Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials

295 Pages·1993·7.03 MB·English

by  J. F. Allard (auth.)

PROPAGATION OF SOUND
IN POROUS MEDIA
ModeUing Sound Absorbing Materials
PROPAGATION OF
SOUND IN POROUS
MEDIA
Modelling Sound Absorbing
Materials
J. F. Allard
Laboratoire d'Acoustique
de l' Universite du Maine,
Le Mans, France
ELSEVIER APPLIED SCIENCE
LONDON and NEW YORK
ELSEVIER SCIENCE PUBLISHERS LTD
Crown House, Linton Road, Barking, Essex IG 11 8JU, England
WITH 15 TABLES AND 133 ILLUSTRAn ONS
© 1993 ELSEVIER SCIENCE PUBLISHERS LTD
British Library Cataloguing in Publication Data
Allard, J .-F.
Propagation of Sound in Porous Media:
Modelling Sound Absorbing Materials
I. Title
620.2
ISBN 1-85166-887-x
Library of Congress Cataloging-in-Publication Data
Allard, J .-F.
Propagation of sound in porous media: modelling sound absorbing
materials/ J .-F. Allard.
p. cm.
Includes bibliographical references and index.
ISBN 1-85166-887-X
1. Porous materials-Acoustic properties-Mathematical models.
2. Absorption of sound. 3. Sound-Transmission. I. Title.
TA418.9.P6A42 1993 92-18626
620.1' 1694'015118--dc20 CIP
No responsibility is assumed by the Publisher for any injury and/or damage to persons
or property as a matter of products liability, negligence or otherwise, or from any use or
operation of any methods, products, instructions or ideas contained in the material
herein.
Special regulations for readers in the USA
This publication has been registered with the Copyright Clearance Centre Inc. (Ccq,
Salem, Massachusetts. Information can be obtained from the CCC about conditions
under which photocopies of parts of this publication may be made in the USA. All other
copyright questions, including photocopying outside the USA, should be referred to the
publisher.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system, or transmitted in any form or by any means, electronic, mechanical, photo
copying, recording, or otherwise, without the prior written permission of the publisher.
Preface
This book has grown out of the research activities of the author in the
fields of sound propagation in porous media and modelling of acoustic
materials. It is assumed that the reader has a background of advanced
calculus, including an introduction to differential equations, complex
variables and matrix algebra. A prior exposure to theory of elasticity
would be advantageous.
Chapters 1-3 deal with sound propagation of plane waves in solids
and fluids, and the topics of acoustic impedance and reflection
coefficient are given a large emphasis. The topic of flow resistivity is
presented in Chapter 2.
Chapter 4 deals with sound propagation in porous materials having
cylindrical pores. The topics of effective density, and of tortuosity, are
presented. The thermal exchanges between the frame and the fluid,
and the behaviour of the bulk modulus of the fluid, are described in
this simple context.
Chapter 5 is concerned with sound propagation in other porous
materials, and the recent notions of characteristic dimensions, which
describe thermal exchanges and the viscous forces at high frequencies,
are introduced.
In Chapter 6, the case of porous media having an elastic frame is
considered in the context of Biot theory, where new topics described
in Chapter 5 have been included.
Chapter 7 deals with multilayered media including fluid, elastic and
porous layers. A modelling of porous layers with transfer matrices is
calculated in the context of the Biot theory. These transfer matrices
are used to predict the surface impedance of porous materials covered
with porous and impervious screens, and the transmission coefficient
of plate-porous-Iayer and plate-porous-Iayer-plate structures.
Chapter 8 introduces a description of axis ymmetrica I fields in
layered porous media.
In Chapter 9, the different methods which have been used to
v
vi Preface
measure the acoustic impedances presented in the book are described.
Chapter 10 is concerned with the effect of perforated facings at
normal and oblique incidence on the impedance of porous stratified
layers.
In Chapter 11, reciprocity and antireciprocity are studied in the
context of sound propagation through stratified materials. Impedance
matrices, evaluated from the transfer matrices of elastic and porous
layers, are used to show that the transmission loss is the same for
opposite directions of propagation in stratified porous materials. The
similarity between the impedance matrices of elastic and porous layers
is highlighted.
Most of the programs which have been used to calculate impedance
and transmission coefficients in illustrations can be obtained by writing
to the author.
During the preparation of this book, the author has enjoyed a close
collaboration with several colleagues. Much of this book has been
influenced by an association with Prof. Y. Champoux and Prof. J.
