On Scattering and Reflection of Sound by Rough Surfaces

We consider random distributions of arbitrary identical protuberances on free or rigid base planes, and derive the approximate reflection coefficients and differential scattering cross sections per unit area . Here , where f(s,k_i) is the scattering amplitude of an isolated protuberance. k_i, k₀, and s are the directions of incidence, specular reflection (with respect to the base plane), and observation; n is the normal of the base; ρ is the average number of scatterers in unit area, and k = 2π/λ. (For a two‐dimensional “striated” surface we divide Z by π/k, and σ by π/2k.) We find for example, that if the horizon angle β approaches zero (i.e., near grazing incidence), then the reflection coefficients approach unity minus terms of the order β, while the back scattering vanishes like β¹, β² for a free, rigid base. Explicit forms are given for arbitrary hemispheres and circular semicylinders, and for the limiting cases of free and rigid surfaces with scatterer radius very small and very large compared to wavelength. The analysis is based on a Green's function formulation of the problem of a single configuration; R and σ follow from an approximation of the ensemble averaged energy flux which takes account of multiple coherent scattering. (The transmission problem of a “random screen” is treated simultaneously.) An elementary derivation is also given, and extensions to distributions of nonidentical scatterers are made.