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.

A new method for the computation of conserved densities of nonlinear differentialdifference equations is applied to Toda lattices and discretizations of the Korteweg-de Vries and nonlinear Schrödinger equations. The algorithm, which can be implemented in computer algebra languages such as Mathematica, can be used as an indicator of integrability.

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We consider the problem of scattering a plane wave from a periodic rough surface. The scattered field is evaluated once the field on the boundary is calculated. The latter is the solution of an integral equation. In fact, different integral equation formulations are available in both coordinate and spectral space. We solve these equations using standard numerical techniques and compare the results to corresponding solutions of the equations using wavelet transform methods for sparsification of the impedance matrix. Using an energy check, the methods are shown to be highly accurate. We limit the discussion in this paper to the Dirichlet problem (scalar) or TE-polarized case for a one-dimensional surface. The boundary unknown is thus the normal derivative of the total (scalar) field or equivalently the surface current. We illustrate two conclusions. First, sparsification (using thresholded wavelet transforms) can significantly reduce accuracy. Second, the wavelet transform did not speed up the overall solution. For our examples the solution time was considerably increased when thresholded wavelet transforms were used.