On the derivation of boundary integral equations for scattering by an infinite one-dimensional rough surface

Abstract

A crucial ingredient in the formulation of boundary-value problems for acoustic scattering of time-harmonic waves is the radiation condition. This is well understood when the scatterer is a bounded obstacle. For plane-wave scattering by an infinite, rough, impenetrable surface S, the physics of the problem suggests that all scattered waves must travel away from (or along) the surface. This condition is used, together with Green’s theorem and the free-space Green’s function, to derive boundary integral equations over S. This requires careful consideration of certain integrals over a large semicircle of radius r; it is known that these integrals vanish as r→∞ if the scattered field satisfies the Sommerfeld radiation condition, but that is not the case here—reflected plane waves must be present. The integral equations obtained are Helmholtz integral equations; they must be modified for grazing incident waves. As such integral equations are often claimed to be exact, and are often used to generate benchmark numerical solutions, it seems worthwhile to establish their validity or otherwise.