Multiple point scattering to determine the effective wavenumber and effective material properties of an inhomogeneous slab

Abstract

In this article we attempt to clarify various notions regarding multiple point scattering. We consider several predictions for the effective material properties of an inhomogeneous slab region which can be derived from classical multiple scattering theories. In particular we are interested in the point scattering limit when wavelengths λ0 ≫ l ∼ a where l is the characteristic length-scale of the distance between inclusions and a is the characteristic length-scale of inclusions. In this limit we are able to derive effective properties which are physically valid for any volume fraction φ, except in the sound-soft scatterer case where there is a condition on the size of φ. We shall confine attention to random distributions of inclusions and employ the Quasi-Crystalline Approximation to yield results. In particular we discuss the different scenarios of acoustics and antiplane elasticity and stress the reciprocity between these two problems which means that they can be solved simultaneously. We make various statements regarding the efficacy of the various multiple scattering theories in the prediction of effective material properties in the quasi-static limit.