Some negative results on the use of Helmholtz integral equations for rough-surface scattering

where p and q are points on S, G is the free-space Green’s function, ∂/∂nq denotes normal differentiation at q, and uinc is the incident field. The Helmholtz integral equations are familiar boundary integral equations, for which there is a complete theoretical framework (Kleinman and Roach, 1974; Colton and Kress, 1983). Many examples of their numerical treatment, usually using boundary elements, can also be found. Suppose now that S is unbounded. A typical problem is the reflection of a plane wave by an infinite two-dimensional rough surface. This is a classical problem, going back to Lord Rayleigh. Standard texts usually treat the problem using approximate techniques, such as perturbation theory or Kirchhoff theory. More recently, there has been an effort to validate these approximate techniques by comparing them with ‘exact’ methods. In particular, comparisons have been made with numerical solutions of Helmholtz integral equations, such as (1). However, the derivation of a Helmholtz integral equation for an infinite rough surface is not straightforward. Indeed, we are not aware of any correct derivations in the literature (even though such equations have been the subject of extensive computational studies). In fact, we can show that such an integral equation is definitely not valid in certain cases! These will be illustrated using some explicit examples.