10.7.2 Acoustic Perturbations and Dispersion Relation . . . . . . . 358 10.7.3 A Comparison With Visco-Thermal Attenuation . . . . . . . 359 A Transmission Matrices A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . A.1.1 Choice of Variables . . . . . . . . . . . . . . . . A.2 Application of Matrices . . . . . . . . . . . . . . . . . . A.2.1 Impedance . . . . . . . . . . . . . . . . . . . . A.2.2 Reflection and Absorption Coefficients . . . . . A.2.3 Transmission Coefficient and Transmission Loss A.2.4 Insertion Loss . . . . . . . . . . . . . . . . . . A.2.5 Noise Reduction . . . . . . . . . . . . . . . . . A.3 Commonly Used Matrices . . . . . . . . . . . . . . . . A.3.1 Porous Screen . . . . . . . . . . . . . . . . . . A.3.2 Area Discontinuities . . . . . . . . . . . . . . . A.3.3 Duct Element . . . . . . . . . . . . . . . . . . A.3.4 Contracted Duct Section, Perforated Plate . . . A.3.5 ‘Expansion Chamber’ and Elbow . . . . . . . . A.3.6 Lined Duct . . . . . . . . . . . . . . . . . . . . A.3.7 Side-Branch Tube . . . . . . . . . . . . . . . . A.3.8 Side-Branch Helmholtz Resonator . . . . . . . A.3.9 Parallel Channels . . . . . . . . . . . . . . . . . A.3.10 Rigid Porous Layers . . . . . . . . . . . . . . . A.3.11 Flexible Layer . . . . . . . . . . . . . . . . . . A.3.12 Thin Porous Plate . . . . . . . . . . . . . . . .

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361 361 362 362 362 363 364 367 369 370 370 372 375 376 377 378 379 380 381 383 383 387

B Flow Resistance Measurements B.1 Simple Method for Steady Flow . . . . . B.1.1 Equations of Motion . . . . . . . B.1.2 Nonlinearity of Flow Resistance . B.2 Simple Method for Oscillatory Flow . . . B.2.1 Some Experimental Results . . . B.2.2 Other Materials . . . . . . . . . B.2.3 A Supplementary Note . . . . .

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389 389 392 393 395 398 400 401

C Historical Notes and References, Absorbers C.1 ‘The Absorption Coefficient Problem’ . . . . . . . C.1.1 Sound Absorption in Porous Materials . . C.1.2 Regarding the Lists of Publications . . . . C.2 Lists of References . . . . . . . . . . . . . . . . . C.2.1 Sound Absorption, Concepts and Analysis C.2.2 Measurements, Methods, and Data . . . . C.2.3 Anechoic Wedges and Rooms . . . . . . . C.2.4 Resonators and Related Matters . . . . . C.2.5 ‘Functional’ or ‘Volume’ Absorbers . . . .

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403 403 404 406 407 407 412 420 424 426

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xiv D Historical Notes and References, Ducts 429 D.1 Brief Historical Note . . . . . . . . . . . . . . . . . . . . . . . . . . 429 D.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 Index

439

List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22

The Normalized Propagation Constant for Waves in a (Narrow) Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity and Temperature Distribution in a Narrow Channel . . . Complex Compressibility . . . . . . . . . . . . . . . . . . . . . . . Impedance Per Unit Length in a Channel . . . . . . . . . . . . . . Example of ‘Micro-Perforate’ Characteristics . . . . . . . . . . . . Acoustic Impedances of Perforated Plate and Screen . . . . . . . . Absorption Characteristics, Perforated Plate-Screen Combination Absorption Coefficient of A Rigid Wall Due to Visco-Thermal Losses Function F Used in Propagation in Narrow Channels . . . . . . . Single Porous Sheet Cavity Absorber . . . . . . . . . . . . . . . . ‘Universal’ Rigid Sheet Cavity Absorption Spectra . . . . . . . . . 1/3 and 1/1 Octave Band Absorption Spectra . . . . . . . . . . . . Absorption Spectra, Discussion of Partition Spacing . . . . . . . . Rigid Sheet/Cavity Absorber, Optimum NRC . . . . . . . . . . . . Nonlinearity of Cavity Absorber . . . . . . . . . . . . . . . . . . . Equivalent Impedance of a Limp, Porous Sheet . . . . . . . . . . Limp Porous Sheet Cavity Normal Incidence Absorption Spectra . Limp Porous Sheet Cavity Diffuse Field Absorption Spectra . . . Absorption Spectra for a Perforated Plate/Screen Cavity Absorber Low Frequency Limp Sheet Cavity Resonance . . . . . . . . . . . Sheet Absorber, Alpha vs Resistance . . . . . . . . . . . . . . . . . Periodic Lattice Absorber . . . . . . . . . . . . . . . . . . . . . . Absorption Spectra of a Uniform Sheet Lattice . . . . . . . . . . . Absorption Spectra of Nonuniform Lattice of Limp Sheets . . . . Absorption Spectra of Nonuniform Lattice of Sheet Absorbers . . ‘Surface’ and ‘Volume’ Absorbers . . . . . . . . . . . . . . . . . . Reflection, Transmission, and Absorption: Plane Wave Incident on a Thin Porous Sheet . . . . . . . . . . . . . . . . . . . . . . . . . Coefficients of Reflection, Transmission, and Absorption . . . . . . Absorption Area or Cross Section of a Sheet in a Diffuse Field . . Effect of Diffraction on Sound Absorption by a Sheet . . . . . . . Two Different Unit Cells in Periodic Lattice . . . . . . . . . . . . xv

16 18 21 22 26 27 30 33 45 54 55 61 62 64 65 67 70 71 72 75 76 78 79 80 81 82 82 84 85 87 94

xvi 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15

Radiation Impedance of a Resonator in a Wall . . . . . . . . . Absorption and Scattering Cross Sections . . . . . . . . . . . . Helmholtz Resonator . . . . . . . . . . . . . . . . . . . . . . . Propagation Constant in Three-Dimensional Resonator Array . Experimental Demonstration of Transient Resonator Response Acoustically Driven Vortex Rings . . . . . . . . . . . . . . . . Nonlinear Orifice Resistance . . . . . . . . . . . . . . . . . . . Nonlinear Absorption, Perforated Plate Resonator . . . . . . . Nonlinear Absorption Characteristics of Resonator . . . . . . . Orifice Whistle . . . . . . . . . . . . . . . . . . . . . . . . . . Tone Generation in Industrial Driers . . . . . . . . . . . . . . Stability Diagram of a Flow Excited Resonator . . . . . . . . . Flow Excited Resonances of a Side-Branch Cavity in a Duct . . Mode Coupling in Flow Excited Resonators . . . . . . . . . . Flow Excitation of a Slanted Resonator in a Duct . . . . . . . .

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14

Maximum Possible Absorption Coefficient of a Rigid Porous Layer The ‘Slot Absorber’ . . . . . . . . . . . . . . . . . . . . . . . . . . Input Impedance of a Slot Absorber . . . . . . . . . . . . . . . . . Normal Incidence Absorption Spectra of a Slot Absorbers . . . . . Influence of Porosity on Sound Absorption of a Slot Absorber . . . Effect of Heat Conduction on the Absorption in a Slot Absorber . Angle of Refraction in a Slot Absorber . . . . . . . . . . . . . . . . Diffuse Field Absorption Characteristics of a Slot Absorber . . . . Propagation Constant in a Rigid Porous Material . . . . . . . . . . Penetration Depth of Sound in a Porous Layer . . . . . . . . . . . Absorption Spectra of a Locally Reacting Rigid Porous Layer . . . Angular Dependence of Absorption Coefficient . . . . . . . . . . Absorption Characteristics of a Locally Reacting Rigid Porous Layer Absorption Characteristics of a Nonlocally Reacting Rigid Porous Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Absorption Coefficient of Porous Layer vs Flow Resistance . . . . Effect of a Perforated Facing on Absorption Characteristics . . . . Effect of Resistive Screen on a Porous Layer . . . . . . . . . . . . Absorption Characteristics, Three Porous Layers . . . . . . . . . . Comparison, Uniform Porous Layer with Nonuniform . . . . . . . Porous Layer with Air Backing, a Comparison of Absorption Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Comparison of Sheet Absorbers and Rigid Porous Layers . . . . View Angle vs Emission Angle in a Wind Tunnel . . . . . . . . . . Refraction at a Flow Boundary . . . . . . . . . . . . . . . . . . . . Vortex Sheet Model of Boundary Layer . . . . . . . . . . . . . . . Effect of Boundary Layer Flow on Sound Absorption . . . . . . . Effect of Refraction on Sound Absorption for Stiffness Reactive Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Region of Total Reflection in a Wind Tunnel . . . . . . . . . . . .

5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23 5.24 5.25 5.26 5.27

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106 108 110 114 115 117 119 120 121 124 126 128 129 130 131 145 146 149 150 151 152 153 154 160 160 161 162 163 164 165 167 168 169 170 171 172 175 176 177 178 180 181

xvii 5.28 5.29

Polar and Azimuthal Angles, Slot Absorber . . . . . . . . . . . . . 184 Angle of Refraction in a Porous Material . . . . . . . . . . . . . . 189

6.1 6.2 6.3 6.4 6.5 6.6 6.7

Dispersion Relation, Flexible Porous Material . . . . . . . . . Velocity Distributions in Flexible Porous Layer . . . . . . . . . Pressure Amplitude Distributions in a Flexible Porous Layer . Dissipation Functions in a Flexible Porous Layer . . . . . . . . Absorption Spectra of a Flexible Porous Layer . . . . . . . . . Effect of Perforated Plate Bonded to a Flexible Porous Layer . Absorption Spectra of a Flexible Porous Layer with Perforated Facing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flexible Porous Layer with Perforated/Screen Facing . . . . . Shock Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . Amplitude Dependence on Wave Speed . . . . . . . . . . . . Shock Wave Reflection Coefficients . . . . . . . . . . . . . . . Shock Wave Reflections from Flexible Porous Layers . . . . . Compression of Flexible Layer by a Shock Wave . . . . . . . . Apparatus for Measuring Compliance of a Porous Material . .

6.8 6.9 6.10 6.11 6.12 6.13 6.14 7.1 7.2 7.3 7.4 7.5 7.6 7.7 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15

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200 202 203 204 205 207

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208 210 212 214 215 216 216 218

Generation of a Higher Acoustic Mode in a Duct . . . . . . . . . . Measurement of Insertion Loss . . . . . . . . . . . . . . . . . . . Comparion of TL, TL0, IL, and NR of Lined Duct . . . . . . . . . Attenuation Spectrum . . . . . . . . . . . . . . . . . . . . . . . . Comparison of Insertion Loss and Transmission Loss . . . . . . . Acoustically Equivalent Duct Configurations for the Fundamental Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Comparison of TL, TL0, and NR . . . . . . . . . . . . . . . . .

238 242 243 247 247

Attenuation Spectra of a Rectangular Duct, Local Reaction . . . . Optimum Design of Lined Duct . . . . . . . . . . . . . . . . . . . Attenuation Characteristics of Rectangular Duct, Nonlocally Reacting Liner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Porous Plug Characteristics . . . . . . . . . . . . . . . . . . . . . A High Frequency Muffler for Special Application . . . . . . . . . Locally vs Nonlocally Reacting Duct Liner, an Example . . . . . . Attenuation in a Rectangular Duct vs Flow Resistance/Inch of Liner Proposed Air Induction Muffler for an Automobile . . . . . . . . . Rectangular Duct Lined on all Four Sides . . . . . . . . . . . . . Attenuation in a Rectangular Duct with Four Sides Lined . . . . . Attenuation Spectra of a Square Duct Lined with a Locally Reacting Porous Liner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Circular and Annular Lined Ducts . . . . . . . . . . . . . . . . . . Attenuation Spectra of a Circular Duct with a Locally Reacting Porous Liner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attenuation in a Circular Lined Duct . . . . . . . . . . . . . . . . Attenuation in an Annular Duct . . . . . . . . . . . . . . . . . . .

250 253 258 260 262 263 264 265 266 267 268 268 269 270 271 272 273

xviii 8.16 8.17 8.18 8.19 8.20 8.21 8.22 8.23 8.24 8.25 8.26 8.27 8.28 8.29 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10

Nonuniform Duct . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Open Area of a Perforated Facing on Attenuation . . . . Effect of Nonlinearity and Induced Motion of a Perforated Facing on Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transmission Loss, Duct with Multilayer Liner . . . . . . . . . . . Attenuation in Duct with Slotted Duct Liner . . . . . . . . . . . . Concerning Optimum Partition Spacing in a Duct Liner . . . . . . Geometrical Acoustics and Duct Attenuation . . . . . . . . . . . . Effect of Refraction in a Duct . . . . . . . . . . . . . . . . . . . . Example of the Effect of Flow on Insertion Loss . . . . . . . . . . Effect of Temperature on Attenuation in a Lined Duct . . . . . . Friction Factor in Pipe Flow . . . . . . . . . . . . . . . . . . . . . Propagation in a Liquid Pipe Line . . . . . . . . . . . . . . . . . . Air Layer Attenuator in a Liquid Pipe Line . . . . . . . . . . . . . Transmission Characteristics in a Water Line with Air Layers . . .

277 279 280 281 283 286 286 289 291 295 296 296

9.13 9.14 9.15 9.16 9.17

Insertion Loss and Input Impedance of a Straight Pipe Section . Transmission Loss of an Expansion Chamber . . . . . . . . . . . Insertion Loss of an Expansion Chamber . . . . . . . . . . . . . Transmission Loss of a Contraction Chamber . . . . . . . . . . . Insertion Loss of the Contraction Chamber . . . . . . . . . . . . Side-Branch Resonator in a Duct . . . . . . . . . . . . . . . . . Transmission Loss of a Side-Branch Resonator . . . . . . . . . . Insertion Loss of Side-Branch Tube, High Source Impedance . . Automotive Labyrinth Resonator for Air Induction . . . . . . . . Insertion Loss of a Side-Branch Resonator, Low Source Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Absorption, Reflection, and Transmission Coefficients of a Perforated Plate with Flow . . . . . . . . . . . . . . . . . . Nonlinear Reflection, Transmission, and Absorption Coefficient of Perforated Plate . . . . . . . . . . . . . . . . . . . . . . . . . Shock Wave Reflection from an Orifice Plate . . . . . . . . . . . Nonlinear Attenuation . . . . . . . . . . . . . . . . . . . . . . . Air Induction into a Cylinder of a Combustion Engine . . . . . . Regarding Acoustic Supercharging . . . . . . . . . . . . . . . . Regarding Acoustic Supercharging . . . . . . . . . . . . . . . .

10.1 10.2

Rectangular Duct Lined on Two Sides with Different Liners . . . 332 Liquid Pipe Line with Slightly Compliant Walls . . . . . . . . . . 351

A.1 A.2 A.3 A.4 A.5 A.6

Four-Pole Network . . . . . Acoustic ‘Barrier’ . . . . . . Area Expansion in a Duct . Area Contraction in a Duct . Side-Branch Tube in a Duct Parallel Duct Attenuator . .

9.11 9.12

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275 276

302 303 304 306 307 308 309 310 312

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316 317 320 322 325 326

361 363 372 374 379 381

xix B.1 B.2 B.3 B.4 B.5

Simple Apparatus for Steady Flow Resistance Measurement Flow Speed Dependence on Flow Resistance . . . . . . . . Flow Resistance Apparatus for Oscillatory Flow . . . . . . . Measured Frequency Dependence of Resistance . . . . . . . Flow Resistance and Reactance of Screens . . . . . . . . . .

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390 394 396 399 399

Part I

Absorbers

Chapter 1

Introduction 1.1 GENERAL COMMENTS Normally, noise is not only a nuisance, but it can have more serious adverse effects on both human beings and machines. It is frequently referred to as a form of ‘pollution’ and has been the object of environmental regulations. Noisy facilities, in violation of such regulations, in some cases have been forced to close until compliance is achieved.1 Furthermore, high level noise and vibration often give rise to structural failures as occasionally occur in control valves as a result of flow induced instability (through feedback) and resulting violent oscillations. In practice the efforts to reduce noise often involve a ‘quick fix’ attempt based on engineering ‘common sense’ and on trial and error approaches. If this turns out to be unsatisfactory, however, a ‘basic’ study of the problem is often called for. This involves identifying and trying to understand the physical mechanisms involved in the generation and transmission of sound and vibration and then finding feasible means of noise reduction. This can become a tedious and lengthy effort often because of the constraints that the measures undertaken must not adversely affect the function of the machine or process involved. As the subtitle of this book indicates, it is divided into two parts, noise reduction by means of 1. absorbers and 2. duct attenuators.2 The First Part, Absorbers, starts out with a discussion of basic mechanisms of absorption by which acoustic energy is converted into heat in viscous and thermal boundary layers in a sound field. Analysis of sound propagation through narrow channels is then used as a basis for applications that follow. Thus, extensive treatments are given of the absorption characteristics of porous sheets, both rigid and flexible, of acoustic resonators, and of porous material in bulk, both rigid and flexible. Multilayer absorbers are also analyzed to seek means of improving and ‘shaping’ the absorption characteristics. 1 Occasionally, noise is generated intentionally for the purpose of masking irritating sounds; the masking noise often is referred to as acoustic ‘perfume.’ 2 The term ‘Noise Control’ has not been used as the title of the book since it implies a broader scope including such topics as noise sources, atmospheric acoustics, human response to noise, and considerable emphasis on noise regulations.

3

4


The bulk of these studies deal with ‘ordinary’ conditions, where linear acoustic theory is applicable, but a section is devoted to nonlinear effects including a discussion of some experimental results with shock waves. The Second Part of the book, Duct Attenuators, starts out with a description of how their performance is described, calculated, and measured. We distinguish between ‘dissipative’ and ‘reactive’ ducts. The dissipative ducts are lined with one or more layers of porous material, and the chapters that follow present extensive analyses of the attenuation characteristics of different duct configurations. Of particular interest is the comparison of ducts with locally and nonlocally reacting liners. In the ‘reactive’ duct, reflection plays a major role. In addition to the chapters dealing directly with absorbers and ducts, there are four appendices. The first of these is a fairly substantial discussion of transmission matrices as they are of considerable value in facilitating the analysis of both absorber and duct problems. The second appendix describes simple methods (and apparatus) for measuring the flow resistance of porous materials for both steady and oscillatory flow. The two remaining appendices present brief historical reviews of absorbers and duct attenuators and categorized lists with a fairly large number of references. The present lists could provide a start for anyone who is intent on producing a database on everything published in the fields considered here. No attempts have been made to assign credit for contributions to various aspects; it would have required a considerable additional effort, which was not deemed essential in this context. The main part of each chapter is essentially descriptive and contains numerical results, which should be of direct interest for design work. The mathematical analysis is generally placed in an analytical supplement at the end of each chapter and is included for readers who want to go deeper into the subject matter. To strictly follow this scheme of separation turns out to be awkward at times, and some deviations are to be found here and there. Actually, important final equations are presented in the descriptive part and are framed for emphasis. Usually they are the basis for the numerical results shown in numerous graphs, but can be skipped in a first reading without lack of continuity.

1.2 TERMINOLOGY AND NOTATION This section deals with definitions of acoustically relevant physical quantities and a few commonly used terms. Harmonic motion refers to a time dependence expressed by a periodic function, A cos(ωt − φ), where A is the amplitude, ω the angular frequency, and φ, the phase angle. The angular frequency is ω = 2πf and the frequency is f = 1/T , where T is the period. Frequently, both of these quantities will be referred to simply as frequency when there is no risk for confusion. In defining a complex amplitude, we have used cos(ωt) = {exp(−iωt)} with the time factor exp(−iωt) rather than exp(iωt), so that in going from the time domain to the frequency domain, ∂/∂t → −iω. Sound pressure p(t) is the time dependent variation in pressure caused by compressions and rarefactions of a gas (or whatever material is involved) and the corresponding time dependent velocity of the fluid will be denoted by u(t). Harmonic time

INTRODUCTION

5

dependence is generally assumed and the complex amplitudes of these quantities are p(ω) and u(ω). Unless there is risk for confusion, the argument ω will be omitted. The fluid velocity u should not be confused with the speed of propagation of a sound wave, which will be referred to as the speed of sound, c, or sound speed. The fluid velocity u is sometimes referred to as particle velocity. The material used for sound absorption is usually porous. The difference between a porous sheet and a porous layer in bulk in general is that the sheet has to be very thin in comparison with a wavelength, typically a small fraction of an inch, so that the compression of the material and the air in it can be neglected. The terms screen and sheet will be regarded as synonyms. By an acoustically rigid sheet or porous material in this book is meant that it does not move when it interacts with sound. Normally, this usage would be a bit of a misnomer since any object will be induced to move to some extent by a sound wave. By contrast, a flexible material is mobile (even though the term normally implies that the material can be bent or flexed, which is not necessary for it to be mobile). A limp porous sheet is one without bending stiffness and a limp porous layer in bulk has no stiffness. A rigid wall, which is also impervious to air, will sometimes be called acoustically hard. The impedance is the ratio of the complex amplitudes of pressure and fluid velocity in harmonic time dependence at a given frequency. In some presentations, but not here, the volume velocity (i.e., the product of velocity and the flow area involved) is used as a variable instead of velocity. To distinguish the impedance thus obtained and the impedance used here, our impedance is sometimes referred to as ‘specific’ to indicate that a unit area is involved for the volume flow. We shall use the term impedance only, without the qualifier ‘specific.’ (The distinction between the use of velocity and volume velocity as variables is slightly more than cosmetic when we come to the definition and use of transmission matrices, as discussed in Appendix A.) The notation used here for the impedance is normally a lowercase Roman letter, such as z, with the resistive and reactive (real and imaginary) parts typically denoted by r and x. The impedance in a plane traveling wave is ρc, the wave impedance (resistance),3 which is approximately 420 MKS units (42 CGS) for air at room temperature. In most of the equations in this book the normalized impedance, resistance, and reactance, will be used, normalized with respect to ρc and designated usually by Greek letters, such as ζ , θ , and χ , where ζ = z/ρc, etc. Thus, a quantity ζ , will be referred to simply as ‘impedance’ without the qualifier ‘normalized,’ since it is already implied by the use of Greek letters. The use of ‘rayl’ as a name for a unit of resistance is not used here, CGS and MKS are sufficient. Furthermore, since most of the time we deal with normalized quantities, the question of units is secondary. The absorption coefficient is denoted by α and the pressure reflection coefficient by R. In a diffuse field, where all directions of sound propagation are equally probable, approximately realized in a large reverberation room, it is often the average absorption

3 ρ, density, and c, sound speed.

6


coefficient over all angles of incidence which is of interest, and we have usually used the term diffuse field absorption coefficient rather than the frequently used ‘statistical average’ to designate this quantity. To differentiate between normal incidence and diffuse field values one may use α0 for normal incidence and α1 and α2 for the diffuse field values corresponding to a locally and nonlocally reacting absorber, respectively. In practice, an average absorption coefficient in a 1/3 or 1/1 octave band, is often used; no special notations are given to these quantities. The Noise Reduction Coefficient, NRC, is the arithmetic mean of the octave band absorption coefficients in the octave bands at 250, 500, 1000, and 2000 Hz, and the corresponding values for normal incidence and diffuse field have been denoted by NRC0, NRC1 (local reaction), and NRC2 (nonlocal reaction). The propagation constant of a plane wave in free field is k = ω/c = 2π/λ, where c is the sound speed. When dealing with other materials, we have generally used the notation q for the propagation constant with the normalized value Q = q/k. It is difficult to avoid conflicts in notation, and we have allowed that to happen several times in regard to the use of the symbol q or Q. The ‘quality’ of a resonator has been denoted by Q as have been the mass flow rate and heat flow rate; only the context saves us from confusion. The term acoustically compact refers to an object with a characteristic linear dimension small compared to a wavelength. In the discussion of local and nonlocal reaction of absorbers we have used the term ‘honeycombed’ to indicate that the (air) layer in an absorber is partitioned into acoustically compact cells. In most figure captions, English units have been used. However, whenever numerical computations are involved, conversion to CGS or MKS is usually made.

1.2.1 Chapter Organization Each chapter is divided into two parts; the first, the main part, is mainly descriptive and the second, the supplement, contains derivations and mathematical details. Results of numerical analysis and related graphs are shown and discussed in the main part, which should be sufficient for the reader who has no interest or time to devote to mathematical details. Actually, some important equations, which have been used for graphs, have been included in the main part and placed within boxes for emphasis, but they can be skipped if desired without lack of continuity. Although this scheme appears to be a good one at first sight, it becomes a bit awkward at times if it is to be followed strictly, and some flexibility in this matter has been allowed.

Chapter 2

Sound Absorption Mechanisms As indicated in Section 1.2.1 on chapter organization, we have attempted to place most of the mathematical details and derivations of each chapter in a separate section which can be skipped at the first reading or skipped altogether by the reader who is interested mainly in numerical results. In this chapter, most of this mathematical analysis is summarized in Section 2.7. Some of the most important results, often the basis for the numerical results presented in graphs, are duplicated (and often framed) in the main part of the chapter.

2.1 BRIEF SUMMARY The most important mechanism of sound absorption in porous materials is due to viscosity and, to some extent, heat conduction. Absorption materials generally are porous structures, rigid or flexible, in which the pore size is quite small, typically of the order of a few thousandths of an inch. This chapter deals in some detail with viscous and thermal effects at solid boundaries in a sound field, particularly in narrow channels. It is essential to the understanding of sound absorption in porous materials and a basis for many parts of this book. The macroscopic physical parameters of a porous material that can readily be measured include porosity, flow resistance, weight, and quantities that describe the flexibility of the material. On the microscopic level, the geometry and distribution of the pores or voids in the material or the size distribution of fibers and their arrangement need to be specified, but in general, such details are not available. In Chapter 2, on sheet absorbers, flow resistance is defined operationally and considered to be a known quantity in terms of which the absorption spectra of sheet-cavity absorbers can be determined. Now the flow resistance will be pursued a bit further. For a porous material in general with its complex structure, it appears intractable to calculate the flow resistance in terms of geometrical parameters that describe the micro-structure of the material, even if such a description were available. Unless specific simple models are used, the best one can do is to use dimensional considerations to express the dependence of the flow resistance on basic geometrical and physical parameters, such as pore size distribution, porosity, and shear viscosity. 7

8


The flow resistance per unit thickness of the porous material is then found to be proportional to the coefficient of shear viscosity of the fluid involved and inversely proportional to the square of the characteristic pore size of the material. For a fibrous material with a given porosity, this means that the flow resistance per unit thickness is inversely proportional to the square of the fiber diameter. To gain further insight, it is useful to study the behavior of a structure that can be described in simple geometrical terms and analyzed from first principles. Sound absorption by capillary tube absorbers was studied already by Rayleigh (what else is new!) in his classic Theory of Sound (§348–351), where he extended the also classic Kirchhoff analysis of the visco-thermal attenuation in a tube to include the case when the acoustic boundary layer is not necessarily thin compared to the width of the tube. In this chapter we extend the analysis of this problem, including studies of the temperature and axial velocity distributions, and express the results in such a manner as to make them readily applicable to porous materials in Chapter 5. Steady flow. The first part of the chapter is devoted to steady flow and expressions for the steady flow resistance per unit length in narrow channels are obtained. It is found, as expected from dimensional considerations, that the flow resistance per unit length is proportional to the shear viscosity and inversely proportional to the square of the channel width. The shear viscosity is temperature dependent, and this has to be accounted for in many applications. It is then important to realize that the sound absorption coefficient of a material depends on the ratio of the resistance and the wave impedance ρc of the fluid involved and both these quantities are temperature dependent. Oscillatory viscous flow. In steady flow through a channel, only viscous forces are involved in the interaction between the flow and the channel wall, and the drag force on the wall and the corresponding reaction force on the fluid are proportional to the mean velocity in the channel. In acoustically driven oscillatory flow, inertial forces also play a role, and the velocity profile in the channel is changed from the steady flow parabolic to a flatter profile with a larger velocity gradient at the walls. Because of the effect of inertia, the interaction force on the wall and the corresponding reaction force on the fluid no longer are proportional to the velocity only, but there is also a contribution proportional to the acceleration. The velocity profile in a channel for oscillatory flow is characterized by a velocity gradient at the wall, which increases with frequency. The change in tangential velocity from zero at the wall to the free stream value far away from the wall defines the (acoustic) viscous boundary layer, the thickness of which decreases with increasing frequency. It plays an important role throughout the discussion of porous materials. Thermal effects. A sound wave in free field, far away from boundaries, produces a temperature fluctuation due to the isentropic (adiabatic) compressions and rarefactions in the sound wave. Close to a solid boundary, however, the large heat conduction of a solid compared to the heat conduction of air, prevents such a fluctuation from occurring and the conditions will be isothermal. The transition from isothermal conditions at the boundary to isentropic away from the boundary defines a thermal boundary layer and a corresponding thickness. It is of the same order of magnitude as that of the viscous boundary layer, and, like the viscous boundary layer thickness,

SOUND ABSORPTION MECHANISMS

9

decreases with frequency. The layer is usually quite thin. For example, at 100 Hz, it is about 0.022 cm in air, and it varies as the inverse of the square root of frequency. At frequencies low enough so that the boundary layer fills the entire space in the pores of a porous material, the conditions are isothermal, and this has an important effect on the low frequency absorption characteristics, since the compressibility and the sound speed will be reduced, as will be shown shortly. The present chapter is devoted to a detailed study of these visco-thermal effects in the sound field within a capillary tube. It will enable us to compute the attenuation of sound in such tubes and gain insight into the process of sound absorption in a porous material.

2.2 STEADY FLOW THROUGH A (NARROW) CHANNEL 2.2.1 Flow Resistance The nature of the flow through a channel is characterized by the Reynolds number, which is the ratio of the inertial and the viscous forces in the fluid. The inertial forces are of the order of ρu2 , where u is the velocity and ρ the density, and the viscous forces of the order of μu/d, where d is the characteristic dimension of the cross section and μ the coefficient of shear viscosity (the velocity gradient is of the order of u/d). The Reynolds number is then R = ρu2 /(μu/d) = ud/ν, where ν = μ/ρ is the kinematic viscosity. It is known from experiments that the transition from laminar to turbulent flow in a channel occurs at a Reynolds number of approximately 2000. Since we are here interested in laminar flow, the Reynolds number which we deal with has to be considerably less than 2000. The kinematic viscosity of air is ≈0.15 CGS units and in a tube with a diameter of 1 cm, the flow velocity has to be smaller than 300 cm/sec in order for laminar flow to be expected. With a diameter of 1 mm, the velocity can be 10 times larger. This will give us some idea about the velocity range we deal with here. The velocity is not uniform in the channel. The profile is parabolic, with zero velocity at the walls and maximum in the center. The flow resistance is defined in terms of the average velocity uav in the channel as the ratio of the pressure drop and uav and the flow resistance per unit length, as derived in Section 2.7. Steadyflow channel resistance per unit length 3μ/a 2 = 12μ/d 2 , parallel plates r0c = 8μ/a 2 = 32μ/d 2 , circular tube

(2.1)

μ: Coeff. of shear viscosity ≈ 2 × 10−4 CGS (air). d = 2a: Channel width (diameter). See also Eq. 2.33. The subscript c in r0c refers to ‘channel’ and 0 to steady flow (zero frequency). Elementary √ kinetic theory of gases shows that the coefficient of viscosity is proportional to T , where T is the absolute temperature, but in reality the temperature √ dependence is a bit more complicated. However, it is a good approximation to use T dependence for our temperature scaling purposes. In this context we note that sound

10


absorption depends on the normalized flow resistance r0c /ρc. The sound speed is pro√ portional to T and, at a√ constant static pressure, the density is inversely proportional to T . Therefore, using a T -dependence for viscosity, the normalized flow resistance will be proportional to T . It is very important to keep this in mind in design applications. Sound absorbers and attenuators are frequently used at temperatures of about 1000◦ F, i.e., ≈ 811 K, (in gas turbine exhaust stacks, for example), and the normalized flow resistance of a porous material is then about 2.8 times larger than at room temperature. This can have a large effect on sound absorption and attenuation characteristics. It is sometimes of interest to get an idea of the pore size in a porous material directly from the measured flow resistance per unit length, which is often expressed in terms of ρc units per inch. Thus, if we express the channel width d in mil (1 mil = 0.001 inch = 0.00254 cm) and r0c /ρc in inch−1 , the following numerical relations between√normalized flow resistance per inch and the channel width in mil results, d ≈ 4.6/ r0c /ρc. Thus, a flow resistance of 1 ρc per inch corresponds to an average channel width of 4.6 mil (≈ 0.012 cm).

2.3 ACOUSTIC BOUNDARY LAYERS Sound involves oscillatory flow, and, as for steady viscous flow, there is a viscous boundary layer in which the tangential velocity amplitude goes to zero at the boundary. We shall find that the viscous interaction with a solid boundary not only involves a force component proportional to the velocity, as for steady flow, but also a component proportional to the fluid acceleration. This must be accounted for in a study of sound propagation in narrow channels. There is also a thermal boundary layer in which there is a transition from the temperature fluctuation in the sound in free field, away from the boundary, to zero at the boundary. It will be shown that the general character of sound propagation in a narrow channel depends intimately on the ratio of the channel width and the boundary layer thickness. As a preliminary to such sound propagation studies, we proceed with a discussion of the viscous and thermal acoustic boundary layers at a plane solid wall.

2.3.1 The Viscous Boundary Layer First, let us consider the shear flow generated by a flat infinite plate, which oscillates in harmonic motion in the plane of the plate. Due to friction, this induces a harmonic motion also in the surrounding fluid in which the velocity is the same as that of the plate at the plate contact surface. However, the velocity is found to decrease exponentially with the distance y from the plate (Eqs. 2.36 and 2.37). The distance from the plate at which the amplitude is 1/e ≈ .37 times the amplitude of the plate is called the viscous boundary layer thickness, dv . As shown in Eq. 2.2, it decreases with frequency and density and increases with shear viscosity. Associated with the decay in amplitude, there is also a time delay in the motion corresponding to a phase (lag) angle y/dv . The ‘transmission’ of the motion from the plate out into the fluid is governed by a diffusion process and not a wave motion. The quantity that ‘diffuses’ is the vorticity in the fluid.

SOUND ABSORPTION MECHANISMS Viscous boundary layer thickness √ √ dv = 2ν/ω ≈ 0.22/ f cm. (normal air)

11

(2.2)

ν = μ/ρ: kinematic viscosity (≈ 0.15, standard air). μ: Coeff. of shear viscosity. ρ: Density. f : Frequency, in Hz. ω = 2πf (see also Section 2.7: Eq. 2.37). With f = 100 Hz, the boundary layer thickness is ≈ 0.022 cm in air at room temperature. Surface Impedance For Shear Flow From the velocity field in the fluid, we can determine the (shear) force per unit area of the plate that is required to drive the oscillating flow. The ratio of the complex amplitudes of this stress and the velocity outside the boundary layer is defined as a surface impedance per unit area. The resistive and reactive parts of the impedance turn out to be equal and the magnitude can be expressed as (kdv )ρc/2, where k = ω/c. From the frequency dependence of dv (Eq. 2.2), it follows that the surface impedance is proportional to the square root of frequency. Surface impedance (shear flow) per unit area √ Zs ≡ Rs + iXs = F /u0 = (1 − i) μρω/2 = (1/2)(1 − i)(kdv )ρc

(2.3)

dv : Boundary layer thickness (Eq. 2.2). k = ω/c. μ: Coeff. of shear viscosity. ρ: Density. (See also Section 2.7: Eq. 2.38.) We can interpret the mass reactance in terms of the total kinetic energy of the oscillatory flow in the boundary layer. Integrating the kinetic energy density from 0 to ∞, and expressing the result as (1/2)m|u0 |2 , we find that the corresponding normalized mass reactance ωm/ρc agrees with the expression given in Eq. 2.3. The reverse situation, when the plate is stationary and the velocity of the fluid far away from the plate has harmonic time dependence, the fluid velocity goes to zero at the plate. The transition from the ‘free stream’ velocity to zero occurs in a boundary layer, which has the same form as above. Obviously, it is only the relative motion of the fluid and the plate that matters. There will be an oscillatory force on the plate and a corresponding surface impedance Zs per unit area for the flow with resistive and mass reactive parts, the same as before. We can use this impedance as a good approximation also for a curved surface as long as the radius of the curvature of the surface is much larger than the acoustic boundary layer thickness in which case the surface can be treated locally as a plane. Using this approximation, we can determine the total surface impedance for oscillatory flow in a channel of arbitrary cross section as long as the transverse dimensions are large compared to the boundary layer thickness. Thus, if the perimeter of the channel is S and its area A, the total

12


surface impedance per unit length of the channel will be (S/A)Zs . In addition, there is the mass reactance ωρ of the air itself. When combined with the reactive part of the surface impedance, the total reactance can be expressed as ωρe , where ρe is an equivalent mass density and ρe /ρ a viscous contribution to a ‘structure factor,’ which will be discussed later.

