Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials, Second Edition

361 Pages·2004·5.77 MB·English

by  Jean F. Allard

Propagation of Sound
in Porous Media
Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials, Second Edition J. F. Allard and N. Atalla
© 2009 John Wiley & Sons, Ltd. ISBN: 978-0-470-74661-5
Propagation of Sound
in Porous Media:
Modelling Sound Absorbing Materials,
Second Edition
Jean F. Allard
Universite´ le Mans, France
Noureddine Atalla
Universite´ de Sherbrooke, Qc, Canada
A John Wiley and Sons, Ltd., Publication
Thiseditionfirstpublished2009
©2009,JohnWiley&Sons,Ltd
Registeredoffice
JohnWiley&SonsLtd,TheAtrium,SouthernGate,Chichester,WestSussex,PO198SQ,United
Kingdom
Fordetailsofourglobaleditorialoffices,forcustomerservicesandforinformationabouthowtoapply
forpermissiontoreusethecopyrightmaterialinthisbookpleaseseeourwebsiteatwww.wiley.com.
Therightoftheauthortobeidentifiedastheauthorofthisworkhasbeenassertedinaccordancewiththe
Copyright,DesignsandPatentsAct1988.
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,or
transmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,recordingorotherwise,
exceptaspermittedbytheUKCopyright,DesignsandPatentsAct1988,withoutthepriorpermissionof
thepublisher.
Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprintmay
notbeavailableinelectronicbooks.
Designationsusedbycompaniestodistinguishtheirproductsareoftenclaimedastrademarks.Allbrand
namesandproductnamesusedinthisbookaretradenames,servicemarks,trademarksorregistered
trademarksoftheirrespectiveowners.Thepublisherisnotassociatedwithanyproductorvendor
mentionedinthisbook.Thispublicationisdesignedtoprovideaccurateandauthoritativeinformationin
regardtothesubjectmattercovered.Itissoldontheunderstandingthatthepublisherisnotengagedin
renderingprofessionalservices.Ifprofessionaladviceorotherexpertassistanceisrequired,theservices
ofacompetentprofessionalshouldbesought.
LibraryofCongressCataloguing-in-PublicationData
Allard,J.-F.
Propagationofsoundinporousmedia:modellingsoundabsorbingmaterials/JeanF.
Allard–2nded./NoureddineAtalla.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-470-74661-5(cloth)
1. Porousmaterials–Acousticproperties–Mathematicalmodels. 2. Absorptionofsound. 3.
Sound–Transmission. I. Atalla,Noureddine. II. Title.
TA418.9.P6A42 2009
620.1(cid:2)1694015118–dc22
2009028758
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
ISBN: 978-0-470-746615-0
Typesetin10/12ptTimesbyLaserwordsPrivateLimited,Chennai,India.
PrintedandboundinSingaporebyMarkonoPrintMediaPte,Ltd.
Contents
Preface to the second edition xiii
1 Plane waves in isotropic fluids and solids 1
1.1 Introduction 1
1.2 Notation – 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 5
1.6 Equations of motion 8
1.7 Wave equation in a fluid 10
1.8 Wave equations in an elastic solid 11
References 13
2 Acoustic impedance at normal incidence of fluids. Substitution
of a fluid layer for a porous layer 15
2.1 Introduction 15
2.2 Plane waves in unbounded fluids 15
2.2.1 Travellingwaves 15
2.2.2 Example 16
2.2.3 Attenuation 16
2.2.4 Superpositionoftwowavespropagatinginopposite
directions 17
2.3 Main properties of impedance at normal incidence 17
2.3.1 Impedancevariationalongadirectionofpropagation 17
2.3.2 Impedanceatnormalincidenceofalayeroffluidbacked
byanimperviousrigidwall 18
2.3.3 Impedanceatnormalincidenceofamultilayered
fluid 19
vi CONTENTS
2.4 Reflection coefficient and absorption coefficient at normal incidence 19
2.4.1 Reflectioncoefficient 19
2.4.2 Absorptioncoefficient 20
2.5 Fluids equivalent to porous materials: the laws of Delany and Bazley 20
2.5.1 Porosityandflowresistivityinporousmaterials 20
