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Propagation of sound in porous media : modelling sound by Jean Allard, Noureddine Atalla

By Jean Allard, Noureddine Atalla

Preface to the second one variation. 1 airplane waves in isotropic fluids and solids. 1.1 advent. 1.2 Notation - vector operators. 1.3 pressure in a deformable medium. 1.4 tension in a deformable medium. 1.5 Stress-strain family members for an isotropic elastic medium. 1.6 Equations of movement. 1.7 Wave equation in a fluid. 1.8 Wave equations in an elastic good. References. 2 Acoustic impedance at common occurrence of fluids. Substitution of a fluid layer for a porous layer. 2.1 creation. 2.2 aircraft waves in unbounded fluids. 2.3 major houses of impedance at common prevalence. 2.4 mirrored image coefficient and absorption coefficient at common prevalence. 2.5 Fluids resembling porous fabrics: the legislation of Delany and Bazley. 2.6 Examples. 2.7 The advanced exponential illustration. References. three Acoustic impedance at indirect occurrence in fluids. Substitution of a fluid layer for a porous layer. 3.1 advent. 3.2 Inhomogeneous aircraft waves in isotropic fluids. 3.3 mirrored image and refraction at indirect occurrence. 3.4 Impedance at indirect prevalence in isotropic fluids. 3.5 mirrored image coefficient and absorption coefficient at indirect occurrence. 3.6 Examples. 3.7 airplane waves in fluids akin to transversely isotropic porous media. 3.8 Impedance at indirect prevalence on the floor of a fluid resembling an anisotropic porous fabric. 3.9 instance. References. four Sound propagation in cylindrical tubes and porous fabrics having cylindrical pores. 4.1 creation. 4.2 Viscosity results. 4.3 Thermal results. 4.4 powerful density and bulk modulus for cylindrical tubes having triangular, oblong and hexagonal cross-sections. 4.5 excessive- and low-frequency approximation. 4.6 review of the powerful density and the majority modulus of the air in layers of porous fabrics with exact pores perpendicular to the outside. 4.7 The biot version for inflexible framed fabrics. 4.8 Impedance of a layer with exact pores perpendicular to the skin. 4.9 Tortuosity and circulation resistivity in an easy anisotropic fabric. 4.10 Impedance at general occurrence and sound propagation in indirect pores. Appendix 4.A very important expressions. Description at the microscopic scale. powerful density and bulk modulus. References. five Sound propagation in porous fabrics having a inflexible body. 5.1 advent. 5.2 Viscous and thermal dynamic and static permeability. 5.3 Classical tortuosity, attribute dimensions, quasi-static tortuosity. 5.4 types for the powerful density and the majority modulus of the saturating fluid. 5.5 less complicated versions. 5.6 Prediction of the potent density and the majority modulus of open cellphone foams and fibrous fabrics with the various types. 5.7 Fluid layer such as a porous layer. 5.8 precis of the semi-phenomenological types. 5.9 Homogenization. 5.10 Double porosity media. Appendix 5.A: Simplified calculation of the tortuosity for a porous fabric having pores made from an alternating series of cylinders. Appendix 5.B: Calculation of the attribute size LAMBDA'. Appendix 5.C: Calculation of the attribute size LAMBDA for a cylinder perpendicular to the path of propagation. References. 6 Biot thought of sound propagation in porous fabrics having an elastic body. 6.1 creation. 6.2 rigidity and pressure in porous fabrics. 6.3 Inertial forces within the biot concept. 6.4 Wave equations. 6.5 the 2 compressional waves and the shear wave. 6.6 Prediction of floor impedance at basic occurrence for a layer of porous fabric sponsored via an impervious inflexible wall. Appendix 6.A: different representations of the Biot thought. References. 7 aspect resource above inflexible framed porous layers. 7.1 creation. 7.2 Sommerfeld illustration of the monopole box over a airplane reflecting floor. 7.3 The advanced sin theta aircraft. 7.4 the tactic of steepest descent (passage course method). 7.5 Poles of the mirrored image coefficient. 7.6 The pole subtraction approach. 7.7 Pole localization. 7.8 The changed model of the Chien and Soroka version. Appendix 7.A overview of N. Appendix 7.B review of p r by means of the pole subtraction approach. Appendix 7.C From the pole subtraction to the passage course: in the community reacting floor. References. eight Porous body excitation by means of aspect assets in air and via pressure round and line assets - modes of air saturated porous frames. 8.1 advent. 8.2 Prediction of the body displacement. 8.3 Semi-infinite layer - Rayleigh wave. 8.4 Layer of finite thickness - converted Rayleigh wave. 8.5 Layer of finite thickness - modes and resonances. Appendix 8.A Coefficients r ij and M i,j. Appendix 8.B Double Fourier rework and Hankel rework. Appendix 8.B Appendix .C Rayleigh pole contribution. References. nine Porous fabrics with perforated facings. 9.1 creation. 9.2 Inertial impact and movement resistance. 9.3 Impedance at basic prevalence of a layered porous fabric lined by way of a perforated dealing with - Helmoltz resonator. 9.4 Impedance at indirect occurrence of a layered porous fabric lined via a dealing with having cirular perforations. References. 10 Transversally isotropic poroelastic media. 10.1 advent. 10.2 body in vacuum. 10.3 Transversally isotropic poroelastic layer. 10.4 Waves with a given slowness part within the symmetry airplane. 10.5 Sound resource in air above a layer of finite thickness. 10.6 Mechanical excitation on the floor of the porous layer. 10.7 Symmetry axis diverse from the conventional to the outside. 10.8 Rayleigh poles and Rayleigh waves. 10.9 move matrix illustration of transversally isotropic poroelastic media. Appendix 10.A: Coefficients T i in Equation (10.46). Appendix 10.B: Coefficients A i in Equation (10.97). References. eleven Modelling multilayered structures with porous fabrics utilizing the move matrix approach. 11.1 creation. 11.2 move matrix approach. 11.3 Matrix illustration of classical media. 11.4 Coupling move matrices. 11.5 Assembling the worldwide move matrix. 11.6 Calculation of the acoustic symptoms. 11.7 purposes. Appendix 11.A the weather T ij of the move Matrix T ]. References. 12 Extensions to the move matrix approach. 12.1 advent. 12.2 Finite dimension correction for the transmission challenge. 12.3 Finite measurement correction for the absorption challenge. 12.4 aspect load excitation. 12.5 element resource excitation. 12.6 different purposes. Appendix 12.A: An set of rules to guage the geometrical radiation impedance. References. thirteen Finite point modelling of poroelastic fabrics. 13.1 creation. 13.2 Displacement established formulations. 13.3 The combined displacement-pressure formula. 13.4 Coupling stipulations. 13.5 different formulations when it comes to combined variables. 13.6 Numerical implementation. 13.7 Dissipated strength inside a porous medium. 13.8 Radiation stipulations. 13.9 Examples. References. Index

