propagation of sound in porous media modelling sound
TRANSCRIPT
PROPAGATION OF SOUND IN POROUS
MEDIA Modelling Sound Absorbing
Materials
J. F. Allard
Laboratoire d'Acoustique de Г Universite du Maine,
Le Mans, France
ELSEVIER APPLIED SCIENCE LONDON and NEW YORK
Contents
Preface v
Chapter 1 Plane Waves in Isotropie 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 propagation 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 exponential 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 . . . 3 1 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 perpendicular 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 al. . 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
5.8.2 5.8.3
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 К . . . 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 Other materials . Fibrous materials .
5.9 Simple models 5.9.1 The model of Attenborough 5.9.2 The model of Allard et al. .
5.10 Surface impedance . . . . Appendix 5.A Kinetic energy and tortuosity
Simplified calculation of the tortuosity for a porous material having pores made up of an alternating sequence of cylinders Flow in a slit Calculation of the characteristic dimension Л' 115 Calculation of the characteristic dimension A for a cylinder perpendicular to the direction of propagation . . . 1 1 5
Appendix 5.В
Appendix 5.С Appendix 5.D
Appendix 5.E
93 96 96 105 105 106 107 111
113 114
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 Surface 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 [Г] 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 Ttj of the transfer matrix
[T] 177 Appendix 7.В Transmission through a plate-porous-
material-plate structure . . . . 181
xii Contents
Chapter 8 Acoustic Field Created by Monopole and Dipole Sources in the Presence of Porous Layers
8.1 Introduction 186 8.2 Sommerfeld decomposition for monopole sources and
AT-waves 187 8.3 Acoustic field created by a point source in air above a
layer of rigid framed porous material 190 8.4 Axisymmetrical field in a porous layer having an elastic
frame 194 8.4.1 Velocities and stresses 194 8.4.2 The matrix representation 197
8.5 Acoustic field created by monopole and dipole sources above stratified porous media 201
Chapter 9 Methods of Measuring the Acoustic Impedance 9.1 The Kundt tube 207
9.1.1 Introduction 207 9.1.2 Sound propagation in the Kundt tube . . . 207 9.1.3 Measurement of reflection coefficient and im
pedance with the Kundt tube 210 9.1.4 The TMCT method 210 9.1.5 Limitations 211
9.2 Free-field measurement in a plane acoustic field . . 212 9.3 Limitations of the free-field method—The use of a point
source close to the material 213 9.4 The use of two-dimensional spatial transforms . . 215
Chapter 10 Porous Materials with Perforated Facings 10.1 Introduction 220 10.2 Inertial effect and flow resistance 220
10.2.1 Inertial effect 220 10.2.2 Calculation of the added mass and the added
length 221 10.2.3 Flow resistance 225 10.2.4 Apertures having a square cross-section . . 226
10.3 Impedance at normal incidence of a layered porous material covered by a perforated facing—Helmholtz Resonator 228 10.3.1 Evaluation of the impedance for the case of
circular holes 228
Contents X I I I
10.3.2 Evaluation at normal incidence of the impedance for the case of square holes 233
10.3.3 Examples 234 10.3.4 Design of stratified porous materials covered by
perforated facings 238 10.3.5 Helmholtz resonators 240
10.4 Impedance at oblique incidence of a layered porous material covered by a facing having circular perforations 244 10.4.1 Evaluation of the impedance in a hole at the
boundary surface between the facing and the material 244
10.4.2 Evaluation of the external added length at oblique incidence 249
10.4.3 Evaluation of the impedance of a faced porous layer at oblique incidence 251
10.4.4 Evaluation of the surface impedance at oblique incidence for the case of square perforations. 252
Chapter 11 Impedance and Admittance Matrices, Sound Transmission Through Layered Porous Media and Reciprocity
11.1 Introduction 254 11.2 Simple layers 257 11.3 Transmission through layered materials with the same
fluid in contact with the front and the rear face. . 261 11.3.1 Layered elastic solid 261 11.3.2 Layered elastic solid in contact with an inviscid
fluid 262 11.3.3 Several porous layers in contact . . . . 264 11.3.4 Porous layered medium in air 266 11.3.5 Porous material bonded on to an elastic layer in
an inviscid fluid 268 11.3.6 Porous material bonded onto a plate . . 274
Index 279