Use of specific Green's functions for solving direct problems involving a heterogeneous rigid frame porous medium slab solicited by acoustic waves

Abstract

A domain integral method employing a specific Green's function (i.e., incorporating some features of the global problem of wave propagation in an inhomogeneous medium) is developed for solving direct and inverse scattering problems relative to slab-like macroscopically inhomogeneous porous obstacles. It is shown how to numerically solve such problems, involving both spatially-varying density and compressibility, by means of an iterative scheme initialized with a Born approximation. A numerical solution is obtained for a canonical problem involving a two-layer slab. IntroductionThis work was initially motivated by two problems: i) the design problem connected with the determination of the optimal profile of a continuous and/or discontinuous spatial distribution of the material/geometric properties of porous materials for the absorption of sound [7] and ii) the retrieval of the spatially-varying mechanical and geometrical parameters of bone for the diagnosis of diseases such as osteoporosis [10].Such inverse problems [20] can be decomposed into two sub-problems: i) the determination of the constitutive and conservation relations linking the various spatially-variable mechanical parameters of the porous medium to its response to an acoustic solicitation, and ii) the resolution of the * Correspondence to: J.-P. Groby, Akoestieke en Thermische Fysica, KULeuven, Celestijnenlaan 200D , 3001 Heverlee, Belgium † E-mail: jeanphilippe.groby@fys.kuleuven.be 1 wave equation in an inhomogeneous porous medium (for instance, within the Biot, or rigid frame approximations). Here we focus on the second point.In [17], it is shown that the wave equation describing the propagation in a macroscopicallyinhomogeneous porous medium in the rigid frame approximation can formally take the form of the usual acoustic wave equation in a macroscopically-inhomogeneous fluid (in which the microscopic features of the porous medium are homogenized) with spatial (and frequency) dependent compressibility κ e (x, ω) and density ρ e (x, ω).The present work deals with a method of resolution of direct problems involving acoustic wave propagation in a macroscopically-inhomogeneous fluid medium, whose density and compressibility are both space dependent, this being a prerequisite to the resolution of related inverse problems.This topic is also of great interest in quantum physics (inverse potential scattering [2,37,38,39]), ocean acoustics [9,8,32,11,36,12] (detection of inhomogeneities, sediment exploration, influence of seawater and seafloor composition and heterogenity on the long-range propagation of acoustic waves in the sea, ...), seismology [1,40,42] (determination of the internal structure and composition of the Earth via seismic waves,...), geophysics [42,25,48,44] (characterization of soil, detection of geological features such as hydrocarbon reservoirs, ...), optics and electromagnetism [47,41] (design and characterization of materials having specified response to waves, detection of flaws,...).The wave equation in an inhomogeneous medi...

Propagation of acoustic waves in a one-dimensional macroscopically inhomogeneous poroelastic material

“…10 and solved in the case of rigid frame inhomogeneous porous materials via the Wave Splitting method and "transmission" Green's functions approach or via an iterative Born approximation procedure based on the specific Green's function of the configuration. 11 The recovery of several profiles of spatially varying material parameters by means of an optimization approach, was then achieved in Ref. 2 still in the rigid frame approximation.…”

Surface acoustic wave depth profiling of a functionally graded material

“…[8][9][10] These methods are usually simpler to implement than methods based on a Green's function approach. [11][12][13][14] The accuracy can be set to any desired level, by choosing a sufficiently large number of layers.…”

Reconstruction of material properties profiles in one-dimensional macroscopically inhomogeneous rigid frame porous media in the frequency domain

“…7 The model of rigid frame has been recently extended to macroscopically inhomogeneous porous media. 9,10,12 In the frequency domain, the Helmholtz equation in terms of the fluid pressure p inside the equivalent inhomogeneous fluid is 9,10…”

“…9,10,12 The theory of wave propagation in homogeneous porous materials was initially given by Biot. [1][2][3][4] In most of the plastic foams saturated by a light fluid like air, the rigid frame assumption is valid so that an acoustic excitation impinging on a porous sample induces wave propagation only in the fluid phase.…”

“…3,4,9,10 In particular, when the assumption of rigid frame is valid, [5][6][7]11 i.e., when the saturating fluid is light, such as air, and the frame is not moving, the equations of motion reduce to those of an equivalent inhomogeneous fluid with spatial and frequency dependent effective density and bulk modulus. 9,10,12 The frequency band suitable to this approximation is bounded at high frequency by the diffusion limit ͑when the wavelength is of the order of, or smaller than the pore size͒, and at low frequency by the Biot characteristic frequency, below the which the skeleton may vibrate. Under these conditions, the reflected and transmitted pressure fields as well as the internal pressure field can be numerically determined 9,10 by means of the wave splitting and transmission Green's functions method.…”