2.3.2 The Thermal Boundary Layer By analogy with the discussion of the viscous boundary layer, consider next the temperature field produced by a plane boundary with a temperature which has a harmonic time dependence about its mean value. The surrounding fluid will be heated periodically. Temperature rather than vorticity is now diffused into the fluid, and the temperature field takes the place of the velocity field in the shear motion discussed above. Thus, the temperature amplitude decreases exponentially with distance from the plate and the thermal boundary layer thickness dh is defined in an analogous manner. It is determined by the coefficient of heat conduction rather than the coefficient of shear viscosity and is slightly larger than the viscous boundary layer thickness (by about 10 percent). The expression for it can be obtained from the viscous boundary layer thickness given above by replacing the coefficient of shear viscosity μ by K/Cp , where K is the coefficient of heat conduction and Cp the specific heat per unit mass at constant pressure. Thermal boundary layer thickness K √ cm. (normal air) dh = 2K/ρCp ω = μC dv ≈ 0.25 f p

(2.4)

f : Frequency, Hz. dv : Viscous boundary layer thickness (Eq. 2.2.). K: Heat conduction coeff. Cp : Spec. heat, unit mass. (See also Section 2.7: Eq. 2.43.) The reverse situation when a temperature fluctuation far from the plate varies harmonically and the temperature fluctuation at the plate is zero is analogous to the case of an oscillating fluid above a stationary plate. The temperature fluctuation goes to zero at the plate through the thermal boundary layer. The reason why the temperature fluctuation at the solid boundary can be considered to be zero is that the thermal conduction (and heat capacity) of a solid is much greater than for air. In the free field, far from the boundary, the temperature fluctuations are caused by the periodic compressions and rarefactions in the sound field since the conditions there are very nearly isentropic (adiabatic). An important influence of heat conduction on the sound field is the variation of the compressibility from the isentropic value in free field, 1/γ P , to the isothermal value, 1/P , at the boundary (γ is the specific heat ratio, ≈ 1.4 for air, and P the static pressure). In both these regions, a compression of a fluid element is in phase with the pressure increase. This means that the velocity of the surface of a volume element will be 90 degrees out of phase so that there will be no net work done on the element in one period of harmonic motion.


13

The situation is different within the boundary layer where the conditions are neither isothermal, nor isentropic. A compression leads to a delayed leakage of heat as a result of the diffusion in the air into the boundary and the pressure and velocity will not be 90 degrees out of phase. Thus, within the boundary layer, a net energy transfer will take place from the sound field into the gas and then via conduction into the boundary. The maximum transfer per unit volume of the gas occurs at a distance from the boundary approximately equal to the boundary layer thickness. This is the nature of the acoustic losses caused by heat conduction. Formally, it can be accounted for by means of a complex compressibility κ˜ in the thermal boundary layer which goes from the isothermal value at the boundary to the isentropic value outside the boundary layer. The loss rate per unit volume is proportional to the imaginary part of κ. ˜ There is some heat conduction also in the free field, far away from the plate which leads to a slight deviation from purely isentropic conditions. However, the heat flow is now a result of a gradient in which the characteristic length is the wavelength λ of the sound wave rather than the boundary layer thickness dh , and with λ >> dh , this effect can be neglected in the present discussion.

2.3.3 Power Dissipation, Visco-Thermal Boundary Layer With reference to the discussion in Section 2.7, the acoustic power dissipation at a boundary is the sum of two contributions. The first is due to the shear stresses in the viscous boundary layer and is proportional to the squared velocity amplitude tangential to the boundary. The second is due to the heat conduction in the thermal boundary layer and is proportional to the squared sound pressure amplitude at the boundary. They are obtained by integrating the viscous and thermal losses per unit volume over the boundary layers. Visco-thermal acoustic losses per unit area of a boundary Ls = Lv + Lh = (k/2)[dv |u|2 ρc + (γ − 1)dh |p|2 /ρc] √ ≈ 2 × 10−5 f [|u|2 ρc + 0.45|p|2 /ρc]

(2.5)

dv : Viscous boundary layer thickness, Eq. 2.2. dh : Thermal boundary layer thickness, Eq. 2.4. f : Frequency in Hz. |p|: Sound pressure amplitude (rms). |u|: Tangential velocity amplitude just outside the boundary layer. (See also Section 2.7: Eq. 2.53.) Since the velocity and pressure amplitudes are simply related, the total viscothermal power dissipation per unit area at the boundary can be expressed in terms of either the pressure amplitude or the velocity amplitude. The result obtained for a plane boundary can be used also for a curved boundary, if the local radius of curvature is much larger than the thermal boundary layer thickness.

14


Example: The Q-Value of a Cavity Resonator For a simple mass-spring oscillator with relatively small damping, the sharpness of its resonance is usually expressed as 1/(2π) times the ratio of the total energy of oscillation (twice the kinetic energy) and the power dissipated in one period. This relation is valid also for an acoustic cavity resonator. The total energy of oscillation is now obtained from the known pressure and velocity fields in the resonator and by dividing it with the total visco-thermal losses at the boundaries, the Q-value can be determined since both quantities are proportional to the maximum pressure amplitude in the resonator. The constant of proportionality for the total losses contains a visco-thermal boundary layer thickness dvh = dv + (γ − 1)dh , where dv and dh are the viscous and thermal boundary layer thicknesses and γ = Cp /Cv ≈ 1.4 (for air) is the specific heat ratio. If this scheme is used for a circular tube (quarter wavelength resonator), the Q-value turns out to be simply ≈a/dvh , where a is the radius of the tube. By introducing the frequency dependence of the boundary layer thickness, this can be expressed as √ ≈3.11 a f , where a is expressed in cm and f is the frequency in Hz. (The expression for a parallel plate cavity is the same if a stands for the separation of the plates.) Thus, a circular resonator with a radius of 1 cm and a resonance frequency of 100 Hz has a Q-value of 31.1. In this context we should be aware of the fact that the boundary layer thickness depends on the kinematic viscosity ν = μ/ρ and that it will decrease with increasing static pressure at a given temperature (μ is essentially independent of density). Thus, if a very high Q-value is desired in an experiment, a high pressure and a high density gas should be used. In a typical steam turbine in a nuclear power plant, the static pressure typically is of the order of 1000 atmospheres and the Q-value of acoustic resonances typically will be very high and the damping very low. This has a bearing on the problem of acoustically induced flow instabilities and their impact on key components in such planes; for example, control valves and related structures.

2.4 SOUND PROPAGATION IN A NARROW CHANNEL In the idealized case of a gas with no viscosity and heat conduction, the analysis of propagation of sound in a channel with acoustically hard walls is relatively simple, since the boundary condition which is needed for a solution is simply that the normal velocity at the boundary be zero. The governing mathematical description of the field is the ordinary scalar wave equation. The solution which is then obtained will have a tangential velocity and a temperature fluctuation at the boundary, the latter corresponding to the isentropic pressure fluctuation. In a real fluid, however, such a solution cannot be valid since the tangential velocity must be zero at the boundary as imposed by viscosity (on the basis of the standard assumption of no ‘slip’). Furthermore, since the heat conductivity and the heat capacity of a solid is much greater than for a gas, the temperature fluctuation at the boundary can be assumed to be zero.


15

In order to meet these conditions, it is necessary to start from the acoustic equations for a real fluid. In the linear approximation, it turns out that the solution can be composed of a linear superposition of the solution of an ordinary wave equation and two diffusion equations, one for vorticity and one for temperature. For a discussion of this problem, see, for example, Morse and Ingard, Theoretical Acoustics, Section 6.4, in particular, Problem 6.13, where propagation in the channel between two parallel walls is considered in detail. We shall extend this analysis here and add some observations and numerical results. In the channel under consideration, the two walls are placed at y = ±a and the x-axis is chosen to be the direction of propagation. The sound field is assumed to be independent of z. As mentioned, the wave field in the channel can be shown to be a linear combination of three wave modes, a ‘propagational,’ a ‘thermal,’ and a ‘viscous’ mode, using the terminology in the reference above. The pressure in the propagational mode satisfies an ordinary wave equation and the temperature and velocity in the thermal and viscous modes satisfy diffusion equations. These equations have to be solved subject to the appropriate boundary conditions for velocity and temperature in the total field.

2.4.1 Propagation Constant The temperature fluctuation in the viscous mode and the axial velocity component in the thermal mode turn out to be negligible. Thus, the boundary condition for temperature requires the temperature fluctuations in the propagational and thermal mode to cancel each other. On the basis of this condition, the amplitude constant in the thermal mode can be expressed in terms of the propagational mode amplitude. Similarly, the axial velocity components of the propagational and viscous modes have to cancel each other at the boundary in order to get zero total velocity, and this determines the amplitude of the viscous mode in terms of the propagational mode amplitude. All the modes have velocity contributions in the transverse direction and the boundary conditions of zero normal total velocity at the walls establish a relation from which the transverse propagation constant is obtained, which in turn determines the axial propagation constant q = qr + iqi . Axial propagation constant q −1)Fh Q ≡ q/k ≡ (Qr + iQi )/k = 1+(γ 1−Fv √ 3γ /4 (1 + i)a/dv for a/dv , a/dh > 1

(2.6)

dv , dh : Viscous and thermal boundary layer thicknesses (Eq. 2.2, Eq. 2.4.). a: Half-width of channel (Eq. 2.79). k = ω/c = 2π/λ. Functions Fv and Fh are defined in Eqs. 2.43 and 2.79. The real part, Qr , is the ratio of the free field sound speed, and the phase velocity in the channel and the imaginary part, Qi , yields the attenuation per wavelength.

16


These quantities are plotted vs a/dv in Figure 2.1. The value a/dv = 1 can be considered to represent the transition between the friction dominated and the inertia dominated wave propagation regimes. For values less than 1, the real and imaginary parts of Q approach the same value, which is inversely proportional to a/dv , and hence inversely proportional to the frequency. Since the frequency dependence is expressed in terms of a dimensionless parameter a/dv , it is not immediately clear what the plot means in terms of numerical values. To get a reference, let us consider a separation of the channel wall of 0.1√cm, so that a = 0.05 cm. The boundary layer thickness (Eq. 2.37) is dv ≈ 0.22/ f cm, so that at 100 Hz it is dv ≈ 0.022 cm, i.e., a/dv ≈ 2.3. At this value the real part of Q is only slightly larger than 1 so that the phase velocity is slightly smaller than the free field value. The imaginary part is ≈ 0.2, which means that the attenuation is 54.8 × 0.2 ≈ 11 dB per wavelength, which at 100 Hz is 340 cm (11.2 ft). Thus, at this channel separation and frequency, the propagation is inertia dominated, and the channel surfaces do not have much effect on the wave speed. In order to get into the friction dominated regime at this frequency, the channel width has to be reduced at least by a factor of 2.3, i.e., to a value less than 0.043 cm. The attenuation per wavelength, 11 dB at 100 Hz (about 1 dB/ft) in the example √ above, increases with decreasing frequency approximately as √ 1/ f . This means that the attenuation per unit length will increase approximately as ω. With a wavelength of 340 cm in our example, the attenuation √ per cm is 0.032 dB, and this value will increase with frequency approximately as f/100. The results obtained apply to the fundamental acoustic mode; there is no restriction on the value of λ/a. It is left for the reader to show that for the wavelength to be of the order of the boundary layer thickness in normal air, the frequency must be of the order of 10 mega Hz. In this context it should be pointed out that at sufficiently high frequencies, the visco-thermal and molecular relaxation effects in the bulk of the fluid (away from boundaries) must be accounted for.

Figure 2.1: The real and imaginary parts of the normalized propagation constant in a channel between two parallel plates with a separation d = 2a. The real part is the ratio of the free field sound speed and the wave speed (phase velocity) in the channel and the imaginary part, when multiplied by 54.8, gives the attenuation in dB per wavelength. The result is approximately valid for a circular tube with a diameter equal to twice the channel width (see Eq. 2.80).


17

Kirchhoff Attenuation These results are approximately valid for a circular tube with a diameter which is twice the channel separation so that in the high frequency regime a/dv >> 1 and √ Qi ≈ dvh /4a, where dvh = dv + (γ − 1)dh ≈ 0.31/ f cm, with f in Hz. As before, dv and dh are the viscous and thermal boundary layer thicknesses (Eqs. 2.37 and 2.43), and γ ≈ 1.4 is the specific heat ratio. For the circular tube this expression for Qi is often referred to as the ‘classical’ or Kirchhoff attenuation. The attenuation in dB per unit length is then given by 20 log(e)Qi k ≈ 8.7Qi k, where k = ω/c = 2π/λ, the sound speed c and the wavelength λ referring to free field. In this high frequency region where the boundary layer thickness is small compared to the transverse dimensions of a channel, the attenuation in a channel with an arbitrary cross section of area A and perimeter S can be expressed in the same manner, i.e., Qi = dvh /4D, where D = A/S is the ‘hydraulic’ diameter of the channel (A, area, and S, perimeter). For a circular tube of diameter d, D = d/4, and for the channel between two parallel plates, D = d/2, where d is the width of the channel. It should be mentioned that the attenuation can be obtained to a good approximation by starting from the loss-free wave field in a duct and then using the power dissipation in Section 2.3.3 to express the loss per unit length in a duct. The loss, βw, is proportional to the acoustic power w in the duct and the exponential spatial decay is then obtained from dw/dx = −βW . This approach can be used for both the fundamental and a higher mode. Penetration Depth The amplitude reduction in a distance x is exp(−Qi kx). The distance at which the reduction is 1/e is then Qi /k, which is defined as the penetration depth dp , i.e., dp = Qi λ/2π of a wave in the channel, where λ is the free field wavelength. In the low frequency friction controlled or capillary regime, a/dv 1), on the other hand, the compressibility becomes isentropic, and again, there are no losses. The losses are zero in both the low and high limits of a/dh (low and high frequencies); there is a maximum at a/dh = 1.12, i.e., close to 1, where the imaginary part of the normalized complex compressibility is 0.16. We shall take this conditionto represent the transition from isothermal to isentropic conditions. Since dh = 2K/(ρCp ω), this means that for a given channel width, the maximum occurs at a characteristic thermal transition frequency ωh ≈ 1.25(2K/ρCp a 2 ) and a corresponding relaxation time, the same as the relaxation frequency defined in Section 2.7. With 2K/Cp ≈ 1.3ν, as given earlier, it follows that ωh ≈ 3.25ν/a 2 = 1.08ωv . In other words, this thermal relaxation frequency is about the same as the viscous relaxation frequency in Eq. 2.8 and it can then be obtained from the known steady flow resistance per unit length of a porous material.

2.5 IMPEDANCES We can treat wave propagation in the channel approximately as a one-dimensional transmission line by introducing the average values of the amplitude of sound pressure

22


and the axial velocity over the channel. In the discussion of the acoustical characteristics of such a channel, it is convenient to introduce some impedances.

2.5.1 Impedance Per Unit Length The first is the channel impedance per unit length, i.e., the ratio of the drop in the (complex) sound pressure amplitude per unit length and the average velocity amplitude. Both the resistive and reactive parts of this impedance will be frequency dependent. As the frequency goes to zero, the resistance is found to go to the steady flow value 3μ/a 2 , where μ is the shear velocity and a the half-width of the channel, and does not change noticeably until a/dv becomes greater than 1, as shown in Figure 2.4. Channelimpedance per unit length 3μ/a 2 − i1.2ωρ for ξv > 1

(2.10)

Fv = tan(kv a)/kv a: kv = a/dv , dv : Viscous boundary layer thickness (Eq. 2.2.), a: half-width of channel. (See also Section 2.7: Eq. 2.83.) The other curve shows the frequency dependence of the reactance, normalized with respect to the reactance ωρ per unit length in free field. This curve approaches 1.2 as the frequency goes to zero. In other words, the air in the channel appears to have an equivalent inertial mass density ρe , which is 1.2 times the actual density ρ. Thus, curve (1) in the figure shows the frequency dependence of ρe /ρ, and this ratio sometimes is referred to as a viscous contribution to a structure factor. There is another way to understand the apparent increase in the inertial mass, as follows. Direct calculation of the average of the kinetic energy density ρu2x /2

Figure 2.4: (1): The normalized reactive part of the impedance per unit length of a channel

between two parallel plates, ωρe /ωρ = ρe /ρ = c , where ρec = is the effective mass and c a ‘structure factor.’ The abscissa is the ratio of the half-width of the channel and the viscous boundary layer thickness. (2): The normalized resistance rc /r0c per unit length, where r1c , is given in Eq. 2.83 and r0c is the resistance for steady flow in Eq. 2.32.


23

by integration over the channel using the known parabolic velocity distribution and 2 /2 indeed yields the value ρ = 1.2ρ. Actually, it is expressing the result as Aρe Uxav e sometimes convenient to use this point of view in computing the equivalent inertial mass density for other channels. For example, using the velocity distribution for the steady viscous flow in a circular duct (see Eq. 2.31) leads to ρe ≈ 1.33ρ at low frequencies. √ The flow resistance in the channel increases with a/dv , and hence as ω for a/dv >> 1. The normalized resistance per unit length in the channel approaches kdv /d for a channel of width d between parallel plates and the value 2k/dv for a circular tube of diameter d. The physical reason for the increase in resistance is simply that the velocity gradient at the boundary increases with frequency, as shown in Figure 2.2. In the same high frequency regime, the equivalent mass reactance decreases from the low frequency value 1.2ωρ to ωρ but the reactance ωρ increases faster with frequency than the resistance so that the reactance/resistance ratio will be proportional to the square root of frequency. This is the reason for the increase with frequency of the Q-value of a tube resonator, as derived earlier and which will be discussed later. In the low frequency regime, the resistive and reactive parts of the impedance per unit length are equal at a frequency given by ωv = r0c /1.2ρ, where r0c is the steady flow resistance per unit length. This frequency is not much different from that which makes the viscous boundary layer thickness equal to the half-width of the channel. Both these frequencies can be expressed in terms of r0c /ρ, which has been called the viscous relaxation frequency in the channel.

2.5.2 Complex Density and Wave Impedance In the description of wave motion in a channel or a porous material in which viscous interaction with a boundary is involved, it is convenient to introduce a complex quantity for the equivalent density. In this manner, the friction interaction can be included in the density. It is analogous to the complex mass in an ordinary mass-spring oscillator with friction. The equation of motion is Mdu/dt − Ru = F , where M is the mass, u the velocity, R the friction coefficient, and F the driving force. In harmonic motion with angular frequency ω, the corresponding equation for the complex amplitudes ˜ is (−iωM − R)u(ω) = F (ω). If we introduce a complex mass M(ω) = M − iR/ω ˜ the equation can be written −i M(ω)u(ω) = F (ω), which has the same form as the ˜ friction free oscillator with M replaced by M. Thus, in terms of a complex density ρ, ˜ the impedance per unit length is expressed as −iωρ, ˜ i.e., ρ˜ = zc /(−iω), where zc is given in Eq. 2.10. In this manner, we can readily carry over well-known expressions from wave propagation in a loss-free channel to the corresponding expressions for a lossy channel merely by replacing the ordinary density by the complex density and the ordinary compressibility by the complex compressibility κ. ˜ As examples of the use of the complex density and √ compressibility, consider the well-known–expression for the speed of sound, c = 1/(ρκ), where κ is the isentropic compressibility 1/γ P , γ the specific heat ratio, and P the static pres√ sure. The corresponding propagation constant for a plane wave is k = ω/c = ω ρκ. These expressions are valid also in a lossy channel if we replace ρ with a complex

24


density and κ with a complex compressibility κ. ˜ The propagation constant now becomes complex, the real part determines the phase velocity, and the imaginary √ part the attenuation of the wave in the channel. Similarly, the wave impedance ρ/κ becomes complex and it equals the input impedance of an infinitely long channel including both the resistive and reactive parts. The impedance per unit length, of course, is −i ρω. ˜ For the expression for the complex density we refer to Section 2.7.

2.5.3 Perforated Plate In the analysis in Chapter 2, it was implied that the impedances of the porous sheets or screens could be regarded as experimentally known quantities. It is of interest in design work, however, to be able to calculate the impedance, at least approximately, in terms of geometrical parameters. Thus, guided by the results obtained earlier in this chapter, we shall develop semi-empirical expressions for the impedances of a perforated plate and a wire mesh screen, two elements of considerable practical interest in acoustical applications. In this subsection we consider the first of these elements. We have shown (Eq. 2.33) that the steady flow resistance per unit length is 12μ/d 2 in a channel between two parallel plates and 32μ/d 2 in a circular tube, where μ is the shear viscosity and d the channel width or tube diameter. These expressions are approximately valid for oscillatory flow at frequencies √ below the value at which the acoustic viscous boundary layer thickness dv = 2μ/ω is about half the channel width. At high frequencies, with a boundary layer thickness substantially smaller than the channel width d, the resistance per unit length approaches ρckdv /d for parallel plates and 2ρckdv /d for the circular tube (Eq. 2.83). Actually, this high frequency resistance can be expressed in a form which is valid for any shape of the channel cross section if we use as a characteristic dimension the hydraulic diameter D, which is the ratio of the area and the perimeter of the channel. For two parallel plates we have D = d/2 and for the circular tube, D = d/4. The general expression for the specific resistance per unit length is then ρckdv /2D. Furthermore, we have found that the reactive part of the channel impedance per unit length can be expressed in terms of an effective mass density, which at low frequencies (dv >> d) is (6/5)ρ for a channel between two parallel plates and (4/3)ρ for a circular tube. At high frequencies, dv > 1, the second term within the bracket can be neglected, and θ1 goes to the value for steady flow. For dv /d dh , the conditions in the fluid are isentropic and the compressions and rarefactions in the sound wave produce a harmonic temperature fluctuation with the amplitude τ0 =

γ −1 p T. γ P

(2.47)

Quantity p is the sound pressure amplitude, γ = Cp /Cv , the specific heat ratio, P the ambient pressure, and T the absolute temperature. The acoustic wavelength of interest is large compared to the boundary layer thickness so that we need not be concerned about any change of the sound pressure with position across the boundary layer. However, the compressibility varies, going from the isentropic value 1/γ P to the isothermal, 1/P , as the boundary is approached. These values refer to an ideal gas. Complex Compressibility To determine the complex compressibility throughout the boundary layer, we start with the density ρ(P , T ) being a function of both pressure P and temperature T (not only of pressure alone) so that

dρ =

∂ρ ∂P

dP + T

∂ρ ∂T

dT .

(2.48)

P

From the gas law, P = rρT , we have (∂ρ/∂P )T = ρ/P and (∂ρ/∂T )P = −ρ/T . Then, the quantities dP = p, dρ and dT = τ (y, ω) are treated as complex amplitudes, where τ is given in Eqs. 2.46 and 2.47 in terms of the sound pressure amplitude p, and the compressibility κ˜ = (1/ρ)(dρ/dP ) =

1 [1 + (γ − 1) e−kh y eikh y ]. γP

(2.49)

The tilde symbol is used to indicate that the compressibility is complex and different from the normal isentropic compressibility κ = 1/γ P = 1/ρc2 . For y = 0, κ˜ = 1/P equals the isothermal value, and for y = ∞, κ˜ = 1/γ P , the isentropic value; in the transition region, κ˜ is complex. The imaginary part can be written (2.50) κi = κ(γ − 1)e−y/dh sin(y/dh ). It has a maximum 0.321κ at y/dh = π/4.


39

2.7.4 Power Dissipation in the Boundary Layer The power dissipation per unit area due to viscosity in the acoustically driven oscillatory shear flow over a solid wall has already been expressed in Eq. 2.39. To determine the dissipation due to heat conduction, we start from the conservation of mass equation for the fluid ∂ρ/∂t +ρdiv u = 0. For harmonic time dependence and with the relation between the complex amplitudes of density and pressure (δ and p) expressed as δ = ρ κp ˜ in terms of a complex compressibility κ, ˜ this equation becomes −iωκp ˜ + div u = 0. After integration of this equation over a small volume V with surface area A, and replacing the volume integral of div u by a surface integral over A, we can express the time average power {un p∗}A transmitted through A into the volume element as {(−iω)κ|p| ˜ 2 }V , where un is the inward normal velocity component of the velocity at the surface, |un | and |p| being rms values to avoid an additional factor of 1/2. Thus, the corresponding power dissipation per unit volume becomes Dh = ωκi |p|2 .

(2.51)

The integral of this expression over the boundary layer yields the corresponding acoustic power loss per unit area of the wall. The integration can be taken from 0 clear out to infinity. The contribution to the integral comes mainly from y-values less than a couple of boundary layer thicknesses and quickly goes to zero with increasing y outside the boundary layer. The pressure amplitude |p| can be taken to be constant throughout the layer since the wavelength of interest is much larger than the boundary layer thickness. After insertion of the expression for the compressibility in Eq. 2.49, the loss due to heat conduction per unit area of the wall can be expressed as Lh = (1/2)(γ − 1)kdh |p|2 ,

(2.52)

which is the counterpart of the expression for the viscous power dissipation Lv in Eq. 2.39. The total visco-thermal power dissipation per unit area of the wall then becomes Ls = Lv + Lh = (k/2)[dv |u|2 ρc + (γ − 1)dh |p|2 /ρc] √ ≈ 2 × 10−5 f [|u|2 ρc + 0.45|p|2 /ρc]

(2.53)

where dv : Eq. 2.37. dh : Eq. 2.43. f : Frequency in Hz. |u|: Tangential velocity outside the boundary layer. |p|: Pressure amplitude at the wall, both rms magnitudes. Q-Value of a Cavity Resonator For a simple harmonic oscillator (spring constant K, mass M, and resistance constant R) driven by a harmonic force with frequency independent amplitude, the frequency dependence (response) of the velocity by the familiar amplitude u(ω) is characterized √ 2 2 resonance at the frequency ωr = ω0 − γ , where ω0 = K/M and γ = R/2M. For small damping, ωr ≈ ω0 and the sharpness of the resonance curve is often

40


expressed in terms of the Q-value, Q = ω0 M/R. It can be interpreted as the ratio of the resonance frequency and the total width of the response curve at the ‘half-power point,’ defined by |u(ω)/u(ω0 )|2 = 1/2. Q = ω0 M/R = ω0 M|u|2 /R|u|2 can be interpreted also as ω0 times the ratio of the time average of the energy of oscillation (being twice the kinetic energy average) and the dissipation rate or, apart from a factor of 2π, as the ratio of the energy of oscillation and the dissipation in one period. This relation is valid also for an acoustic cavity resonator in the vicinity of a resonance. The cavity under consideration is a straight tube of length L, area A, perimeter S, open at one end, and terminated by a rigid wall at the other (at x = L). With the pressure amplitude at the wall being p(L), the amplitude at a distance x from the wall is p(x) = p(L) cos(kx), where k = ω/c. Then, if the ‘driving pressure’ at the open end (x = 0) is p0 , we get p(x) = p0 cos[k(L − x)]/ cos(kL). Similarly, the velocity amplitude distribution is |u(x)| = |u0 | sin[k(L − x)]/ cos(kL), where |u0 | = |p0 |/ρc. Integrating the kinetic and potential energy densities ρ|u|2 /2 and |p|2 /2ρc2 over the volume of the tube gives the total energy E = (AL/2)|p0 |2 /ρc2 .

(2.54)

Using these expressions for |u| and |p| in Eq. 2.53 and integrating over the tube walls and accounting for the thermal losses at the wall, we get for the total loss rate W = (SL/2)[|u0 |2 (kdv ρc/2) + (|p0 |2 /ρc)(γ − 1)kdh /2] + A(|p0 |2 /ρc)(γ − 1)kdh /2 = ω(SL/2)(|p0 |2 /ρc2 )(dvh /2)[1 + (A/SL)(γ − 1)dh /dvh ] ≈ ω(SL/2)(|p0 |2 /ρc2 )(dvh /2), where

dvh = dv + (γ − 1)dh = dv [1 + (γ − 1)/ Pr ] ≈ 1.46dv

(2.55) (2.56)

is the ‘visco-thermal’ boundary layer thickness and Pr = μCp /K ≈ 0.77 (air), the Prandtl number, defined in Eq. 2.45. The term in Eq. 2.55, which contains the area A expresses the heat conduction loss at the rigid wall termination, which is usually small compared to the rest.2 Without this contribution and with ω = ω0 , the Q-value of the resonator becomes 2A Q = ω0 E/W ≈ , (2.57) Sdvh which can be interpreted as twice the ratio of the volume of the tube and the volume occupied by the visco-thermal boundary layer. As an example, consider a circular tube of radius a. With A = π a 2 and S = 2π a we get Q-value of a circular tube resonator (2.58) √ Q = a/dvh ≈ 3.11a f 2 There is no tangential velocity at the end wall and no viscous losses.

41


where a: Tube radius, cm. f : Frequency, Hz. dvh : Visco-thermal boundary layer thickness (Eq. 2.56). Thus, Q is simply the ratio of the radius and the boundary layer thickness. The approximate numerical expression was obtained by using dvh ≈ 1.46dv and dv ≈ √ 0.22/ f . Thus, a 100 Hz quarter wavelength circular tube resonator with a one-inch diameter will have a Q-value of ≈39.5. For the channel between two parallel plates, separation d and width w, the area dw and S = 2(w + d) ≈ 2w. Therefore, Q ≈ d/dvh

(parallel plates).

(2.59)

In Chapter 4, we reconsider this problem without making the assumption that the cross sectional dimensions of the tube are large compared to the boundary layer thickness. The Q-value thus obtained is found to be consistent with that obtained here.

2.7.5 Sound Propagation in a Narrow Channel The two walls bounding the channel under consideration are placed at y = ±a, and the x-axis is chosen to be the direction of propagation. The sound field is assumed to be independent of z. The wave field in the channel can be shown to be a linear combination of three wave modes, a ‘propagational,’ a ‘thermal,’ and a ‘viscous’ mode, using the terminology in the reference above. The pressure in the propagational mode satisfies an ordinary wave equation and the temperature and velocity in the thermal and viscous modes satisfy diffusion equations. These equations have to be solved subject to the appropriate boundary conditions for velocity and temperature. The Propagational Mode The pressure field is contributed almost exclusively by the propagational mode, which is a solution to the ordinary wave equation ∇ 2 p + (ω/c)2 p = 0,

(2.60)

where c is the isentropic (free field) sound speed. Thus, the complex amplitude of the sound pressure p(x, y, z, ω) is a superposition of waves of the form exp(iqx + iqy y + iqz z), where q 2 + qy2 + qz2 = (ω/c)2 . With the sound field being independent of z, i.e., qz = 0, so that q 2 + qy2 = (ω/c)2 .

(2.61)

The fundamental acoustic mode (wavelength much longer than the width of the channel) is symmetrical with respect to the center of the channel. Thus, since y = 0 is placed at the center of the channel (with the boundaries at y = ±a, as already mentioned), it follows that pp (x, y, ω) = A cos(qy y)eiqx ,

(2.62)

where qy , yet to be determined, is the propagation constant in the transverse direction of the channel. In the absence of visco-thermal effects, qy = 0 but in general it is not,

42


and after having determined it, the propagation constant q, which is the quantity of primary interest, is obtained from Eq. 2.61. The velocity amplitude components corresponding to the pressure field in the propagational mode are Velocity amplitude, Propagational mode upx = A(q/ωρ)eiqx cos(qy y) upy = A(iqy /ωρ) sin(qy

(2.63)

y)eiqx

where qy : Eq. 2.77. q: Eq. 2.61. The subscript p indicates ‘propagational mode.’ The temperature field associated with the propagational pressure mode is due to the isentropic compression-rarefaction involved. The corresponding complex temperature amplitude is τp = (γ − 1)pp T /γ P ,

(2.64)

where γ is the specific heat ratio Cp /Cv , T the absolute temperature, and P the ambient pressure. The Thermal Mode With the boundary assumed to have a heat conduction coefficient and heat capacity much larger than the fluid, the temperature fluctuation can be assumed to be zero at the boundary. To satisfy this boundary condition, there must be a contribution from a heat conduction mode, which cancels the temperature fluctuation in the propagational mode at the boundary (the temperature fluctuation in the viscous mode is negligible). The temperature τh in the thermal mode is a solution to the diffusion equation ∂ 2 τh ∂τh = (K/ρCp )∇ 2 τ ≈ (K/ρCp ) 2 ∂t ∂y

(2.65)

subject to the condition that at the boundaries, the temperature τh cancels the temperature fluctuation τp in the propagational mode given in Eq. 2.68. For harmonic time dependence and with the y-dependence of the complex temperature amplitude of the form exp(ikh y), it follows that kh = (1 + i)/dh , (2.66) where dh = 2K/(ρCp ) is the thermal boundary layer thickness. The temperature fluctuation caused by the propagational (pressure) mode can be written γ − 1 pp γ −1A τp = (2.67) T = T cos(qy y)eiqx . γ P γ P The appropriate solution to the diffusion equation for the temperature amplitude in the thermal mode must be such that it cancels the contribution from the pressure mode at the boundaries. This means that τh = −τp

cos(kh y) , cos(kh a)

(2.68)


43

which becomes −τp at the boundaries, y = ±a, where it will cancel τp in the propagational mode. The y-dependence of the total temperature amplitude is then

cos(kh y) τ (ω) = τp 1 − . (2.69) cos(kh a) Although the sound pressure in the thermal mode is negligible, its gradient and the corresponding fluid velocity component in the y-direction are not. The velocity component in the thermal mode can be shown to be uh = (γ P /ρc)(1/T )h grad τ , where h = K/ρCp c = iω/ckh2 , where h is of the order of the mean free path. From kinetic theory, h ≈ K/ρCp ct , where ct is the average molecular thermal speed, which is approximately the sound speed. The x-component of this velocity is negligible, since the acoustic wavelength is much greater than the boundary layer thickness, and the significant velocity is the y-component, uhy . From Eqs. 2.62 and 2.67 and the expressions for h and τh , given above, it follows that Transverse velocity amplitude, Thermal mode iqx cos(q a) h y) uhy = i(A/ρc)(γ − 1)(ω/c) khsin(k y sin(kh a) e

(2.70)

where kh : Eq. 2.66, qy : Eq. 2.77, q: Eq. 2.61. The Viscous Mode and the Total Velocity Profile The total velocity components, both in the x- and y-directions, must be zero at the boundaries and we start with the construction of the axial velocity distribution, which satisfies this condition as well as the equations of motion. Thus, the x-component of the velocity in the propagational mode in Eq. 2.63 must be canceled at the boundary by the velocity component in viscous mode, which is a solution to ρ

∂ 2 uvx ∂uvx = μ∇ 2 uvx ≈ μ ∂t ∂y 2

(2.71)

encountered earlier. It is of the same form as the diffusion equation for temperature with μ/ρ corresponding to K/ρCp . Thus, with the y-dependence for the complex velocity being exp(±ikv y) (or cos(kv y) or sin(kv y)), it follows that kv = (1 + i)/dv ,

(2.72)

where dv = sqrt2ν/ω is the viscous boundary layer thickness. The x-component uvx of the velocity in the viscous mode must be such as to cancel the velocity amplitude qP /ωρ of the propagational mode at the boundaries y = ±a. By analogy with the solution for the temperature equation we realize that the appropriate solution is Axial velocity amplitude, Viscous mode v y) iqx uvx = −(Aq/ωρ) cos(k cos(qy a) cos(kv a) e where kv : Eq. 2.72, qy : Eq. 2.77, q: Eq. 2.61.

