Statistical theory of scattering and propagation through a rough boundary–The Bethe–Salpeter equation and applications

Abstract

A systematic theory is given for a general class of scalar waves by introducing a surface Green’s function, which is a 2×2 matrix function governed by boundary equations, transferred onto two reference boundary planes enclosing the real boundary inside. It is subjected to several symmetries, including a relation eventually leading to optical relations. Governing equations of statistical surface Green’s functions of first and second orders are obtained unperturbatively in exactly the same way as for a random medium, on replacing the medium to a surface impedance. Two operator methods are introduced to obtain the surface impedance and integral equations of reflection–transmission coefficients exactly for a given boundary change. The space Green’s functions outside the boundary are obtained by a simple continuation of the surface Green’s functions, and scattering cross sections are obtained from their asymptotic expressions at large distances. Various quantities and equations associated with the incoherent waves are written exactly by the introduction of a scattering matrix, as contrasted with the conventional one for a coherent scatterer. A slightly rough boundary is investigated, with the cross sections for both reflected and transmitted waves, where obtained equations are significant also for higher-order effects, including multiple scattering. An application to boundary-value problems in layer transport is suggested.