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 Travelling waves ................................ 15
2.2.2 Example ......................................... 16
2.2.3 Attenuation ..................................... 16
2.2.4 Superposition of two waves propagating in
opposite directions ............................. 17
2.3 Main properties of impedance at normal incidence ....... 17
2.3.1 Impedance variation along a direction of
propagation ..................................... 17
2.3.2 Impedance at normal incidence of a layer of
fluid backed by an impervious rigid wall ........ 18
2.3.3 Impedance at normal incidence of a
multilayered fluid .............................. 19
2.4 Reflection coefficient and absorption coefficient at
normal incidence ....................................... 19
2.4.1 Reflection coefficient .......................... 19
2.4.2 Absorption coefficient .......................... 20
2.5 Fluids equivalent to porous materials: the laws of
Delany and Bazley ...................................... 20
2.5.1 Porosity and flow resistivity in porous
materials ....................................... 20
2.5.2 Microscopic and macroscopic description of
sound propagation in porous media ............... 22
2.5.3 The Laws of Delany and Bazley and flow
resistivity ..................................... 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 Impedance variation along a direction
perpendicular to an impedance plane ............. 33
3.4.2 Impedance at oblique incidence for a layer of
finite thickness backed by an impervious rigid
wall ............................................ 34
3.4.3 Impedance at oblique incidence in a
multilayered fluid .............................. 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 Effective density and bulk modulus in
cylindrical pores having a circular cross-
section ......................................... 57
4.6.2 Effective density and bulk modulus in slits ..... 59
4.6.3 High-and low-frequency limits of the effective
density and the bulk modulus for pores of
arbitrary cross-sectional shape ................. 60
4.7 The Biot model for rigid framed materials .............. 61
4.7.1 Similarity between Gс and Gs .................... 61
4.7.2 Bulk modulus of the air in slits ................ 62
4.7.3 Effective density and bulk modulus of air in
cylindrical pores of arbitrary cross-sectional
shape ........................................... 64
4.8 Impedance of a layer with identical pores
perpendicular to the surface ........................... 65
4.8.1 Normal incidence ................................ 65
4.8.2 Oblique incidence - locally reacting
materials ....................................... 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 Effective density ............................... 69
4.10.2 Impedance ....................................... 71
Appendix 4.A Important expressions .......................... 71
Description on the microscopic scale ........... 71
Effective density and bulk modulus ............. 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 Direct measurement of the static
permeabilities .................................. 76
5.3 Classical tortuosity, characteristic dimensions,
quasi-static tortuosity ................................ 78
5.5.7 Classical tortuosity ............................ 78
5.3.2 Viscous characteristic length ................... 79
5.3.3 Thermal characteristic length ................... 80
5.3.4 Characteristic lengths for fibrous materials .... 80
5.3.5 Direct measurement of the high-frequency
parameters, classical tortuosity and
characteristic lengths .......................... 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 Pride et al. model for the effective density .... 83
5.4.2 Simplified Lafarge model for the bulk modulus ... 83
5.5 Simpler models ......................................... 84
5.5.1 The Johnson et al. model ........................ 84
5.5.2 The Champoux-Allard model ....................... 84
5.5.3 The Wilson model ................................ 85
5.5.4 Prediction of the effective density with the
Pride et al. model and the model by Johnson
et al. .......................................... 85
5.5.5 Prediction of the bulk modulus with the
simplified Lafarge model and the Champoux-
Allard model .................................... 85
5.5.6 Prediction of the surface impedance ............. 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 Comparison of the performance of different
models .......................................... 88
5.6.2 Practical considerations ........................ 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 Orders of magnitude for realistic double
porosity media .................................. 96
5.10.3 Asymptotic development method for double
porosity media .................................. 97
5.10.4 Low permeability contrast ....................... 98
5.10.5 High permeability contrast ...................... 99
5.10.6 Practical considerations ....................... 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 Λ ... 104
Appendix 5.C: Calculation of the characteristic length Λ
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-strain relations in the Biot theory:
The potential coupling term .................... 112
6.2.3 A simple example ............................... 115
6.2.4 Determination of P, Q and R .................... 116
6.2.5 Comparison with previous models of sound
propagation in porous sound-absorbing
materials ...................................... 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 The two compressional waves .................... 120
6.5.2 The shear wave ................................. 122
6.5.3 The three Biot waves in ordinary air-
saturated porous 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 Prediction of the surface impedance at normal
incidence ...................................... 126
6.6.3 Example: Fibrous material ...................... 129
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 Planes waves associated with the poles ......... 146
7.5.3 Contribution of a pole to the reflected
monopole pressure field ........................ 150
7.6 The pole subtraction method ........................... 151
7.7 Pole localization ..................................... 153
7.7.1 Localization from the r dependence of the
reflected field ................................ 153
7.7.2 Localization from the vertical dependence of
the total pressure ............................. 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 pr by the pole subtraction
method ................................................ 161
Appendix 7.С 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 Excitation with a given wave number component
parallel to the faces .......................... 168
8.2.2 Circular and line 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 Modes and resonances for an elastic solid
layer and a poroelastic layer .................. 111
8.5.2 Excitation of the resonances by a point
source in air .................................. 179
Appendix 8.A Coefficients rij and Mi,j ..................... 182
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 Inertial effect ................................ 187
9.1 Calculation of the added mass and the added length .... 188
