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We discuss the scattering of acoustic or electromagnetic waves from one dimensional rough surfaces. We restrict the discussion in this report to perfectly reflecting Dirichlet surfaces (TE-polarization). The theoretical development is for both infinite surfaces and periodic surfaces, the latter equations derived from the former. We include both derivations for completeness of notation. Several theoretical developments are presented. They are characterized by integral equation solutions for the surface current or normal derivative of the total field. All the equations are discretized to a matrix system and further characterized by the sampling of the rows and columns of the matrix which is accomplished in either coordinate space (C) or spectral space (S). The standard equations are referred to here as CC equations of either first kind (CC1) or second kind (CC2). Mixed representation equations or SC type are solved as well as SS equations fully in spectral space. Computational results are presented for scattering from various periodic surfaces. The results include examples with grazing incidence, a very rough surface and a highly oscillatory surface. The examples vary over a parameter set which includes the geometrical optics regime, physical optics or resonance regime, and a renormalization regime. The objective of this study was to determine the best computational method for these problems. Briefly, the SC method was the fastest but did not converge for large slopes or very rough surfaces for reasons we explain. The SS method was slower and had the same convergence difficulties as SC. The CC methods were extremely slow but always converged. The simplest approach is to try the SC method first. Convergence, when the method works, is very fast. If convergence doesn't occur then try SS and finally CC.

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A crucial ingredient in the formulation of boundary-value problems for acoustic scattering of time-harmonic waves is the radiation condition. This is well understood when the scatterer is a bounded obstacle. For plane-wave scattering by an infinite, rough, impenetrable surface S, the physics of the problem suggests that all scattered waves must travel away from (or along) the surface. This condition is used, together with Green’s theorem and the free-space Green’s function, to derive boundary integral equations over S. This requires careful consideration of certain integrals over a large semicircle of radius r; it is known that these integrals vanish as r→∞ if the scattered field satisfies the Sommerfeld radiation condition, but that is not the case here—reflected plane waves must be present. The integral equations obtained are Helmholtz integral equations; they must be modified for grazing incident waves. As such integral equations are often claimed to be exact, and are often used to generate benchmark numerical solutions, it seems worthwhile to establish their validity or otherwise.