Exact boundary integral equations for scattering of scalar waves from perfectly reflecting infinite rough surfaces

Abstract The derivation of integral equations for scattering from an infinite rough surface is not a straightforward procedure. The usual formalism involves a single plane wave incident on the surface and the free-space Green’s function. Problems arise due to waves propagating horizontal to the surface and to the inability to evaluate in a simple way the Helmholtz–Kirchhoff or Green’s surface integral on the upper hemisphere. One approach is to change the usual Sommerfeld radiation condition to exclude horizontal waves. In this paper we take a different approach. Instead of the single incident plane wave as the Born term we choose the flat surface result of incident plus reflected field. The additional field scattered from the rough surface satisfies the Sommerfeld condition. The full Green’s function is chosen as a combination of the free-space Green’s function and its image. The main result is that the Helmholtz–Kirchhoff integral using this new Born term and the image Green’s function can be evaluated exactly (and simply) on the hemisphere. This leads directly to integral equations on the total field (flat surface plus scattered fields) for the Dirichlet and Neumann problems. Simple coordinate-space representations for the kernels of these equations are also presented. Since we use image functions we have standing waves in the z -direction, so that the result is not strictly categorized as a diffraction result. Nevertheless, the integral relations induced by their choice yield an exact and simple result for the solution of an otherwise difficult problem.