Abstract
Wave propagation in macroscopically inhomogeneous porous materials has received much attention in recent years. The wave equation, derived from the alternative formulation of Biot's theory of 1962, was reduced and solved recently in the case of rigid frame inhomogeneous porous materials. This paper focuses on the solution of the full wave equation in which the acoustic and the elastic properties of the poroelastic material vary in one-dimension. The reflection coefficient of a one-dimensional macroscopically inhomogeneous porous material on a rigid backing is obtained numerically using the state vector (or the so-called Stroh) formalism and Peano series. This coefficient can then be used to straightforwardly calculate the scattered field. To validate the method of resolution, results obtained by the present method are compared to those calculated by the classical transfer matrix method at both normal and oblique incidence and to experimental measurements at normal incidence for a known two-layers porous material, considered as a single inhomogeneous layer. Finally, discussion about the absorption coefficient for various inhomogeneity profiles gives further perspectives.
Highlights
The study of wave propagation in macroscopically inhomogeneous porous media was initially motivated by (1) the design of sound absorbing porous materials with optimal material and geometrical property profiles1 and (2) the retrieval of the spatially varying material parameters of industrial foams.2 These, and other inverse problems, are of great importance in connection with the characterization of the mechanical properties of naturally occurring macroscopically inhomogeneous porous materials, such as bones
The literature on inhomogeneous media is extensive in several fields of physics, from optics and electromagnetism,3,4 to acoustics,5,6 and geophysics7 and granular media
Similar methods are used in electromagnetism to model the propagation of electromagnetic waves in anisotropic or gyrotropic stratified materials.17,18. It consists in the rewriting of the constitutive linear stressstrain relations and the momentum conservation law in terms of the state vector, whose components can be chosen arbitrarily as long as they are continuous along the inhomogeneity of the material
Summary
The study of wave propagation in macroscopically inhomogeneous porous media was initially motivated by (1) the design of sound absorbing porous materials with optimal material and geometrical property profiles and (2) the retrieval of the spatially varying material parameters of industrial foams. These, and other inverse problems, are of great importance in connection with the characterization of the mechanical properties of naturally occurring macroscopically inhomogeneous porous materials, such as bones. It consists in the rewriting of the constitutive linear stressstrain relations and the momentum conservation law in terms of the state vector, whose components can be chosen arbitrarily as long as they are continuous along the inhomogeneity of the material. This leads after spatial Fourier transform to a first-order ordinary differential system of equations whose unknown is the state vector. Applications in material design for engineering applications are treated, by comparing the absorption coefficient of a macroscopically inhomogeneous porous plate with various inhomogeneity profiles that are either continuous or discontinuous
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