Diffraction of two-dimensional high-frequency electromagnetic waves by a locally perturbed two-part impedance plane

During the second half of the last century mixed boundary-value problems had been an appealing research subject for both mathematicians and engineers. Among this kind of problems those connected with wave propagation in half-spaces or slabs bounded by sectionally homogeneous boundaries took an important place because they were motivated by microwave applications. The simplest problem of this kind is the classical two-part problem which can be reduced to a functional equation involving two unknown functions, say Ψ+(ν) and Ψ−(ν), which are regular in the upper and lower halves of the complex ν-plane, respectively. This functional equation can be rigorously treated by the Wiener-Hopf technique. When the boundary consists of three (or more) parts, the resulting functional equation involves also an entire function, say P(ν), in addition to Ψ+(ν) and Ψ−(ν), which makes the problem not solvable exactly. A local (non-homogeneous) perturbation on a two-part boundary, which is of extreme importance from engineering point of view, gives also rise to a problem of this type. The known methods established to overcome the difficulties inherent to the three-part problems are based on the elimination of the entire function P(ν) first to obtain a linear system of two singular integral equations for Ψ+and Ψ−. After having determined the functions Ψ+(ν) and Ψ−(ν) by solving this system of integral equations numerically, the function P(ν) is found from the functional equation in question. Numerical solutions to the aforementioned system, which need rather hard computations, cannot provide results which are suitable to physical interpretations. The aim of the present paper is to establish a new method which is based, conversely, on the elimination of the unknown functions Ψ+(ν) and Ψ−(ν) first to obtain a linear integral equation of the first kind for the entire function P(ν), which can be solved rather easily by regularized numerical methods. Then the functions Ψ+(ν) and Ψ−(ν) are determined through the classical Wiener-Hopf technique. The result to be obtained by this approach seems to be more suitable to physical interpretations and permits one to reveal the effect of the perturbation on the scattered wave. Some illustrative examples show the applicability and effectiveness of the method.