Nicolas, of the University of Sherbrooke (Quebec, Canada); Prof. A.
Cops and Dr W. Lauriks, of the Catholic University of Leuven
(Belgium); and Prof. C. Depollier and Dr D. Lafarge, of the
Laboratoire d' Acoustique de I'Universite du Maine, Unite de
Recherche Associee, au Centre National de la Recherche Scientifique,
Le Mans (France).
Part of the book was written while the author was an invited
Professor at the University of Salford (UK), and at the Institute for
Microstructural Research of the National Research Council in Ottawa
(Ontario, Canada). The author would like to thank Prof. P. Lord and
Dr G. Daigle for their hospitality during these periods, and the
support of the Centre National de la Recherche Scientifique during the
preparation of the book.
Several students, researchers and professors have read and criticized
versions of the manuscript. The author would like to thank S. Gorog,
B. Brouard, Dr K. Attenborough, Dr D. L. Johnson, Dr D. Lafarge,
Dr D. Waddington, Dr M. Tamura, Prof. J. I. Dunlop and Prof. P.
Lord for their comments.
Finally, the author expresses thanks to Miss Janette Le Moine for
her contribution to typing the various drafts of the book.
J. F. ALLARD
Contents
Preface v
Chapter 1 Plane Waves in Isotropic Fluids and Solids
1.1 Introduction 1
1.2 Notations-Vector operators. 1
1.3 Strain in a deformable medium 2
1.4 Stress in a deformable medium 4
1.5 Stress-strain relations for an isotropic elastic medium 6
1.6 Equation of motion . 9
1.7 Wave equation in a fluid. 10
1.8 Wave equations in an elastic solid 12
1.9 Potential and kinetic energy in continuous media 14
Chapter 2 Acoustic Impedance at Normal Incidence of Fluids, and
Highly Porous Materials
2.1 Introduction 16
2.2 Plane waves in unbounded fluids 16
2.2.1 Travelling waves . 16
2.2.2 Complex notation 17
2.2.3 Example 18
2.2.4 Attenuation. 18
2.2.5 Superposition of two waves propagating in oppo-
site directions 18
2.3 Main properties of impedance at normal incidence . 19
2.3.1 Impedance variation along a direction of propaga-
tion. 19
2.3.2 Impedance at normal incidence of a layer of fluid
backed by an impervious rigid wall . 20
2.3.3 Impedance at normal incidence of a multilayered
fluid 21
vii
viii Contents
2.4 Reflection coefficient and absorption coefficient at normal
incidence . 21
2.4.1 Reflection coefficient . 21
2.4.2 Absorption coefficient. 22
2.5 Fluids equivalent to porous materials with high porosity:
The laws of Delany and Bazley 23
2.5.1 Porosity and flow resistivity in porous materials. 23
2.5.2 Microscopic and macroscopic description of sound
propagation in porous media 24
2.5.3 The laws of Delany and Bazley, and flow resis-
tivity 25
2.6 Examples . 26
2.7 The complex exponeQtial representation 27
Chapter 3 Acoustic Impedance at Oblique Incidence in Fluids, and
Highly Porous Materials
3.1 Introduction 31
3.2 Inhomogeneous plane waves in isotropiC fluids 31
3.3 Reflection and refraction at oblique incidence 34
3.4 Impedance at oblique incidence in isotropic fluids 35
3.4.1 Impedance variation along a direction perpen-
dicular to an impedance plane . 35
3.4.2 Impedance at oblique incidence for a layer of fluid
of finite thickness backed by an impervious rigid
wall. 37
3.4.3 Impedance at oblique incidence in a multilayered
fluid 38
3.5 Reflection coefficient and absorption coefficient at
oblique incidence 39
3.6 Examples 40
3.7 Plane waves in fluids equivalent to anisotropic highly
porous media 42
3.8 Impedance at oblique incidence at the surface of a fluid
equivalent to an anisotropic porous material 45
3.9 Example 46
Chapter 4 Sound Propagation in Cylindrical Tubes and Porous
Materials Having Cylindrical Pores
4.1 Introduction 48
4.2 Viscosity effect in a cylindrical tube 48
Contents ix
4.3 Thermal effects. 53
4.4 Effective density and bulk modulus for cylindrical tubes
having triangular, rectangular and hexagonal cross-
sections 58
4.5 High and low frequency approximation 59
4.6 Evaluation of the effective density and the bulk modulus
of the air from flow resistivity and porosity, in layers of
porous materials with identical pores perpendicular to the
surface 62
