Embedding formulae for Laplace–Beltrami problems on the sphere with a cut

Abstract

We consider the problem of diffraction of a plane wave by a quarter-plane (a flat cone) with Dirichlet boundary conditions. The most efficient approach to this problem is the technique of Smyshlyaev’s formulae or a modified Smyshlyaev’s formula, both of which are representations of the diffraction coefficient as contour integrals over a complex parameter. These representations have been proven independently. Here we are demonstrating a link between these classes of formulae. The link is established by developing an embedding procedure (in the very special sense of diffraction theory) on the unit sphere.