(2.73)

44


The total axial velocity is obtained by adding the contribution upx from the propagational mode in Eq. 2.63,

cos(kv y) cos(qy a) cos(kv y) ux = upx 1 − ≈ upx 1 − . cos(kv a) cos(qy y) cos(kv a)

(2.74)

It will be established shortly that qy a 1

(2.79)

where√F , ξv , ξh : Eq. 2.77 and 2.78, dvh : Eq. 2.80, k = ω/c, ξv = a/dv , ξh = a/dh , dv = 2μ/ρω, dh = 2K/Cp ρω, dvh = dv + (γ − 1)dh . For a circular tube with a diameter d, the high frequency approximation of the propagation constant is Q ≈ 1+i[dv +(γ −1)dh ]/d = 1+idvh /d

(dvh = dv +(γ −1)dh 1, the resistance agrees with that for oscillatory (incompressible) flow in Eq. 2.40. Wall Stress and Viscous Interaction Impedance The impedance in Eq. 2.83 contains a mass reactance, which for ξ 1

(2.85)

where F , ξ : Eqs. 2.77 and 2.78, Q: Eq. 2.79. Quantity Q ≡ Qr + iQi ≡ q/k is the normalized propagation constant. The qualitative difference between the wave impedance at low and high frequencies, both in terms of magnitude and phase, reflects the change in character of the interaction force from viscous (diffusive) to inertial (propagational) dominance as the frequency increases. As ξ increases, the conditions in the channel approach those in free field and zwc /ρc → 1, as expected. In the diffusion regime, ξ 1, and it has a maximum value of the imaginary part of (γ − 1)/2 (equal to 0.2 for air) obtained for ω = ωh or a/dh ≈ 1.12. In Figure 2.3 we have plotted κ˜ av1 as a function of a/dh , and it can be seen to be in rather good agreement with κ˜ av1 given in Eq. 2.93. The expression for κav2 can be useful in the study of the rigid porous absorber since it is a good approximation to replace the thermal by the viscous relaxation time, which is determined by the known flow resistance of the material. For a channel we have ωv = roc /ρ and with r0c = 3μ/a 2 it follows that ωv = 3μ/a 2 ρ = 3Pr (K/ρCp a 2 ) ≈ 2.3(K/ρCp a 2 ) ≈ 0.92ωh ,

(2.99)

where we have used Pr = μCp /K ≈ 0.77 for the Prandtl number of air. In other words, the viscous and thermal ‘relaxation’ frequencies ωv and ωh are almost the same numerically so that ωv can also be used as an indicator of the transition between isothermal and isentropic conditions. For porous material with a normalized flow resistance θ = r/ρc per inch, we may use (see Eq. 2.99) Ratio of viscous and thermal relaxation frequencies fh ≈ fv = r/(2πρ) = cθ/2π ≈ 2140θ Hz

(2.100)

For example, a material with θ = 0.5 per inch yields fv ≈ 1070 Hz at room temperature, with c ≈ 1120 ft/sec. In other words, for most materials used in practice, the conditions within the porous material are isothermal over a substantial range of frequencies.

Chapter 3

Sheet Absorbers As indicated in Section 1.2.1 on chapter organization, we have attempted to gather most of the mathematical details and derivations in a separate section, which can be skipped at the first reading or skipped altogether by the reader who is interested mainly in results. In this chapter, most of this mathematical analysis is summarized in Section 3.6. Some of the most important results, often the basis for the numerical results presented in graphs, are duplicated in the main part of the chapter.

3.1 INTRODUCTION AND BRIEF SUMMARY Before proceeding with this (or any other) chapter, it is advisable that the reader become familiar with the discussion of terminology and notation in Chapter 7. This, chapter deals with a sound absorber consisting of a porous sheet or screen backed by an air layer and a rigid wall, which is parallel with the sheet; the air layer then forms an important part of the absorber. In another configuration, the sheet is simply hung from the ceiling in a room without any direct relation to any particular wall. These two configurations are sometimes referred to as ‘surface’ and ‘volume’ absorber, respectively (see Figure 3.17). When sound interacts with a sheet, some of the incident acoustic energy is lost to heat as a result of the friction drag between the sheet and the acoustic velocity field, i.e., sound is absorbed.

3.1.1 Single Sheet Surface Absorber Sheet absorbers are alternatives to conventional uniform porous layers in bulk and can have some advantages in regard to several nonacoustical factors having to do with erosion, water containment, ease of cleaning, hostile environments (high temperature and flow), contamination of the air by fibers, etc. When a sheet is used as a ‘volume’ or ‘functional’ absorber, ease of mounting and the possibility for creative architectural designs are other factors which may favor the use of sheets. When used as a surface absorber, the air backing or cavity between the sheet is largely responsible for the frequency dependence of the absorption. If the sheet is in direct contact with the wall, the absorption is essentially zero, and in order for the absorber to be effective, the depth of the air backing normally should not be less than 53

54


Locally reacting

Nonlocally reacting

Incident sound wave

L

L

Figure 3.1: Porous sheet cavity absorber. Left: Locally reacting. Right: Nonlocally reacting.

a quarter wavelength. With a quarter wavelength backing, the absorption coefficient can be made to be 100 percent at the peak and in excess of about 80 percent over a frequency band of approximately one octave. Narrower absorption bands with about the same peak value occur at higher frequencies corresponding to cavity depths of an odd number of quarter wavelengths. The cavity can be empty or ‘honeycombed’ (Figure 3.1) with acoustically compact honeycombs, which make the absorber locally reacting. If it is empty, the absorber is nonlocally reacting. A simple example of the nonlocally reacting kind is a sheet hung as a curtain or drape in front of a wall. For normal incident sound, the performance of the two configurations will be the same, of course, but in a diffuse sound field, the locally reacting absorber is better, except at the frequencies for which the backing depth is an integer number of half wavelengths. At these frequencies the absorption coefficient dips to zero for both normal incidence and diffuse field; this does not occur for the nonlocally reacting absorber. The dips are quite pronounced in a narrow band spectrum but less pronounced in broader bands, say 1/3 or 1/1 octaves, as will be demonstrated shortly. By absorption spectrum we mean the relationship between the absorption coefficient and frequency. The complete absorption spectra, for locally and nonlocally reacting sheet absorbers, are shown in Figure 3.2 for a rigid and in Figure 3.8 for a flexible sheet, which demonstrate the special characteristics mentioned above. For normal incidence, the absorption coefficient α0 becomes 1.0 at a quarter wavelength resonance if the flow resistance of the sheet is one ρc, but in a diffuse field, a flow resistance of ≈2 ρc yields the best overall performance. With proper choice of sheet parameters, the acoustically induced motion of the sheet can be utilized to lower the resonance frequency of the absorber substantially below the regular quarter wavelength resonance, and 100 percent absorption can be obtained at this new low frequency resonance if the normalized flow resistance of the sheet is chosen to equal the ratio mr of the sheet mass and the mass of the air

SHEET ABSORBERS

55

Figure 3.2: The absorption coefficient of a rigid sheet-cavity absorber vs L/λ, where L is cavity depth and λ, the wavelength. Normalized flow resistances of the sheet: 1, 2, 4. (a): Normal incidence. (b) Diffuse field, local reaction. (c): Diffuse field, nonlocal reaction. Bandwidth: ≈1/60 OB. For curves with explicit frequency dependence, see the next figure and Appendix C.

layer (assuming mr >> 1). The corresponding resonance frequency is then reduced √ by a factor, which is approximately equal to (π/2) mr . However, the reduction in resonance frequency is achieved at the expense of a decrease in the width of the resonance.

3.1.2 Multisheet Absorber An absorber consisting of several parallel porous sheets in the form of a lattice in front of and parallel with a rigid wall can yield an absorption which can come close to and even exceed that of a uniform porous layer as can be seen in Figure 5.21. Actually, at sufficiently low frequencies, typically below 200 Hz, a single sheet absorber can be better than a uniform layer of the same thickness and the same total flow resistance.

3.1.3 Single Sheet as a ‘Volume’ Absorber In this case, the appropriate measure of the performance is the absorption area or absorption cross section per unit area of sheet material. Under the idealized assumption of an immobile, infinite sheet, the absorption is frequency independent. However, due to the induced motion of the sheet, the low frequency performance is reduced as

56


shown in Figure 3.20. An additional, and generally even more important, reduction at low frequencies is caused by diffraction, which is considered in the last section (Figure 3.21). As for the surface absorber, there is an optimum flow resistance for each frequency. If the wavelength is smaller than the dimensions of the sheet, the best overall performance of the volume absorber in a diffuse field is obtained with a normalized flow resistance of 3.2. The corresponding high frequency limit of the absorption area is then almost equal to the physical area of the sheet material, corresponding to an absorption coefficient of almost 48 percent for the sound, which is incident on each of the two surfaces of the sheet. For normal incidence on a rigid infinite sheet under optimum conditions (flow resistance 2), 50 percent of the sound is absorbed by the sheet, 25 percent is reflected, and 25 percent transmitted through the sheet. It should be realized, however, that if the wavelength is large compared to the sheet dimensions, the optimum flow resistance and the absorption cross section per unit sheet area decrease with the size of the sheet because of diffraction, which tends to reduce the difference between the sound pressure amplitudes on two sides of the sheet and the corresponding flow velocity through the sheet. This effect is also included in the analysis and can be regarded as a low frequency approximation for the absorption, as shown in Figure 3.21.

3.2 RIGID SINGLE SHEET WITH CAVITY BACKING 3.2.1 Flow Resistance and Impedances In a harmonic sound wave, the drop in sound pressure amplitude across a sheet and the corresponding fluid velocity through the sheet are both oscillatory and in the idealized case of a purely resistive sheet, these oscillations are in phase. The ratio of the amplitudes of the pressure drop and the velocity is then a flow resistance, which is the same as for steady flow. However, in reality, the velocity lags behind the pressure by some phase angle due to an inertial reactance of the sheet. The phase angle goes to zero as the frequency goes to zero, but it can be substantial at sufficiently high frequencies. The amplitudes of the pressure drop and the velocity are now described by complex amplitudes which contain both the magnitudes and phase angles. Their ratio is also a complex quantity, a complex impedance, with a magnitude and a phase angle, which expresses the combined effects of the resistance and the reactance of the sheet. The measurements of the steady flow resistance as well as the impedance of the sheet are described in Appendix A, where some experimental data are shown and discussed. If the sheet is not moving, the velocity through the sheet, i.e., the relative velocity of the air with respect to the sheet, will be the same as the absolute velocity of the air just in front and just behind the sheet, and a measurement of the ratio of the amplitudes of the pressure drop and the air velocity then yields, by definition, the interaction impedance. If the sheet is not rigid but induced to oscillate as a result of the interaction with the sound, the velocity amplitude of the air outside the sheet will not be the same as the amplitude of the relative motion of the air and the sheet, and an impedance defined in terms of the absolute rather than the relative velocity amplitude will contain

SHEET ABSORBERS

57

an inertial contribution resulting from the mass of the screen, as discussed later in this chapter. The corresponding impedance, based on the absolute rather than the relative velocity, will be called the equivalent impedance. By making the mass of the screen large enough, the induced motion can be made negligible; the screen becomes immobile or ‘rigid.’ The presence of an inertial component in the equivalent impedance of the sheet is easy to understand because of the induced motion of the sheet, but an inertial component of the interaction impedance (the sheet is stationary) is a more subtle matter. A detailed analysis of an oscillatory flow (sound) in a porous material shows that there are actually two contributions to the inertial part of the impedance. One is due to the reaction force between the fluid and the structure as the fluid is forced to change direction and velocity along a tortuous path through the material. This force is proportional to the acceleration of the fluid, and thus acts like an inertial reactance and an apparent increase in the inertial mass density of the fluid. The flow resistive force is proportional to the velocity. Actually, there is a second contribution to the inertial reaction force on the fluid. It stems from the oscillatory friction force on the material, which is not quite in phase with the air velocity. The corresponding reaction force on the fluid from the boundary then contains a component proportional to the acceleration, i.e., an inertia component, which can be interpreted as a contribution to the induced mass of the fluid. Of course, there is also the inertia of the air itself within the sheet but in the present treatment of the sheet as a very thin layer, it will be neglected. For a rigid sheet, the assumption of being purely resistive is often quite good over the frequency range of most interest in noise control applications. One finds, for example, that for cloth-like sheets, the resistance typically exceeds the mass reactance at frequencies below 3000 Hz, but this frequency limit depends on the fiber diameter and the structure of the cloth. For a flexible sheet, however, the acoustically induced motion often contributes a significant inertial component to the impedance, particularly if the flow resistance is large.

3.2.2 Resonances and Anti-Resonances The resistance of a porous sheet varies only weakly with frequency, typically as the square root of frequency. The sheet reactance always represents an inertia. However, the reactance of the air layer between the sheet and the rigid backing wall can be either inertia- or stiffness-like. If the sheet is purely resistive, the reactance of the absorber is contributed only by the air layer and is zero when its thickness L is an odd number of quarter wavelengths yielding the ‘quarter wavelength resonances’ of the absorber. The sheet then finds itself in the plane, where the velocity amplitude is a maximum in the standing wave, which is established in the air layer. At resonance, the input impedance of the absorber is then the impedance of the sheet alone, and if the sheet resistance is chosen to equal the wave impedance ρc of a plane wave at normal incidence, impedance matching, 100 percent absorption results, and no sound will be reflected (at an angle of incidence φ, the corresponding

58


value is ρc/ cos φ). If the sheet itself contributes a reactive impedance component, the resonance occurs at a frequency slightly lower than the quarter wavelength value; the stiffness-like reactance of the air layer is then canceled by the inertial reactance of the sheet. On the other hand, at frequencies for which the layer thickness is an integer number of half wavelengths, the sheet is located in a plane where the velocity amplitude is zero and the pressure amplitude a maximum, and the layer impedance is infinite. No absorption occurs and we have an anti-resonance. For sound at oblique incidence, the impedance of the air backing depends on whether or not it is honeycombed, as indicated in the figure. By ‘honeycombed’ is meant that the air backing is divided into cells by partitions and that the cell size is much smaller than the wavelength (acoustically compact). With honeycombed backing, the air motion in the layer is forced to be perpendicular to the wall, and the input impedance will be the same as for normal incidence. The air motion in a cell depends only on the sound pressure at the surface of this cell and is not affected by the sound pressure at other cells (there is no acoustic communication between them). Therefore, the input impedance of the boundary, i.e., the ratio of the sound pressure amplitude at the boundary and the normal component of the velocity amplitude, will be independent of the angle of incidence. The surface is then said to be locally reacting. Sheet absorber, Input impedance, Local reaction ζi ≡ θi + iχi = ζ + i cot(kL)

(3.1)

k = ω/c = 2π/λ. ζ = θ + iχ : Normalized sheet impedance. If purely resistive: χ = 0. L: Air layer thickness (Figure 3.1). (See also Section 3.6, Eq. 3.19.) For an absorber which is not honeycombed, the motion of the air in the backing layer will be a superposition of waves traveling in the same directions as the incident and reflected wave outside the absorber, and it will depend not only on the local sound pressure amplitude but on the distribution over the entire absorber. The absorber is then called nonlocally reacting. It will perform like a locally reacting absorber at normal incidence, of course, but at oblique incidence important differences occur. As for the locally reacting absorber, there still will be a standing wave in the air backing, but the distance between adjacent pressure minima (or maxima), i.e., the spatial half period in the direction normal to the boundary, will be greater than the half wavelength of the incident sound. It will be half a wavelength for normal incidence, but it will go to infinity at grazing incidence when the wave fronts are perpendicular to the boundary so there is no periodic variation in a direction normal to the boundary, i.e., parallel with the wave fronts. At an arbitrary angle of incidence, the spatial period perpendicular to the boundary varies as λx = λ/ cos φ, where φ is the angle of incidence (the angle between the direction of propagation and the normal to the boundary, Figure 3.1) and λ is the wavelength of the incident sound. The corresponding x-component of the propagation constant is kx = k cos φ = 2π/λx . As far as the sound is concerned, it ‘measures’ the thickness of the air layer in units of the spatial period λx , and the thickness of the absorber thus appears to the incident

59

SHEET ABSORBERS

sound to decrease with increasing angle of incidence. The first quarter wavelength resonance now occurs when L = λx /4, the condition for maximum absorption, and if L = λx /2, we have an anti-resonance and no absorption. Note that the corresponding frequencies will be higher than for normal incidence. For a given L, the anti-resonance can only happen at one angle of incidence; the absorption at other angles will not be zero and this means that, unlike the locally reacting absorber, the nonlocal absorber cannot have a vanishing diffuse field absorption coefficient (which is an average over all angles of incidence) and the input impedance is dependent on the angle of incidence. Sheet absorber, Input impedance, Nonlocal reaction ζi ≡ θi + iχi = ζ + i(1/ cos φ) cot(kx L)

(3.2)

L: Layer thickness. φ: Angle of incidence, Figure 3.1. k = ω/c = 2π/λ, kx = k cos φ. ζ = θ + iχ : Normalized sheet impedance, Eq. 3.1. (See also Section 3.6, Eq. 3.20.) If the cell size in the honeycomb is not much smaller than the wavelength, the calculation of the input impedance becomes considerably more complicated and will not be considered here. The absorption spectrum then falls between those for the locally and nonlocally absorber, closer to the former at low frequencies and to the latter, at high.

3.2.3 Absorption Spectra Recall that the ratio of the amplitudes of sound pressure p and fluid velocity u in a plane traveling wave is ρc, where ρ is the density and c the sound speed, the wave impedance of the fluid. At an angle of incidence φ, the velocity normal to the boundary is ux = u cos φ, and the corresponding ‘component’ of the wave impedance is p/ux = ρc/ cos φ. If the sound encounters a boundary with an input impedance, which is different from this impedance, the boundary does not ‘feel’ like free field to the incoming sound and reflection will occur. The reflection coefficient R is the ratio of the reflected (complex) pressure amplitude at the boundary and the incident amplitude and, like the impedance, is a complex quantity. The corresponding absorption coefficient, the ratio of the reflected and incident intensities, is shown in Eq. 3.22, and is repeated here. Pressure reflection coefficient, Absorption coefficient R = (ζi cos φ − 1)/(ζi cos φ + 1) α(φ) = 1 − |R|2 =

(3.3)

4θi cos φ (1+θi cos φ)2 +(χi cos φ)2

ζi = θi + iχi : Normalized input impedance (Eqs. 3.1, 3.2). φ: Angle of incidence, Figure 3.1. see Eq. 3.22. (See also Section 3.6, Eq. 3.21.) As mentioned earlier, the input impedance of a sheet absorber at a quarter wavelength resonance is simply the sheet resistance, and if this is chosen to be ρc/ cos φ, 100 percent absorption will occur at resonance and an angle of incidence φ, as can be

60


seen for normal incidence (φ = 0) in Figure 3.2. This is the case of ‘impedance matching.’ At other frequencies and/or values of the sheet resistance, reflection occurs. Averaging the absorption coefficient over all angles of incidence in a diffuse field, we obtain the Diffuse field absorption coefficient π/2 α(φ) cos φ sin φ dφ αst = 0 π/2 = 2 0 α(φ) cos φ sin φ dφ π/2 0

(3.4)

cos φ sin φ dφ

α(φ): Absorption coefficient, Eq. 3.3. φ: Angle of incidence, Figure 3.1. (See Eq. 3.23.) The computed absorption spectra of a rigid, purely resistive sheet-cavity absorber with and without partitioned air backing are shown in Figure 3.2 for both normal incidence and diffuse field and for normalized sheet resistances θ of 1, 2, and 4. Note that the frequency parameter is expressed in normalized form as L/λ, where L is the layer thickness and λ the free field wavelength. We refer to this nondimensional presentation loosely as ‘universal,’ and it is chosen here so that the quarter wavelength resonances and the half-wavelength anti-resonances readily can be spotted. If instead an explicit frequency dependence is desired, with the frequency in Hz rather than the nondimensional parameter L/λ, one curve is needed for each layer thickness L, whereas here, one curve will do (hence, the designation ‘universal’). With a normalized sheet resistance θ = 1, the resonance absorption coefficient at normal incidence is 1, as we have seen, but it will be less than 1 if θ > 1. However, for oblique incidence, on the other hand, it will be 1 for waves with an angle of incidence given by cos φ = 1/θ . With θ < 1, the absorption coefficient is less than unity for all angles of incidence, and it follows that to optimize the diffuse field absorption coefficient, θ should be >1. If the cavity depth is an integer number of half wavelengths, the absorption coefficient becomes zero for the locally reacting absorber for both normal incidence and in diffuse field. In either case, the input impedance is infinite. The appearance of the spectra in the vicinity of these anti-resonances depends on the bandwidth used in the presentation, i.e., on the number of frequencies used in computing the graph. To get a ‘clean’ zero of the absorption coefficient in the graph requires a very narrow bandwidth. In this case, the bandwidth is approximately 1/60th of an octave (400 data points in L/λ range from 0.01 to 1, which covers 6.6 octaves). At a given frequency away from resonance so that the impedance is not purely resistive, there is an optimum resistance for maximum absorption. As shown in Section 3.6, this occurs when the resistance is approximately equal to the reactance. 1/3 and 1/1 Octave Band Average Absorption The ‘universal’ absorption spectra in Figure 3.2 are useful in general discussions of absorption since they are valid for all values of the thickness L of the air layer by the use of the frequency parameter L/λ. In practice, however, it is more convenient to have frequency in Hz as a variable.

SHEET ABSORBERS

61

Figure 3.3: Absorption spectra of a rigid resistive sheet backed by a 4 inch air layer and rigid wall. The sheet resistance is 1 ρc. The graphs refer to 1/12, 1/3, and 1/1 octave bands, respectively. In each graph, the three curves, starting from the top (at about 800 Hz), correspond to normal incidence, diffuse field local reaction, and diffuse field, nonlocal reaction. In the top graph in Figure 3.3, which concerns a rigid sheet-cavity absorber with a normalized flow resistance of 1 and a cavity depth of 4 inches, 121 values of the frequency have been used evenly distributed logarithmically over the frequency range 10 to 10000 Hz, approximately 10 octaves. This corresponds to an approximate bandwidth of 1/12th octave in the top graph, which is narrow but not as narrow is in Figure 3.2. In engineering applications, however, the average absorption in an even wider frequency band, typically 1/3 and 1/1 octaves, is generally of more interest, and the overall appearance of the absorption curves can then be markedly different, as can be seen in Figure 3.3. The average in a band was calculated from the absorption coefficients at 7 equally spaced (logarithmically) frequencies in a 1/3 octave band and at 13 frequencies in a full octave band (see Eq. 3.29). In the example in Figure 3.3, the graphs, starting from the top, refer to bandwidths 1/12, 1/3, and 1/1 octaves; the center frequencies of the bands are marked. Notice that only the narrow band curves in the top graph show clean zeroes at the anti-resonances. Incidentally, to check these, we note that at the first anti-resonance the wavelength is twice the air layer thickness, i.e., λ = 8 inches and the frequency 1680 Hz, which checks. The frequency at the quarter wavelength resonance is half of this value, i.e., 840 Hz, consistent with the location of the first absorption peak in the top graph. Notice that with a bandwidth of 1/3 octave, there are no clean zeroes at the antiresonance frequencies and for the 1/1 octave absorption spectrum, these frequencies cannot be identified.

62


3.2.4 Wire Screens In the analysis so far in this chapter, the absorption coefficient has been expressed in terms of the impedance of the sheet or screen involved, and it has been implied that this impedance is an experimentally known quantity. Furthermore, a purely resistive sheet has often been assumed and used in numerical examples. It is of interest to check this assumption by using the computed impedance of a wire mesh screen as given in Chapter 3, and it is of general interest in design work to be able to calculate the impedance, at least approximately, in terms of geometrical parameters of the screen or sheet.

3.2.5 Effect of Honeycomb Cell Size Two main kinds of absorbers have been considered in the text, locally and nonlocally reacting, i.e., honeycombed and not honeycombed. However, a liner can be nonlocally reacting even if honeycombed if the cell size in the honeycomb is not small compared to the wavelength. Measurements of the diffuse field absorption coefficient of sheet absorbers with air backing, partitioned into square cells, 2 ft by 2 ft, have shown that the absorption spectrum generally falls between those of locally (partition spacing small compared to a wavelength) and nonlocally reacting absorbers (no partitions), closer to the former at low frequencies and to the latter at high frequencies. An example is shown in the left graph in Figure 3.4. The absorber in this case consists of a resistive sheet in contact with a perforated facing of thickness 0.15 inches, hole diameter 3/16 inches, and hole spacing 1/2 inches (square pattern). The flow resistance of the combination was 1.7 ρc. The thickness of the air backing was 2 inches. The solid line is the measured and the dashed lines, the computed. The curve with the dots between the dashes refers to local reaction. The trend mentioned above is apparent; low frequency performance approaching that of local reaction and high frequency performance that of nonlocal reaction. As a reference, we note that the frequency at which the partition spacing is half a wavelength is 280 Hz, which normally is the frequency below which local reaction can be assumed. 1.0

1.0

0.8

0.8

0.6

0.6

f0

AV

0.4

0.4

f 0 = 870 CPS k L = 0.79 0 =1.7 MEASURED

0.2

1AV

f0 = 270 CPS k0 L = 1.26

0.2

= 1.5

f0

CALCULATED

2AV

0 100

2

3

4

5 6 7 8 9 1000 ν CPS

MEASURED 2 AV

2

3

4

5

6000

0 100

2

3

4

5

6

7

8 9

ν CPS

1000

2

CALCULATED 3

4

5

6000

Figure 3.4: Measured absorption spectra of a resistive sheet/perforated facing combination with air backing compared with computed values for local and nonlocal reactions. Left: Backing depth: 2 inches, Right: 10 inches.

63

SHEET ABSORBERS

The spectrum on the right in the figure refers to an absorber with a relatively low resonance frequency, with an air backing of 10 inches but with the same partition spacing as before. Now, the absorption is high at relatively low frequencies for which the wavelength is large enough (in comparison with the partition spacing) to qualify the absorber as nearly locally reacting. Indeed, the calculated absorption spectrum for a locally reacting absorber agrees well with the measured.

3.2.6 Examples and Comments 1. Effect of cell size in a partitioned air backing Consider first an absorber, such as a resistive sheet with air backing with partitions only in one direction, perpendicular to the yx-plane and with a spacing which is not necessarily small compared to a wavelength. In other words, except for the sheet, the boundary is like a diffraction grating or a slot absorber. Try to calculate the frequency dependence of the reflection and absorption coefficient of the boundary at a given angle of incidence using whatever approximations you feel reasonable. This is not a simple problem, and, as far as we know, has not been analyzed. For someone who has a mathematical inclination and wants to sharpen his or her mathematical tools and physical insights, this could be an interesting project. 2. Rigid sheet absorber for maximum NRC A rigid resistive sheet is backed by a 4 inch air layer and a rigid wall. By repeated computations (iterations), find the (normalized) flow resistance of the sheet, which gives the highest possible diffuse field noise reduction coefficient, NRC, when the absorber is locally reacting (partitioned air backing). For this optimum flow resistance, show the narrow band and octave band absorption curves in the frequency range from 10 to 10,000 Hz. SOLUTION The NRC is the average of the octave band absorption coefficients in the bands 250, 500, 1000, and 2000. We have to distinguish between 3 different values corresponding to normal incidence (NRC0), diffuse field for local reaction (NRC1), and diffuse field for nonlocal reaction (NRC2). The dependence of NRC on the resistance is not very strong in the vicinity of the optimum and, to within one percent, a maximum value of the diffuse field value, local reaction, is found to be NRC1 = 0.81 for values of the resistance between 1.6 and 2.3. The corresponding NRC values for normal incidence and diffuse field, nonlocal reaction, are 0.74 and 0.63, respectively. The narrow and octave band absorption spectra that correspond to a sheet resistance of 2 are shown in the Figure 3.5. 3. The noise reduction coefficient (a) Find the optimum resistance for maximum NRC0, NRC1, and NRC2. (b) In all cases make plots of the NRC-values vs the flow resistance.(c) Explain

64


Figure 3.5: Left: OB absorption spectra. Right: 1/12 OB spectra.

why at high frequencies the diffuse field octave band absorption coefficient is almost independent of frequency for the nonlocally reacting absorber, but not for the locally reacting. (d) What is the smallest weight of the sheet for which it can be regarded as acoustically rigid when the normalized flow resistance is 2? 4. Double sheet absorber (a) The outer screen in a double screen absorber is 8 inches from the rigid backing wall and the inner screen, 4 inches. The screens are identical, each with a normalized flow resistance of 1 and a weight of 0.2 lb/ft2 . Determine the noise reduction coefficients. Compare these values with those obtained if both screens are put together and placed 8 inches from the wall. (b) Is it possible to improve the NRC1 value by using another placement of the screens within the constraint of a total absorber thickness of 8 inches? If so, give these locations and the corresponding NRC1 value. SOLUTION (a) With the screens 4 inches apart, as indicated, the NRC values for normal incidence, diffuse field local reaction, and diffuse field, nonlocal reaction, are 0.90, 0.90, and 0.78, respectively. With the screens put together, 8 inches from the wall, the corresponding values are 0.75, 0.81, and 0.71. In the latter case they are treated as a single screen with twice the flow resistance and twice the weight. (b) Yes. By choosing the screen distances from the wall 8 and 6 inches, we get NRC1 = 0.91. The other values are then NRC0 = 0.88 and NRC2 = 0.78. 5. Optimization of the absorption spectrum (a) In some applications it is not always the NRC that is relevant. Often in noise control problems it is at low and mid frequencies where absorption is needed. Optimize the design for the 125 Hz octave band. (b) Carry out the analogous optimization for screen placement also for NRC0 and NRC2.

65

SHEET ABSORBERS

(c) Given the total thickness of a two screen (limp) absorber, determine the optimum resistances, weights, and placements to yield maximum NRC1. This is an ambitious undertaking and probably does not lead to a unique answer. However, it might be worth playing around with to get some design guidelines. (d) Extend the study to include 3 and 4 sheets with the same total thickness of the absorber as above and do some computer experiments to find optimum sheet parameters. 6. Effect of temperature and nonlinearity on sheet absorber Use the absorber described in the sample run, i.e., a perforated plate/sheet absorber. Cavity depth: 4 inch. Perforated plate: Open area: 5 percent. Thickness and hole diameter: 0.05 inch. Weight: 2 lb/ft2 . Repeat the determination of the octave band absorption spectra, now with a temperature of 1000◦ F. SOLUTION The results obtained are shown in Figure 3.6. (Compare the results obtained at 70◦ F.) At 1000◦ F, the flow resistance is larger than at 70◦ F, since both the shear viscosity and the wave impedance of air depends on temperature. As shown in Chapter 5, the normalized flow resistance increases approximately as T , where T is the absolute temperature (Kelvin). Thus, at 1000◦ F, T ≈ 811 K and at

Figure 3.6: Level dependence of absorption spectra at 1000◦ F.

66

NOISE REDUCTION ANALYSIS 70◦ F, ≈ 294, and the increase in the normalized flow resistance due to the temperature increase will be by a factor of 811/294 ≈ 2.76. The increase provides an improved impedance match of the absorber at 80 dB and the absorption is improved. At 1000◦ F, the resistance is already high enough at 80 dB for a good impedance match, and the increase in sound pressure does not alter the absorption substantially. The example serves to support what is already expressed in the text that temperature can play an important role in the design of absorbers.

3.3 FLEXIBLE POROUS SHEET WITH CAVITY BACKING The acoustically induced motion of a sheet can have a pronounced effect on absorption particularly when the flow resistance is larger than the mass reactance of the sheet. The effect will be studied for a limp sheet, but the results obtained can readily be extended to apply to a sheet with its own resonances.

3.3.1 The ‘Equivalent’ Impedance When the sheet is mobile, the interaction between a sound wave and the sheet results in an acoustically induced velocity u of the sheet so that the air velocity amplitude through the sheet (relative to the sheet) will be different from the absolute amplitude u of the air at the two sides of the sheet. The velocity through the sheet is the relative velocity ur = u − u . Based on these three velocities, ur , u , and u, we can define three different impedances of the sheet as the ratio of the pressure amplitude drop across the sheet and each of these velocities. The interaction impedance and the corresponding flow resistance, which were introduced in the previous section, refer to the relative velocity ur , the structural impedance refers to the velocity u of the sheet, and the equivalent impedance to the absolute air velocity amplitude u. The equivalent impedance will be denoted by z with the real and reactive parts being r and x . Even for a purely resistive sheet, the equivalent impedance will contain a mass reactive part due to the induced motion of the sheet. The prime, as used for the velocity of the sheet, serves as a reminder that z accounts for the induced motion of the sheet. In terms of an electrical circuit analogy, the equivalent impedance z can be thought of as the parallel combination of the interaction impedance z = r + ix and the structural impedance zs ≈ −iωm. Flexible sheet absorber, Equivalent sheet impedance z ≡ r + ix ≡ p/u = zzs /(z + zs )

(3.5)

z = ρcζ : Sheet impedance (Eq. 3.1). zs : ‘Structural’ impedance zs = −iωm for a limp sheet. m: Sheet mass per unit area. z : Accounts for induced motion. (See Section 3.6, Eq. 3.33.) The structural impedance depends on the elastic properties, tension, and mounting of the sheet but for a limp sheet without tension its magnitude is simply the mass

SHEET ABSORBERS

67

reactance ωm, where m is the mass of the sheet per unit area and ω the angular frequency. Most of our discussion will concern a limp sheet for which the degree of induced motion depends on the ratio r/ωm of the flow resistance r and the reactance ωm. In that case, the induced motion becomes most pronounced at low frequencies. At a sufficiently high frequency or sheet mass the sheet becomes essentially immobile and behaves as discussed in the previous section. The induced motion affects both the equivalent resistance r and reactance x of the sheet. Figure 3.7 shows the computed frequency dependence of r /r and |x |/r, where r is the flow resistance of the sheet. The frequency is normalized with respect to the characteristic angular frequency fm = (1/2π )r/m at which the flow resistance r (the interaction resistance) equals the mass reactance of the sheet. As the frequency goes to zero, the equivalent resistance goes to zero because the induced velocity of the sheet approaches the absolute velocity of the air and the entire pressure drop across the sheet is used up for the inertial reactance of the sheet, i.e., for the acceleration of the sheet. The air velocity is then 90 degrees out of phase with the pressure drop. The pressure drop caused by the in phase component of the velocity is essentially zero. As the frequency increases, the inertia of the sheet reduces the induced motion and the sheet response approaches that of the rigid sheet. In this (high frequency) limit, the equivalent flow resistance r approaches the flow resistance r of the sheet so that r /r approaches 1, as shown. At the characteristic frequency fm , the equivalent resistance r is half of the flow resistance r. The equivalent inertial reactance is represented by the bell-shaped curve in the figure. It is zero in the limit of both low and high frequencies. In the low frequency end this is because the frequency goes to zero and in the high end because the sheet is essentially immobile so that the velocity through the sheet is in phase with the pressure drop with no 90 degrees out of phase reactive component. The largest possible equivalent mass reactance of a limp porous sheet is half of its flow resistance, i.e., r/2, and it occurs at the characteristic angular frequency ωm = r/m.

