Statistical theory of scattering and propagation over a random surface

Abstract

First, a systematic theory is developed for a deterministic surface upon introduction of a surface Green function defined on a reference plane of the rough surface and determined by an equivalent surface impedance. The surface impedance is obtained by an exact transformation of a given boundary equation on the real boundary surface onto the reference plane and is given explicitly in an operator form, in terms of given surface displacement. An exact integral equation is also derived for the reflection coefficient and is applied to a rough surface of large scale, with the aid of an entirely new tangent-plane method. The scattering cross-section per unit area thus obtained is consistent with the power conservation, shadowing and multiple scattering, in contrast with those given in the literature. Governing equations of the statistical-surface Green functions of first and second orders are derived unperturbatively in form of the Dyson and Bethe-Salpeter equations, respectively, by following exactly the procedure established for a random medium. In particular the power conservation and associated optical relations are investigated in detail, in view of a complete lack of theories on this aspect. Results of specific problems are extensively summarised, including scattering cross-section per unit area and detailed optical relations inherent between various quantities involved, with several examples not only for a scalar wave but also for electromagnetic waves.