Integral equations and scattering diagram in the problem of diffraction by two wedges shifted along the line of contact with polygonal boundary

In this work we study the acoustic problem of diffraction by two wedges with different wave velocities. 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 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. We apply Kontorovich–Lebedev transform in order to separate radial and angular variables and to reduce the problem at hand to integral equations of the second kind for the so called spectral functions. The kernel of the integral equations given in the form of integral of the Macdonald functions product 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. Exploiting an alternative integral representation of the Sommerfeld type, expressions for the scattering diagram is then given in terms of the spectral functions.