Figure 3.7: The frequency dependence of the equivalent resistance r and reactance x of a limp sheet (accounting for the induced motion). These quantities are normalized with respect to the flow resistance r of the sheet. The frequency is normalized with respect to the characteristic frequency fm = (1/2π)r/m at which r = ωm m (see Eq. 3.33).

68


In the discussion above, it was implied that the interaction impedance z = r + ix is purely resistive, i.e., x = 0. In reality, there is also a mass reactive component (negative x), which is normally insignificant at low and middle range frequencies. If this reactance is accounted for, the equivalent impedance becomes a bit more complicated (see Eq. 3.33). Effect of Bending Stiffness and Structural Resonances of the Sheet For a limp sheet, the magnitude of the structural reactance is simply ωm. If the sheet is under tension or if it has bending stiffness, it will have structural resonances at which the structural impedance is very low so that the velocity amplitude of the sheet will be almost the same as that of the air. The equivalent resistance of the sheet then will be small and the absorption will have a (narrow) dip in the vicinity of a resonance. However, this is usually of little practical interest. As indicated in Section 3.6, the mathematical analysis becomes somewhat involved but the outcome generally can be said to be of relatively little importance.

3.3.2 A Low Frequency Resonance From a practical standpoint, the importance of the limpness of a porous sheet is that in a sheet-cavity absorber, the resonance absorption occurs at a frequency lower than the quarter wavelength resonance familiar from the discussion of the rigid sheet absorber in the previous section. Furthermore, the equivalent resistance of the sheet can be considerably lower than the flow resistance of the sheet, as illustrated in Figure 3.7, and, depending on the magnitude of the flow resistance, this can lead either to an increase or a decrease in sound absorption, as will be shown shortly. With a proper combination of flow resistance and mass of the sheet, 100 percent resonance absorption can be obtained at normal incidence. In many noise control applications, the absorption of low frequency sound is the most difficult to achieve, and the possibility of utilizing the flexibility of a sheet for the purpose of bringing the absorption maximum to a lower frequency can be of practical importance. For a qualitative understanding of this resonance, let us start with an impervious limp sheet. The air backing behind the sheet acts like a spring, and the resonance of the system is that of an ordinary mass-spring oscillator. Thus, the resonance frequency, in principle, can be made as low as we wish by making the mass sufficiently large. The problem is that if the sheet is impervious, there will be no sound absorption. However, with a sheet with a high but finite flow resistance r it is expected to have approximately the same resonance frequency as the impervious sheet. Due to the induced motion of the sheet, the equivalent resistance r will be lower than the flow resistance r (see Figure 3.7) and with a proper choice of parameters, it should be possible to make the equivalent flow resistance equal to ρc so that 100 percent resonance absorption will result. Rule of Thumb for 100% Absorption at the Lowest Resonance The analysis in Section 3.6 shows that for a sheet weight much larger than the weight of the air in the backing layer (at least by a factor of 5), an approximately 100 percent

SHEET ABSORBERS

69

normal incidence resonance absorption is obtained at the low frequency resonance if the normalized flow resistance of the sheet equals the ratio of the weight of the sheet and the weight of the air layer. The corresponding resonance frequency turns out to be lower than the ordinary quarter wavelength resonance by a factor approximately equal to π/2 times the square root of the weight ratio. Thus, with a weight ratio of 10, this factor is approximately 5. The penalty for the reduction in the resonance frequency is a narrower absorption peak. These observations are illustrated by the computed absorption spectra discussed in the following section (see Eq. 3.33).

3.3.3 Absorption Spectra For an absorber consisting of a rigid resistive sheet backed by an air layer and a rigid wall, the lowest resonance occurs when the thickness of the air layer backing is one quarter wavelength. For example, with a 4 inch cavity, this frequency is ≈ 840 Hz at room temperature, and 100 percent absorption at resonance is obtained if the flow resistance of the sheet is chosen to be ρc, i.e., 420 MKS. For the limp sheet, as indicated above, the first resonance can be made to occur at a frequency considerably lower than the quarter wavelength resonance frequency and if the parameters are chosen properly, the absorption coefficient at resonance can be 100 percent. As a start, the proper design parameter can be obtained from the ‘weight ratio’ rule of thumb, given above, but numerical parametric studies can easily be performed to establish the range of validity of this simple rule. The absorption coefficient is then computed in the same way as for the rigid sheet absorber (Eqs. 3.19, 3.20, and 3.23) but with the equivalent impedance of the sheet used instead of the interaction impedance, thus accounting for the induced motion of the sheet. Some computed absorption coefficients vs the layer thickness-to-wavelength ratio L/λ for a limp porous sheet absorber are shown in Figures 3.8 and 3.9. The graphs in each figure refer to sheet-to-air layer weight ratios of 4, 8, and 16, and the curves in each graph correspond to different flow resistances of the sheet, from 1 to 16. Figure 3.8 applies to sound at normal incidence and it is of particular interest to note the appearance of the low frequency resonance we discussed above. It occurs for relatively large weight ratios. These results are consistent with the rule of thumb which says that if the normalized flow resistance equals the weight ratio, 100 percent absorption is obtained at the resonance. For example, for the weight ratio mr = 8 and a normalized flow resistance of θ = mr = 8, the peak absorption is ≈ 100 percent and it occurs at L/λ ≈ 0.058, which is lower than the quarter wavelength resonance value 0.25 by a factor of about 4.3. According to the rule, this ratio is √ ≈ (π/2) mr ≈ 4.4. In a diffuse field, the locally reacting absorber (Figure 3.9) yields results, which are similar to those for normal incidence, exhibiting zero absorption when the thickness of the air backing is an integer number of half wavelengths. A notable feature is that for a sheet resistance of 1, the absorption is less than for a resistance of 2 at practically all frequencies and for this reason the absorption curve for unit resistance has been included only for the mass ratio 4. Generally, the best overall absorption is obtained when the normalized resistance lies between one quarter to one half of the mass ratio.

70


Figure 3.8: Normal incidence absorption spectra of a limp resistive sheet-cavity absorber. Weight ratios, sheet/air layer: 4, 8, and 16. Normalized sheet (flow) resistance values: 1 to 16, as indicated. (For explicit frequency dependence, see Appendix C.)

For example, with a mass ratio of 16, a resistance of 4 gives an almost flat absorption curve at about 80 percent between L/λ = 0.05 and 0.4. For the nonlocally reacting absorber, the diffuse field absorption peaks are markedly lower than for normal incidence and for a locally reacting absorber in a diffuse field but the absorption coefficient does not go to zero at any frequency (Figure 3.9). As mentioned earlier, this is due to the fact that the input impedance is angle dependent and an anti-resonance cannot occur at the same frequency for all angles of incidence. Even so, the performance of the nonlocally reacting absorber is generally inferior to that of the locally reacting absorber.

3.3.4 Examples and Comments 1. Limp sheet absorber for maximum NRC Consider the same situation as in Problem 2, in Section 3.2.6, but the sheet is now limp with a weight of 0.2 lb/ft2 . (a) Again, determine the maximum NRC1 that can be obtained with this absorber by varying the flow resistance of the sheet. (b) What would be the resonance frequency of the absorber if the sheet were impervious?

71

SHEET ABSORBERS

Figure 3.9: Diffuse field absorption spectra of a limp resistive screen-cavity absorber. Left: Locally reacting. Right: Nonlocally reacting. Weight ratios, screen/air layer: 4, 8, and 16. Normalized screen resistance values: 1 to 16, as indicated.

SOLUTION (a) We find from that for a flow resistance between 2.2 and 2.3, the NRC1 is 0.85, i.e., somewhat larger than for the rigid sheet in Problem 2. The corresponding values NRC0 and NRC2 for normal incidence and diffuse field, nonlocal reaction, are 0.78 and 0.67, respectively. (b) Resonance occurs when the total input reactance of the absorber is zero, i.e., when the mass reactance of the sheet is canceled by a stiffness reactance from the air layer. If the mass per unit area of the sheet is m, the depth of the air

72


Figure 3.10: Left: Hard contact (laminate). Right: Loose contact. layer, L, the angular frequency, ω = 2πf , and the sound speed, c. This leads to the equation for the resonance frequency ωm/ρc = cot(ωL/c).

(3.6)

However, rather than solve this equation numerically, we can use a computer program to simulate the impervious membrane by making the flow resistance large enough so that the sheet for practical purposes is impervious. Thus, running the program with a resistance of 200, say, we get a resonance at ≈ 190 Hz. We can confirm that this indeed satisfies Eq. 3.6. A weight of 0.2 lb/ft2 corresponds to a mass m = 0.098 g/cm2 . Then, with L = 10.2 cm, c = 34000 cm/sec and ρc = 42, the left-hand side of the equation becomes 2.78 and the righthand side 2.67. This means that the resonance frequency is somewhat below 190 Hz. It should be noted also that if we had assumed the wavelength at resonance to be much larger than the cavity depth, so that ωL/c > L and f > 1, the absorption cross section becomes frequency independent and depends only on the sheet resistance. As was the case for normal incidence, there is an optimum value of the flow resistance for maximum absorption also in a diffuse field. In the high frequency limit, where the sheet is essentially immobile, this value is found to be ≈ 3.2. The corresponding maximum absorption cross section is ≈ 0.95 per unit area of sheet material. The corresponding one-sided diffuse field absorption coefficient of the sheet is half of this value, i.e., 0.475. These values, 3.2 for resistance and 0.475 for the absorption coefficient, should be compared with 2 and 0.5 for normal incidence on one side of the rigid sheet in Figure 3.19.

SHEET ABSORBERS

85

Figure 3.20: Absorption area per unit area of sheet material for a purely resistive limp sheet in a diffuse sound field, accounting for the fact the sound is incident on both sides of the sheet. The frequency is normalized with respect to fm = r/2πm, where r is the flow resistance and m the mass per unit area of the sheet material. (An absorption cross section of unity then corresponds to an absorption coefficient of 0.5 for the sound that is incident on one side.) Normalized flow resistance: 0.4, 0.8, 1.6, 3.2, and 6.4 (see Eq. 3.75).

3.5.3 Finite Sheet, Effect of Diffraction For a sheet of finite dimensions, the effect of diffraction can reduce the absorption significantly at wavelengths large compared to the dimensions of the sheet. Qualitatively, this can be understood by considering steady, inviscid flow about an object such as a circular, impervious, immobile sheet. At normal incidence, the velocity and pressure distributions over the disk will be the same on both sides so that there will be no pressure difference across the sheet that tends to force the flow through the sheet. For oscillatory flow, the pressure difference will not be zero, although it does go to zero as the frequency goes to zero. The reason why there will be a pressure difference and a net force amplitude on the sheet in oscillatory flow has to do with the distortion of the flow by the sheet and is dynamically equivalent to an increase of the inertial mass of the fluid in the region of the sheet. This ‘induced’ mass is similar to that experienced when moving an object (your hand, for example) back and forth in water. With the induced mass denoted by mi , the force amplitude on the disk will be fi = ωmi u, where u is the amplitude of the relative velocity of the disk and the fluid. Our approximate analysis along these lines2 involves a circular porous disk at wavelengths, λ, large compared to the disk diameter d and can be regarded as a low frequency approximation in the calculation of the absorption cross section to supplement the previous analysis, where the sheet dimension was assumed large compared

2 P. M. Morse and K. U. Ingard, Linear Acoustic Theory, Volume XI/1 in Handbuch der Physik, Springer-

Verlag, 1961.

86


to the wavelength. The absorption cross section given here accounts for absorption of sound impinging on both sides of the disc. Absorption cross section, Low frequency approximation σa ≈ A|ur /u|2 (64/π 3 ) (ka)2 θ/|ζ + iχi |2

(3.16)

a: Radius of porous disc (sheet). k = ω/c. A: Area of disc. m = mass per unit area of disc. r = θρc = flow resistance. ur /u: = 1/(1 + ir/ωm) (see Eq. 3.85). ζ : Equivalent normalized disc impedance (see Eq. 3.33). χi = −i(8/(3π ))ka: (k = ω/c) (see Section 3.6, Eqs. 3.86 and 3.87). Whether or not the flow will go through the disc rather than around it depends on the ratio of the normalized average mass reactance ωmi /π a 2 per unit area and the equivalent impedance of the sheet. For a purely resistive and immobile sheet, the condition for a substantial flow-through to occur is ωmi /ρc > θ, where θ is the normalized flow resistance of the sheet. The induced mass of the finite sheet plays about the same role as the mass per unit area of the infinitely extended sheet as far as the reduction of absorption cross section at low frequencies is concerned. Actually, for a sheet with a diameter of 1 ft, the induced mass per square foot corresponds to a weight of 0.08·4/3π ≈ 0.035 lb/ft2 (≈ 5 oz/yd2 ), where we have used 0.08 lb/ft3 for the density of air. This is about the same as the weight of a typical woven sheet, and the induced motion of the sheet then plays the same role as diffraction in reducing the low frequency absorption. Thus, in this case, no significant improvement in the low frequency absorption would result by making the sheet heavier than 0.035 lb/ft2 since the absorption would be dictated by effect of diffraction. For more details, we refer to Section 3.6. The optimum values of the flow resistance discussed in connection with the results for infinite sheets are no longer valid for a finite sheet. The optimum normalized resistance in the low frequency regime for a sheet of radius a is approximately equal to ka = 2πa/λ, i.e., directly proportional to the diameter of the sheet. Figure 3.21 shows an example of the calculated frequency dependence of absorption cross section per unit area of sheet material for a finite sheet, and in the same figure is also shown the corresponding result for an infinite sheet. At low frequencies, the lower curve applies and at high frequencies, the upper. These curves have to be joined smoothly to yield the complete absorption curve. The weight is 0.05 lb/ft2 , typical for a woven cloth, and the normalized flow resistance of the material is 1.0. The diffraction limits the absorption at frequencies below ≈ 800 Hz. In order for the effect of diffraction to be relatively small, the diameter of the sheet in this case should not be less than 3 ft. It is interesting to compare the performance of a thin sheet with that of a solid porous layer with an impervious barrier in the center so as to create two porous half layers backed by the rigid barrier. With a total layer thickness of 1 inch, the two sided absorption area will not be much different from that of the sheet.

87

SHEET ABSORBERS

Figure 3.21: Two-sided absorption cross section per unit area of a circular porous disc with a diameter of 2 ft, a weight of 0.05 lb/ft2 , and a normalized flow resistance of 1.0. The lower curve is the low frequency approximation (see Eq. 3.16), in which diffraction limits the absorption at low frequencies. The upper curve refers to the infinite sheet considered in Figure 3.20 (see Eq. 3.75).

A rigorous analysis of the absorption and scattering from a finite porous limp sheet is not available, as far as we know, and it represents a challenge in mathematical acoustics; a challenge which is not only interesting for its own sake but because it also happens to be of some practical importance.

3.6 MATHEMATICAL SUPPLEMENT 3.6.1 Rigid Single Sheet Cavity Absorber Impedances The (normalized) interaction impedance ζ of the sheet is considered to be known either from measurements (Appendix A) or from analysis (Chapter 3). For a plane wave at normal incidence, wavelength λ = c/f , the pressure field in the cavity behind the sheet will be a standing wave with the pressure maximum at the rigid wall. The complex amplitude distribution will be of the form p(x, ω) = A cos(kx),

(3.17)

where x is the distance from the wall. The sheet is located at x = −L. The corresponding velocity field is obtained from ρ∂ux /∂t = −∂p/∂x, which for harmonic time dependence yields3 ux (ω) = iAk sin(kx)/ωρ. 3 The time factor used in this book in the definition of a complex amplitude is exp(−iωt).

(3.18)

88


The ratio of the amplitudes of pressure and velocity at x = −L is then iρc cot(kL) with the normalized value i cot(kL). Adding the impedance ζ of the sheet yields the total input impedance ζi ≡ θi + iχi = ζ + i cot(kL),

(3.19)

where k = ω/c = 2π/λ, ζ = θ + iχ : For a resistive sheet, χ = 0. L: Layer thickness. The input reactance χi is zero (resonance) at the frequencies determined by χ + cot(kL) = 0. For a purely resistive sheet, χ = 0, and hence cot(kL) = 0, which means that kL = (2n − 1)π/2 or L = (2n − 1)λ/4 (n = 1, 2 . . .), the layer thickness being an odd number of quarter wavelengths at the nth resonance. The input resistance θi is then the same as the sheet resistance θ if we neglect any losses within the cavity. The assumption of a purely resistive screen is normally quite good over a wide frequency range. (As mentioned before, one finds, for example, that for cloth-like sheets, the magnitude of the normalized reactance typically is |χ | ≈ (f/5000)θ, where f is the frequency in Hz.) For a wire mesh screen the frequency dependence of θ and χ , in terms of the geometrical parameters involved, is discussed in Chapter 3. For a locally reacting boundary, the impedance is independent of the angle of incidence (Figure 3.19) but for a nonlocally reacting boundary, it is not, ζi ≡ θi + iχi = ζ + i(1/ cos φ) cot(kx L),

(3.20)

where (see Eq. 3.19), φ: Angle of incidence (see Figure 3.1), kx = k cos φ, ζ : Sheet impedance. For nonlocal reaction, the directions of the waves incident on the wall and reflected from the wall will be the same as those of the waves incident and reflected from the absorber. The distance between adjacent maxima in the standing wave within the air layer in the normal direction will be λx = λ/ cos φ and the corresponding propagation constant is kx = 2π/λx = k cos φ. The factor 1/ cos φ = k/kx in front of cot(kx L) in Eq. 3.20 is the normalized wave impedance ‘in the x-direction’ of the incident plane wave, (1/ρc)p/ux = 1/ cos φ. Absorption Coefficient Because of the change in impedance encountered by the sound wave as it reaches the absorber, reflection will occur. If the x-dependence of the complex amplitudes of the incident and reflected pressure waves are expressed as pi = exp(ikx x) and pr = R exp(−ikx x), the corresponding velocity fields are uix = (1/ cos φ) exp(ikx x) and uxr = −(1/ cos φ) exp(−ikx x). With the screen at x = 0, the total pressure amplitude at the screen will be p = pi + Rpr and the total velocity ux = uix + urx = (1/ρc cos φ)(pi − pr ). From these relations it follows that p/ux = (ρc/ cos φ)(1 + R)/(1 − R). Equating this with the known input impedance, the reflection coefficient can be expressed in terms of ζi , R = (ζi cos φ − 1)/(ζi cos φ + 1).

(3.21)

89

SHEET ABSORBERS The corresponding absorption coefficient is α(φ) = 1 − |R|2 =

4θi cos φ , (1 + θi cos φ)2 + (χi cos φ)2

(3.22)

where ζi = θi + iχi : Eqs. 3.19 and 3.20, φ: Angle of incidence. In a diffuse sound field, the intensity of the sound is the same in all directions. For an angle of incidence φ (with respect to the normal, the polar angle), the acoustic power, which strikes a surface element of unit area will be proportional to cos φ, the projected area normal to the incident intensity. The probability of having intensity striking the boundary in an angular interval dφ about the angle φ is proportional to the solid angle 2φ sin φdφ on a unit sphere with the center at the surface element. π/2 Thus, the total power striking the element will be proportional to 0 cos φ sin φ dφ. The absorbed power is obtained by inserting the absorption coefficient α(φ) as a factor in the integrand. Thus, the average absorption coefficient in a diffuse sound field becomes π/2 π/2 α(φ) cos φ sin φ dφ 0 =2 αst = π/2 α(φ) cos φ sin φ dφ, (3.23) 0 cos φ sin φ dφ 0 where α(φ): Eq. 3.22. It is often called the statistical average or, as in this book, the diffuse field absorption coefficient. The integration over angle of incidence, which is involved in computing the diffuse field absorption coefficient, is based on the assumption that the material is isotropic so that the impedance is independent of the azimuth angle of the incident wave. For a nonisotropic material, as obtained if the air backing is partitioned only in one direction, the integration involves also the azimuth angle, as for the slot absorber in Chapter 3 and the general anisotropic material discussed in Chapter 5. For the locally reacting absorber, the integral in Eq. 3.23 can be expressed in closed form in terms of the normalized input impedance ζi = θi + iχi , 2 − χ2 θ θi 1 8θi χ i 1− ln((1 + θi )2 + χi2 ) + i 2 i arctan( ) , (3.24) αst = |ζi |2 |ζi |2 |ζi | χi 1 + θi where ζi = θi + iχi : Eq. 3.19. As already indicated, the sheet interaction impedance ζ often can be considered to be purely resistive. Then, for the locally reacting absorber, the resonance frequencies are given by cot(kL) = 0, independent of the angle of incidence, and for the nonlocally reacting, by cot(kL cos φ) = 0. If the mass reactance of the sheet is not negligible, which is often the case when the sheet includes a perforated thin plate or ‘facing,’ the resonance frequencies will be reduced accordingly. It follows from Eq. 3.22 that at resonance, with χi = 0, the absorption coefficient is (α)max =

4θ cos φ . (1 + θ cos φ)2

(3.25)

90


With θ = 1, the resonance absorption coefficient at normal incidence is 1. With θ > 1, the normal incidence absorption coefficient will be less than 1 but for oblique incidence, on the other hand, it will be unity for waves with an angle of incidence given by cos φ = 1/θ . With θ < 1, the absorption coefficient will be less than unity for all angles of incidence. It follows that to optimize the diffuse field average absorption coefficient, the input resistance should be greater than 1 (for local reaction, it is about 1.8). At a given frequency and χi , there is an optimum resistance for maximum absorption. This optimum and the corresponding maximum absorption coefficient for the locally reacting absorber is given by √ θi cos φ = 1 + χi cos φ αmax = √ 2 . 2 1+

(3.26)

1+(χi cos φ)

For normal incidence and at wavelengths much greater than the cavity depth so that kL 1 so the term 1 can be neglected compared to mr in the denominator. Then, recalling that the quarter wavelength resonance frequency f1/4 = c/4L, the new low frequency resonance is expressed in terms of it. f0 ≈

1 c/L 2 f1/4 = √ , √ 2π mr π θ

where f1/4 = c/4L, θ ≈ mr = m/ρL.

(3.48)

94


For a rigid sheet, the lowest resonance frequency is the quarter wavelength resonance at f1/4 and the lowest resonance of the limp sheet resonator is approximately √ (π/2) mr times lower. Absorption Coefficient The absorption coefficient for the flexible sheet absorber follows from the expressions for the absorption coefficient of the rigid sheet absorber by replacing the interaction impedance ζ by the equivalent sheet impedance ζ given in Eq. 3.33.

3.6.3 Uniform (Periodic) Lattice This section can be considered to be an exercise in wave propagation in a periodic structure or lattice. It turns out to yield some interesting results about sound absorption by a multisheet absorber. The absorber consists of N identical, equidistant, limp, resistive sheets placed in front of a rigid wall, as shown in Figure 3.13. Each sheet is specified acoustically by the equivalent sheet impedance ζ , defined in the previous section (see Eq. 3.33). The separation of adjacent sheets is much larger than the acoustic thermal boundary layer so that the compressibility of the air in the lattice can be assumed to be isentropic. This should be kept in mind when the absorption spectra of a lattice absorber are compared with those of uniform porous layers in which heat conduction leads to isothermal compressibility at low frequencies. Unit Cells The analysis starts with a choice of a unit cell of the lattice, and in Figure 3.22 two possibilities are indicated. In the first, the beginning of the cell is at a point just in front of one sheet at x = nd, and the end of the cell is just in front of the adjacent sheet at x = (n + 1)d, where n is an integer. The complex amplitudes of sound pressure and velocity at these locations are denoted by p(xn ), u(xn ) and p(xn+1 ), u(xn+1 ), respectively. This unit cell is asymmetrical in the sense that it is not the same for sound waves traveling in the positive and negative directions. It contains two elements, an air layer and a sheet. Asymmetrical cell ζ′

ζ′

pn

pn+1

Symmetrical cell ζ′

ζ′

2

2

pn

ζ′ 2

ζ′ 2

pn+1

Figure 3.22: The asymmetric cell consists of two elements, one sheet and an air layer. The symmetric cell has three elements, a half-sheet, an air layer, and a second half-sheet.

95

SHEET ABSORBERS

The second unit cell is symmetrical, however. Each sheet is considered to be a combination of two sheets, each with the acoustic impedance equal to half of that of one sheet, and the reference planes are placed between these sheets. Thus, in this case the unit cell consists of three elements, two half-sheets and an air layer. Propagation Constant and Wave Impedance For a single harmonic traveling wave, with the time dependence expressed by the factor exp(−iωt), the space dependence of the field variables at the sheet locations is expressed by the factor exp(iqx xn ). The ratio p(xn+1 )/p(xn ) = exp(iqx d) then defines the propagation constant qx in the x-direction of the lattice. The same ratio applies also to the complex velocity amplitude of the fluid. In general, qx is complex and contains both a phase shift and an amplitude decay per cell. To calculate the propagation constant qx , a relationship between the complex amplitudes of the field variables at the two ends of the unit cell must be derived from the dynamics of the components of the cell, i.e., the sheet and the air column between the sheets. The linear relationship between the field variables at the two ends of the lattice can be expressed as p(xn ) = T11 p(xn+1 ) + T12 ρcu(xn+1 )

(3.49)

ρcu(xn ) = T21 p(xn+1 ) + T22 ρcu(xn+1 ),

(3.50)

where u is the x-component of the complex velocity amplitude and Tij the elements of the transmission matrix of the unit cell of the lattice, which have to be determined in terms of the frequency ω and the lattice parameters. The wave impedance is Zρc = p(xn )/u(xn ) = p(xn+1 )/u(xn+1 ) and with p(xn+1 )/p(xn ) = u(xn+1 )/u(xn ) = exp(iqx d), it follows from Eqs. 3.49 and 3.50 that e−iqx d = T11 + T12 /Z = T22 + T21 Z Z = (T11 − T22 )/2T21 ∓ (i/T21 ) 1 − [(T11 + T22 )/2]2 ,

(3.51)

where, in the last expression, the elements Tij are not all independent but related by T11 T22 − T12 T21 = 1, a general property of a passive ‘four-pole,’ in the language of circuit analysis. From these relations it follows that (3.52) e−iqx d = (T11 + T22 )/2 − (i/T21 ) 1 − [(T11 + T22 )/2]2 . Expressing exp(iqx d) in a similar manner yields cos(qx d) = (eiqx d + e−iqx d )/2 =

T11 + T22 . 2

(3.53)

The propagation constant and the wave impedance can be expressed explicitly in terms of the lattice parameters after the sheet impedance has been specified and the matrix elements Tij for a unit cell of the lattice have been calculated. For the first

96


of the unit cells in Figure 3.22, they are obtained as the product of the matrix of an air layer of thickness d and the matrix for the sheet. For a sound wave with an angle of incidence φ on the lattice (the angle measured from the normal to the surface), kx = (ω/c) cos φ and the matrix elements for the air layer are A12

A11 = A22 = cos(kx d) = −i(1/ cos φ) sin(kx d), A21 = −i cos φ sin(kx d)

(3.54)

and for the sheet S11 = S22 = 1 S12 = ζ , S21 = 0,

(3.55)

where ζ is the normalized equivalent impedance discussed earlier. From the product of these matrices we obtain the matrix elements for the unit cell, Tij = k Sik Akj , i.e., T11 = cos(kx d) − iζ cos φ sin(kx d) T12 = −i(1/ cos φ) sin(kx d) + ζ cos(kx d) T21 = −i cos φ sin(kx d) T22 = cos(kx d).

(3.56)

The corresponding expressions for the propagation constant qx and the normalized wave impedance Z, as given by Eqs. 3.53 and 3.51, are then cos(qx d) =

T11 + T22 ζ cos φ = cos(kx d) − i sin(kx d) 2 2

and Z± =

ζ

sin(qx d) ± , 2 sin(kx d) cos(φ)

(3.57)

(3.58)

where sin(qx d) = 1 − cos2 (qx d) has been used together with Eq. 3.57. The plus and minus signs in the last expression refer to propagation in the positive and negative x-directions, respectively. The reason for the difference in the wave impedances for propagation in the two directions is that the unit cell is not symmetrical. For propagation in the positive direction, the cell starts just in front of a sheet and the impedance refers to the ratio of pressure and velocity amplitudes at this point. For propagation in the negative direction, the cell starts with the air column between two sheets. The ratio of the pressure amplitude and the negative velocity amplitude at this point is [sin(qx d)/ sin(kd) cos(φ)]−ζ /2. The difference between this impedance and the wave impedance in Eq. 3.58 is ζ , as it should be. If we had chosen the symmetrical unit cell in Figure 3.22, the expression for the wave impedance in Eq. 3.58 would have contained only the second term, with the magnitude of the impedance being independent of the direction of wave travel. The propagation constant would have been the same as before. The wave impedance becomes infinite if kx d = nπ (cot(kx d) = ∞), i.e., if d cos φ = nλ/2, which is the well-known Bragg condition for constructive interference

97

SHEET ABSORBERS

of the waves reflected from the different layers in a lattice as mentioned in the main text in the discussion of the absorption spectrum for a periodic lattice. Input Impedance and Absorption Coefficient In terms of the wave impedances Z± , the complex amplitudes of pressure and velocity at the nth sheet become p(xn ) = Aeiqx xn + Be−iqx xn ρcu(xn ) = (A/Z+ )eiqx xn + (B/Z− )e−iqx xn .

(3.59)

(In the second term of the last equation, a negative sign is contained in the expression for the wave impedance Z− .) If the lattice is backed by a rigid wall at x = L = N d, where N is the number of sheets in front of the wall, the boundary condition is u(L) = 0, which establishes the relation B/A = −(Z− /Z+ ) exp(i2qx L). After a little algebra, the amplitudes of pressure and velocity at the nth sheet can be expressed as ζ

x d) + p(xn ) = C i cot[qx (L − xn )] cossin(q φ sin(kx d) 2 sin[qx (L − xn )] ρcu(xn ) = C sin[qx (L − xn )],

(3.60)

where C = −(2iA/Z+ ) exp(iqx L). The first sheet in the lattice is located at x1 = 0, and it follows that the input impedance is ζi ≡ θi + iχi =

p(0) u(0)ρc

=

ζ

2

x d) + i cossin(q φ sin(kx d) cot(N qx d),

cos(qx d) = cos(kx d) − i ζ

cos φ

2

sin(kx d),

(3.61)

where kx = k cos φ, ζ : 3.33, d: Figure 3.13, φ: Angle of incidence, N : Number of sheets, qx : Eq. 3.57. Since qx has an imaginary part, the factor cot(N qx d) → −i as N → ∞, and the input impedance becomes the same as for an infinite lattice as given in Eq. 3.58, i.e., ζi → Z+ . As another test, consider the case of a single sheet, i.e., N = 1. From Eq. 3.57 for cos(qx d), the input impedance becomes the familiar ζi = ζ + i[1/ cos(φ)] cot(kx d). Having obtained ζi , the absorption coefficients follow from Eqs. 3.22 and 3.23. Some computed results are shown in Figure 3.14. Field Distribution Within a Cell To compute the field distribution in the region xn < x < xn+1 between two sheets at the location x + ξ , we merely apply the transmission matrix for the air column between xn + ξ and xn + d of thickness d − ξ to obtain p(xn + ξ ) = p(xn+1 ) cos[kx (d − ξ )] − i(ρc/ cos φ)u(xn+1 ) sin[kx (d − ξ )] ρcu(xn + ξ ) = −ip(xn+1 ) cos φ sin[k(d − ξ )] + ρcu(xn+1 ) cos[kx (d − ξ )]. (3.62)

98


The input impedance in Eq. 3.61 contains sin(qx d) = 1 − cos2 (qx d), which can be computed with the help of Eq. 3.57 and cot(N qx d) can be calculated from cot(qx Nd) = i[exp(iN qx d) + exp(−iN qx d)]/[exp(iN qx d) − exp(−iN qx d)] with the use exp(iqx d) either as cos(qx d) + i sin(qx d) or directly from Eq. 3.52. Alternate Choice of Unit Cell The symmetrical unit cell begins at the midpoint between two sheets and ends at the midpoint between the adjacent sheet pair, as indicated in Figure 3.22. In this case, the wave impedance should be the same (except for sign) and independent of the direction of wave travel. It is left as an exercise to show that the elements of the transmission matrix elements are (for normal incidence)

S11 = cos(kx d) − i ζ2 sin(kx d)

S12 = −i sin(kx d) + ζ cos2 (kx d/2) S21 = −i sin(kx d) + ζ sin2 (kx d/2)

S22 = cos(kx d) − i ζ2 sin(kx d),

(3.63)

where kx = ω/c (normal incidence) and c the sound speed. Similarly, the wave impedance becomes (see first part of Eq. 3.58) Z± = ∓

sin(qx d) sin(qx d) =± . S21 sin(kx d) + i(ζ1 /2) sin2 (kx d/2)

(3.64)

The magnitude of this impedance is independent of the direction of wave travel, as it should be. The expression for the propagation constant is the same as that obtained for the asymmetrical unit cell considered above.

3.6.4 Nonuniform Lattice The sheets need no longer be identical and the distance between them need not be the same so that a nonuniform (nonperiodic) lattice is formed. Transmission Matrix Each cell of the lattice consists of two elements, an air column and a sheet. The transmission matrix Cn of the nth cell (starting the count from the rigid wall) can be written as the matrix product of the transmission matrices A and S of the nth air column and the nth sheet Cn = (SA)n , (3.65) and the transmission matrix T of the entire multisheet absorber of N sheets is then

N T11 T12 = Cn , (3.66) T ≡ T21 T22 1

where n = 1 is the cell at the rigid wall.

99

SHEET ABSORBERS

The transmission matrices of the air layer and the sheet and have already been calculated (Eq. 3.55). Input Impedance and Absorption Coefficient In terms of the matrix elements Tij in Eq. 3.66, the relations between the sound pressure p and normal velocity component u at locations (1) and (2) at the beginning of the absorber and at the wall are p(1) = T11 p(2) + T12 ρcu(2) ρcu(1) = T21 p(2) + T22 ρcu(2),

(3.67)

where ρc is the wave impedance of the ambient fluid (air). Since the backing wall is rigid, i.e., u(2) = 0, the normalized input impedance of the lattice becomes ζi ≡ θi + iχi =

p(1) T11 . = ρcu(1) T21

(3.68)

Having obtained ζi , the absorption coefficients can be computed as for the singlesheet absorber.

3.6.5 Sheet as a Volume Absorber Plane Wave Reflection, Transmision, and Absorption, Infinite Sheet With reference to Figure 3.18, consider a harmonic plane wave incident in the xy-plane of an infinite sheet located at x = 0. The angle of incidence is φ. The spatial dependence of the wave is determined by the factors X(x) = eikx cos φ ,

Y (y) = eiky sin φ ,

(3.69)

where k = ω/c. The complex amplitudes of the incident, reflected, and transmitted pressure waves are then AX(x)Y (y), BX(−x)Y (y), and CX(x)Y (y), respectively, and the x-components of the corresponding velocity amplitudes are A cos φX(x)Y (y)/ ρc, −B cos φX(−x)Y (y)/ρc, and C cos φX(x)Y (y)/ρc. The pressure and velocity fields to the left and the right of the sheet are p1 = pi + pr = [AX(x) + BX(−x)]Y (y) p2 = pt = CX(x)Y (y) u1 = ui + ur =

1 ρc

cos φ [AX(x) − BX(−x)] Y (y)

u2 = ut =

C ρc

cos φX(x)Y (y).