4.6.1 Effective density in cylindrical pores having a
circular cross-section . 62
4.6.2 Effective density in slits 64
4.6.3 The Biot model for rigid framed materials 65
4.6.4 Bulk modulus of the air in slits. 66
4.6.5 Effective density and bulk modulus of air in
cylindrical pores of arbitrary cross-sectional shape 67
4.7 Impedance of a layer with identical pores perpendicular
to the surface 69
4.7.1 Normal incidence. 69
4.7.2 Oblique incidence-Locally reacting materials 70
4.8 Tortuosity and flow resistivity in a simple anisotropic
material 71
4.9 Impedance at normal incidence and sound propagation in
a material with oblique pores 73
4.9.1 Effective density. 73
4.9.2 Bulk modulus of the air in the material 75
4.9.3 Impedance 75
4.9.4 Summary of Section 4.9 76
Chapter 5 Sound Propagation in Porous Materials Having a Rigid
Frame
5.1 Introduction 79
5.2 The concept of tortuosity in the work by Johnson et ai. 79
5.3 Characteristic dimension for viscous forces. 82
5.4 Characteristic dimension for the bulk modulus of the air
in a porous material. 84
5.5 General expression of the effective density. 87
5.6 Frequency dependence of the bulk modulus of the air in
porous materials 90
5.6.1 Materials with cylindrical pores 90
x Contents
5.6.2 Other porous materials 90
5.6.3 Shape factors and dimension factors 92
5.7 Summary of the two equivalent formulations for the
effective density p and the bulk modulus K 92
5.8 Examples . 93
5.8.1 Porous material having pores made up of an
alternating sequence of circular cross-sectional
shaped cylinders of two different diameters 93
5.8.2 Other materials . 96
5.8.3 Fibrous materials. 96
5.9 Simple models . 105
5.9.1 The model of Attenborough 105
5.9.2 The model of Allard et al. . 106
5.10 Surface impedance . 107
Appendix 5.A Kinetic energy and tortuosity 111
Appendix 5.B Simplified calculation of the tortuosity
for a porous material having pores made
up of an alternating sequence of cylin-
ders 113
Appendix 5.C Flow in a slit 114
Appendix 5.D Calculation of the characteristic dimen-
sion A' . 115
Appendix 5.E Calculation of the characteristic dimen-
sion A for a cylinder perpendicular to
the direction of propagation 115
Chapter 6 Biot Theory of Sound Propagation in Porous Materials
Having an Elastic Frame
6.1 Introduction 118
6.2 Stress and strain in porous materials 119
6.2.1 Stress 119
6.2.2 Strain 119
6.2.3 Stress-strain relations in the Biot theory. The
potential coupling term 119
6.2.4 A simple example 122
6.2.5 Determination of P, Q and R 124
6.2.6 Comparison with previous models of sound prop-
agation in porous sound absorbing materials . 124
6.3 Inertial forces in the Biot theory . 125
Contents xi
6.4 Wave equations 127
6.5 The two compressional waves and the shear wave 129
6.5.1 The two compressional waves . 129
6.5.2 The shear wave . 131
6.5.3 The three Biot waves in ordinary air-saturated
porous materials . 132
6.5.4 Example 133
6.6 Prediction of surface impedance at normal incidence for a
layer of porous material backed by an impervious rigid
wall 136
6.6.1 Introduction. 136
6.6.2 Prediction of the surface impedance at normal
incidence 137
6.6.3 Example 139
Chapter 7 Prediction of Sudace Impedance and Sound Transmission
for Multilayered Porous Media
7.1 Introduction 145
7.2 Sound propagation in a porous layer at oblique incidence 145
7.2.1 The acoustic field in a layer of porous material at
oblique incidence. 145
7.2.2 The matrix representation. 147
7.2.3 Evaluation of the matrices [f] and [T] 148
7.3 Transfer matrix representation of a layered material 151
7.4 Surface· impedance at oblique incidence of materials
consisting of several porous layers 153
7.4.1 Evaluation of the surface impedance from the
transfer matrix elements 153
7.4.2 Examples: Materials with porous screens 155
7.5 Materials with impervious screens. 161
7.6 Sound propagation in materials including layers of elastic
solids and fluids 165
7.7 Sound transmission of plane waves through layered
materials . 168
Appendix 7.A The elements Tij of the transfer matrix
[no In
Appendix 7.B Transmission through a plate-porous-
material-plate structure 181