The method of parabolic equation in application to Weinstein's problems

摘要

The Weinstein's problems are the problems of diffraction of a high-frequency grazing wave by a grating consisting of absorbing half-infinite screens. Recently, the authors have developed a novel approach to Weinstein's problems [2-5]. This approach consists of the following steps. First, the parabolic approximation is introduced, which means that the Helmholtz equation is replaced by the parabolic equation. Second, the embedding formula is derived. The formula expresses reflection coefficients in terms of the edge Green's functions. Then the ordinary differential equation for the directivities of the Edge Green's functions is derived. This equation is called the spectral equation. Unfortunately, its coefficient remains unknown. For some particular cases authors obtained an analytical expression for the coefficient and, therefore, for the solution of the original problem. For the general case a numerical procedure to find unknown coefficient of the spectral equation is introduced. The procedure is based on the method of “ordered exponential” equation. The paper presents an overview of these methods.