(3.70) (3.71) (3.72) (3.73)

The boundary conditions at the sheet, p1 − p2 = ζ ρcu1 and u1 = u2 , then lead to the relations A + B − C = Cζ cos φ A − B = C, where z = ρcζ is the equivalent sheet impedance, as defined earlier.

(3.74)

100


Addition of the two equations gives C = 2A/(2 + ζ cos φ), the pressure transmission coefficient is τ = C/A, and the corresponding power transmission coefficient |τ |2 . The pressure reflection coefficient is R = B/A = 1 − τ and the corresponding coefficient for power is |R|2 . From conservation of energy it follows that the fraction of the incident power, which is absorbed by the sheet is α = 1 − |R|2 − |τ |2 . Thus, to summarize, τ=

C A

=

2 2+ζ cos φ

R =1−τ = α(φ) = 1 − |R|2 − |τ |2 =

ζ cos φ 2+ζ cos φ

(3.75)

4θ cos φ , (2+θ cos φ)2 +(χ cos φ)2

(3.76)

where ζ = θ + iχ : Eq. 3.33, φ: Angle of incidence. This coefficient can be determined also directly from a calculation of the friction losses in the sheet by first determining the x-component of the velocity amplitude at the sheet C 2A cos φ u= cos φ = , ρc ρc 2 + ζ cos φ and then the power loss per unit area (1/2)|u|2 ρcθ = |A|2 /2ρc α(φ) α(φ) =

4θ cos φ , (2+θ cos φ)2 +(χ cos φ)2

(3.77)

where α is the absorption coefficient. For a sheet with a purely resistive impedance (χ = 0) and a sufficiently large mass so that the sheet can be regarded as immobile as far as the interaction with sound is concerned, the normal incidence coefficients of reflection, transmission, and absorption depend on the sheet resistance as shown in Figure 3.19. Infinite Sheet, Diffuse Field, Effect of Induced Motion The sheet is assumed to be purely resistive. As was mentioned earlier, this is normally a good approximation. With sound incident only on one side of the sheet, the average absorption coefficient is α1 = 2

π/2

α(φ) sin φ cos φ dφ.

(3.78)

0

With the expression for α(φ) given in Eq. 3.77 and with η ≡ cos φ, the average absorption coefficient can be written 1 2 α1 = 2θ 0 (1+θ η/2)η2 +(χ η/2)2 dη

= |ζ8θ |2 1 − |ζ2θ |2 log(1 + θ + 14 |ζ |2 )

χ

+ |ζ8θ |2 (θ 2 − χ 2 ) χ |ζ2 |2 arctan( 2+θ ) ,

(3.79)

101

SHEET ABSORBERS

where ζ = θ + iχ : Eq. 3.33, η = cos φ, φ: Angle of incidence, m: Sheet mass per unit area. The corresponding expressions for the average value of the transmission and reflection coefficients |τ |2 and |R|2 based on acoustic power are π/2 |τ |2 = 2 0 |τ |2 sin φ cos φ dφ

χ

= |ζ4 |2 log(1 + θ + |ζ |2 /4) − 2θ χ arctan( 2+θ ) |R|2 = 1 − α1 − |τ |2 .

(3.80)

Effect of Diffraction An approximate analysis for a circular porous disk4 shows that for wavelengths large compared to the disk diameter, the drop of the sound pressure amplitude across the disk (radius a) is given by √ k a2 − r 2 p = −ip0 (4/π) cos φ , (3.81) 1 − (8/3π )ikaβ where p0 is the pressure amplitude of the incoming wave, β = 1/ζ , the normalized equivalent admittance of the screen, ζ , the corresponding equivalent sheet impedance, and φ the angle of incidence. Integration over an impervious rigid disk (β = 0) results in the amplitude of the acoustically induced force on the disk, a 8 f = p2πr dr = −ip0 cos φ (π a 2 ka). (3.82) 3π 0 If we introduce p0 = ρcu0 , where u0 is the velocity amplitude in the incoming wave and consider normal incidence (φ = 0), the force becomes f = (−iωu0 )(8a/3π ) ρπ a 2 ≡ −iωmi , where mi = (πa 2 )(8a/3π )ρ. (3.83) Thus, the induced mass mi created by the disk can be thought of as the mass of a fluid cylinder with the area of the disk as a base and with a length 8a/3π = 4d/3π , where d = 2a is the diameter of the disk. Whether or not the flow will go through the disk rather than around it depends on the ratio of the normalized average mass reactance ωmi /π a 2 per unit area and the impedance of the sheet. For a purely resistive and immobile screen, this condition is ωmi /ρc > θ , where θ is the normalized flow resistance of the sheet. Introducing the expression for the induced mass in Eq. 3.83, we can write the condition as (4/3π)kd > θ or d > (3λ/8)θ, (3.84) where we have used k = ω/c = 2π/λ. In other words, for values of the flow resistance of the order of unity, the diameter of the sheet should exceed approximately one-third 4 P. M. Morse and K. U. Ingard, Linear Acoustic Theory, Volume XI/1 in Handbuch der Physik, Springer-

Verlag, 1961.

102


of a wavelength in order that the flow will begin to have a preference for going through the disk rather than around it. Thus, a relatively heavy disk, such that the induced motion can be ignored, with a diameter of 1 ft should begin to respond like an infinitely extended sheet for wavelengths less than ≈ 8/(3θ) ft. For a sheet resistance of the order of unity, this corresponds to frequencies above ≈ 420 Hz. It should be noted that the induced mass of the finite sheet plays about the same role as the mass per unit area of the infinitely extended sheet as far as the reduction of absorption cross section at low frequencies is concerned. Actually, for a sheet with a diameter of 1 ft, the induced mass per square foot corresponds to a weight of 0.08 · 4/3π ≈ 0.035 lb/ft2 (≈ 5 oz/yd2 ), where we have used 0.08 lb/ft3 for the density of air. This is about the same as the weight of a typical woven sheet, and the induced motion of the sheet then plays the same role as diffraction in reducing the low frequency absorption. Thus, in this case, no significant improvement in the low frequency absorption would result by making the sheet heavier than 0.035 lb/ft2 since the absorption would be limited by the effect of diffraction. To investigate in more detail the frequency dependence of the absorption cross section of the disk, we use the expression for the pressure drop across the sheet in Eq. 3.81. The pressure drop produces an oscillatory flow relative to the disk, and if this velocity amplitude is denoted by ur , the energy absorption per unit area becomes |ur |2 θ ρc. Integration over the area of the disk and over the angle of incidence in a diffuse sound field yields the average power absorbed per unit area of one side of the disk. The amplitude of the relative velocity through the disk is ur = u − u , where u

is the amplitude of the induced velocity of the disk and u = p/(ρcζ ) the absolute velocity amplitude of the air just outside the disk. Quantity ζ is the equivalent disk impedance. Furthermore, for a purely resistive limp disk so that ζ = θ, we get u /u = θ/(θ − iωm/ρc) (see discussion of the single limp sheet cavity absorber). The amplitude of the relative velocity ur = u − u can then be expressed as ur /u =

−iωm r f/fm , = 1/(1 + ir/ωm) = r − iωm i − f/fm

(3.85)

where f is the frequency and fm = r/(2πm), the characteristic frequency. The power absorbed per unit area of the disk is |ur |2 θρc. From the expressions for ur /u, u = p/(ρcζ ) and p in Eq. 3.81, integration over the disk and the angle of incidence yields the average power absorption of a disk of area A in a diffuse sound field (for one-sided exposure to the sound) |p0 |2 ur 2 32 (ka)2 θ , A| | 2

ρc u π |ζ + iχi |2

(3.86)

where χi = −(8/3π)ka is the average normalized induced mass reactance per unit area of the disk. In order to relate this to an absorption cross section or absorption area in a diffuse sound field, we have to introduce the average intensity in a diffuse field, which is known to be Iav = |pav |2 /4ρc, where |pav |2 = 4π |p0 |2 . Thus, the factor |p0 |2 /ρc in Eq. 3.86 can be expressed as Iav /π . Thus, if we replace π 2 in the denominator of

103

SHEET ABSORBERS

Eq. 3.86 by π 3 , we obtain the average absorption cross section of the disk for onesided exposure in a diffuse field. For two-sided exposure, we replace the factor 32 by 64 in the equation to obtain the total absorption cross section σa σa ≈ A|

ur 2 64 (ka)2 θ | , u π 3 |ζ + iχi |2

(3.87)

where |ur /u|: Eq. 3.85, ζ : Eq. 3.33, θ: Normalized flow res. χi : Eq. 3.86. In the low frequency approximation, as expressed by Eq. 3.87, there will be an optimum flow resistance for a purely resistive immobile disk (ζ = θ) and a fixed value of ka given by θ = χi = (8/3π)ka. The corresponding two-sided maximum absorption cross section is (8/π 2 )(ka)πa 2 . The optimum value of the flow resistance for the infinite sheet, discussed in connection with Figure 3.20, is not valid for the finite sheet; the optimum resistance is now approximately equal to ka = 2π a/λ, i.e., directly proportional to the diameter of the sheet. Figure 3.21 shows an example of the calculated frequency dependence of the absorption cross section per unit area of sheet material, as obtained from Eq. 3.87 (with the modifications explained in the text following the equation).

Chapter 4

Resonators 4.1 INTRODUCTION AND SUMMARY Normally, an acoustic resonator refers to a cavity (bottle) resonator rather than a ‘mechanical’ one, such as a plate or a bell, and this convention applies also to this chapter. Generally, all dimensions of a cavity resonator are assumed large compared to the visco-thermal boundary layer thickness but small compared to the wavelength of the incident sound (‘acoustical compactness’). The walls are acoustically hard so that sound absorption is due solely to visco-thermal losses at the walls. Then, the absorption will be significant only in the vicinity of the resonance frequencies of the cavity unless porous materials, such as screens, are added. Actually, additional damping is obtained without such material if the incident sound pressure is high enough to make nonlinear effects significant. Then, the oscillatory flow in an orifice or at a (sharp) corner will separate so that vorticity is created. The corresponding energy will be drawn from the sound wave and create a damping, which often is larger than the visco-thermal effects. Similarly, if steady flow is present, the coupling between the flow and the sound will produce acoustic losses. The simplest acoustic resonator, at least geometrically, is a straight tube. If the tube is closed at one end and open at the other, the fundamental resonance frequency occurs at a wavelength which is 4 times the acoustic length of the tube. Higher resonances occur whenever the tube length is an odd number of quarter wavelengths. If a resonator is exposed to an incident sound wave, as shown in Figure 4.1, a certain portion of the incident acoustic power will be absorbed and the rest will be scattered. The amounts of absorbed and scattered power are conveniently expressed in terms of the absorption and scattering cross sections of the resonator, Aa and As , respectively. This means that if the intensity of the incident sound is I , the absorbed and scattered powers will be I Aa and I As , respectively, as discussed in some detail in the next section. The maximum possible absorption cross section of a resonator at resonance is λ2 /2π when the resonator is set in a rigid wall (baffle) and λ2 /4π without a wall, where λ is the wavelength of the incident sound. It should be remembered that the resonator is acoustically compact so that the opening is small compared to the wavelength and

105

106


Figure 4.1: Tube resonator in a wall (infinite baffle) and normalized radiation impedance (resistance and reactance) of a piston source in a wall vs ka, where k = 2π/λ and a is the radius of the resonator orifice. the absorption and scattering cross sections (as defined here) are independent of the angle of incidence of the sound. In the tube resonator, the potential and kinetic energies are distributed throughout the tube, but in the Helmholtz or bottle resonator, the two forms of energy can be considered to be spatially separated or ‘lumped,’ the kinetic energy residing in the neck of the resonator and the potential energy in the cavity. The Helmholtz resonator is then analogous to an ordinary mass-spring oscillator with the mass being the mass of the air in the neck and the spring being provided by the stiffness of the air in the cavity. If a pulse in the form of a harmonic wave train strikes a resonator, the resonator mode will be excited and then decays exponentially with time. The frequency in this decay or reverberation is the resonance frequency of the resonator. If the frequency of the incident sound is close to but not equal to the resonance frequency, distinct beats will occur between the incident and decaying fields. These can be clearly audible, as discussed in connection with the experiments referred to in Figure 4.5. A (tube) resonator is often used as side-branch in a duct to attenuate sound in narrow bands around the resonance frequencies of the tube but the discussion of it is deferred to Chapter 8, which deals with sound propagation in ducts. The transmission and insertion loss measures of such a side-branch resonator are then illustrated by specific examples. To cover a broad frequency range, several resonators can be used in parallel. Interaction of a fluid flow with a resonator can lead to either damping or excitation. Both can be of considerable practical importance.

4.2 ABSORPTION AND SCATTERING A single resonator in the form of a tube and mounted in a wall is shown schematically in Figure 4.1. Most of what we have to say about it applies equally well to any other type of cavity resonator. The tube resonator is convenient, since its acoustical properties, such as the input impedance, can readily be obtained from the material in Chapter 3. In our analysis of the sound absorption and scattering by a resonator, it has been

107

RESONATORS

assumed that the air layer in the resonator opening moves with uniform velocity so that it can be regarded as a plane, mass-less piston. It will generate sound both inside and outside the resonator, and the motion of the piston will be impeded by the radiation forces from the two sides. The resulting force is proportional to the sum of the input and radiation impedances of the resonator. At low frequencies, the radiation resistance increases as the square of the frequency but levels off to the constant value of unity (normalized) at high frequencies. The normalized reactance first increases linearly with frequency, reaches a maximum of 0.5 at ka = 1, where k = 2π/λ and a is the radius of the resonator opening, and then decreases to zero with increasing frequency, as shown in Figure 4.1. The frequency dependence of the input reactance is stiffness-controlled below the resonance frequency and mass-controlled above. At resonance the reactance is zero and the input impedance is purely resistive and, as shown in Section 4.5, proportional to the square root of the resonance frequency. This follows from the frequency dependence of the viscous boundary layer thickness. With all the walls of the resonator being acoustically hard, only the visco-thermal boundary losses contribute to sound absorption. The viscous losses along the tube walls dominate. With only the fundamental acoustic mode involved in the tube, there will be no viscous losses on the rigid wall at the end of the tube. However, there will be a thermal boundary layer on this wall, which contributes a small amount to the overall losses. Actually, as shown in Section 2.6, the boundary layer can be accounted for acoustically by replacing it with an equivalent acoustic admittance so that an acoustically hard wall in effect is replaced by one which is somewhat softened. In terms of this equivalent admittance ηt , the input impedance takes the form Tube resonator in a wall, input and radiation impedance ζi ≡ θi + χi =

cot(qL)−iηt −i+ηt cot(qL)

≈ i cot(qL)

ζr = θr + χr ≈

(at resonance ≈ π dvh /4a)

(4.1)

(ka)2 /2 − i(8/3π )ka

where q: see Eq. 4.21, ηt : End wall admittance ( >> 1. λ 2π M/ρ

(6.19)

For example, with R = 32 and M/ρ = 50 we get L/λ >> 0.10. Formally, from Eq. 6.13 and by analogy with what was done in Chapter 5, we can now proceed to calculate the absorption characteristics of a limp porous layer backed by a rigid wall. The results can be used for a flexible material in general as long as we are in the mass controlled region of the material, i.e., if any resonance determined by the mass and the stiffness of the material falls well below the region of interest. However, as will be seen in the next section, the lowest resonance of a typical flexible absorber usually does not fall below the region of interest, contrary to the assumption for the validity of the results of the analysis of the limp material. For this reason, numerical results obtained in this section have not been shown.

6.8.2 Equations for Coupled Waves From the idealized cases of rigid and limp materials treated so far, we now turn to the more realistic structure with finite compliance different from zero. A general description of the elastic properties of the structure can be quite involved, but must be used if we wish to account for the excitation of shear and surface waves by an incident sound wave. However, we shall assume that these waves in the structure are unimportant in the present context, and treat the porous material like an isotropic ‘fluid.’ In any event, the analysis will be applicable without qualifications for sound of normal incidence on a porous layer. Even so, we need two elastic constants (which we shall express as ‘compressibilities’ in our analysis) in much the same way as in the description of a coil spring for which one compressibility (although quite small) refers to the material (steel) from which the spring is made and the other to the compressibility of the spring (structure), which is much larger. We shall denote the compressibility of the material in the structure by κ and the normalized (compliance) of the porous frame by K realizing that generally κ is much smaller than K(1/ρc2 ). Each of these constants are assumed to have been determined for harmonic excitation of the material and expressed as complex numbers to account for internal damping.

222


If the cells or interstices of the material are all open, the mass per unit volume of the structure will be M = (1 − H )ρ , where ρ is the mass density of the material from which the structure is made. For a urethane foam, the specific density ρ is of the order of 1.2 and for glass fiber about 2.5. The porosity H is the ratio of the air volume in the open cells and the total volume. If the material contains portions with closed fluid filled cells, the mass density ρ

and the compressibility κ are changed accordingly for those portions. Mass Balance As before, we define the average velocity u of the air in the open cells of the material in such a way that ρu is the average mass flow rate. The linearized mass conservation equations for the air and the structure are then, with H = 1 − H , ∂H H ∂ρ ∂t + ρ ∂t + ρdiv u = 0

H

∂ρ

∂t

+ ρ

∂H

∂t

+ H ρ div u = 0.

(6.20) (6.21)

It has been tacitly assumed that there is no steady (i.e., time independent) flow through the porous material. In the presence of such a flow, additional terms must be included in the linearized equations. In free field, with the sound pressure in the fluid denoted by p, we can express (1/ρ)∂ρ/∂t as κ∂p/∂t, where κ is the (isentropic) compressibility. Similarly, with V = 1/H , the term (1/H )∂H /∂t = −(1/V )∂V /∂t can be written K∂p /∂t, where p is the ‘pressure’ in the structure (the negative of the average stress in the structure) and K the compressibility of the porous structure defined by K = −(1/V )∂V /∂P = (1/H )∂H /∂P . (The definition K = (1/M)∂M/∂P is not quite the same because of the possible variation in ρ in addition to the variation in H , and the volume based definition is more in accord with what is measured.) Quantity (1/ρ )∂ρ /∂t becomes ≈ κ ∂p/∂t, assuming that the pressure in the fluid is the main cause of the compression of the structural material itself. As it turns out, this effect generally can be neglected. We shall deal with harmonic time dependence and internal damping will be accounted for in terms of complex compressibilities of the fluid and of the structure. As in Chapter 5, the complex compressibility of the fluid per unit volume of the structure is written κ˜ = H κ˜ 1 , where κ1 is the complex compressibility per unit volume of the fluid. The complex amplitude equations corresponding to Eqs. 6.20 and 6.21 then become ˜ ) (with ∂/∂t → −iω and dH = −dH = −H Kp ˜ − iωκp ˜ = −div u iH ωKp

˜ −iωKp − iωκ˜ p = −div u .

(6.22) (6.23)

As before, the tilde symbol has been used as a reminder that the compressibilities are complex.

223

FLEXIBLE POROUS MATERIALS Momentum Balance

Using the same notation as in the corresponding equation for a rigid material, we have ∂ρu ∂ρGs (u − u ) = −zv (t)(u − u ) − − grad p (6.24) ∂t ∂t with the corresponding complex amplitude equation −iωρu ˜ = zu − grad p,

(6.25)

where, as in Chapter 3, the interaction impedance is z = zv − iGs ωρ,

(6.26)

ρ/ρ ˜ = 1 + iz/ωρ = s + izv /ωρ,

(6.27)

and the complex density ratio

where z is the interaction impedance, s the structure factor, and zv , the viscous contribution to the interaction impedance. In the numerical analysis to be discussed later, we base zv on the result for an equivalent single channel, as was discussed in Chapter 3. The analogous equation for the structure is ˜ = zu − grad p , −iωMu

(6.28)

M˜ = M(1 + iz/ωM).

(6.29)

where

In most cases of interest, it is a good approximation to neglect the first term on the left side in Eq. 6.22 and the second term in Eq. 6.23. Then, to summarize, the linearized equations for mass and momentum balance take the form iωκ˜ p = div u iωK˜ p = div u

−iωρ˜ u = zu − grad p −iωM˜ u = zu − grad p .

(6.30)

Taking the divergence of the momentum equations and eliminating the terms involving the velocity amplitude, making use of the mass conservation equations, we obtain the coupled wave equations below. ∇ 2 p + ka2 p = ca p

∇ 2 p + ks2 p = cs p with ka2

=

ω2 κ˜ ρ, ˜

˜ ca = iωzK,

ks2

=

ω2 M˜ K˜

cs = iωzκ, ˜

(6.31)

224


where ρ: ˜ Eq. 6.27, κ˜ = H κ1 , κ1 : Eq. 6.14. If the terms, which were dropped are retained, there will be corrections. Thus, ˜ ca , a factor ka2 will contain a factor (1 − izκ /ωρκ), ˜ ks2 , a factor (1 + iH z/ωM),

˜ δa = (1 − iH ωρ/z), ˜ and cs , a factor δs = (1 + iωM κ˜ /zκ). ˜ These corrections, however, usually have insignificant effects. Dispersion Relation As in Chapter 3, the spatial dependence of the complex amplitude of the incident and refracted pressure wave are expressed as exp(ikx x + iky y + kz z) and exp(iqx x + iqy y + qz z), where kx = k cos φ, ky = k sin φ sin ψ), kz = k sin φ cos ψ, and k = ω/c. Quantities qx , qy , and qz are the components of the propagation constant q in the material with the magnitude q =

qx2 + qy2 + qz2 .

To obtain the dispersion relation for a wave in the porous material we introduce the spatial dependence given above (∇ 2 p = −(qx2 + qy2 + qz2 )p = −q 2 p) into Eq. 5.2 and obtain [−q 2 + (ω/c)2 ka2 ]p = ca p

[−q 2 + (ω/c)2 ks2 ]p = cs p,

(6.32)

where ka , ks , ca , and cs are given in Eq. 6.31. From these relations follows the equation for q, the dispersion relation, in terms of the normalized propagation constant Q = q/(ω/c), (q 2 − ka2 )(q 2 − ks2 ) = ca cs .

(6.33)

In the numerical analysis it is convenient to normalize the propagation constant, and we introduce Q = q/k, Ka = ka /k, Ks = ks /k, Ca = ca /k 2 , and Cs = cs /k 2 , where k = ω/c and c are the sound speed in the fluid in free field. In terms of the isentropic ˜ and compressibility, κ = 1/ρc2 , we can rewrite Ca and Cs as Ca = [z/(−iωρ)]K/κ ˜ The factor z/(−iωρ) can be considered to be the ratio of the Cs = [z/(−iωρ)]κ/κ. interaction force and the inertia force on the fluid. In terms of these normalized quantities, the dispersion relation 6.33 and its solution can be expressed as (see Eq. 6.31) Q4 − (Ka2 + Ks2 )Q2 + Ka2 Ks2 + C = 0 Q21,2 = 12 (Ka2 + Ks2 ) ± 12 (Ka2 − Ks2 )2 − 4C, where 2 ˜ ˜ Ka = (ρ/ρ)( ˜ κ/κ) ˜ Ks2 = (M/ρ)( K/κ) ˜ C = −Ca Cs = (z/ωρ)2 (K/κ)( κ/κ), ˜

(6.34)

where the subscripts 1 and 2 refer to the two characteristic modes of the system, corresponding to the plus and minus signs, respectively. If no terms had been omitted in the original mass conservation equations, the coupling constant C would have contained a factor δa δs but its effect is generally insignificant.

FLEXIBLE POROUS MATERIALS

225

Special Case: Rigid Material A rigid material is characterized by K˜ = 0. We then obtain ˜ ρ/ρ) ˜ Q1 = Ka = (κ/κ)( Q2 = 0

(6.35)

for the two modes. The first is the fluid-borne wave with a propagation constant Q1 , which is consistent with the value obtained for a wave in a rigid material. The second wave mode is the structure-borne; it has Q2 = 0, which corresponds to an infinite wave speed (as expected) since we are dealing with the idealization of zero compressibility. Special Case: Limp Material In the other limit, with K˜ → ∞, Ks2 and C both go to infinity as K˜ does. Then, after having brought out the factor (Ks2 − Ka2 ) from the radical, we get upon expansion of the radical in terms of quantities ∝ 1/K, Q212 = (1/2)(Ka2 + Ks2 ) ± (1/2)(Ks2 − Ka2 )[1 − 2C/(Ks2 − Ka2 )2 ].

(6.36)

Insertion of the expressions for Ka2 , Ks2 , and C in Eq. 6.34 shows that the terms involving (z/ωρ)2 cancel each other and we get z(−iωM) Q21 ≈ Ka2 + C/Ks2 = (κ/κ) ˜ 1 + i ωρ(z−iωM) = (κ/κ)(1 ˜ + ize /ωρ) Q22 ≈ Ks2 − C/Ks2 = ∞,

(6.37)

where the equivalent interaction impedance ze = z(−iωM)/(ze − iωM). This is consistent with the result obtained for the limp material in the previous section. The structure-borne wave now has an infinite propagation constant, which means zero wave speed and wavelength. It will also have infinite attenuation and will disappear as soon as it has been excited so that only the fluid-borne wave remains. Nevertheless, in some cases, for example when the surface is covered with an impervious membrane, the structure-borne wave has to be included to satisfy boundary conditions at the surface.

6.8.3 Pressure and Velocity Fields For each of the propagation constants obtained from the dispersion relation there is a corresponding wave mode in which the spatial variations of pressure and velocity are the same for the fluid and the structure. The general expression for the wave field in the porous material is then a linear superposition of the two modes. To determine the excitation of these modes by an incident plane sound wave, we proceed as follows. The plane wave incident on the absorber is of the form exp(ikx x + iky y + ikz z), where kx = k cos φ, ky = k sin φ sin ψ, kz = k sin φ cos ψ, and k = ω/c, φ and ψ being the polar and azimuthal angles, respectively (Figure 6.32). The field within the material is the sum of one wave traveling in the positive x-direction and one in

226


the negative direction. The wave in the positive x-direction has the same form as the incident wave with k replaced by q, kx replaced by qx , etc. Trace matching of the wave front at the boundary requires qy = ky and qz = kz and we get for Qx = qx /k, Qx =

Q2 − sin2 φ,

where Q = q/k, q 2 = qx2 + qy2 + qz2 , and k = ω/c, as discussed in Chapter 3. We place x = 0 at the wall and x = −L at the surface. The velocity and pressure fields will be linear combinations of contributions from the two characteristic modes of the system, identified by their propagation constant q1 and q2 . Since the velocity components ux and u x of the air and structure are both zero at x = 0 (rigid wall) the characteristic modes for the velocity field will be sine functions, i.e., ux = [U1 sin(q1x x) + U2 sin(q2x x)]eiqy y+iqz z u x = [V1 U1 sin(q1x x) + V2 U2 sin(q2x x)]eiqy y+iqz z ,

(6.38)

where q1 and q2 are the propagation constants for the two modes and V1 = U1 /U1 and V2 = U2 /U2 are the ratios of the velocity amplitudes in the structure and the air for the two modes. The corresponding expressions for the pressure field are p = [P1 cos(q1x x) + P2 cos(q2x x)]eiqy y+iqz z p = [1 P1 cos(q1x x) + 2 P2 cos(q2x x)]eiqy y+iqz z ,

(6.39)

where 1 = P1 /P1 as the ratio of the amplitudes in the structure and in the fluid for mode 1, with an analogous expression for mode 2. The pressure ratio is simplest obtained directly from Eq. 6.32 having obtained the propagation constants for the two modes from Eq. 6.36. Thus, in terms of the normalized propagation constant, we obtain 1,2 ≡

Q21,2 − Ka2 P

(−iz/ωρ)(κ/κ) ˜ . = = 2 − K2 ˜ P Q (−iz/ωρ)(K/κ) s 1,2

(6.40)

/U Similarly, the ratios V1,2 = U1,2 1,2 of the amplitudes of the velocity modes in the structure and in the fluid are obtained by insertion of the velocity and pressure fields in the momentum equations in Eq. 6.30 for the fluid and the structure. For each of the characteristic wave modes the spatial dependence of p and p is the same so that the ratio of their gradients will be the same as the ratio of the pressure amplitudes. Thus, we obtain for the corresponding ratio of the velocity amplitudes

V1,2 ≡

U1,2

U1,2

=

iωρ ˜ 1,2 − z , iωM˜ − z1,2

(6.41)

where 1,2 = (P /P )1,2 is given in Eq. 6.39. Finally, the relation between the pressure and velocity amplitudes P1,2 and U1,2 is obtained by insertion of the velocity and pressure fields for the fluid in the momentum

227

FLEXIBLE POROUS MATERIALS equation for the fluid in Eq. 6.30, P1 = −iρc Z1 U1 P2 = −iρc Z2 U2 Z1 =

1 ˜ Q1x (ρ/ρ

− izV1 /ωρ)

Z2 =

1 ˜ Q2x (ρ/ρ

− izV2 /ωρ).

(6.42)

The complex amplitudes of sound pressure and the x-component of fluid velocity just outside the boundary are denoted by p0 and u0 . From our definition of average fluid velocity within the material explained earlier, continuity of mass flow through the surface of the material at x = −L and sound pressure in the fluid require that u0 = ux (−L) and p0 = p(−L). Furthermore, the stress in the structure at the surface must be zero, p (−L) = 0, which means P2 cos(q2x L) = −(1 /2 )P1 cos(q1x L).

(6.43)

This boundary condition, p (−L) = 0, deserves a comment. In our formulation of the coupling between the air and the structure, the interaction force per unit length on the structure is expressed as being proportional to the relative velocity between the air and the structure; the constant of proportionality is the interaction impedance. Consequently, the interaction force on an element of the structure at the surface of the layer goes to zero as the thickness of the element goes to zero, and there is no sound pressure or sound pressure gradient in the air that contributes to the force. Therefore, the strain and the stress (pressure) in the structure are zero at the surface (otherwise the element would have infinite acceleration) unless it is in mechanical contact with another (porous) structure. The situation is analogous to that of a coil spring that interacts through viscous drag with the surrounding air. The strain and stress of the free end of the spring are zero. With P2 /P1 given by Eq. 6.47, the corresponding velocity ratio is obtained from Eq. 6.42, U2 /U1 = (Z1 /Z2 )(P2 /P1 ),

(6.44)

where Z1 and Z2 are given in Eq. 6.42. With this expression for U2 /U1 , the velocity amplitude distribution in the air and in the structure can now be computed from Eq. 6.38. Dissipation Function Having obtained the pressure and velocity distributions in the layer, we can readily determine the distribution of the dissipation of acoustic energy within the porous layer. With the velocity amplitudes being rms values, the distribution of the time average dissipation per unit volume due to the friction drag can be written θρc|u−u |2 , where

228


θρc is the real part of the interaction impedance. We normalize this expression with respect to the dissipation at the surface of the layer, w/w0 = |u − u |2 /|u0 − u 0 |2 w0 = θρc|u0 − u 0 |2 ,

(6.45)

where the subscript 0 refers to the conditions at the surface of the absorber. As explained in Chapter 2, the power dissipation due to the compression of the porous structure can be written as ωK2 κ|p |2 , where K2 is the imaginary part of the complex compressibility K˜ of the structure, and p = P1 [1 cos(Q1 kx) + 2 (P2 /P1 ) cos(Q2 kx)], is given in Eqs. 6.42 and 6.43. The factor κ = 1/ρc2 accounts for the fact that K2 is normalized with respect to air compressibility κ. To normalize this expression with respect to w0 (Eq. 6.45), we obtain the velocities at the surface from Eq. 6.39 with x = −L. Then, combining with the velocity ratio U2 /U1 , given in Eq. 6.44, and the ratio P1 /U1 in Eq. 6.42, we obtain for the normalized compressional dissipation function w /w0 =

kL K2 |p |2 , R|u0 − u 0 |2

(6.46)

where k = ω/c and R = θ L is the total normalized resistance of the layer. Examples of the velocity and pressure induced dissipation functions w/w0 and w /w0 , as well as their sum, are plotted in Figure 6.39 vs x/L.

6.8.4 Absorption Coefficients Open Surface We consider first the case when the surface of the porous layer is open; later, the effect of a thin impervious surface film will be studied. Recall Eq. 6.44 for the velocity ratio U2 /U1 = (Z1 /Z2 )(P2 /P1 ).

(6.47)

The sound pressure and velocity u0 at x = −L can be expressed as (see Eqs. 6.38 and 6.39) p0 = P1 (1 − 1 /2 ) cos(q1x L) = P2 (1 − 2 /1 ) cos(22x L) u0 = −U1 sin(q1x L) − U2 sin(q2x L) = −i(P1 /ρcZ1 ) sin(q1x L) − i(P2 /ρcZ2 ) sin(q2x L).

(6.48)

The normalized input admittance of the flexible porous layer is then Normalized input admittance, Flexible porous layer, Open surface u(0) = −iη1 tan(Q1x kL) − iη2 tan(Q2x kL) ηi ≡ βi + iσi = ρc p(0) η1 = 2 /Z1 (2 − 1 )

(6.49)

η2 = 1 /Z2 (1 − 2 ) where Z1 and Z2 : see Eq. 6.43, corresponding normalized input impedance: ζi = 1/ηi .

229

FLEXIBLE POROUS MATERIALS With reference to Chapter 2, the absorption coefficient is α(φ) =

4βi cos φ . (βi + cos φ)2 + σi 2

(6.50)

Closed Surface Some porous materials, such as urethane foams, are processed in such a way as to leave an impervious skin of negligible mass on the surface. The surface can also be made impervious by application of a membrane or plate in contact with the surface. We consider first the case when the mass of the impervious layer can be neglected in which case the boundary conditions at the surface are p0 = p + p

u0 = u = u .

(6.51)

From the second of these relations it follows that U2 (1 − V2 ) sin(q2x L) = −U1 (1 − V1 ) sin(q1x L),

(6.52)

and the velocity at the surface can be expressed as u0 = −

V1 − V2 V1 − V2 U1 sin(q1x L) = U2 sin(q2x L). 1 − V2 1 − V1

(6.53)

With the pressure at the surface being p0 = P1 (1 + 1 ) cos(q1x L) + P2 (1 + 2 cos(q2x L),

(6.54)

it follows that the normalized input impedance p0 /ρcu0 can be written Normalized input impedance, Flexible porous layer, Closed surface ζi ≡ θi + iχi = iζ1 cot(Q1x kL) + iζ2 cot(Q2x kL) ζ1 = ζ2 =

1−V2 V1 −V2 1−V1 V2 −V1

(1 + 1 )Z1

(6.55)

(1 + 2 )Z2

where Z1 and Z2 are given in Eq. 6.43. With reference to Chapter 2, the absorption coefficients can now be computed in terms of the input impedance (Eqs. 3.22 and 3.23). Surface with a Resistive Screen If a screen is attached to the open surface, the boundary condition at the surface is that the velocity of the structure be the same as the velocity of the screen. Furthermore, the force acting on the screen is the sum of the interaction force due to the relative motion of the air and the screen and the force due to the contact with the porous layer.