2.5.2 Microscopicandmacroscopicdescriptionofsoundpropagation
inporousmedia 22
2.5.3 TheLawsofDelanyandBazleyandflowresistivity 22
2.6 Examples 23
2.7 The complex exponential representation 26
References 26
3 Acoustic impedance at oblique incidence in fluids. Substitution
of a fluid layer for a porous layer 29
3.1 Introduction 29
3.2 Inhomogeneous plane waves in isotropic fluids 29
3.3 Reflection and refraction at oblique incidence 31
3.4 Impedance at oblique incidence in isotropic fluids 33
3.4.1 Impedancevariationalongadirectionperpendicular
toanimpedanceplane 33
3.4.2 Impedanceatobliqueincidenceforalayeroffinite thickness
backedbyanimperviousrigidwall 34
3.4.3 Impedanceatobliqueincidenceinamultilayeredfluid 35
3.5 Reflection coefficient and absorption coefficient at oblique incidence 36
3.6 Examples 37
3.7 Plane waves in fluids equivalent to transversely isotropic porous
media 39
3.8 Impedance at oblique incidence at the surface of a fluid equivalent
to an anisotropic porous material 41
3.9 Example 43
References 43
4 Sound propagation in cylindrical tubes and porous materials
having cylindrical pores 45
4.1 Introduction 45
4.2 Viscosity effects 45
4.3 Thermal effects 50
4.4 Effective density and bulk modulus for cylindrical tubes having
triangular, rectangular and hexagonal cross-sections 54
4.5 High- and low-frequency approximation 55
4.6 Evaluation of the effective density and the bulk modulus of the air
in layers of porous materials with identical pores perpendicular
to the surface 57
4.6.1 Effectivedensityandbulkmodulusincylindricalpores
havingacircularcross-section 57
4.6.2 Effectivedensityandbulkmodulusinslits 59
CONTENTS vii
4.6.3 High-andlow-frequencylimitsoftheeffectivedensityand
thebulkmodulusforporesofarbitrarycross-sectionalshape 60
4.7 The Biot model for rigid framed materials 61
4.7.1 SimilaritybetweenG andG 61
c s
4.7.2 Bulk modulusoftheairinslits 62
4.7.3 Effectivedensityandbulkmodulusofairincylindrical
poresofarbitrarycross-sectionalshape 64
4.8 Impedance of a layer with identical pores perpendicular to the surface 65
4.8.1 Normalincidence 65
4.8.2 Obliqueincidence–locallyreactingmaterials 67
4.9 Tortuosity and flow resistivity in a simple anisotropic material 67
4.10 Impedance at normal incidence and sound propagation in oblique
pores 69
4.10.1 Effectivedensity 69
4.10.2 Impedance 71
Appendix 4.A Important expressions 71
Descriptiononthemicroscopicscale 71
Effectivedensityandbulkmodulus 71
References 72
5 Sound propagation in porous materials having a rigid
frame 73
5.1 Introduction 73
5.2 Viscous and thermal dynamic and static permeability 74
5.2.1 Definitions 74
5.2.2 Directmeasurementofthestaticpermeabilities 76
5.3 Classical tortuosity, characteristic dimensions, quasi-static tortuosity 78
5.3.1 Classicaltortuosity 78
5.3.2 Viscouscharacteristiclength 79
5.3.3 Thermalcharacteristiclength 80
5.3.4 Characteristiclengthsforfibrousmaterials 80
5.3.5 Directmeasurementofthehigh-frequencyparameters,
classicaltortuosityandcharacteristiclengths 81
5.3.6 Static tortuosity 82
5.4 Models for the effective density and the bulk modulus of the saturating
fluid 83
5.4.1 Prideetal.modelfortheeffectivedensity 83
5.4.2 SimplifiedLafargemodelforthebulk modulus 83
5.5 Simpler models 84
5.5.1 TheJohnsonetal.model 84
5.5.2 TheChampoux–Allardmodel 84
5.5.3 TheWilsonmodel 85
5.5.4 PredictionoftheeffectivedensitywiththePrideetal.model
andthemodelbyJohnsonetal. 85
5.5.5 PredictionofthebulkmoduluswiththesimplifiedLafarge
modelandtheChampoux-Allardmodel 85
viii CONTENTS
5.5.6 Predictionofthesurfaceimpedance 87
5.6 Prediction of the effective density and the bulk modulus of open cell
