Integral equation methods for scattering by in nite rough surfaces

In this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non-locally perturbed half-plane. These boundary value problems arise in a study of timeharmonic acoustic scattering of an incident eld by a sound-soft, in nite rough surface where the total eld vanishes (the Dirichlet problem) or by an in nite, impedance rough surface where the total eld satis es a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined doubleand single-layer potential and a Dirichlet half-plane Green’s function. For the impedance problem we propose two boundary integral equation formulations, both using a half-plane impedance Green’s function, the rst derived from Green’s representation theorem, and the second arising from seeking the solution as a single-layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident elds including an incident plane wave, the impedance boundary value problem for the scattered eld has a unique solution under certain constraints on the boundary impedance. Copyright ? 2003 John Wiley & Sons, Ltd.