230


We refer to Chapter 1 for a discussion of the screen. We use the notation z ≡ ρcζz for the interaction impedance of the screen and the mass per unit area is m. The boundary condition requires that the velocity of the screen must equal the velocity amplitude of the porous structure at x = −L, i.e., u x (−L). The velocity amplitude u0 of the air just outside the screen is the same as the amplitude ux (−L) at the entrance to the porous layer. From the definition of interaction impedance z, the flow induced force on the screen is then z(u − u ), and it follows that the equation of motion of the screen is z(u − u ) = −iωmu + p .

(6.56)

This equation now serves as the boundary condition from which we can determine the amplitude ratio U1 /U2 of the two modes in the system. Thus, with zt = z − iωm and Vs ≡ z/zt , it follows from Eqs. 6.41 and 6.42 that [(Vs − V1 )U1 sin(qx1 L) + (Vs − V2 )U2 sin(q2x L)] = (1/zt )[1 P1 cos(q1x L) + 2 P2 cos(q2x L)].

(6.57)

Expressing the pressure amplitudes P1 , P2 in terms of the velocity amplitudes U1 , U2 from Eq. 6.43, we obtain for the velocity ratio U2 (Vs − V1 ) sin(qx1 L) + i(1 Z1 /ζt ) cos(qx1 L) =− . U1 (Vs − V2 ) sin(qx2 L) + i(2 Z2 /ζt ) cos(qx2 L)

(6.58)

With p0 being the sound pressure amplitude in front of the screen it follows from the definition of the interaction impedance that p0 − p = zs (u − u ) = −iωmu + p

(6.59)

p0 = (1 + 1 )P1 cos(q1x L) + (1 + 2 )P2 cos(qx2 L) −(−iωm)[V1 U1 sin(q1x L) + V2 U2 sin(q2x L)].

(6.60)

or

The velocity amplitude in front of the screen is, from Eq. 6.41, u0 = −[U1 sin(qx1 L) + U2 sin(qx2 L)].

(6.61)

From these expressions for p0 and u0 and the velocity ratio in Eq. 6.58, we obtain the input impedance ζ0 = z0 /ρc ζ0 =

i(1+1 )Z1 U1 cos(q1x L)+i(1+2 )Z2 U2 cos(q2x L) U1 sin(q1x L)+U2 sin(q2x L) ωm V1 U1 sin(q1x L)+V2 U2 sin(q2x L) −i( ρc ) U1 sin(q1x L)+U2 sin(q2x L) .

(6.62)

To check this expression, we consider first the limiting case when the interaction impedance is zero. We should then recover the result for the open surface in Eq. 6.49. With zs = 0 and m = 0 (zt = 0) it follows from Eq. 6.58 that 2 Z2 cos(q2x L)U2 = −1 Z1 cos(q1x L)U1 .

(6.63)

231

FLEXIBLE POROUS MATERIALS

Using this relation in the numerator in Eq. 6.62, we obtain for the admittance 1/ζ0 a result which agrees with that obtained earlier in Eq. 6.49. Actually, we can obtain the result for a closed surface if we put zs = zt = ∞ (to make the screen impervious) and m = 0 to simulate a skin with negligible mass. This results in Vs = 1 and Eq. 6.58 yields (1 − V2 ) sin(q2x L)U2 = (1 − V1 ) sin(q1x L)U1 .

(6.64)

If this relation is used in the denominator of Eq. 6.62, we find the impedance to be in agreement with the result in Eq. 6.55. It is important to note that to account for a screen cover attached to an open porous layer it is not merely a matter of adding the impedance of the screen to the input impedance of the bare porous layer. Such an addition is valid only if the screen is placed close to but not in hard contact with the screen. Use of 4 × 4 Matrices An alternate and more straightforward calculation of the input impedance in the presence of a screen can be made by using the 4 × 4 matrices of the screen and the porous layer. Whenever a structure is comprised of flexible layers in mechanical contact with one another, a matrix formulation of a study of the acoustical characteristics of the structure involves 4 × 4 matrices. This obtains where the dynamical description of a structural element can be made with only two field variables, which, when taken together with the two variables for the sound field leads to a total of four variables. As above, we denote by subscript 0 the field variables just in front of the screen and by subscript 2 the variables at the rigid backing wall. Then, if Tij denote the matrix elements of the product of the matrices for the screen and the porous layer, we have, accounting for u2 = u 2 = 0, p0 = T11 p2 + T13 p2

ρcu0 = T21 p2 + T23 p2

(6.65)

so that the normalized input impedance is ζi =

T11 + T13 (p2 /p2 ) . T21 + T23 (p2 /p2 )

(6.66)

The ratio p2 /p2 is obtained by the requirement that p0 = 0 at the front surface where there is not mechanical contact with the screen (see the discussion in connection with Eq. 6.47). Thus, p0 = T31 p2 + T33 p2 = 0

(6.67)

yields p2 /p2 = −T31 /T33 so that the input impedance becomes ζi =

T11 T33 − T13 T31 . T21 T33 − T23 T31

(6.68)

232


From the product of the matrices of the screen and the porous layer, we can express matrix elements Tij of the combination in terms of ζe and Mij to obtain T11 = M11 + ζs (M21 − M41 ) T13 = M13 + ζs (M23 − M43 ) T21 = M21

(6.69)

T23 = M23 T31 = −ζs M21 + M31 + ζt M41 T33 = −ζs M23 + M33 + ζt M43 . The input impedance then follows from Eq. 6.68. The matrix multiplication can, of course, be done directly by a computer without the need for explicit expressions for the matrix elements and in a problem involving several layers and screens, this is the only tractable method. General Comments In order to include the frequency dependence of the interaction impedance (flow resistance and structure factor), we again use the results from the equivalent slot absorber and proceed to determine the equivalent parameters a/dv and a/dh from the measured steady flow resistance as discussed in Chapter 3 and express them in terms of the frequency ratio f/fv , where fv = r0 /(2πρ). These parameters determine Fv = F [(1 + i)a/dv ] and Fh = F [(1 + i)a/dh ] (where F is given in Eq. 2.78), which in turn are used to determine the viscous interaction impedance and the complex compressibility. Thus, for the viscous interaction impedance zv , the interaction impedance z, the ˜ complex density ratios for the fluid and the solid, ρ/ρ ˜ and M/ρ, and the complex compressibility of the fluid we have used the expressions Fv zv = (1/H )(−iωρ) 1−F v

z = zv − iωGs ρ ρ/ρ ˜ = 1 + iz/ρ ˜ M/ρ = M(1 + iz/ωM).

(6.70)

The complex compressibility of the fluid per unit volume of the structure is κ˜ = H κ˜ 1 , where H is the porosity and κ˜ 1 /κ = 1 + (γ − 1)Fh .

(6.71)

The induced mass factor Gs and the corresponding structure factor s = 1 + Gs as well as the complex compressibility of the porous structure are empirical constants, which have to be obtained from experiments. The propagation constants Q1,2 (Eq. 6.36) for the two modes can now be calculated together with the corresponding pressure and velocity ratios, input admittance or impedance, and the absorption coefficient, as outlined in this chapter.

Part II

Duct Attenuators

Chapter 7

Duct Acoustics 7.1 PRELIMINARIES As defined here, duct acoustics deals with the propagation of sound in ducts. It is of practical interest in a variety of different contexts ranging from speech and hearing (vocal tract, ear canal) to music (wind instruments), internal combustion and jet engines, wind tunnels and air handling systems in buildings. In some of these applications, the related problem of noise reduction by means of lined ducts and silencers is a small but an essential part; actually, it is the main topic of these notes. When it comes to noise reduction in the ear canal, most of us merely use a plug of cotton or some soft tissue paper or, to be fancier, a commercial brand of ear plugs or muffs, without worrying much about the physics involved. For industrial duct systems, this ear plug approach no longer is possible in general, since the duct usually has to carry flow with only a small pressure drop allowed. Then, a more analytical method is called for since cost and effectiveness of noise reduction play a significant role in the design. Noise reduction in a duct silencer thus obtained is caused in part by sound absorption and in part by reflections and interference effects. Normally, these mechanisms are interrelated and cannot be strictly separated. As a rough classification, however, a silencer in which sound absorption dominates is usually referred to as dissipative and when reflection dominates, as reactive. The latter typically is used for low frequency sound and pulsations. The dissipative silencer is often simply a duct lined with an appropriate (porous) material. The designations ‘silencer’ and ‘muffler’ apply to both dissipative and reactive duct attenuators. Although conceptually, sound transmission through a duct is not much different from sound transmission through a hole in a wall, when it comes to a detailed quantitative analysis, however, additional difficulties arise. Actually, the mathematical analysis of sound transmission through a duct with a complicated shape generally requires the use of tedious finite element numerical procedures. They are comparatively slow and leave much to be desired in terms of providing general insight. Fortunately, there are duct configurations of great practical interest, which do lend themselves to direct mathematical analysis from first principles, and considerable insight can be gained

235

236


and general design principles established by a thorough study of the corresponding solutions. This is the main emphasis in this book. Before proceeding to the ‘nuts’ and ‘bolts,’ we summarize briefly some of the concepts and effects that will be discussed at some length in the rest of the book. Wave modes. We start by illustrating wave modes in a duct with rigid, nonporous walls by a simple experiment. The bulk of subsequent analysis is devoted to the simplest of these modes, the fundamental mode. Although the basic characteristics of higher modes are discussed, inclusion of them in practical cases is often intractible analytically and semi-empirical corrections are used to yield higher order mode corrections. This approach holds true also for the effects of refraction of sound in a nonuniform mean flow and temperature distribution in a duct. Description of silencer performance. A discussion of the quantities used in the description of the performance of a lined duct is next; attenuation, transmission loss (TL and TL0), insertion loss (IL), and noise reduction (NR), and procedures for the measurement of these quantities. The difference between them is apparent particularly at low frequencies and unlined (reactive) duct elements such as straight pipe sections, contraction and expansion chambers, and side-branch resonators. Duct liners, local and nonlocal reaction. To achieve sound absorption and a related attenuation of sound in a duct, the wall is normally lined with a porous material. If the material is placed in compartments separated by closely spaced transverse solid partitions, the sound field in such compartments is determined only by the sound pressure in the duct at the entrance to the compartment, and not at other locations. Such a liner is called locally reacting. The sound pressure produces a ‘pumping’ of air in and out of the compartment, and this leads to energy absorption. Axial motion in the liner is prevented by the partitions. It should be clear that if the liner (and hence the partitions) occupy the entire duct, the transmission of sound is prevented by the partitions (if assumed rigid). With the partitions removed, there will be sound transmission in the axial direction within the liner and the corresponding sound field at a given axial position depends not only on the sound pressure in the duct at that location but on the entire axial distribution of pressure along the duct. Such a liner is said to be nonlocally reacting. In reality, transverse partitions are normally used but their separation is not always so small that the liner qualifies as locally reacting. The corresponding measured duct performance then falls between the values obtained for locally and nonlocally reacting liners. Actually, at low and high frequencies the predicted values for the two types of liners are essentially the same but in the mid-frequency range, local reaction yields a somewhat higher predicted value than nonlocal. It should be added that the mathematical analysis of the locally reacting duct is considerably simpler than for the nonlocally reacting. At high frequencies, higher order modes will affect the experimental data and yield higher values than predicted for the fundamental mode. Fortunately, the problem of noise reduction at high frequencies is typically not critical since the low, and to some extent, the mid-frequencies typically dictate the design. Acoustically equivalent ducts. Considerable attention is paid to parametric studies of parallel baffle type silencers or attenuators (sometimes referred to loosely as ‘splitters’), see Figure 7.6, and the acoustically equivalent rectangular lined ducts or

DUCT ACOUSTICS

237

channels. Their attenuation spectra are shown in Figures 8.1 and 8.3 for locally and nonlocally reacting liners respectively. The effect of perforated facings and screens on such liners have also been determined. Actually, results for other multilayer liners of porous materials, screens, air layers, and perforated plates are presented in this book and can readily be determined with a computer program. The effects of slots in a porous liner are also discussed. From the family of attenuation curves obtained for different parameter values, optimum design and the corresponding maximum attenuation values have been obtained. The results are summarized in Figure 8.2. Effects of mean flow. Normally, a duct carries a mean flow which has at least five effects on sound transmission. The first is convection; the wave speed is the sum of the local sound speed and the flow velocity and is increased or decreased by the flow depending on the direction of sound propagation. The flow speed is nonuniform across the duct with the maximum at the center of the duct. This leads to refraction of sound, toward the boundary downstream and away from the boundary upstream with a corresponding increase and decrease, respectively, of the attenuation. It is particularly important at high frequencies. A semi-empirical correction for this effect is proposed. Another effect of flow is a modification of the boundary impedance caused by soundflow interaction. This is of particular importance when the liner consists merely of a perforated plate backed by an air layer or when cavity resonators are involved. The resistive component of the boundary impedance is increased by the flow and this leads to broadening of the resonances. In addition, flow generated noise in a duct normally is increased somewhat by insertion of a silencer; this tends to decrease its insertion loss somewhat. Still another flow effect is the sound attenuation in turbulent duct flow. An analysis of it is deferred to Chapter 9 and an interesting result is a comparison with visco-thermal attenuation, which shows that the flow induced attenuation usually is dominant, at least at low frequencies. Effects of temperature. The major effect of temperature is its influence on the sound speed (and hence the wavelength at a given frequency) as the performance of a duct liner normally is intimately related to the ratio of the thickness of the liner and the wavelength. This (typically) results in a shift of the attenuation spectrum toward higher frequencies with increasing temperature. Like the flow velocity distribution, the temperature distribution in a duct is not uniform. A higher temperature at the center of the duct leads to refraction of sound toward the boundary, but unlike the effect of flow, it is independent of the direction of the sound. Thus, the refractions by flow and temperature cooperate for sound in the flow direction but are opposed in the opposite direction. The temperature can also affect the boundary impedance of a duct liner since the flow resistance of a liner increases with temperature. Liquid pipe lines. A question often asked is ‘To what extent can the results obtained for air ducts be applied to liquid pipe lines?’ An answer is given in Chapter 9 by two examples, one involving a pipe line with a slightly resilient wall and the other with a highly resilient one.

238


7.2 WAVE MODES 7.2.1 Simple Illustration The sound wave in a duct with rigid walls can have many different forms. The simplest is the fundamental wave mode in which the sound pressure is uniform across the duct, and it behaves like a plane wave in free field. We can imagine such a wave as being generated by an oscillating plane piston (approximated by a loudspeaker) at the beginning of the duct. This wave will travel unattenuated along the duct at all frequencies if we neglect visco-thermal effects at the boundaries and absorption within the gas itself. If the piston is simulated by the two loudspeakers in Figure 7.1, the speakers have to be driven in phase to produce a plane wave. If the speakers are driven 180 degrees out of phase, so that one pushes when the other pulls, the average axial velocity amplitude in the duct will be zero, and there will be no plane wave generated. Sound will still be produced, though, not in the plane wave mode but in the form of ‘higher order’ modes. Unlike the plane wave, however, the character of a higher mode depends strongly on frequency. The wave contributions from the individual speakers travel out into the duct, one with a positive and the other with a negative sound pressure. If there were no phase difference between them proportional to the path difference of wave travel to the point of observation (receiver), they would cancel each other. This is the case in the mid-plane of the duct where the net sound pressure will be zero. Loudspeakers

(0,0)-mode

Oscillator Amplifier

D

Speakers in phase: Plane wave propagation at all frequencies (0,1)-mode

Speakers 180 deg out of phase

Figure 7.1: If the two loudspeakers operate in phase (push-push), a plane wave will be generated. If they are 180 degrees out of phase (push-pull), the (0, 1) higher acoustic mode will propagate if the frequency exceeds the cut-on frequency c/2D, where c is the sound speed and D is the duct width.

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At a point in the duct away from the mid-plane, however, there will be a source-to-receiver path difference so that the elementary waves at the receiver are not completely out of phase. In fact, if the difference is half a wavelength, the waves will arrive in phase so that constructive interference results. We then get a wave that travels through the duct with a node at the mid-plane and, unlike the fundamental mode (i.e., the plane wave), it is characterized by zero average oscillatory axial flow in the duct (the flows above and below the mid-plane are 180 degrees out of phase). The wave is called a higher order mode but could also be termed a ‘push-pull’ wave. With reference to the example in the figure, the pressure amplitude variation in the vertical direction shows a nodal plane in the horizontal plane at the center. The pressure amplitude in the x-direction (horizontal) is uniform, i.e., no nodal plane. At higher frequencies than that in the figure, more than one nodal plane can occur. In general, there will be vertical nodal planes also, and in terms of the number of nodal planes m and n in the two directions, the wave is designated as an (m, n)-mode. It can be generated by several pairs of loudspeakers like those in the figure. The plane wave then is the (0, 0)-mode and the wave in the figure is the (0, 1)-mode. For the (0, 1)-mode to be propagated through a hard walled duct without attenuation due to destructive interference between the waves from the two speakers, requires that the wavelength be short enough so that a path difference between these two waves of half a wavelength can be obtained; at sufficiently low frequencies and correspondingly long wavelengths, this is not possible, and the interference between the push and pull contributions to the sound pressure leads to a decrease of the resulting amplitude with distance. The reason is that the path difference decreases with increasing distance from the source so that destructive interference will be more pronounced with increasing distance. As a result, it turns out that the wave amplitude will decrease exponentially with distance. Such a wave is called evanescent. The largest path difference is at the beginning of the duct in the plane of the source where it equals the width D of the duct (from the top of one speaker to the bottom of the other). Then, if half a wavelength equals the duct width D, there will be constructive interference between the elementary wave from the top of one speaker and the bottom of the other, and λ/2 = D signifies the condition of ‘cut-on’ of the higher mode. The corresponding frequency, f0,1 = c/λ = c/2D, is called the ‘cut-on’ frequency of the (0, 1)-mode (it is also called the ‘cut-off’ frequency, the choice depending on from what direction the frequency is approached, we suppose). At this frequency the mode represents a standing wave perpendicular to the duct axis. The arrangement shown in Figure 7.1 is useful as a simple table top demonstration of higher mode generation. The sound source consists of two identical loudspeakers mounted in the wall at the beginning of the duct. They are driven by an oscillatoramplifier, as indicated. To change the speakers from in-phase to 180 degrees out-ofphase operation (from ‘push-push’ to ‘push-pull’) simply involves switching the leads from the amplifier to the speakers, as shown. In a particular experiment, the duct height was D = 25 cm corresponding to a cut-on frequency of 684 Hz. With the speakers operating out-of-phase, increasing the frequency through the cut-on value brings us out of the exponential decay regime and a marked change in sound pressure emitted from the duct is observed. The duct can be said to act like a high-pass filter for the (0, 1)-mode.

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Because of the evanescence (exponential decay with distance) of the wave below the cut-on frequency, the sound that radiates from the end of the duct is feeble. It is due to what is left of the evanescent wave when it reaches the end. It is also possible that a weak plane wave component may be present because of an unavoidable difference in the speakers so that the average velocity over the total source surface is not exactly zero. In any event, if one of the speakers is turned off in this push-pull operation, a substantial increase in sound pressure is observed because the wave, which is now generated by the remaining speaker, is not destructively interfered with. As will be clear in our subsequent discussion, the wave field in the (0, 1)-mode can be thought of as a superposition of plane waves, which travel at a certain angle with respect to the duct axis, being repeatedly reflected from the duct walls to build up a traveling (0, 1)-mode wave. At the cut-on frequency, these waves are normal to the axis of the duct, but at a higher frequency, the angle φ with the axis is given by sin φ = (λ/2)/D. The phase velocity of this mode will be the speed of the intersection point of a wave front with the boundary (or the duct axis), and this speed is c/ sin φ, i.e., greater than the free field sound speed and the (0, 0)-mode in the duct. Because of the difference in phase velocities, the superposition of a plane wave and a higher mode results in a wave field that varies with position along the duct. Similar arguments show that if the wavelength is smaller than D/2n, where n is an integer, a mode, the (0, n)-mode with n nodal planes and a cut-on frequency f0,n = nf0,1 , can propagate. Again, the wave field in the duct can be regarded as a superposition of plane waves which are reflected back and forth between the boundaries and traveling in a direction which makes an angle φ with the duct axis, where sin φ = λ/(2nD). The phase velocity of a higher mode is always greater than the sound speed in free field and, like the angle φ, is frequency dependent. If the duct wall has an absorptive liner, the reflections from the walls involve not only a reduction of the amplitude but also a phase change and the superposition of the waves no longer leads to ‘clean’ nodal planes and there will be no sharply defined cut-off frequency. The sound field still can be expressed as a superposition of fundamental and higher order modes corresponding to the (0, 0) and (m, n) modes. In most problems of noise control, the fundamental mode is of main interest and most of the numerical results given in these notes refer to it.

7.3 MEASURES OF SILENCER PERFORMANCE 7.3.1 Attenuation If a microphone is moved along the air channel in a lined duct in which the fundamental mode is dominant, we find that the recorded sound pressure level decreases linearly with distance from the source except for some irregularities at the entrance and at the end of the duct. These irregularities are due to higher order modes and reflection. (If the frequency is below the cut-on values for the higher modes, the contribution from these modes to the total sound field, the dominant contribution to the field, is normally from the fundamental mode.) The slope of the linear decrease of the sound pressure level of this mode is, by definition, the attenuation per unit length in the duct, and it is this and related quantities described below that are computed in

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the mathematical analysis in Chapter 10. The (total) attenuation of the fundamental mode is the product of the slope and the length of the duct.

7.3.2 Transmission Loss, TL and TL0 Two-Room Method The concept of transmission loss, TL, generally refers to a partition wall between two (reverberation) rooms, a source room and a receiving room. It is normally defined as the ratio, expressed in dB, of the incident acoustic power on the partition wall from the source room and the power transmitted into the receiving room. On the assumption of diffuse sound fields in the rooms, the incident and transmitted powers, and hence the TL, can be expressed in terms of the measured average sound pressure levels in the two rooms, the area of the partition, and the absorption area (determined from measured reverberation time) in the receiver room. In principle, the measurement of the transmission loss of a duct silencer could be carried out in the same way with the duct inserted into an opening of a heavy wall between the two rooms. The transmission loss of the duct then could be determined from the average sound pressures in the two rooms in the same way as for the partition wall. This method does not seem to have been used for ducts, however. One reason is, no doubt, that a silencer normally is not used in this manner, but the method should be seriously considered as an option. ‘Standard’ Method The silencer to be studied is placed in a test duct, as shown schematically in Figure 7.2. Typically, the sound field is produced by one of the two options shown in the figure, which also allows for the generation of a mean flow through the duct. The sound field in the test duct at the entrance of the silencer is a superposition of an incident wave and a wave reflected from the silencer. Similarly, the sound field at the exit contains a transmitted wave and a wave reflected from the end of the test duct. The extraction of the primary incident and transmitted powers (in the test duct) from these fields for the determination of TL as the ratio (in dB) of these powers is by no means obvious and requires further measurements and analysis. This is one of the hidden difficulties with TL, which is seldom thought of. In this context, one might think that if the incident wave is in the form of a short pulse (which has a broad frequency spectrum), the separation of the incident and reflected waves should be straight forward. However, since the pulse generally contains higher modes in addition to the plane wave, the overall pulse shape will change with position because the different modes in the pulse have different phase velocities, and the interpretation of the data can be complicated. Thus, the measurement of TL is not simple in comparison with that of other quantities, such as the insertion loss (see below). However, the calculation of TL is relatively simple, however, and whenever data of TL are presented, they often are calculated rather than directly measured (see Section A.2.3) or inferred from insertion loss measurements.

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Figure 7.2: Top: Sound is injected from a plenum chamber. Reflections from both ends of the main duct will be significant at low frequencies. Bottom: Sound is injected from the sides of the duct, which contains a parallel baffle or wedge absorber to make reflection from the source end negligible. In both cases, air passages are made available for mean flow to enter the test duct.

The great advantage of TL is that it depends only on the characteristics of the silencer and not on system parameters of the test facility, such as the location of the silencer in the test duct, the reflections from the ends of the test duct, and the impedance of the sound source. For this reason, TL is frequently used, as in this book, in comparing (calculated) performance of different silencers and should, in principle, be used in specifying performance in silencer manufacturers’ literature. As if the various measures of silencer performance were not enough, we introduce an additional one, a transmission loss TL0 which, unlike TL, readily can be measured. The incident power is now replaced by the net value, i.e., the difference between the direct incident power from the source and the power reflected from the silencer. The net transmitted power is defined in an analogous manner as the difference between the power transmitted by the silencer and the power reflected from the end of the test duct. The corresponding transmission loss is then defined as the ratio, expressed in dB, of these net powers. We denote this transmission loss by TL0 to distinguish it from TL. This new transmission loss, unlike TL, can be measured by standard intensity probes through integration of the intensity over the area of the test duct on the inlet and exit of the silencer. The difference between the incident and transmitted net powers is absorbed by the silencer. The quantitative analysis of TL and TL0 is given in Section A.2.3. TL0 depends on the termination impedance of the test duct, whereas TL is independent of all system parameters. We refer to Figure 7.3 where various measures of silencer performance for a particular silencer are illustrated. If the ends of the test duct are reflection free, the differences in this example are rather small. The transmission loss is always positive if no sound, such as flow noise, is generated within the silencer. For a source which delivers the same net acoustic power regardless of the acoustical load, TL0 would be the ideal quantity to use in describing silencer

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Figure 7.3: 1/1 and 1/3 OB spectra of TL, TL0, IL, and NR of a channel lined on one side with a porous layer. Liner thickness: 5 inches. Flow resistance (normalized): 0.5 per inch. Channel length: 4 ft. Air channel width: 5 inches. L1 = L2 = 10 ft (see Figure 7.2). The first four graphs refer to a reflection-free source and termination (of main duct). For the last two, the main duct is open ended, as indicated. The dashed line in the TL spectrum includes semi-empirical correction for higher modes.

performance. Actually, for many sources, including jet engines, the only source data that is available is the power level PWL. This level is generally assumed to be independent of the acoustic load. Then, based on the PWL of the source, and the TL0 of a silencer, the output power level of the silencer will be PWL-TL0. For a purely reactive silencer, such as an unlined duct section or expansion chamber, there is no power absorbed and TL0 will be zero. The calculated and frequently

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reported TL for such a silencer is generally not zero but it has limited value since it is rarely measured and difficult to apply in practice. It is then better to use the insertion loss, to be described next.

7.3.3 Insertion Loss, IL In our discussion of silencer performance, we have in mind an experimental arrangement shown in Figure 7.2, where a silencer, in the form of an acoustically lined duct section, is located in a larger duct, which we have called the ‘test duct’ or ‘main duct.’ Sound is injected into it at one end and a fan provides a mean flow, which can be reversed in direction. The noise from the fan is reduced to an insignificant level by means of an appropriate silencer. The insertion loss, IL, is defined as the change of the sound pressure level at a fixed location or as the change of the level of the total radiated acoustic power caused by the insertion of the silencer. The power can be determined by integrating the intensity over the exit end of the test duct by means of an intensity probe or by measuring the average sound pressure level in a reverberation room, which terminates the test duct. Thus, unlike TL, it is simple to measure and is of more direct practical value than TL. The calculation of IL, however, is more involved than for TL since it requires knowledge of system parameters such as the source impedance, the location of the silencer in the test duct, and the reflection coefficients at both ends of the test duct. In mathematical modeling, the source can be characterized by an ‘internal pressure’ and an equivalent source impedance (by analogy with electrical circuits). For example, a fan has a low source impedance, whereas a positive displacement pump or compressor has a high source impedance. Maximum power from the source is obtained when the input resistance of the silencer equals the source resistance and when the corresponding reactive parts cancel each other. Unlike the transmission loss, the insertion loss can be positive or negative depending on the degree of impedance mismatch caused by the silencer (see examples in Figure 7.3). Thus, it is possible (typically at low frequencies) that the insertion of the silencer will improve the impedance match and hence cause increased power output from the source. The input impedance of the test duct is often referred to loosely as the ‘acoustic load’ on the source. Our mathematical analysis of silencer performance, starting from first principles, considers only the fundamental mode, and higher modes are accounted for by semiempirical corrections. The open area of a lined duct silencer normally is smaller than the area of the test duct so that the sound encounters a (sudden) contraction in the area at the entrance of the silencer and a sudden expansion at the exit. These area changes affect TL and IL but have nothing to do with attenuation within the silencer, which, as we have seen, deals only with what happens after the sound has entered the silencer. Thus, it should not be surprising to find that the TL and the attenuation per unit length multiplied by the length of the silencer are not equal. For example, if the absorptive liner is removed in such a silencer, the attenuation (and TL0) will be zero but TL (and IL) will not be because of the effects of reflections. A typical test duct in a laboratory may have a cross section of 4 ft by 4 ft and a length of about 60 ft. It is connected to a fan via a large silencer and a plenum chamber,

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which contains the sound source(s), as illustrated schematically in Figure 7.2. The frequency dependence of the insertion loss typically is expressed in terms of 1/1, 1/3, and sometimes 1/12 octave band averages over a frequency range from 31.5 to 8000 Hz. In one of the source configurations in Figure 7.2, the sound is injected into the duct from one or more loudspeakers placed in an acoustically treated plenum chamber. Both ends of the test duct then will be highly reflective at low frequencies, and the equivalent source impedance is approximately equal to the impedance of an open ended duct. In the second arrangement in the figure, the source end of the duct contains a wedge or parallel baffle absorber designed to give high absorption over the range of frequencies involved and arranged in such a way as to permit air flow through the test duct. The normalized source impedance of such a ‘reflection free’ source will be approximately 1 under ideal conditions. The sound can be injected into the duct from loudspeakers mounted in the wall of the duct, as shown. The arrangement in the figure applies specifically to the kind of silencer used in air handling applications in buildings. However, what has been said about the relation between transmission loss and insertion loss applies in general. A silencer stack, as in a test cell for jet engines, can contain many elements, lined ducts, area transition elements, expansion chambers, side branches, elbows, etc. The test cell typically is U-shaped with an air inlet and a discharge stack. The insertion loss, IL, often refers to the insertion of the entire test cell with all its acoustical interior treatments, the reference power then being the power emitted by the bare jet engine. Another insertion loss may refer to the insertion of an interior acoustical treatment in an existing cell; the reference power is then the power emitted from the untreated cell. If the insertion loss of the untreated cell is IL0 (the reference power being that of the bare jet engine) and the insertion loss of the treated cell is IL, the insertion loss of the interior acoustical treatment is IL-IL0. In the particular case of a reflection-free source (normalized internal impedance = 1), the insertion loss of a straight empty hard duct is zero (see Section 9.1). Thus, in the bottom sketch in Figure 7.2, assuming the source to be reflection-free, the insertion loss of the entire duct is the same as the insertion loss of the interior lined duct section. Multi-Source Environment Effect of ‘Background Noise’ Another, but sometimes overlooked factor, is that with the normally used noise measurement equipment, the measured noise level includes the contribution from all sources including the ‘background’ noise, not only the primary noise source under consideration. Therefore, if the background noise is high enough the measured (total) level will be practically the same after insertion of a silencer, i.e., the insertion loss of the silencer will be practically zero. A typical example concerns the insertion loss of an exhaust muffler for an automobile. Assuming that the car is not moving but with the engine running, the total noise is contributed by several sources, the air inlet, the exhaust, valves, and other engine components. The reference level in this case is the level obtained for the exhaust pipe without the muffler installed. To determine the effect of a muffler on the exhaust noise, the microphone has to be placed sufficiently close to the exhaust so that the

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exhaust noise dominates. As the distance from the exhaust is increased, the noise from other sources (background noise) will be relatively more significant and the insertion loss will decrease with the distance from the exhaust. If a silencer itself produces flow noise, it is possible that the insertion loss will be negative. In one case, the noise from a jet engine test facility was too high to be acceptable and a muffler was added to the top of the exhaust stack. However, no reduction of the total noise output was achieved. The muffler itself produced enough additional flow noise to make its overall insertion loss zero and even negative.

7.3.4 Noise Reduction, NR Noise reduction is the difference between the sound pressure level at the beginning and at the end of an acoustical element or silencer. Unlike TL, this quantity depends not only on the element itself but on system parameters. For this reason, the quantity can be misleading, but it is sometimes used to check the performance of a silencer in situ in an exhaust stack, for example. By measuring the sound pressure levels at the bottom and top of the silencer, L1 and L2 , the noise reduction, NR = L1 − L2 , is obtained. At high and middle frequencies this measure is frequently not too different from TL or IL, but considerable deviations can occur. For example, if the top of the silencer happens to fall in a pressure node in the standing wave between the silencer and the top of the stack, the NR thus obtained can be much larger than the TL or IL. Another example in which NR will be somewhat absurd is in a standing wave field in a bare duct. Depending on the locations of the measured values of L1 and L2 , the NR of the bare duct section can be anything, positive, negative, very large or very small, including zero. Therefore, the concept should be used with care.

7.3.5 Numerical Examples For comparison of the different performance measures of a silencer, we have shown in Figure 7.3 the computed spectra of TL, TL0, IL, and NR of a silencer consisting of a rectangular duct lined on one side with a porous layer. In each graph the 1/1 OB (octave band) and 1/3 OB spectra are shown and the 1/1 OB spectrum for TL also includes the spectrum with a semi-empirical correction for higher order modes.1 As can be seen, there is not much difference in this example between TL, TL0, IL, and NR when the source and the termination of the main duct are reflection-free.2 With the main duct open ended and the reflection from the source assumed to be the same as from the open end of the main duct, the IL and the NR, can differ markedly from the TL at low frequencies and can be negative at some frequencies, as shown in the last two graphs in the figure. 1 It is recommended that the reader use a computer program to produce similar graphs for other values of the silencer parameters. 2 Recall that TL refers to incident and transmitted power and TL0 to the net values of these quantities. Also recall that NR is the difference in sound pressure level at the entrance and exit of the silencer and that TL = IL when the source and termination are reflection-free.

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Figure 7.4: The attenuation per unit length (the air channel width) for the lined duct referred to in Figure 7.3.

Figure 7.5: Transmission loss (TL) and insertion loss (IL) of a circular lined duct. L =

5 ft. L1 = 10 ft, L2 = 10 ft (see Figure 7.2). Inner diameter: 24 inches. Liner thickness: 8 inches. Flow resistance: 0.25 ρc/inch. Perforated plate: Thickness: 0.1 inch. Hole diameter: 0.1 inch. Mat: Normalized flow resistance: 0.1. Weight: 0.1 lb/ft2 . Left: Transmission loss. Right: Insertion loss. Reflection-free source. Open ended main duct.

The attenuation per unit length (the channel width) is shown in Figure 7.4. The peak value of the attenuation is approximately 4.2 dB per unit, with the unit length being the channel width 5 inches and the duct length 4 ft, i.e., 9.6 unit lengths, and the total attenuation will be 40.3 dB. This is consistent with the peak value of the 1/3 OB TL of about 40 dB, and the frequency of the maximum, approximately 500 Hz, is the same for both. Actually, in this case, there is not much difference between the total attenuation and the 1/3 OB TL spectrum. Another example of typical computed IL and TL spectra refer to a circular lined duct as shown in Figure 7.5 in which the source is reflection-free but the main duct is open ended. The porous liner is covered with a resistive mat and a perforated plate. The length is L = 5 ft, the diameter of the air channel 24 inches, and the thickness of the liner 8 inches. There is no mean flow. The distance from the sound source to

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the beginning of the lined duct is L1 = 10 ft and the distance from the end of the duct to the end of the test duct is L2 = 10 ft. The source is assumed to be reflection free, i.e., the normalized source impedance is 1. The mathematical analysis of duct performance is conveniently done by means of transmission matrices (see Appendix A) upon which computer programs are based. In regard to IL, one has to make clear how the reference power is defined. In the examples given above, it refers to the bare test duct of length L + L1 + L2 . The fluctuations in the insertion loss depend on the frequency dependence of the acoustic input impedance at the source and the corresponding impedance mismatch and power output for the test duct both with and without the silencer inserted. Note that the insertion loss can be negative which means, of course, that the power radiated from the end of the test duct is larger with the silencer than without. The transmission loss, on the other hand, is always positive. If both source and termination of the test duct are reflection-free, TL and IL are equal but TL0 will be slightly different. In measurements of duct performance, particular care should be taken in making clear which quantity is measured3 so that a measurement of NR is not taken to mean IL, TL, or TL0.

