Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials 2e

1 Plane waves in isotropic fluids and solids.

1.1 Introduction.

1.2 Notation – vector operators.

1.3 Strain in a deformable medium.

1.4 Stress in a deformable medium.

1.5 Stress–strain relations for an isotropic elastic medium.

1.6 Equations of motion.

1.7 Wave equation in a fluid.

1.8 Wave equations in an elastic solid.

References.

2 Acoustic impedance at normal incidence of fluids. Substitution of a fluid layer for a porous layer.

2.1 Introduction.

2.2 Plane waves in unbounded fluids.

2.3 Main properties of impedance at normal incidence.

2.4 Reflection coefficient and absorption coefficient at normal incidence.

2.5 Fluids equivalent to porous materials: the laws of Delany and Bazley.

2.6 Examples.

2.7 The complex exponential representation.

References.

3 Acoustic impedance at oblique incidence in fluids. Substitution of a fluid layer for a porous layer.

3.1 Introduction.

3.2 Inhomogeneous plane waves in isotropic fluids.

3.3 Reflection and refraction at oblique incidence.

3.4 Impedance at oblique incidence in isotropic fluids.

3.5 Reflection coefficient and absorption coefficient at oblique incidence.

3.6 Examples.

3.7 Plane waves in fluids equivalent to transversely isotropic porous media.

3.8 Impedance at oblique incidence at the surface of a fluid equivalent to an anisotropic porous material.

3.9 Example.

References.

4 Sound propagation in cylindrical tubes and porous materials having cylindrical pores.

4.1 Introduction.

4.2 Viscosity effects.

4.3 Thermal effects.

4.4 Effective density and bulk modulus for cylindrical tubes having triangular, rectangular and hexagonal cross-sections.

4.5 High- and low-frequency approximation.

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.

4.7 The biot model for rigid framed materials.

4.8 Impedance of a layer with identical pores perpendicular to the surface.

4.9 Tortuosity and flow resistivity in a simple anisotropic material.

4.10 Impedance at normal incidence and sound propagation in oblique pores.

Appendix 4.A Important expressions.

Description on the microscopic scale.

Effective density and bulk modulus.

References.

5 Sound propagation in porous materials having a rigid frame.

5.1 Introduction.

5.2 Viscous and thermal dynamic and static permeability.

5.3 Classical tortuosity, characteristic dimensions, quasi-static tortuosity.

5.4 Models for the effective density and the bulk modulus of the saturating fluid.

5.5 Simpler models.

5.6 Prediction of the effective density and the bulk modulus of open cell foams and fibrous materials with the different models.

5.7 Fluid layer equivalent to a porous layer.

5.8 Summary of the semi-phenomenological models.

5.9 Homogenization.

5.10 Double porosity media.

Appendix 5.A: Simplified calculation of the tortuosity for a porous material having pores made up of an alternating sequence of cylinders.

Appendix 5.B: Calculation of the characteristic length Λ'.

 Appendix 5.C: Calculation of the characteristic length Λ  for a cylinder perpendicular to the direction of propagation.

References.

6 Biot theory of sound propagation in porous materials having an elastic frame.

6.1 Introduction.

6.2 Stress and strain in porous materials.

6.3 Inertial forces in the biot theory.

6.4 Wave equations.

6.5 The two compressional waves and the shear wave.

6.6 Prediction of surface impedance at normal incidence for a layer of porous material backed by an impervious rigid wall.

Appendix 6.A: Other representations of the Biot theory.

References.

7 Point source above rigid framed porous layers.

7.1 Introduction.

7.2 Sommerfeld representation of the monopole field over a plane reflecting surface.

7.3 The complex sinθ plane.

7.4 The method of steepest descent (passage path method).

7.5 Poles of the reflection coefficient.

7.6 The pole subtraction method.

7.7 Pole localization.

7.8 The modified version of the Chien and Soroka model.

Appendix 7.A Evaluation of N.

Appendix 7.B Evaluation of pr by the pole subtraction method.

Appendix 7.C From the pole subtraction to the passage path: Locally reacting surface.

References.

8 Porous frame excitation by point sources in air and by stress circular and line sources – modes of air saturated porous frames.

8.1 Introduction.

8.2 Prediction of the frame displacement.

8.3 Semi-infinite layer – Rayleigh wave.

8.4 Layer of finite thickness – modified Rayleigh wave.

8.5 Layer of finite thickness – modes and resonances.

Appendix 8.A Coefficients rij and Mi,j.

Appendix 8.B Double Fourier transform and Hankel transform.

Appendix 8.B Appendix .C Rayleigh pole contribution.

References.

9 Porous materials with perforated facings.

9.1 Introduction.

9.2 Inertial effect and flow resistance.

9.3 Impedance at normal incidence of a layered porous material covered by a perforated facing – Helmoltz resonator.

9.4 Impedance at oblique incidence of a layered porous material covered by a facing having cirular perforations.

References.

10 Transversally isotropic poroelastic media.

10.1 Introduction.

10.2 Frame in vacuum.

10.3 Transversally isotropic poroelastic layer.

10.4 Waves with a given slowness component in the symmetry plane.

10.5 Sound source in air above a layer of finite thickness.

10.6 Mechanical excitation at the surface of the porous layer.

10.7 Symmetry axis different from the normal to the surface.

10.8 Rayleigh poles and Rayleigh waves.

10.9 Transfer matrix representation of transversally isotropic poroelastic media.

Appendix 10.A: Coefficients Ti in Equation (10.46).

Appendix 10.B: Coefficients Ai in Equation (10.97).

References.

11 Modelling multilayered systems with porous materials using the transfer matrix method.

11.1 Introduction.

11.2 Transfer matrix method.

11.3 Matrix representation of classical media.

11.4 Coupling transfer matrices.

11.5 Assembling the global transfer matrix.

11.6 Calculation of the acoustic indicators.

11.7 Applications.

Appendix 11.A The elements Tij of the Transfer Matrix T ].

References.

12 Extensions to the transfer matrix method.

12.1 Introduction.

12.2 Finite size correction for the transmission problem.

12.3 Finite size correction for the absorption problem.

12.4 Point load excitation.

12.5 Point source excitation.

12.6 Other applications.

Appendix 12.A: An algorithm to evaluate the geometrical radiation impedance.

References.

13 Finite element modelling of poroelastic materials.

13.1 Introduction.

13.2 Displacement based formulations.

13.3 The mixed displacement–pressure formulation.

13.4 Coupling conditions.

13.5 Other formulations in terms of mixed variables.

13.6 Numerical implementation.

13.7 Dissipated power within a porous medium.

13.8 Radiation conditions.

13.9 Examples.

References.

Index.