Scattering by infinite one-dimensional rough surfaces

We consider the Dirichlet boundary-value problem for the Helmholtz equation in a non-locally perturbed half-plane. This problem models time-harmonic e lectromagnetic scattering by a one-dimensional, infinite, rough, perfectly conducting surface; the same problem arises in acoustic scattering by a sou nd soft surface. Chandler-Wilde & Zhang have suggested a radiation conditio n for this problem, a generalization of the Rayleigh expansion condition fo r diffraction gratings, and uniqueness of solution has been established. Re cently, an integral equation formulation of the problem has also been propo sed and, in the special case when the whole boundary is both Lyapunov and a small perturbation of a flat boundary, the unique solvability of this inte gral equation has been shown by Chandler-Wilde & Ross by operator perturbat ion arguments. In this paper we study the general case, with no limit on su rface amplitudes or slopes, and show that the same integral equation has ex actly one solution in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of inc ident fields including the incident plane wave, the Dirichlet boundary-valu e problem for the scattered field has a unique solution.