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Similar arguments about the reconstruction of a real signal can be used to demonstrate that the bulk moduli K in both representations are related by complex conjugation; the same is true for the density ρ. References Allard, J. , Bourdier, R. and L’Esp´erance, A. (1987) Anisotropy effect in glass wool on normal impedance at oblique incidence. J. , 114, 233–8. REFERENCES 27 Attenborough, K. (1971) The prediction of oblique-incidence behaviour of fibrous absorbents. J. , 14, 183–91. Bies, D. A. H.

29). 7. 8) has been calculated. 3. 16) can be used, Z(M1 ) being the impedance of the air gap. 17). 29) for Zc and k for the fibrous material have been used. 6 A layer of porous material fixed on a rigid impervious wall. 7 The impedance Z at normal incidence of a layer of fibrous material of thickness d = 10 cm, of normal flow resistivity σ = 10 000 N m−4 s, calculated according to the laws of Delany and Bazley. 8 A layer of fibrous material with an air gap between the material and the rigid wall. 10.

The air gap increases significantly the absorption at low frequencies. This is explained by the fact that sound absorption is mainly due to the viscous dissipation, related to the velocity of air in the porous medium. When the material is bonded onto a hard wall, the particle velocity at the wall is zero, and thus the absorption deteriorates rapidly at low frequencies. When backed by an air gap, the particle velocity at the rear face of the material oscillates and reaches a maximum at the quarter-wavelength of the lowest frequency of interest, thus increasing the absorption.

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