Reflection and transmission by randomly spaced cylindrical inclusions in a solid slab-like region: Twersky’s theory

V. Twersky developed in the sixties a theory describing the reflection and transmission process of scalar plane waves from slab-like regions of randomly distributed scatterers. The theory, which is based on Foldy’s approximation and neglects “hole corrections”, is an alternative approach to an exact calculation of multiple scattering that fails when the number of scatterers is important. Twerky’s theory furnishes the properties of the coherent plane wave propagating inside the slab. From the point of view of the coherent wave, the slab may be described as an effective fluid medium; its reflection and transmission coefficients may be formally written as those of a fluid plate. In this work, Twersky’s theory is extended to random distributions of cylindrical inclusions in elastic media. Before examining the reflection and transmission process of such multiple scattering media, our first objective is to theoretically validate the results predicted by the theory. With this aim in view, as an exact calculation of multiple scattering fails for purely random distributions, the idea consists in adapting Twersky’s theory to distributions of scatterers periodic in one direction and random in the other one. Indeed, the reflection and transmission coefficients of such half-periodic media can be calculated by using a method originally developed for purely periodic media, i.e. phononic crystals. In our case, it consists in considering each half-periodic medium as a random distribution of linear periodic arrays of scatterers. This method furnishes exact reflection and transmission coefficients which can be successfully compared with those predicted by Twersky’s theory.