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

TL;DR

This paper develops explicit formulas for effective wave numbers and reflection coefficients of P and SV waves in an elastic medium with random cylindrical scatterers, extending scalar wave models and highlighting wave coupling effects.

Contribution

It introduces a multiple scattering analysis for elastic waves in random media, providing explicit formulas without renormalization and comparing high frequency and dilute approximations.

Findings

Effective wave numbers derived for dilute scatterer distributions.

Coupling between P and SV waves at low scatterer concentrations.

Expressions for reflection coefficients depending on frequency and scatterer concentration.

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.