A two-scale model from the point of view of the small-slope approximation

Abstract

General results for the scattering cross section following from the small-slope approximation (SSA) are applied to the case of two-scale surface roughness which can be represented as a superposition of small-scale and large-scale components. The purpose of the paper is to obtain analytically tractable results with obvious physical meaning which can be used for estimates prior to undertaking extensive numerical calculations according to exact unambiguous expressions of the SSA. The general case of vector (electromagnetic) or scalar (sound) waves is considered. The statistics of small-scale roughness is not assumed to be Gaussian (in any sense) or space-homogeneous, and the possible dependence of the statistics of small-scale roughness on a large-scale undulating surface is taken into account. As a result, a modified local spectrum of small-scale components of roughness enters into corresponding expressions for the scattering cross section. It is demonstrated that under appropriate conditions, the resulting formulae for the scattering cross section reduce to the conventional two-scale model.