7.3.6 Pressure Drop and Flow Noise (Self-Noise, SN) The pressure drop in flow ducts and silencers typically is dominated by the contribution at discontinuities in the flow area. For a silencer, the major contribution comes from the exit flow, where flow separation and associated turbulence occurs. Similarly, most of the flow generated noise in a duct typically results from flow separation and related turbulence at area discontinuities. As for the pressure drop, it is the exit flow from a silencer which normally is the dominant contributor to the ‘self-noise’ (SN). However, manufacturers’ recommended limits of the flow speed to be used in silencers usually are sufficiently low so that SN is not much of a problem. However, if SN does become a problem, the insertion loss of the silencer can be compromised. It follows also that a silencer with high SN will yield higher insertion loss when used in conjunction with sources of high acoustic power or when placed close to a given source where the noise level is high. For example, if the sound pressure level is 100 dB and the insertion loss of the silencer is 30 dB, the level after the silencer will be reduced to approximately 70 dB if the self-noise power level is 65 dB, say. If, on the other hand, the noise level to be reduced is only 90 dB (rather than 100 dB), the insertion loss will be limited to 25 dB, although the silencer is capable of 30 dB.4 This problem of self-noise has frequently led to disappointing results in efforts to improve the insertion loss of an existing facility, such as an exhaust silencer in a gas turbine power plant. If an original installation does not meet the specified requirements, an additional length of silencer has frequently been added as a ‘fix,’ but often with little or no effect. In several such cases, the failure most likely was due to flow generated noise, although transmission through the walls is a possibility. 3 Confusion in this regard is frequently encountered in practice.

4 Some manufacturers of silencers include the self-noise power in the description of their products.

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Actually, a self-noise limited insertion loss can be encountered even if there is no mean flow in the duct. Such was the case in some experiments we carried out with high amplitude acoustic pulses (weak shock waves). The study involved simply the insertion of a 1 ft long porous baffle in the extension of a shock tube; half of the area of the tube was occupied by the baffle. The spectra of the pulses emitted from the tube with and without the baffle were compared and the corresponding insertion loss was determined. The insertion loss obtained for an incident wave with a peak pressure of 0.2 atm and a flow resistance of the baffle of 0.3 ρc per cm was consistent with linear theory. The insertion loss for a 0.7 atm wave, however, was considerably lower at high frequencies and a similar result was obtained with a porous material with a flow resistance of 0.6 ρc per inch. This behavior of the insertion loss could be caused either by a wave induced compression of the porous material in the baffle at high incident pulse pressures or by the noise from the (turbulent) exit jet pulse. The flow velocity in this jet is expected to be higher (at least by a factor of 2) when the baffle is present (recall that total acoustic power from a jet is proportional to the 8th power of the velocity). This adds to the transmitted sound in the pulse and the insertion loss will be reduced accordingly. Thus, in such a case, self-noise will be produced by the sound (shock) itself. As an aside, we note that the shock wave used in these experiments has an energy spectrum, which is dominated by low frequencies, controllable mainly by the length of the driver section in the shock tube (remember that this pulse has a ‘DC’ component). Thus, a shock tube should be kept in mind by experimenters as a possible sound source in absorption and transmission measurements when ordinary loudspeakers may not yield sufficient power at low frequencies.

7.4 LINED DUCTS Our analysis of lined ducts in Chapter 8 deals mainly with sound propagation in a rectangular duct having one wall lined with an acoustical material, as indicated on the top left in Figure 7.6. Actually, the results are valid also for the acoustically equivalent configurations shown in the figure. This includes the case with two opposite walls in the duct lined with identical liners (top right in the figure). In the configuration at the bottom left, the two liners are different, and the duct is then acoustically equivalent to the parallel baffle arrangement at bottom right in the figure. We refer to Chapter 10, Figure 10.1, for the analysis of configurations with different liners on the two sides of the duct. Both locally and nonlocally reacting liners are considered. The locally reacting liner contains partitions that force the oscillatory flow in the sound wave to be perpendicular to the liner surface; the velocity amplitude in the liner at any position along the duct is then determined solely by the sound pressure at that location, and the input impedance and the corresponding admittance are then independent of the distribution of sound pressure along the duct; these quantities

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Figure 7.6: Equivalent duct configurations (silencers), which yield the same attenuation for the fundamental acoustic mode. can then be regarded as known a priori either from calculation or from measurements.5 As in the hard-walled duct, there are many wave modes possible, but there can be no mode with a strictly constant pressure amplitude across the duct corresponding to the (0, 0)-mode in the hard duct. The sound pressure at a wall lined with a porous material will not be the same as that at a rigid, impervious wall. Furthermore, as the frequency increases, there are many wave patterns with several minima and maxima that will ‘fit’ across the duct; there will be no pure nodal planes, however, as in the case of the hard-walled duct above. Out of the many possible modes, we shall be particularly interested in analyzing the lowest mode and its spatial decay rate. The method of solution is well established, but we have added some observations and numerical studies, which can be useful. The corresponding analysis for ducts with nonlocally reacting liners yields some new aspects to the attenuation characteristics. The solution to the wave equation for the sound pressure amplitude is composed of wave functions representing ‘standing’ waves in the transverse directions y and z, and a traveling wave in the x-direction along the duct. In what follows, we consider only a wave traveling in the positive x-direction. This is sufficient for the calculation of the attenuation. When it comes to an analysis of the transmission loss and the insertion loss, reflections have to be accounted for and a wave also in the negative direction must be included. When a sound absorptive lining is present, the fundamental mode will decay. The frequency dependence of this decay, which is the main topic of interest, depends on several parameters. For higher order modes, the mode identification is no longer as clear as the hard duct and there is no longer a well defined cut-on frequency signaling a transition from evanescence to propagation. As for the fundamental mode, there will be decay at all frequencies, although the decay of a higher mode will be enhanced below the ordinary hard-wall cut-on frequency. The decay of a higher mode is greater

5 Since for a locally reacting liner the impedance is independent of the angle of incidence, normal

incidence data can be used, obtained, for example, with the two-microphone method in a tube.

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than for the fundamental, and it is the latter that usually dominates the sound that is emitted from the end of a lined duct. However, particularly for short ducts, higher modes can contribute markedly to the overall attenuation. At high frequencies, in the geometric or ray acoustics regime, a higher mode can be visualized as a ray which travels at a certain angle with respect to the duct axis. Every time this wave is reflected from a boundary it loses some of its energy and the amplitude will decay with distance in the duct. An expression for the decay can be derived in terms of the absorption coefficient of the boundary. At least in this regime, there is a unique relation between absorption and sound transmission, which is not the case in general. The same geometrical approach cannot be used for the fundamental mode, however.

7.5 ‘REACTIVE’ SILENCERS The sound reducing mechanisms involved in a silencer are absorption, reflection, and interference. In the previous section dealing with ‘dissipative’ silencers, absorption was dominant; in a ‘reactive’ silencer, reflection and interference dominate. With reference to Figure 7.3 and the related discussion, there is usually relatively little difference between total attenuation, transmission loss, insertion loss, and noise reduction, except at low frequencies. The difference depends on the bandwidth, as indicated, decreasing with increasing bandwidth, and on the source and termination impedance. For a reactive silencer, which, in its purest form contains no absorptive material, the difference between the various measures can be considerable.

7.6 ACOUSTICALLY EQUIVALENT SILENCERS The bulk of the data presented in this book refers to a rectangular duct and to the related and commonly used configurations of lined channels such as parallel baffle or ‘splitter’ silencers as shown in Figure 7.6. Because of the acoustic equivalence of these ducts (equivalent as far as the fundamental acoustic mode is concerned), the numerical work has been concentrated on a rectangular duct with one side lined, as shown at the top left in the figure. The liner is a uniform porous rigid layer of thickness d in a duct with an air channel width, D1 = D/2. The width D is the channel width in the equivalent duct (top right) with two opposite sides lined. Similarly, the two bottom configurations in the figure are acoustically equivalent and refer to the slightly more general case when the thicknesses d1 and d2 of the liners are not necessarily the same. The expression for the attenuation is derived in Chapter 10 for the duct on the left. The figure on the right is an example of a parallel baffle attenuator, which is generally used to accommodate a large volume flow through the attenuator while controlling pressure drop and/or physical size. It should be emphasized that these equivalencies are valid for the fundamental mode because of the symmetry of the pressure amplitude with respect to the dashed planes in the figure. The transverse velocity amplitude is zero in these planes and therefore can be replaced by rigid, impervious walls without altering the sound field. In practice, the focus on the fundamental mode usually does not present much of a limitation since this mode has the lowest attenuation and therefore generally dictates

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the performance of the duct. Semi-empirical corrections for higher modes and flow will be discussed later. The above process of building equivalent duct configurations can be repeated to more than two air channels with the thicknesses of the interior baffles being alternately 2d1 and 2d2 .

7.7 ADDITIONAL COMMENTS ON SILENCER TESTING As already mentioned, a typical laboratory test of a silencer is carried out in a facility illustrated schematically in Figure 7.2 with the open end of the test duct terminating in a calibrated reverberation room.6 The input power to the room is then determined from the average sound pressure level in the room. The transmitted power can be measured also by means of an intensity probe in the duct. The change in the acoustic power in dB resulting from the insertion of the silencer in the test duct is the insertion loss, IL, of the silencer in this particular test facility. Although, normally, insertion loss is considered to be the most relevant from a practical standpoint, it suffers from its dependence on the source impedance and the effects of reflections from both ends of the test duct. This means that the acoustic power output from the source may be different with and without the silencer in the test duct, and the reflections will make the IL depend on the axial position of the silencer; this dependence can be significant, particularly at low frequencies. The crucial difficulty with IL is that the conditions in the laboratory and in the field installation rarely are the same so that the effects of source characteristics and reflections are apt to be different in the laboratory and in the field. A model of a field installation can be made, of course, and tested in the laboratory. This is frequently done in special applications such as jet engine test cells, for example. It is then important that proper scaling parameters are used. The ratios of wavelength and dimensions have to be maintained and the effects of temperature on sound speed, wave impedance, and flow resistance have √ to be accounted for. The sound speed increases with √ the absolute temperature as T , and the wave impedance is proportional to P / T , where P is the static pressure. The flow resistance is proportional to the √ coefficient of shear viscosity, which increases with temperature approximately as√ T and the frequency dependence of the flow resistance is given approximately as f . If the main objective of a silencer test is to make comparisons between different silencers and not to model a field installation, one might consider using performance measures other than IL, such as TL, TL0, or even NR. The effects of reflections from the ends of the test duct can be made small by using properly designed parallel baffle or wedge absorbers at both ends of the test duct. The lower sketch in Figure 7.2 has an absorber only at the source end.

6 The calibration relates the average sound pressure level in the room, and the power input to the room. It depends on total sound absorption in the room, which can be determined by measurement of the reverberation time.

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Figure 7.7: Filled circles: Computed TL (IL). Filled squares: TL0. Open rectangles: NR. Dashed line: TL with semi-empirical correction for higher modes. Refers to a silencer test as shown in Figure 7.2 with both walls of the silencer lined. The ends of the test duct are reflection-free. Length of silencer: 3 ft. Channel width: 4 inches. Thickness of liner: 8 inches. Flow resistance: 0.3 ρc per inch. With both ends of the test duct made reflection-free by means of absorbers, the IL and the TL are identical, independent of the location of the silencer in the test duct. In practice, the reflections can be significant at low frequencies and the useful frequency range of the duct absorbers should be established. IL can be measured as before. The only problem that might be encountered is that if a reverberation room is used for the measurement of power, the sound source has to be powerful enough to compensate for the reduction in power caused by the absorber at the end of the duct. If an intensity probe is used for power measurement, this problem does not occur. TL0 can be measured as described earlier by means of intensity probes for determining the net input power and the net transmitted power. The simplest of all quantities to measure is NR, the difference of the (average) sound pressure levels at the two ends of the silencer, but it is the quantity which usually is most affected by reflections from the end. It is assumed that whenever probes are used within the duct that the signal caused by flow-probe interaction is adequately reduced. The calculated octave band spectra of IL (TL), TL0, and NR in a special case are shown in Figure 7.7. There is only a small difference between TL (IL) and TL0. NR is consistently somewhat higher than TL but has the same general frequency dependence.7 Since the ends of the test duct are reflection-free in this case, each of these quantities is uniquely determined by the silencer characteristics and is independent of system parameters. As mentioned earlier, it should be kept in mind that TL0 will be zero for a reactive silencer (no power absorbed).

7 Recall that for normal incidence on an impervious partition wall the NR is close to 6 dB higher than

the TL because of the pressure doubling at the wall due to reflection.

Chapter 8

Lined Ducts A duct with the interior walls lined with an appropriate porous material is presently the most common form of a general purpose silencer. As far as acoustical characteristics of the liner are concerned, the most important physical property of the liner material is the flow resistance. This depends on the porosity and the ‘structure factor’ of the material. For a compressible material, compliance and other elastic properties are obvious additional quantities of interest, although flow resistance still remains the most important, particularly since the elastic properties are rarely known a priori and seldom measured. It should be realized, though, that in our efforts to improve the low frequency performance of duct liners and silencers, the compliance of the liner material will play a very important role in much the same way as for sound absorption. The effect of compliance on sound absorption is summarized briefly in our chapter on flexible porous materials to explain the sometimes observed high absorption coefficient at low frequencies. It is possible to enhance the low frequency performance of lined duct silencers in a similar manner by proper use of compliant materials.

8.1 ATTENUATION MECHANISMS 8.1.1 Dissipation in Duct Liners As a qualitative description of the mechanisms of sound attenuation in a lined duct, we note that the sound pressure in a wave within a lined air channel produces ‘pumping’ of air in and out of the porous duct liner, and this gives rise to conversion of acoustic energy into heat and hence attenuation of the sound wave. For a locally reacting liner, divided into cells by partitions, the velocity of the air in this pumping action is forced to be perpendicular to the wall. On the other hand, in a nonlocally reacting liner there is an oscillatory velocity component also in the axial direction of the duct and a corresponding wave in the liner. This wave feeds energy back into the air channel of the duct so that, in effect, the liner provides a flanking path. This is what makes the two liners behave differently, particularly for small flow resistances, as can be seen from the results in Figures 8.1 and 8.3. 255

256


The amplitude of the air pumping by the sound wave into the liner is inhibited at low frequencies because of the stiffness reactance of the air in the liner, and this is the main reason for the decrease toward zero of the attenuation at low frequencies. In the high frequency region, the sound field in the duct can be thought of as composed of a free running plane wave grazing the boundary, as discussed in Chapter 7. The interaction with the boundary then will be the same as for a plane wave in free field at grazing incidence yielding a pressure reflection coefficient of −1. The incident and reflected waves then cancel each other at the boundary to produce zero pressure amplitude at the boundary and hence no pumping effect and attenuation. This qualitative explanation is strictly speaking applicable to a higher order mode in the duct, which can be regarded as a plane wave bouncing back and forth between the boundaries, as discussed in Chapter 7. For the fundamental mode, we can use the following (qualitative) explanation for the decrease of the attenuation toward zero at high frequencies. A compression in the wave in the center of the duct travels as a signal toward the boundary where it is reflected. The time required for the signal to return to the middle is D/c, where D is the duct width. If this delay time is longer than the acoustic period T , i.e., D/c > T or D/λ > 1, the reflected signal in effect will not be ‘felt’ (on the average) by the wave during the compression period, exactly as if the boundary were not present and no attenuation results. For further discussion, we refer to Section 10.1. At mid-frequencies, with wavelengths of the order of the width of the air channel, we are close to a resonance of a standing wave between the walls, and this tends to increase the sound pressure at the walls. The attenuation typically goes through a maximum in this frequency region before it turns toward zero with increasing frequency, as explained above; the decrease turns out to be approximately as the inverse square of the frequency, only weakly dependent on the liner impedance. Thus, with the attenuation going to zero both in the low and high frequency limits, it is clear that the attenuation spectrum becomes bell shaped, on the average. In this context it should be remembered that the absorption coefficient of a porous material increases with frequency over the entire frequency range of interest, and this is an explicit demonstration that there is no true one-to-one correspondence between absorption coefficient and attenuation, an assumption made (and often used) in early accounts of the subject.

8.1.2 Interference Interference is often thought of as a result of the superposition of two or more sound fields produced by different sources. Such an interference can give rise to cancellation of the sound from one source by the sound from another. When this technique is utilized in practice, it is usually referred to as ‘active noise control.’ An incident and reflected field can also produce interference, the typical example being the standing wave in a tube. Another example is encountered in Section 8.7.2 dealing with sound propagation in a water line with an air layer as a liner.

257

LINED DUCTS

In general, the attenuation is a combination of the effects of dissipation and interference. Interference is of particular importance in reactive silencers, as discussed in Chapter 9. After these mainly qualitative considerations, we turn to quantitive aspects of the subject dealing with specific duct geometries and start with lined duct silencers with a rectangular cross section.

8.2 RECTANGULAR DUCTS The main purpose of this section is to present and discuss numerical data on lined rectangular ducts. It was considered appropriate to include Eqs. 8.1 and 8.2, which produced the calculated data even though the equations are not essential for this purpose. The mathematical analysis is given in Section 10.2.

8.2.1 Locally Reacting Liner For the fundamental mode in an unlined duct, the axial dependence of the complex sound pressure amplitude is exp(ikx x), where kx = ω/c = 2π/λ. The magnitude of the sound pressure is then constant and there is no attenuation, neglecting visco-thermal effects. With reference to the analytical supplement in Chapter 10, the propagation constant is modified by the finite boundary impedance from kx = ω/c to kx =

(ω/c)2 − ky2 , where ky , a complex quantity, is determined from the admit-

tances of the boundaries by solving (numerically) a transcendental equation. Thus, kx becomes complex, and the imaginary part ki determines the decay rate and attenuation of the sound pressure, i.e., |p(x)| = |p(0)| exp(−ki x). Propagation constant, Locally reacting liner kx ≡ kr + iki = (ω/c)2 − ky2

(8.1)

ky D1 tan(ky D1 ) = −ikD1 η1 where η1 : Liner admittance (normalized), D1 : Channel width, Figure 10.1, ky : see Eq. 10.9. For a porous liner, the impedance or admittance is determined by the thickness of the liner, its flow resistance per unit length, and to some extent, by the porosity and structure factor, which are not independent of the flow resistance. Computed attenuation spectra for a duct lined on one side with a locally reacting porous liner are shown in Figure 8.1. The attenuation is expressed in dB per unit length, where the unit length is the width D1 of the air channel (top left in Figure 7.6). The frequency parameter is the ratio D1 /λ of the channel width D1 and the free field wavelength λ, and the flow resistance of the porous material is expressed in terms of the normalized value of the total flow resistance of the liner. In this manner, the number of parameters needed to describe the attenuation spectrum is reduced. It is in this sense that the curves might be called ‘universal.’ It should be stressed that the attenuation refers to the fundamental mode in the duct.

258


Figure 8.1: Attenuation of the fundamental mode (in dB per length of duct equal to the channel width D1 ) in a rectangular duct with one side lined with a locally reacting rigid porous layer with a total normalized flow resistance (2, 4, ... 32). Liner thickness: d. Fraction open area D1 /(D1 + d) = 20 to 70 percent. The results can be used for a duct with two opposite walls lined with identical liners if the channel width (distance between liners) is 2D1 . In practice, a channel with a width larger than a wavelength normally carries higher wave modes with attenuations higher than for the fundamental mode. Until these higher modes have been attenuated down to the level of the fundamental mode, the average attenuation in the duct for the entire sound field will be larger than for the fundamental mode. Thus, the attenuation of the overall sound field will not be constant along the duct but once the higher modes have been ‘filtered’ out, the

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259

attenuation for the fundamental mode will be valid. Thus, in regard to the total attenuation, the effect of higher modes will be more important for short ducts than for long. It should be noted that the results can be used, again expressed in terms of the attenuation per unit length D1 , for a duct with two opposite walls having identical liners if the distance between the liners is D = 2D1 . For example, if the channel width in such a ‘two-sided’ duct is D = 8 inches, the width that should be used in the figures is 4 inches in both the abscissa and ordinate to obtain the attenuation in a distance of D1 = 4 inches. Attenuation spectra are shown for values of the open or ‘free’ area fraction of the duct, D1 /(D1 + d), from 20 to 70 percent in steps of 10, where D1 is the width of the air channel and d the thickness of the liner, as shown in Figure 7.6. For each of these values, curves corresponding to normalized total flow resistances of the liner, = 2, 4, 8, 16, and 32, are shown. The spectra are bell shaped, consistent with the qualitative observations made earlier, and it should be noted that when the attenuation is expressed in this manner, the maximum value will be essentially the same, ≈ 4 dB per channel width, independent of the fraction open area of the duct for values less than 60 to 70 percent. However, the width of the curve at the maximum and the attenuation at frequencies below the maximum do depend on the open area fraction; the larger the open area, the narrower the spectrum. The lowest normalized total liner resistance considered in the figures is = 2. Lower values generally are not of practical interest because the low frequency attenuation is poor. As the resistance decreases, the duct liner begins more and more to behave like a quarter wavelength resonator with pronounced attenuation peaks when the liner thickness is close to an odd number of quarter wavelengths. The attenuation at the peaks will exceed the 4 dB values mentioned above but the peaks are correspondingly narrow. Optimum Design Notice that for each value of the frequency parameter D1 /λ, there is an optimum total normalized resistance of the liner. For example, it follows from Figure 8.1 that for an open duct area of 30 percent and D1 /λ = 0.02, the optimum total normalized resistance of the liner is ≈ 6 and the corresponding maximum attenuation per unit length (channel width D1 ) is ≈ 0.5 dB. It is shown in Chapter 10 (Eq. 10.18) that if the porous liner is replaced with a resistive screen backed by an air layer, the attenuation curves will be quite similar in the low frequency regime in which the wavelength is much larger than the channel width. The maximum possible attenuation that can be achieved with the screen liner at low frequencies (d 1, the angle of incidence of the reflected waves is close to 90 degrees and a reflection coefficient close to unity. Disregarding the reflections, we have found the simple relation in Eq. 8.4 to agree quite well with experimental data3 obtained for lined ducts ranging in length from 3 to 10 ft and in channel widths from 3.5 inches to 12 inches. This geometrical acoustics result tells us nothing about the frequency dependence of the transmission loss except that the analysis is meaningful only at wavelengths considerably shorter than the duct width D. We have used an empirical frequency dependence of the higher mode correction (to be added to the fundamental mode TL) such that 0 for f < f1 (8.5) δ≈ δmax (1 − exp(−0.35 f/f1 )) for f ≥ f1 , where δmax = 10 · log(π/2 arctan(D/2L)) ≈ 10 log(π L/D) and f1 = c/2D. The correction approaches the upper limit discussed above as the frequency increases, starting from zero at a frequency f1 = c/2D and reaching a value close to the limiting value at a frequency ≈ 10 f1 . This behavior has been found to be in fair agreement with experimental TL data for silencers ranging in length from 3 ft to 10 ft with channel widths from 3 inches to 12 inches. The line source model used here as a simple illustration of the idea can be replaced by one in which the sound enters the duct from a diffuse field in a plenum chamber, but this model is not pursued here.

8.6.2 Convection A mean flow in the duct can influence the sound transmission in several ways. The convection of sound by the flow is the most obvious effect. It leads to a reduction of the attenuation in the downstream direction and an enhancement upstream. The refraction of sound, resulting from the transverse flow gradient in the duct, is important only at high frequencies and leads to an increase in the attenuation in the downstream direction and a decrease upstream. Typically, the two effects cancel each other at a frequency where the attenuation is a maximum. More subtle is the effect of flow on the input impedance of a duct liner. The effect is usually small but can be significant if the liner consists of a perforated plate backed by an air layer or a porous layer with small flow resistance. The effect of normal and grazing flow on the flow resistance of a perforated plate is discussed in the next chapter. In a mean flow with velocity U , a plane sound wave will be convected by the flow so that the resulting wave speed with respect to a stationary frame of reference will be c + U , where c is the sound speed (relative to the fluid). U is positive (negative) for downstream (upstream) propagation. 3 Courtesy of Industrial Acoustics Company, Inc., New York.

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285

The attenuation in dB is proportional to the distance of wave travel expressed in terms of the wavelength (or the number of ‘pumping’ periods into the boundary during the passage of the sound). The wavelength downstream will be longer than upstream by a factor (1 + M)/(1 − M), where M = U/c is the flow Mach number. Thus, the distance of wave travel, as measured in wavelengths, will be greater upstream than downstream, and the corresponding attenuation will be greater by the same factor (1 + M)/(1 − M). As an example, with a flow speed of 5000 fpm, corresponding to M ≈ 0.07 at room temperature, this factor is ≈ 1.16 and an attenuation of 30 dB in the downstream direction corresponds 34.8 dB upstream. However, this result is valid only at relatively low frequencies when the fundamental mode dominates in the duct and convection is the important flow effect. When the wavelength is smaller than the channel width, it turns out that the effect of flow is reversed, leading to an increase of the attenuation in the downstream direction and a decrease in the upstream direction. This is a result of refraction, as will be discussed next.

8.6.3 Refraction The semi-empirical higher order mode correction given above was based on ray acoustics and applied to a uniform gas at rest. Under such conditions, the acoustic rays were straight and the angle φ0 below which rays emerged from the duct without striking the duct wall was simply φ0 = arctan(D1 /(2L)), as shown in Figure 8.22. In a moving hot gas there will be lateral gradients of velocity and temperature and hence of the wave speed. With the maximum temperature and velocity assumed to be at the center of the duct channel, the flow refracts the sound toward the boundary for a wave in the downstream direction and away from the boundary in the upstream direction. The thermal gradient refracts toward the boundary, independent of direction. Thus, in the downstream direction, the flow and temperature gradients cooperate in refracting the sound toward the boundary but oppose each other upstream. Under steady state operating conditions of a silencer, the wall temperature is expected to be essentially the same as the gas temperature and in our computer program only the effect of a flow gradient is accounted for. In turbulent duct flow, the velocity goes to zero at the walls in a non-uniform manner, most of the decrease occurring in a relatively thin boundary layer with a thickness depending on the flow velocity. Outside this layer the decrease of the velocity from the center of the duct can be approximated as linear, for the present purpose. In such a flow field, a sound ray will follow a circular path with a radius of curvature given by4 1/R = −(1/c)d(U + c)/dy, where y is the vertical coordinate. A positive value of the curvature corresponds to a ray that turns upwards. The effect of a temperature gradient depends of the sound speed, (1/c)dc/dy = (1/2T )dT /dy, where T is the absolute temperature. The geometrical considerations in Figure 8.22 in estimating the transmission loss now have to be reexamined to account for curvature of the sound rays.

4 See, for example, Uno Ingard, Waves and Oscillations, Cambridge University Press, 1988, Section 12.6.

286


Figure 8.23: Effect of refraction in a duct due to nonuniform flow, which shows how the limiting ray in Figure 8.19 is now curved, reducing the critical angle below which sound escapes from the duct.

Figure 8.24: An illustration of the effect of flow direction on the transmission loss including semi-empirical corrections for flow and higher modes. Thin lines: No flow. Solid line: Sound in downstream direction. Dotted line: Sound in upstream direction. Flow speed in the silencer channel: 100 ft/sec.

Thus, Figure 8.23 shows the ray (solid line) in the downstream direction emitted under the critical angle φ0 below which emitted rays will escape through the duct without being reflected from the boundary. The dashed, curved line represents the corresponding ray in the upstream direction. This figure shows half of a duct, which is lined on one side and which contains a centered line source at the beginning of the duct. For flow in the downstream direction, the curvature will reduce the critical angle and increase the attenuation. The opposite holds true for sound in the upstream direction. Applying a little geometry, we can express the critical angle in terms of the radius of curvature, and hence in terms of the flow gradient (see Eq. 10.79). Having obtained

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287

φ0 , a higher order mode correction can thus be obtained as described in connection with Figure 8.22. Figure 8.24 shows an example of the effect of flow on the transmission loss. The equivalent average velocity gradient was estimated to be U/(12D). It should be noted that at frequencies below ≈ 2000 Hz, the insertion loss is decreased by the flow for sound propagation in the downstream direction and increased at higher frequencies due to the effect of convection. At higher frequencies, the insertion loss is increased in the downstream direction due to refraction.

8.6.4 Scaling Laws A nonuniform distribution of temperature in a duct gives rise to refraction, as discussed earlier. There are more direct effects of temperature, however. The obvious√and most important is the increase of the sound speed with temperature, c ∝ T , where T is the absolute temperature. At a given frequency, the wavelength then will increase with temperature and the ‘acoustic’ thickness (measured in terms of wavelengths) of a duct liner will be reduced. This generally results in a decrease in the attenuation. Another√effect is the increase in the shear viscosity with temperature (approximately as T ) and the corresponding increase in the flow resistance of a porous material. This effect√is compounded by the decrease of the wave impedance ρc with temperature (∝ 1/ T ) so that a flow resistance normalized with respect to ρc will increase approximately in direct proportion to the absolute temperature T . As a commonly encountered example of duct liner design, we note that a typical temperature in the exhaust stack of a gas turbine power plant can be about 1000◦ F (T ≈ 811 K), and the effect of temperature on the attenuation then can be quite substantial. Thus, it is important that model experiments’ data of duct performance obtained at room temperature be appropriately corrected when used in applications at other temperatures. The static pressure does not influence the sound speed or the coefficient of shear viscosity, but it does affect the wave impedance, and hence the normalized flow resistance and the kinematic viscosity. In a duct with flow, the temperature dependence of the kinematic viscosity, and hence the Reynolds number will influence the static pressure drop. So far it was tacitly assumed that the gas was air at static pressure of 1 atm. In some applications, typically involving compressors, both the gas and the static pressure can be different, and this can have a substantial effect on duct liner performance through the influence on wave impedance, shear viscosity, and hence the normalized resistance. The input values in our computer programs refer to a measured normalized flow resistances in air at room temperature (293 K) and a pressure of 1 atm. Appropriate corrections are made within the programs. The scaling laws of interest for duct design can be summarized as follows using the following notations. Molecular weight: M, density: ρ, absolute temperature: T , static pressure: P , shear viscosity: μ, kinematic viscosity: ν. The reference values of these quantities are given the subscript ‘0’ and refer to air at P0 = 1 atm and T0 = 293 K.

288


Density ρ/ρ0 = (P /P0 )(T0 /T )(M/M0 )

(8.6)

Sound Speed c/c0 =

(T /T0 )(M0 /M)

(8.7)

Wave Impedance ρc/ρ0 c0 = (P /P0 ) (M/M0 )(T0 /T )

(8.8)

For example, with P /P0 ≈ 5, M/M0 ≈ 4, and T /T0 ≈ 1.18 (150◦ F), we get ρc/ρ0 c0 ≈ 5 · 2/1.09 ≈ 9.2. Shear Viscosity The coefficient of shear viscosity μ is approximately independent of density and approximately proportional to the square root of temperature. Thus, for a gas with a reference value μ0 we have μ/μ0 = T /T0 . (8.9)

Kinematic Viscosity The kinematic viscosity, ν = μ/ρ, expresses the ratio of viscous and inertial forces in a sound field and determines, for example, resonance in √ the sharpness of an acoustic √ a cavity. With μ being proportional to (T ) and ρ/ρ0 = (P /P0 ) (T /T0 ), it follows that ν/ν0 = (T /T0 )3/2 (P0 /P ).

(8.10)

Reynolds Number The Reynolds number of the flow with speed U in a duct of width D is R = U D/ν. For a given mass flow rate Q, the velocity is proportional to Q(T /P ) so that R/R0 = T0 /T .

(8.11)

An increase in temperature, and hence a decrease in R, tends to decrease the intensity of instabilities in a flow and it will affect the wall friction in turbulent duct flow.

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Normalized Flow Resistance θ = r/(ρc), where r is the specific flow resistance. Thus, for a gas with a molar mass M, θ/θ0 = μ(T )/μ0 (T0 )(ρ0 c0 )/(ρc) ≈ μ(T0 )/μ0 (T0 ) T /T0 (ρ0 c0 )/(ρc) = μ(T0 )/μ0 (T0 ) M0 /M(P0 /P )(T /T0 ). (8.12) For example, with μ(T0 )/μ0 (T0 ) ≈ 0.75 and with the values of other parameters, M/M0 ≈ 4, etc., given above, we get θ/θ0 ≈ 0.09. In other words, in order for a liner in the muffler to be effective, a flow resistance approximately 11 times the value in normal air would be required. In the computer program, all the input resistances refer to the values obtained for normal air. To be able to apply the program to other gases than air, the quantities, M/M0 , μ/μ0 , in addition to pressure and temperature, are also input parameters. The scaling factors are incorporated in the program. Example As an illustration we consider a rectangular duct lined on one side with an 8 inch thick porous layer, covered with a perforated facing. The flow resistance of the porous layer is 0.5 ρc per inch, measured at room temperature. The perforated plate has an open area of 23 percent, thickness, 0.1 inch, and hole diameter, 0.1 inch. The width of the air channel is 4 inches. Determine the spectrum of the attenuation per unit length (unit length being the channel width) at room temperature, 70◦ F, and at a temperature of 1000◦ F. The result is shown in Figure 8.25. (It is applicable also to a duct with two sides lined (identical liners) and a channel width between the liners of 8 inches, i.e., twice the original channel width.) The result in the figure is expressed in dB per channel width in a duct with one side lined, and if we apply the program to a duct with two walls lined and use the new channel width as unit length, the attenuation per channel width will be twice that in the figure.

Figure 8.25: Effect of temperature on attenuation in a lined duct. For parameter values: see text.

290


The attenuation spectra at the two temperatures are shown. The markers signify octave band average values assuming the source spectrum to be uniform. The spectrum at 1000 degrees is displaced toward higher frequencies in comparison with the spectrum at 70 degrees and the attenuation at mid-frequencies is lower. This is an effect which can be of considerable practical importance. For example, with a duct length of 10 channel widths, an increase in temperature from 70 to 1000◦ F results in a decrease in attenuation at 500 Hz from 32 to 14, i.e., approximately 18 dB, a substantial amount.

