Sommerfeld Diffraction Problems with First and Second Kind Boundary Conditions

Abstract

A class of interface problems is considered where the solutions of one Helmholtz equation in the upper half-plane $x_2 > 0$ and of another in the lower half-plane $x_2 0$ and $x_1 < 0$, respectively. In general, there appears a coupled system of Wiener–Hopf equations. Necessary and sufficient conditions for the correctness of the problem in a Sobolev space setting are presented as well as explicit formulas for a factorization of the Fourier symbol matrix of the one-medium problem, the solution in closed form, and its asymptotics near the origin.