foams and fibrous materials with the different models 88
5.6.1 Comparisonoftheperformanceofdifferentmodels 88
5.6.2 Practicalconsiderations 88
5.7 Fluid layer equivalent to a porous layer 89
5.8 Summary of the semi-phenomenological models 90
5.9 Homogenization 91
5.10 Double porosity media 95
5.10.1 Definitions 95
5.10.2 Ordersofmagnitude forrealisticdoubleporositymedia 96
5.10.3 Asymptoticdevelopmentmethodfordoubleporositymedia 97
5.10.4 Lowpermeabilitycontrast 98
5.10.5 Highpermeabilitycontrast 99
5.10.6 Practicalconsiderations 102
Appendix 5.A: Simplified calculation of the tortuosity for a porous material
having pores made up of an alternating sequence of cylinders 103
Appendix 5.B: Calculation of the characteristic length (cid:2)(cid:2) 104
Appendix 5.C: Calculation of the characteristic length (cid:2) for a cylinder
perpendicular to the direction of propagation 106
References 107
6 Biot theory of sound propagation in porous materials having
an elastic frame 111
6.1 Introduction 111
6.2 Stress and strain in porous materials 111
6.2.1 Stress 111
6.2.2 Stress–strainrelationsintheBiottheory:Thepotential
couplingterm 112
6.2.3 Asimpleexample 115
6.2.4 DeterminationofP,QandR 116
6.2.5 Comparisonwithpreviousmodelsofsoundpropagation
inporoussound-absorbingmaterials 117
6.3 Inertial forces in the Biot theory 117
6.4 Wave equations 119
6.5 The two compressional waves and the shear wave 120
6.5.1 Thetwocompressionalwaves 120
6.5.2 Theshearwave 122
6.5.3 ThethreeBiotwavesinordinaryair-saturatedporous
materials 123
6.5.4 Example 123
6.6 Prediction of surface impedance at normal incidence for a layer
of porous material backed by an impervious rigid wall 126
6.6.1 Introduction 126
6.6.2 Predictionofthesurfaceimpedanceatnormalincidence 126
6.6.3 Example:Fibrousmaterial 129
CONTENTS ix
Appendix 6.A: Other representations of the Biot theory 131
References 134
7 Point source above rigid framed porous layers 137
7.1 Introduction 137
7.2 Sommerfeld representation of the monopole field over a plane
reflecting surface 137
7.3 The complex sinθ plane 139
7.4 The method of steepest descent (passage path method) 140
7.5 Poles of the reflection coefficient 145
7.5.1 Definitions 145
7.5.2 Planeswavesassociatedwiththepoles 146
7.5.3 Contributionofapoletothereflectedmonopolepressurefield 150
7.6 The pole subtraction method 151
7.7 Pole localization 153
7.7.1 Localizationfromthe r dependenceofthereflectedfield 153
7.7.2 Localizationfromthe verticaldependenceofthetotalpressure 155
7.8 The modified version of the Chien and Soroka model 156
Appendix 7.A Evaluation of N 160
Appendix 7.B Evaluation of p by the pole subtraction method 161
r
Appendix 7.C From the pole subtraction to the passage path: locally
reacting surface 164
References 165
8 Porous frame excitation by point sources in air and by stress circular
and line sources – modes of air saturated porous frames 167
8.1 Introduction 167
8.2 Prediction of the frame displacement 168
8.2.1 Excitationwithagivenwavenumbercomponentparallel
tothefaces 168
8.2.2 Circularandline sources 172
8.3 Semi-infinite layer – Rayleigh wave 173
8.4 Layer of finite thickness – modified Rayleigh wave 176
8.5 Layer of finite thickness – modes and resonances 177
8.5.1 Modesandresonancesforanelasticsolidlayeranda
poroelasticlayer 177
8.5.2 Excitationoftheresonancesbyapointsourceinair 179
Appendix 8.A Coefficients r and M 182
ij i,j
Appendix 8.B Double Fourier transform and Hankel transform 183
Appendix 8.C Rayleigh pole contribution 185
References 185
9 Porous materials with perforated facings 187
9.1 Introduction 187
9.2 Inertial effect and flow resistance 187
9.2.1 Inertialeffect 187
x CONTENTS