8.6.5 Static Pressure Drop in Ducts The Reynolds number5 is R = U D/ν, where ν = μ/ρ is the kinematic viscosity, ≈0.15 CGS for normal air (μ is the coefficient of shear viscosity and ρ the density). For a rectangular channel of width W and height H , we get D = 4W H /2(W + H ), which, if W 11076 Hz. Generally, we shall be interested in frequencies below f1 = 11076 Hz, and we shall deal only with the fundamental mode here. For an air layer liner, to be treated in the second example, the various modes of propagation will be discussed in some detail. With only the fundamental mode present in the pipe, it will be driven in a ‘breathing’ mode and its impedance will be that of a resonator with a resonance frequency equal to the ‘ring’ frequency v/(2πa) of the boundary wall, where v is the longitudinal wave speed in the material and a the pipe radius. It is assumed then that the wall thickness is small compared to a. The corresponding period is the roundtrip time of the longitudinal wave in the boundary wall (for details, see Chapter 10). The internal damping in the wall is accounted for by assigning to it a complex elastic modulus and a corresponding loss factor of the material so that the elastic modulus is expressed as E(1 − i). As shown in Chapter 10, the phase velocity and the damping in the water can then be expressed in terms of and the parameter β = (D/d)(ρ0 c02 )/E, where D is the pipe diameter, d, the wall thickness, E ≡ ρ1 c12 , (the real part of) the elastic modulus of the wall material, ρ0 c02 , the inverse of the compressibility of water (ρ0 the density and c0 the wave speed). As an example, consider a steel pipe with D = 3 inches and a wall thickness d = 1/4 inches. Then, the ring frequency is f0 = v/(2π a) ≈ 2.1 × 104 Hz, and with ρ0 /ρ1 ≈ 1/7.8 and c0 /c1 ≈ 1440/5000, we find that the sound speed is decreased by approximately 6.5 percent as a result of the compliance of the wall independent of frequency if ω 0.6, the absorption decreases with increasing M0 . Nonlinear Reflection, Transmission, Absorption We consider again the perforated plate in the previous section, this time focusing on the effect of the acoustic nonlinear resistance of the plate on the reflection, transmission, and absorption of an incident sound wave. A more general liner is one in which the perforated plate is backed by a porous layer. The induced motion influences the velocity amplitude in the orifices of the plate; the velocity amplitude relative to the plate is given by 2δ ζs u0 , (9.7) =

c ζ0 + 2s ζ0 + ζs

Figure 9.11: The power absorption (A), reflection (R), and transmission coefficient (T) of a perforated plate for a plane wave at normal incidence with superimposed mean flow. Plate thickness, 0.2 inches, hole diameter, 0.2 inches, open area fraction, 20 percent. Left: No flow. Right: Mach number in orifice, M = 0.4. In each case, the reflection coefficient is close to 1 at 10,000 Hz.

316


where δ = |pi |/γ P , ζ0 = ζ0 + (1 − s)2 |u0 |/c, ζs = s(−iωm/ρc), γ = 1.4 (specific heat ratio), and ζ0 = ζ0 ζs /(ζs + ζ0 ). The quantity ζ0 is the linear orifice impedance, which is sζ , where ζ is given in Eq. 9.6. The pressure magnitude of the incident wave is |pi | and P is the static pressure. Having solved this equation (numerically) for |u0 |/c, we obtain the reflection and transmission coefficients R and T for pressure and the coefficient α of sound absorption within the plate from the relations R = 2(ζ0 + s)/(ζ0 + 2s) T =1−R α = 1 − |R|2 − |T |2 .

(9.8)

The coefficients for acoustic power reflection and transmission are |R|2 and |τ |2 . For the perforated plate in Figure 9.12, the nonlinearity does not significantly affect the coefficients for incident pressure levels less than ≈ 120 dB. Results obtained for levels of 100 dB (essentially linear regime) and 160 dB are shown in the figure. The absorption coefficient in Figure 9.12 at 160 dB remains constant, essentially independent of frequency at low frequencies. This is due to the fact that the perforated plate was assumed to be rigid. In reality, it is flexible and will be induced to move by the sound, particularly for small open areas and a sufficiently light plate.

9.5.2 Shock Wave Interaction With an Orifice Plate In addition to studies of the interaction of shock waves with porous materials, we considered also the reflection from and the insertion loss of a porous layer inserted as a centered partition in a circular tube and also of an orifice plate used as a termination of the tube. In both instances, the insertion loss was found to decrease with increasing amplitude of the incident wave. Most likely this reduction has to do with the noise generated by the jet flow caused by the incident pulse, which prevents the sound to reach its normally low level thus reducing the insertion loss. With the orifice plate at the end of the tube, the amplitude of the reflected wave is expected to be between what is obtained for a rigid wall and an open end termination,

Figure 9.12: Power reflection (R), transmission (T), and absorption coefficients (A). Orifice plate: Thickness = hole diameter = 0.2 inches. Open area = 20 percent. Right. 160 dB.

317

REACTIVE DUCT ELEMENTS 2

Incident pulse

Pressure, atm Experiment 1 Theory

Time 25 ms

Figure 9.13: Shock wave reflection from an orifice plate with an open area fraction of 0.6. The pressure is measured in the shock tube 1 m from the membrane in the driver section, peak pressure 0.94 atm (≈ 193 dB). Compare the reflections from closed and open ends.

reflection coefficients being +1 and ≈−1, respectively. Measurements were made with orifice plates with open area fractions s = 0.22, 0.42, 0.60 and 0.79. Indeed, the positive portion of the reflected wave dominated the negative portion for small open areas but for S = 0.79, the roles were reversed. At s = 0.6, the positive and negative portions were about the same and the reflected pulse amplitude was at a minimum with a reflection coefficient of ≈ 0.1 (Figure 9.13). This referred to an incident pulse amplitude at the plate of 0.73 atm (0.94 atm at the microphone position 1 m from the plate) and a pressure in the driver section of the shock tube of 3.7 atm.

9.6 ATTENUATION IN TURBULENT FLOW IN DUCTS A mathematical supplement to this section is given Section 10.7. We have already discussed the flow effects of convection and refraction on sound attenuation in a lined duct; refraction has a significant effect at high frequencies and convection is dominant at low frequencies. This section addresses a different question related to flow, namely the attenuation that might result from the direct interaction of sound with the turbulent duct flow. In turbulent flow, the static pressure drop is approximately proportional to the square of the velocity at sufficiently high Reynolds numbers, and when the flow is modulated by the acoustically driven oscillatory flow, there will be a time dependent perturbation in the pressure drop proportional to the product of the acoustic velocity amplitude and the mean velocity. The product of these perturbations corresponds to an acoustic energy loss. The mechanism is simply that during the half cycle when the fluid velocity in the sound wave adds to the steady velocity there will be a larger increase in friction losses than the decrease during the other half of the cycle, when the velocities are in opposition. This leaves a time average increase in the loss, which results in attenuation. Another attenuation in a hard-walled duct is due to the visco-thermal losses at the walls of the duct, but we shall find that turbulent flow usually causes a higher attenuation, at least at low frequencies. There are other effects, which should be considered in a more detailed analysis than that given here. For example, the spatial variation of mean pressure, density, and

318


velocity along the duct leads to a decrease in sound pressure in the flow direction even if the acoustic energy were conserved along the duct. This is due to the pressure loss in the mean flow due to friction. As a result, conservation of mass flux requires the mean flow velocity in the duct to increase with distance as the mean pressure and density decrease. If the duct is long enough, the flow will be choked at the end of the duct. The decrease of sound pressure with altitude in sound propagation in the atmosphere is a similar effect, now resulting from the altitude dependence of density and temperature, and hence of the wave impedance ρc. Since the intensity in a sound wave is p 2 /ρc or u2 ρc, a decrease of the density (or sound speed, or both) requires p2 to decrease and u2 to increase with altitude in order for acoustic energy to be conserved.

9.6.1 Static Pressure Drop See discussion in Chapter 8.

9.6.2 Sound Attenuation As shown in Section 10.7, the attenuation over a travel distance x due to the interaction of a fundamental acoustic mode in a duct with turbulent flow is Attenuation in turbulent duct flow Attenuation ≈ 8.7 ψ

|M| x 1+M D

dB

(9.9)

(M: Mach number, x: Distance, D: Hydraulic diameter) This is valid for sufficiently large Reynolds numbers. The decay in a circular hard duct due to visco-thermal boundary losses at the boundary is known to be exp(−kdvh x/D), √ where dvh = dv + (γ − 1)dh isthe visco-thermal boundary layer thickness, dv = 2μ/ρω, the viscous, and dh = 2K/(ρωCp ) the thermal boundary layer thickness. The quantities μ and K are the coefficients of shear viscosity and heat conduction, Cp , the specific heat per unit mass, and ρ, the density. In air at room √ temperature, the frequency dependence of dvh can be expressed as dvh ≈ 0.31/ f cm, where f is the frequency in Hz. Thus, the decay of the √ amplitude in a distance equal to the diameter D then becomes kdvh ≈ 5.7 × 10−5 f , and the corresponding attenuation in dB is obtained by multiplying by 20 log(e) ≈ 8.7. Visco-thermal attenuation in a duct p(x)/p(0) = exp(−kdvh x/D) √ kdvh ≈ 5.7 × 10−5 f √ Attenuation in dB, air ≈ 49.6 × 10−5 f (x/D) (x: Length, D: Diameter, f : Frequency in Hz)

(9.10)

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REACTIVE DUCT ELEMENTS

To compare this with the flow induced attenuation, we use a typical value for the friction constant ψ ≈ 0.02. We then find that the flow induced attenuation becomes equal to the visco-thermal counterpart at a frequency

f ≈ 1.2 × 10

5

M 1+M

2 .

(9.11)

Below this frequency, the flow induced attenuation dominates. As an example, with M = 0.1, the frequency is ≈ 1200 Hz and the attenuation from each of the contributors is then 0.017 dB for x = D, i.e., negligible in comparison with the attenuation obtained for a lined duct.

9.6.3 A Proposed AeroAcoustic Instability As discussed in the previous section, the pressure gradient in the flow is expressed as dP ρU 2 = −(ψ/D) . dx 2

(9.12)

There is a pressure gradient also in a harmonic traveling sound field, which for a plane wave has the magnitude |

dp | = (2π/λ)|p|, dx

(9.13)

where |p| is the sound pressure amplitude and λ the wavelength. Over half of the temporal period of the wave this gradient will be in opposition to the gradient of the mean flow. This adverse gradient might influence the turbulence in the flow, and hence the sound attenuation. In order for such an effect to occur, the pressure gradient in the sound field must be greater than the gradient of the mean pressure, i.e., |dp/dx| > |dP /dx|. Then if we introduce the expression P = ρc2 /γ for the static pressure and the Mach number M = U/c, it follows from Eqs. 10.128 and 10.129 that |dp/dx| > |dP /dx| corresponds to |p|/P < (γ /4π) ψ(λ/D)M 2 ≈ 0.11(λ/D) ψ M 2 .

(9.14)

As an example, we choose ψ ≈ 0.02 in which case Eq. 10.123 becomes |p|/P T /8 in the first term in Eq. 9.18 is F (t) = Cn =

∞

n=0 Cn cos(nωt) 4 cos(nπ/4) − π(n2 −4) (n = 2, 0)

C0 = 1/2π,

(9.20)

C2 = 1/4

and the expansion of S(t) = sin(nωt − φn ), defined in the same time interval, is +

sin(φn ) S(t) = − sin(nπ/4) nπ q=1 (Sq sin(qωt − φn ) − Tq sin(qωt + φn )

∞

Sq =

sin(q−n)π/4 π(q−n)

Tq =

Sq = 1/4

sin(q+n)π/4 π(q+n)

(q = n).

(q = 0) (n = 0)

(9.21) (9.22)

324


The first term is the time average of S(t) (corresponding to q = 0) over the period T . For q = 0, the dominant term in the expansion of S(t) corresponds to q = n and is Sn = 1/4. All other terms will be neglected. Rewriting the term Sn sin(nωt − φn ) in Eq. 9.21 as Sn cos(nωt − φn − π/2), it follows from Eqs. 9.18 to 9.20 that the expression for the total contribution to the nth harmonic component of U (t) is Un (t) = (Ac /Ap )(2ωd)Cn cos(nωt) − (V /Ap )nωPn Sn cos(nωt − φn − π/2) = {Un (nω)e−inωt } Un (nω) = (Ac /Ap )(2ωD)Cn − i(V nω/4Ap )Pn (nω),

(9.23)

where we have introduced the complex amplitude Un (nω) and corresponding complex pressure amplitude Pn (nω) ≡ Pn exp(iφn ). (It should be noted that the complex amplitude of the nth harmonic component of velocity has been denoted simply by U (nω) and the magnitude by Un with an analogous notation for the pressure.) The pressure in the pipe just after the valve is the pressure in the cylinder minus the pressure drop p over the valve. Then, if the input impedance of the pipe is denoted by ζ (ω)ρc, we have ζρc ≡ [p(ω) − p(ω)]/Ux (ω) = p(ω)/Ux − ζv ρc, where ζv ρc is the valve impedance and x refers to the direction indicated in Figure 9.15. Recalling that the velocity U (t) is counted positive in the negative x-direction and that pn (ω) = Pn (ω)ρc2 , we obtain −ζ (ω)ρcUn (nω) = Pn (nω)ρc2 for the calculation of the complex amplitude Pn (nω) of the normalized sound pressure, where ζ = ζ + ζv . With Un (nω) given by Eq. 9.23, it follows that −[(Ac /Ap )(2ωD)Cn − i(κV nω/4Ap )Pn (nω)]ρcζ (nω) = Pn (nω)ρc2 ,

(9.24)

i.e., Pn (nω) ≡ Pn eiφn =

−(Ac /Ap )(2ωD/c)Cn ζ (nω) . 1 − i(κV nωζ (nω)/4cAp )

(9.25)

We rewrite this equation in a somewhat more compact form by introducing the V = Ac , where is the distance from the center position of the piston and the valve. Furthermore, we express the displacement amplitude as a certain fraction σ of , D = σ . Then, with = (Ac /Ap ) and k = ω/c, Eq. 9.25 can be written Pn (nω) ≡ Pn eiφn = −

2σ (k )Cn ζ (nω) . 1 − i(nk )ζ (nω)/4

(9.26)

(Cn : Eq. 9.20, ζ = ζ + ζv , ζ : Eq. 9.27.) The impedance ζv can be treated as an orifice plate impedance. In the vicinity of a quarter wavelength resonance of the pipe, the input impedance of the pipe will dominate and ζv can be neglected. Of this resonance, particularly in the vicinity of a frequency for which the pipe length is an integer number of half wavelengths, and it will be important. For numerical results, we refer to Figures 9.16 and 9.17.


325

Figure 9.16: (a): Pressure amplitude (rms, μPascal) in the pipe at the valve, (b) radiated power (μW), and (c) supercharge efficiency contribution of each of the first 10 harmonic components of the valve frequency at an engine speed of 2000 rpm. Pipe length: 4 ft. Cylinder diameter: 4 inches. Pipe diameter: 3 inches. Distance between center position of the cylinder and the valve: 3 inches. Piston amplitude: 2 inches.

326


Figure 9.17: The same set of curves as in Figure 9.16 except that the engine speed is now 2800 rpm.

327


9.8.2 Pipe Impedance In the presence of mean flow, the major acoustic losses in the pipe generally occur as a result of the interaction of sound with the separated flow at discontinuities. For example, the discharge of a flow from the end of a pipe at a mean flow Mach number M is equivalent to an acoustic resistance at the discharge end with the normalized value approximately equal to M. Often the Mach number is in the range 0.05 to 0.2 and the visco-thermal losses at the boundary of a straight pipe are then often negligible. Even in the absence of a mean flow, losses can occur at the ends of a pipe or at discontinuities if the amplitude of the oscillatory flow in the sound field is large enough to cause flow separation. The corresponding equivalent resistance at the end of the pipe then has the normalized value |u|/c, where |u| is the local velocity amplitude in the sound field. In the present case, flow separation is likely to occur at the valve port and at the air inlet end of the pipe. In the absence of losses, the termination impedance ζt of the pipe is simply the radiation impedance ζr of the open end of the pipe in free field. When the losses in the pipe are sufficiently small, they can be accounted for as an additional resistive component of the termination impedance. Thus, with a normalized equivalent impedance at the end of the pipe denoted by ζt , the normalized input impedance of a pipe of length L, as seen from the valve port is ζ ≡ θ + iχ = (ζt − i tan(kL))/(1 − iζt tan(kL)) i.e., θ = (θt (1 + tan2 (kL)))/(1 + θt2 tan2 (kL)) χ = −((1 − θt 2 tan(kL))/(1 + θt 2 tan2 (kL)),

(9.27)

where k = ω/c = 2π/λ. This result follows directly from the transmission matrix of a pipe and the expression for the impedance. As the frequency increases, the resistance goes from θ = θt to a maximum value θmax = 1/θt when kL = (2n − 1)π/2, i.e., when the pipe length is an odd number of quarter wavelengths. This corresponds to the resonances of a pipe with one end open and the other closed. For kL > 1. This is consistent with the attenuation spectra in Figure 8.1.

10.2.2 Nonlocally Reacting Liner Duct Lined On One Side The bottom liner in Figure 10.1 is now removed (d1 = 0), and the width of the air channel is denoted by D1 . The partitions within the liner are also removed so that it becomes nonlocally reacting. As we have seen in Section 8.2 there is not much difference in the attenuation spectra for locally and nonlocally reacting liners, and in hindsight this section may seem unnecessary, particularly since it is algebraically much more cumbersome than for the locally reacting liner. It is included nevertheless for the record and for those readers who may want to study the approach used and check the analysis. In constructing the sound field, we note that the sound pressure must have a maximum at the rigid walls of the duct (at y = 0 and y = D1 + d), and the complex amplitudes of the sound pressure in the air space and within the liner then must be given by p(ω) = A cos(ky y)eikx x p (ω)

= B cos(qy

y )eiqx x

(air)

(10.25)

(liner),

(10.26)

where y = D1 + d − y and A and B are constants. For the wave in the air, the components of the propagation constant are kx and ky , kx2 + ky2 = (ω/c)2 . In the liner the corresponding quantities are qx and qy with qx2 + qy2 = q 2 . The propagation constant q in a porous material is ˜ (10.27) q = κ˜ ρ, where κ˜ and ρ˜ are the complex values of the compressibility and the density of the air within the porous material. The corresponding normalized propagation constant is Q = q/k = (κ/κ)( ˜ ρ/ρ). ˜ Since the spatial variation of the fields in the air and in the liner must be the same to satisfy the boundary conditions for all values of x, we must have qx = kx =

(ω/c)2 − ky2 .

The expressions for the y-components of the velocity amplitudes (counted positive in the positive y-direction) in the air and in the porous layer are ikx x uy = (1/ωρ) ∂p ∂y = −A(ky /ωρ) sin(ky y)e

u y = −(1/ωρ) ˜ ∂p ˜ sin(qy y )eiqx x , (10.28) ∂y = B(qy /ωρ) √ where we have made use of ∂/∂y = −∂/∂y. The quantity ρ˜ = ρ s + ir/ωρ is the complex density of air within the porous material, where s is the structure factor and r, the flow resistance per unit thickness of the liner, as mentioned above.

338


The boundary conditions at the surface of the liner at y = D1 and y = d are p = p and uy = u y and from these, the following relation obtains (from Eqs. 10.25 to 10.28) Equation for ky , nonlocal reaction ky D1 tan(ky D1 ) = −(ρ/ρ)q ˜ y D1 tan(qy d) ≡ −ikD1 ηi

(10.29)

˜ tan(qy d). Qy = qy /k. where ηi = −iQy (ρ/ρ) This equation for the determination of ky D1 can be brought into the same form as the corresponding equation for the locally reacting liner in Eq. 10.9 if we introduce the normalized input admittance of the liner in the y-direction. It follows from the equation of motion for the porous material that the normalized wave admittance for a wave traveling in the y-direction in the porous material is Qy (ρ/ρ), ˜ where Qy = qy /k. The corresponding input admittance of the liner of thickness d is then ηi = −iQy (ρ/ρ) ˜ tan(qy d). If this expression is used in Eq. 10.9, we obtain Eq. 10.29, which illustrates that the analyses in the two cases are consistent. Since the wave velocity in the x-direction must be the same in the air and in the material, the wave numbers kx and qx must be the same also. Accounting for this equality and making use of the wave equations in the two regions, we have, with k = ω/c, Axial propagation constant, kx = kxr + ikxi kx2 = qx2 = k 2 − ky2 = q 2 − qy2

(10.30)

within the porous material, q = The quantity q is the (total) propagation constant κ˜ ρ˜ with the normalized value Q = q/k = (κ/κ)( ˜ ρ/ρ). ˜ Combining Eqs. 10.29 and 10.30 we can solve (numerically) for ky and then obtain kx from Eq. 10.30. The imaginary part of kx yields the attenuation with the spatial variation of the pressure amplitude expressed by exp(−ki x). Wave Impedance If one wishes to determine the insertion loss of a finite lined duct element, we need an expression for the wave impedance of the fundamental mode in the duct and we proceed to derive an expression for it. We are interested in a plane wave incident on the lined duct element, and use for the wave impedance the ratio of the average sound pressure amplitude across the duct and the average axial velocity amplitude. With the y-dependence of the sound pressure amplitudes in the air and in the liner given by p = A cos(ky y) and p = B cos(qy y), and the open area fraction by σ = D1 /(D1 + d), the average pressure over the total duct area becomes pav = Aσ

sin(ky D1 ) ky D1

+ B(1 − σ )

sin(qy d) qy d

= A cos(ky D1 )[σ T (ky D1 ) + (1 − σ )T (qy d)] T (x) ≡ tan(x)/x,

(10.31)

where we have made use of A cos(ky D1 ) = B cos(qy d), which follows from the boundary condition p = p for y = D1 (y = d).

MATHEMATICAL SUPPLEMENTS AND COMMENTS

339

Returning to Eqs. 10.25 and 10.26 for the pressure amplitudes in the air channel and in the porous material, we obtain the corresponding axial velocity amplitudes

/∂x. The average values of these ˜ from ux = −(1/ωρ)∂p/∂x and ux = −(1/ωρ)∂p quantities are obtained by integrating cos(ky y) and cos(qy d) over the width D1 of the channel and the thickness d of the porous layer, respectively, and dividing by d + D1 . Then, making use of B/A = cos(ky D1 )/ cos(qy d), obtained from the boundary condition at the porous surface, the average axial velocity amplitude is found to be ˜ − σ )T (qy d)], uav = A(kx /ωρ) cos(ky D1 )[σ T (ky D1 ) + (ρ/ρ)(1

(10.32)

where ρ/ρ ˜ is the complex density ratio discussed earlier. For a wave traveling in the positive x-direction, the ratio ζd = (1/ρc)pav /uav

(10.33)

is then the normalized wave impedance. Low Frequency Approximation At sufficiently low frequencies such that the wavelength is large compared to d and D1 , we expect that ky D1 1, the argument ky D1 will lie in the vicinity of π/2, and we express the corresponding approximate high frequency solution for ky D1 as


341

ky D1 ≈ π/2 + . Insertion into Eq. 10.42 yields tan(ky D1 ) ≈ −1/, and hence ≈ −i(π/2)/qy (ρ/ρ)D ˜ 1 H . Thus, ky D1 ≈

π ˜ 1 ). (1 − i/qy (ρ/ρ)D 2

(10.43)

Thecorresponding high frequency approximation for the propagation constant kx = k 2 − ky2 is then given by kx ≈ k + i

(π/2)2 . 3 qy (ρ/ρ)kD ˜ 1

(10.44)

Then, if we use the expression for qy from Eq. 10.41 in Eq. 10.44 and ρ/ρ ˜ ≈ s we get, with s H ≈ 1, 1/4

kx ≈ k + i

d/D1 1 π 2 ( ) . D1 2 (kD1 )7/4 (H )1/4

(10.45)

Thus, under these conditions, the high frequency attenuation is weakly dependent on the liner resistance, ∝−1/4 , increasing somewhat with decreasing flow resistance. For the locally reacting liner it was independent of the resistance to a first approximation. Two Sides Lined Having established that the normalized admittance to a nonlocally reacting porous layer can be expressed in terms of the propagation constant qy in the porous material, as shown in Eq. 10.29, we can use the result in Eq. 10.9 for the locally reacting liners for determination of the propagation constant in the duct merely by expressing the admittance η1 and η2 in this manner. Thus, with the propagation constants in the

layers denoted by q1 and 2 with the corresponding y-components q 1y =

2 , q12 − q1x

we obtain, with q1x = q2x = kx = k 2 − ky2 , an equation for the determination of the y-component ky of the propagation constant in the air channel, and hence the desired kx =

k 2 − ky2 .

10.3 SUPPLEMENT TO SECTION 8.3, OTHER DUCT TYPES 10.3.1 Rectangular Duct Lined On All Sides The duct under consideration is shown in Figure 8.9. The sides of the rectangular air channel are Dy and Dz . The liners are locally reacting and the normalized admittances are ηy1 , ηy2 , ηz1 , and ηz2 , as shown. We place the origin of the coordinate system at the center of the duct with the y and z being vertical and horizontal, respectively. The x-axis runs into the page along the duct axis.

342


The pressure distribution in the duct is expressed as p = A(eiky y + Ry e−iky y )(eikz z + Rz e−ikz z ),

(6.1.21)

and the corresponding velocity field has the components uy = (1/ − iρω)(−∂p/∂y) and yz (1/ − iρω)(−∂p/∂z). Then, by imposing the boundary conditions of the known admittances at the boundaries y = 0, y = Dy , z = 0, and z = Dz , we can express Ry in terms of ky , ηy1 , ηy2 and Rz in terms of kz , ηz1 , ηz2 , and by eliminating Rz and Rz , we obtain equations for ky and kz . Thus, Ky tan(ky Dy ) = −i

ηy1 + ηy2 , 1 + ηy1 ηy2 /Ky2

(10.46)

where Ky = ky /k and k = ω/c. The analogous equation for Kz = kz /k is obtained by replacing y by z. Generally, these equations have to be solved numerically. In the special case when the admittances of opposite walls are the same, η1y = η2y = ηy and η1z = η2z = ηz , the equation for Ky can be expressed as Ky tan(Ky kDy /2) = −iηy

(6.1.21a)

as explained in the discussion following Eq. 10.8 with an analogous equation for Kz . The propagation constant kx then follows from Eq. 10.9, i.e, (10.47) kx = kr + iki = k 2 − ky2 − kz2 . The corresponding attenuation is 20 log 10(e)ki ≈ 8.7ki dB per unit length. A numerical example has already been given and discussed in Figure 8.14. The extension of the analysis to nonlocally reacting liners follows from the treatment of that in Section 10.2.2.

10.3.2 Circular Duct Locally Reacting Liner With the radial component of the propagation constant in the porous liner denoted by qr (in a locally reacting liner there is no other component), the expression for the sound pressure field in the liner is p(r) = AH0(1) (qr r) + BH0(2) (qr r), (1)

(2)

(10.48)

where H0 = J0 + iY0 and H0 = J0 − iY0 are the Hankel functions of zeroth order, being combinations of the Bessel and Neumann functions J0 and Ys . (We could equally well have expressed the total pressure field as a linear combination of J0 and Y0 , which would have been a bit simpler. Try it!) By analogy with plane waves, these relations correspond to exp(±ikx) = cos(kx) ± i sin(kx), H0(1) representing an (2) outgoing and H0 an incoming cylindrical wave. With qr being the only component, we have qr = q, where q is defined in Eq. 10.27.


343

The corresponding radial velocity amplitude is obtained from the momentum equation −iωρu ˜ r = −∂p/∂r, where, as before, ρ˜ is given in Eq. 10.35, ηw (1) (2) AH1 (qr) + BH1 (qr) . (10.49) ur (r) = i ρc ˜ r /k) (normalized wave admittance), where k = ω/c. The quantity ηw = (ρ/ρ)(q We have used dH0 (z)/dz = −H1 (z), where H1 is the Hankel function of the first order. The boundary condition of zero radial velocity at the outer radius b of the porous layer, which is in contact with the rigid duct wall, yields (1)

(2)

B/A = −H1 (qr b)/H1 (qr b).

(10.50)

Using this relation in combination with Eqs. 10.48 and 10.49, we find for the normalized admittance of the porous liner η = ρc

J1 (qr a)Y1 (qr b) − J1 (qr b)Y1 (qr a) ur (a) = iηw . p(a) J0 (qr a)Y1 (qr b) − J1 (qr b)Y0 (qr a)

(10.51)

For a thin layer, (b − a) > 1 and kr a >> 1, we have J1 /J0 ≈ tan(kr a − π/4), and it follows from Eq. 10.55 that the argument of the tan()-function must be close to π/2, and we put kr a − π/4 = π/2 + . With tan(π/2 + ) ≈ −1/, it follows from the equation that ≈ −i(kr a/kaη) and kr a ≈

3π 1 3π ≈ (1 − i ). 4(1 + i/kaη) 4 kaη

(10.57)

The solution of Eq. 10.55 for kr a over the entire frequency range is obtained numerically and the axial propagation constant kx = kxr +ikxi follows from Eq. 10.52 and the attenuation from p ∝ exp(−kxi x). An example of the computed frequency dependence of the attenuation is shown and discussed in Figure 8.14. Nonlocally Reacting Liner In a liner without transverse partitions (nonlocally reacting), there will be sound propagation within the liner also in the axial direction, and if the corresponding component of the propagation constant is qx , we have qr = q 2 − qx2 , where q is the ‘total’ propagation constant in the porous material as given before in Eq. 10.27. Similarly, in the air channel, we have kr2 = k 2 − kx2 . Since the axial variation in the sound field must be the same in the air channel as in the liner, we have qx = kx . The field matching condition at the boundary of the liner, expressed by Eq. 10.56 in which η is given by Eq. 10.51, is still valid. With kx2 = k 2 − kr2 and qr = q 2 − kx2 ityields an equation for kr from which follows the desired propagation constant kx = k 2 − kr2 in the duct in complete analogy with the procedure for the rectangular duct. The equation for kr has to be solved numerically.

10.3.3 Annular Duct If we add to the circular lined duct a concentric porous core or rigid cylinder with a porous liner, the air channel becomes an annular duct, which is another configuration of practical interest. A sound field with no azimuthal angle dependence, to which the fundamental acoustic mode belongs, will be of the form p = [AJ0 (kr r) + BY0 (kr r)]eikx x = A[J0 (kr r) + RY0 (kr r)]eikx x kx = k 2 − kr2 ,

(10.58)

where k = ω/c and R = B/A. The corresponding radial velocity, obtained from the momentum equation −iωρur = −∂p/∂r, is ur = −

1 A kr kr [AJ1 (kr r) + BY1 (kr r)] = i [J1 (kr r) + RY1 (kr r)]. iωρ ρc k

(10.59)


345

The inner and outer radii of the annulus are denoted by a and b. Both p and ur are continuous at these boundaries, and we denote their normalized admittances by ηa and ηb . Thus, at r = b we have ηb = i

kr J1 (kr b) + RY1 (kr b) . k J0 (kr b) + RY0 (kr b)

(10.60)

Replacing b by a, and ηb by −ηa ,1 we get the corresponding equation at r = a. From these two relations, we can eliminate R and obtain the following equation for determination of the radial propagation constant kr −

i(kr /k)J1 (kr b) − ηb J0 (kr b) i(kr /k)J1 (kr a) + ηa J0 (kr a) = . Y0 (kr a)ηa + i(kr /k)Y1 (kr a) Y0 (kr b)ηb − i(kr /k)Y1 (kr b)

(10.61)

It is instructive, and also useful as starting points for numerical root-finding routines, to consider the form of these relations in the limits of low and high frequencies corresponding to small and large values of the arguments of the Bessel functions. For z 0

where the constant of proportionality C is the integral of (Dz /k)| cos(ky y)|2 from y = 0 to D. Quantities kmr and kmi are the real and imaginary parts of the propagation constant kx of the mth mode, as obtained from Eq. 10.70.

348


The corresponding average attenuation in dB is obtained from the ratio of the powers W (0) and W (L) at the beginning (x = 0) and at the end (x = L) of the duct, or, expressed in decibels, Atten = 10 log[W (0)/W (L)].

(10.74)

In order to what extent this average attenuation might differ from the attenuation of the fundamental mode, we consider a case in which all the amplitudes Am of the first 20 modes are the same and find that there is no significant difference between the average attenuation and the attenuation of the fundamental mode. ‘Y-Modes’ The numerical results in Figures 8.1 and 8.3 apply only to the lowest order solution for ky , corresponding to the fundamental mode. Higher order values can be obtained as additional solutions to Eq. 10.8. The corresponding attenuation will be greater than for the fundamental mode but after they have decayed after a relatively short distance from the entrance of the duct, the fundamental mode attenuation takes over. For a sufficiently long duct, the latter generally determines the overall average attenuation. A more detailed analysis requires not only a determination of the higher mode values of ky but also the amplitude of the various modes at the entrance of the duct. These depend on the nature of the sound field incident sound field. We shall limit ourselves here merely to an estimate of the role of higher order modes in the region of high frequencies where ray acoustics can be used as an approximation. Semi-Empirical Higher Order Mode, TL Correction In order to get an idea of the TL at very high frequencies, we considered a line source placed at the beginning of the duct, as shown in Figure 8.22. The width of the air channel is D and the length L. If the intensity radiated from the source in a direction corresponding to the angle φ (see figure) is I (φ), the total power radiated into the duct is W =

π/2

−π/2

I (φ) dφ.

(10.75)

If the liner is perfectly absorbing, there will be no reflections of the rays that strike the liner, and the only rays that reach the end of the duct are those emitted in the angular range from −φ0 to φ0 , where φ0 = arctan(D/2L). The transmitted power is then We =

φ0

−π0

I (φ) dφ.

(10.76)

In the absence of the liner, the power leaving the duct will be the same as the power W emitted into the left hemisphere of the source. Thus, the transmission loss is T L = 10 log(W/We ). (10.77)


349

In the special case of an omni-directional source, I (φ) = W/π, We = (2φ0 /π )W , and TL = 10 log(π/(2φ0 ). For small angles, φ0 ≈ D/2L and TL ≈ 10 log(π L/D), as given in the text. To get an idea of the role of directivity, we consider as an example I (φ) = A cos(φ) = (W/2) cos(φ). The power escaping without reflections is then W sin(φ0 ), and the corresponding transmission loss becomes about 2 dB smaller than for the omnidirectional source. In reality, the liner is not totally absorbing, and we can improve the expression for We by accounting for the number of reflections suffered by a ray along its path to the exit of the channel. This number depends on the angle of emission φ of the ray. After each reflection, the intensity is reduced by the factor |R|2 , where R is the pressure reflection coefficient of the liner. Then, if the number of reflections is n(φ), the emitted power from the duct will be We =

π/2

−π/2

I (φ)|R|2n(φ) dφ.

(10.78)

With a given duct liner, the angular dependence of the absorption coefficient can readily be obtained. In our case, there will be one reflection from the acoustic liner in the angular range between −φ0 = arctan(D/2L) and φ1 = arctan(2D/3L). With reference to the discussion of Figure 8.23, refraction of a ray in the nonuniform flow in the duct changes the angles φ0 and φ1 , and hence the output power We . The ray will be curved and, as mentioned, the radius of curvature will be R = dM/dy. It follows from Figure 8.23 that (L + R cos φ0 )2 + (R cos φ0 ) − (D1 /2)2 = R 2 from which it follows D1 /2 L 1 − (L2 + D12 /4)/4R 2 − . (10.79) φ0 = 2R L2 + D12 /4 A positive (negative) value of R corresponds to downstream (upstream) propagation√leading to a decrease (increase) of the critical angle φ0 . At a value R = (1/2) L2 + D 2 , the angle will be zero. In this manner, we can get an idea of the effect of refraction on the transmission loss for higher modes in the high frequency limit. The line source model used here for the purpose of illustration can be replaced by one in which the sound enters the duct from a diffuse field in a plenum chamber.

10.5 SUPPLEMENT TO SECTION 8.7, LIQUID PIPE LINES 10.5.1 Liquid Pipe Line With Slightly Compliant Walls Wall Admittance The sound pressure in the water is denoted by p and the induced radial displacement of the pipe wall by ξ . As indicated earlier, local reaction of the boundary is assumed. If the acoustic wavelength is much larger than the pipe diameter, we can assume the

350


displacement to be independent of the azimuthal angle. The strain is ξ/a and the stress in the wall σ = Eξ/a, (10.80) where E is the elastic modulus, and a the unperturbed radius of the pipe. We have assumed that h

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