Integral Equations and the Scattering Diagram in the Problem of Diffraction by Two Shifted Contacting Wedges with Polygonal Boundary

Abstract

The acoustic problem of diffraction by two wedges with different wave velocities is studied. It is assumed that the wedges with parallel edges have a common part of the boundary and the second wedge is shifted with respect of the first one in the orthogonal to the edges direction along the common part of the boundary. The wave field is governed by the Helmholtz equations. On the polygonal boundary, separating these shifted wedges from the exterior, the Dirichlet boundary condition is satisfied. The wave field is excited by an infinite filamentary source, which is parallel to the edges. In these conditions, the problem is effectively two-dimensional. The Kontorovich–Lebedev transform is applied to separate the radial and angular variables and to reduce the problem at hand to integral equations of the second kind for so-called spectral functions. The kernel of the integral equations given in the form of an integral of the product of Macdonald functions is analytically transformed to a simplified expression. For the problem at hand, some reductions of the equations are also discussed for the limiting or degenerate values of parameters. Making use of an alternative integral representation of the Sommerfeld type, expressions for the scattering diagram are then given in terms of spectral functions. Bibliography: 24 titles.