9.2.2 Calculationofthe addedmassandtheaddedlength 188
9.2.3 Flowresistance 191
9.2.4 Apertureshavingasquarecross-section 192
9.3 Impedance at normal incidence of a layered porous material
covered by a perforated facing – Helmoltz resonator 194
9.3.1 Evaluationoftheimpedanceforthe
caseofcircularholes 194
9.3.2 Evaluationatnormalincidenceoftheimpedanceforthecase
ofsquareholes 198
9.3.3 Examples 199
9.3.4 Designofstratifiedporousmaterialscoveredbyperforated
facings 202
9.3.5 Helmholtzresonators 203
9.4 Impedance at oblique incidence of a layered porous material covered
by a facing having circular perforations 205
9.4.1 Evaluationoftheimpedanceinaholeattheboundary
surfacebetweenthefacingandthematerial 205
9.4.2 Evaluationoftheexternaladdedlengthatobliqueincidence 208
9.4.3 Evaluationoftheimpedanceofafacedporouslayerat
oblique incidence 209
9.4.4 Evaluationofthesurfaceimpedanceatobliqueincidence
forthecaseofsquareperforations 210
References 211
10 Transversally isotropic poroelastic media 213
10.1 Introduction 213
10.2 Frame in vacuum 214
10.3 Transversally isotropic poroelastic layer 215
10.3.1 Stress–strainequations 215
10.3.2 Waveequations 216
10.4 Waves with a given slowness component in the symmetry plane 217
10.4.1 Generalequations 217
10.4.2 Wavespolarizedinameridianplane 219
10.4.3 Waveswithpolarizationperpendiculartothemeridianplane 219
10.4.4 Natureofthedifferentwaves 219
10.4.5 Illustration 220
10.5 Sound source in air above a layer of finite thickness 222
10.5.1 Descriptionoftheproblems 222
10.5.2 Planefieldinair 223
10.5.3 Decouplingoftheairwave 226
10.6 Mechanical excitation at the surface of the porous layer 227
10.7 Symmetry axis different from the normal to the surface 228
10.7.1 Predictionoftheslownessvectorcomponentsofthedifferent
waves 228
10.7.2 Slownessvectorswhenthesymmetryaxisisparallelto
thesurface 230
10.7.3 Descriptionofthedifferentwaves 230
CONTENTS xi
10.8 Rayleigh poles and Rayleigh waves 232
10.8.1 Example 234
10.9 Transfer matrix representation of transversally isotropic poroelastic
media 236
Appendix 10.A: Coefficients T in Equation (10.46) 238
i
Appendix 10.B: Coefficients A in Equation (10.97) 239
i
References 240
11 Modelling multilayered systems with porous materials using
the transfer matrix method 243
11.1 Introduction 243
11.2 Transfer matrix method 244
11.2.1 Principleofthemethod 244
11.3 Matrix representation of classical media 244
11.3.1 Fluidlayer 244
11.3.2 Solidlayer 245
11.3.3 Poroelasticlayer 247
11.3.4 Rigidandlimpframelimits 251
11.3.5 Thinelasticplate 254
11.3.6 Imperviousscreens 255
11.3.7 Porousscreensandperforatedplates 256
11.3.8 Othermedia 256
11.4 Coupling transfer matrices 257
11.4.1 Twolayersofthesamenature 257
11.4.2 Interfacebetweenlayersofdifferentnature 258
11.5 Assembling the global transfer matrix 260
11.5.1 Hardwallterminationcondition 261
11.5.2 Semi-infinite fluidterminationcondition 261
11.6 Calculation of the acoustic indicators 263
11.6.1 Surfaceimpedance,reflectionandabsorptioncoefficients 263
11.6.2 Transmissioncoefficientandtransmissionloss 263
11.6.3 Pistonexcitation 265
11.7 Applications 266
11.7.1 Materialswithporousscreens 266
11.7.2 Materialswithimperviousscreens 271
11.7.3 Normalincidencesoundtransmissionthrough
aplate–poroussystem 274
11.7.4 Diffusefieldtransmissionofaplate–foam system 275
Appendix 11.A The elements T of the Transfer Matrix T] 277
ij
References 280
12 Extensions to the transfer matrix method 283
12.1 Introduction 283
12.2 Finite size correction for the transmission problem 283
12.2.1 Transmittedpower 283
12.2.2 Transmissioncoefficient 287
12.3 Finite size correction for the absorption problem 288