Scattering by infinite one-dimensional rough surfaces

Abstract

We consider the Dirichlet boundary-value problem for the Helmholtz equation in a non-locally perturbed half-plane. This problem models time-harmonic electromagnetic scattering by a one-dimensional, infinite, rough, perfectly conducting surface; the same problem arises in acoustic scattering by a sound-soft surface. Chandler-Wilde & Zhang have suggested a radiation condition for this problem, a generalization of the Rayleigh expansion condition for diffraction gratings, and uniqueness of solution has been established. Recently, an integral equation formulation of the problem has also been proposed 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 integral equation has been shown by Chandler-Wilde & Ross by operator perturbation arguments. In this paper we study the general case, with no limit on surface amplitudes or slopes, and show that the same integral equation has exactly one solution in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including the incident plane wave, the Dirichlet boundary-value problem for the scattered field has a unique solution.

Acoustic Scattering by Mildly Rough Unbounded Surfaces in Three Dimensions

“…Thus integral equation methods for the 3D rough surface scattering problem are essentially different from the 2D case studied in [7,9,8,32,3].…”

“…For the two-dimensional (2D) rough surface scattering case much progress has been made in terms of deriving well-posed BIEs for a variety of acoustic, electromagnetic, and elastic wave problems [9,8,32,2]. Surprisingly, none of the analysis for the 2D case extends straightforwardly to three dimensions; indeed most of the 2D analysis appears to be unsuitable in the 3D case.…”

“…Finally, we note that, since our results assume boundary data in the space L 2 (Γ), they do not include the interesting and problematic case of plane wave incidence, which is included in the analogous theory that has been developed for the 2D problem [8,32]. For a partial theoretical justification for BIE methods for 3D rough surface scattering with plane wave incidence, namely a justification, with some provisos, of Green's representation formula, see [16].…”

“…In more detail, in the 2D case bounded integral operators have been obtained by replacing the standard fundamental solution by the Dirichlet or impedance Green's function for a half-plane that contains the domain D of propagation (see, e.g., [8,32]). This modification leads to kernels of boundary integral operators that are weakly singular in their asymptotic behavior at infinity so that the integral operators are bounded on L p (Γ) for 1 ≤ p ≤ ∞ and on BC(Γ), the space of bounded continuous functions on Γ.…”

Inverse wave scattering by unbounded obstacles: uniqueness for the two-dimensional Helmholtz equation

“…The mathematical theory of forward scattering by an unbounded rough surface was mainly established by S. N. Chandler-Wilder and his collaborators over the last fifteen years, using integral equation methods (see, e.g. [7,11,12]) or variational methods ( [8,9]). Concerning uniqueness in inverse rough surface scattering problem, as far as we know, the only reference is due to Chandler-Wilder & Ross [10] who proved that a Dirichlet rough surface in a lossy medium can be uniquely determined by the near-field data above the surface corresponding to only one incident plane wave, which generalizes Bao's result [3] on periodic structures to rough surface scattering.…”