Effective wavenumbers and reflection coefficients for an elastic medium containing random configurations of cylindrical scatterers

Abstract

Propagation of P and SV waves in an elastic solid containing randomly distributed inclusions in a half-space is investigated. The approach is based on a multiple scattering analysis similar to the one proposed by Fikioris and Waterman for scalar waves. The characteristic equation, the solution of which yields the effective wave numbers of coherent elastic waves, is obtained in an explicit form without the use of any renormalisation methods. Two approximations are considered. First, formulae are derived for the effective wave numbers in a dilute random distribution of identical scatterers. These equations generalize the formula obtained by Linton and Martin for scalar coherent waves. Second, the high frequency approximation is compared with the Waterman and Truell approach derived here for elastic waves. The Fikioris and Waterman approach, in contrast with Waterman and Truell's method, shows that P and SV waves are coupled even at relatively low concentration of scatterers. Simple expressions for the reflected coefficients of P and SV waves incident on the interface of the half space containing randomly distributed inclusions are also derived. These expressions depend on frequency, concentration of scatterers, and the two effective wave numbers of the coherent waves propagating in the elastic multiple scattering medium.

A comparison of stochastic and effective medium approaches to the backscattered signal from a porous layer in a solid matrix

“…While the effective wavenumber due to scattering theory is well-established for dilute systems, its application to more concentrated systems is still the subject of development. However, there has been a recent emergence of interest in the effective properties of inhomogeneous materials derived from scattering theory; in particular a number of workers [27][28][29][30][31][32][33] have attempted to obtain properties other than the elastic moduli, such as effective density, effective viscosity and the effective reflection and transmission coefficients of both a semiinfinite half-space, and a slab.…”

“…They follow either the Waterman and Truell 24,27 or Fikioris and Waterman 25,28,29 formulations, expressing the scattered field either as a multipole expansion, based on the Rayleigh partial-wave method, 27,28 or through the transition-matrix description of Varadan. 26,29 While two of the studies apply to nonviscous fluids, 27,29 the other is applicable to cylinders in solids, 28 considering both longitudinal and shear wave propagation. Maurel 30 took a slightly different approach, applying the Born approximation to terminate the scattering field equations, rather than applying a closure assumption, and using a Green's function description rather than the multipole expansion for the scattered fields.…”

“…[31][32][33] Of greater interest in the present work, is the determination of the reflection characteristics of a layer, or slab, of material containing scatterers, which has been derived by a number of the same workers, based on the scattering theory formulations described above. [27][28][29][30]36 In summary, what has been established is as follows: (a) The effective material properties of a slab or layer are the same as the effective material properties of a half-space. (b) The effective (ensembleaveraged) reflected and transmitted fields from the slab have the same form as the summed multiple reflections from a homogeneous layer.…”

Numerical Simulation of Wave Propagation in 3D Elastic Composites with Rigid Disk-Shaped Inclusions of Variable Mass

“…The approximation for the complex wave number (24) can be considered as a special case of the multiple wave scattering models of higher orders [4,5], and it involves only the first order in the inclusion density and is thus only valid for a dilute or small inclusion density. In the case of a large density or high concentration of inclusions, more sophisticated models such as the self-consistent approach or the multiple scattering models should be applied, to take the mutual dynamic interactions between individual inclusions into account.…”

“…For small inclusion concentration or dilute inclusion distribution, their mutual interactions and the multiple wave scattering effects can be neglected approximately. In this case, the theory of Foldy [1], the quasi-crystalline approximation of Lax [2] and their generalizations to the elastic wave propagation [3][4][5] can be applied to determine the effective wave (phase) velocities and the attenuation coefficients in the composite materials with randomly distributed inclusions. In these models, wave scattering by a single inclusion has to be considered in the first step.…”