Reflection and transmission at low concentration by a depth-varying random distribution of cylinders in a fluid slab-like region

This paper deals with multiple scattering by a random arrangement of parallel circular elastic cylinders immersed in a fluid. The cylinders are distributed in a region called “slab” that is located between two parallel planes orthogonal to a given x-direction. The disorder inside the slab depends on the x-variable. The goal is to calculate the reflection and transmission coefficients by this space-varying slab. For low concentrations of cylinders, two methods are developed from Twersky's theory on the propagation of coherent waves in an effective medium. The first method is based upon the discretization of the properties of the space-varying slab. The second one is based on the WKB method. They are successfully compared in the case of a smooth space-varying slab in which the random distribution of cylinders varies slowly along the x-direction. An effective mass density is defined, which allows the derivation of the mean acoustic displacement from the mean pressure field. The continuity of both pressure and normal displacement is thus shown at the interface between two different effective media as well as at the interface between the space-varying slab and a homogeneous fluid.