9.2.3 Flow resistance ................................ 191
9.2.4 Apertures having a square cross-section ........ 192
9.3 Impedance at normal incidence of a layered porous
material covered by a perforated facing - Helmoltz
resonator ............................................. 194
9.3.1 Evaluation of the impedance for the case of
circular holes ................................. 194
9.3.2 Evaluation at normal incidence of the
impedance for the case of square holes ......... 198
9.3.3 Examples ....................................... 199
9.3.4 Design of stratified porous materials covered
by perforated facings .......................... 202
9.3.5 Helmholtz resonators ........................... 203
9.4 Impedance at oblique incidence of a layered porous
material covered by a facing having circular
perforations .......................................... 205
9.4.1 Evaluation of the impedance in a hole at the
boundary surface between the facing and the
material ....................................... 205
9.4.2 Evaluation of the external added length at
oblique incidence .............................. 208
9.4.3 Evaluation of the impedance of a faced porous
layer at oblique incidence ..................... 209
9.4.4 Evaluation of the surface impedance at
oblique incidence for the case of square
perforations ................................... 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-strain equations ........................ 215
10.3.2 Wave equations ................................. 216
10.4 Waves with a given slowness component in the
symmetry plane ........................................ 217
10.4.1 General equations .............................. 217
10.4.2 Waves polarized in a meridian plane ............ 219
10.4.3 Waves with polarization perpendicular to the
meridian plane ................................. 219
10.4.4 Nature of the different waves .................. 219
10.4.1 Illustration ................................... 220
10.5 Sound source in air above a layer of finite
thickness ............................................. 222
10.5.1 Description of the problems .................... 222
10.5.2 Plane field in air ............................. 223
10.5.3 Decoupling of the air wave ..................... 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 Prediction of the slowness vector components
of the different waves ......................... 228
10.7.2 Slowness vectors when the symmetry axis is
parallel to the surface ........................ 230
10.7.3 Description of the different waves ............. 230
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 Ti in Equation (10.46) ......... 238
Appendix 10.B: Coefficients Ai in Equation (10.97) ......... 239
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 Principle of the method ....................... 244
11.3 Matrix representation of classical media .............. 244
11.3.1 Fluid layer .................................... 244
11.3.2 Solid layer .................................... 245
11.3.3 Poroelastic layer .............................. 247
11.3.4 Rigid and limp frame limits .................... 251
11.3.5 Thin elastic plate ............................. 254
11.3.6 Impervious screens ............................. 255
11.3.7 Porous screens and perforated plates ........... 256
11.3.8 Other media .................................... 256
11.4 Coupling transfer matrices ............................ 257
11.4.1 Two layers of the same nature .................. 257
11.4.2 Interface between layers of different nature ... 258
11.5 Assembling the global transfer matrix ................. 260
11.5.1 Hard wall termination condition ................ 261
11.5.2 Semi-infinite fluid termination condition ...... 261
11.6 Calculation of the acoustic indicators ................ 263
11.6.1 Surface impedance, reflection and absorption
coefficients ................................... 263
11.6.2 Transmission coefficient and transmission
loss ........................................... 263
11.6.3 Piston excitation .............................. 265
11.7 Applications .......................................... 266
11.7.1 Materials with porous screens .................. 266
11.7.2 Materials with impervious screens .............. 271
11.7.3 Normal incidence sound transmission through
a plate-porous system .......................... 274
11.7.4 Diffuse field transmission of a plate-foam
system ......................................... 275
Appendix 11.A The elements 7y of the Transfer Matrix T] .... 277
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 Transmitted power .............................. 283
12.2.2 Transmission coefficient ....................... 287
12.3 Finite size correction for the absorption problem ..... 288
12.3.1 Surface pressure ............................... 288
12.3.2 Absorption coefficient ......................... 289
12.3.3 Examples ....................................... 291
12.4 Point load excitation ................................. 295
12.4.1 Formulation .................................... 295
12.4.2 The TMM, SEA and modal methods ................. 297
12.4.3 Examples ....................................... 298
12.5 Point source excitation ............................... 303
12.6 Other applications .................................... 304
Appendix 12.A: An algorithm to evaluate the geometrical
radiation impedance ................................... 305
References ................................................. 306
13 Finite element modelling of poroelastic materials .......... 309
13.1 Introduction .......................................... 309
13.2 Displacement based formulations ....................... 310
13.3 The mixed displacement-pressure formulation ........... 311
13.4 Coupling conditions ................................... 313
13.4.1 Poroelastic-elastic coupling condition ......... 313
13.4.2 Poroelastic—acoustic coupling condition ........ 314
13.4.3 Poroelastic-poroelastic coupling condition ..... 315
13.4.4 Poroelastic—impervious screen coupling
condition ...................................... 315
13.4.5 Case of an imposed pressure field .............. 316
13.4.6 Case of an imposed displacement field .......... 317
13.4.7 Coupling with a semi-infinite waveguide ........ 317
13.5 Other formulations in terms of mixed variables ........ 320
13.6 Numerical implementation .............................. 320
13.7 Dissipated power within a porous medium ............... 323
13.8 Radiation conditions .................................. 324
13.9 Examples .............................................. 327
13.9.1 Normal incidence absorption and transmission
loss of a foam: finite size effects ............ 327
13.9.2 Radiation effects of a plate-foam system ....... 329
13.9.3 Damping effects of a plate-foam system ......... 331
13.9.4 Diffuse transmission loss of a plate-foam
system ......................................... 333
13.9.5 Application to the modelling of double
porosity materials ............................. 335
13.9.6 Modelling of smart foams ....................... 339
13.9.7 An industrial application ...................... 343
References ................................................. 347
Index ......................................................... 351
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