Accurate solution to scattering by a semi-circular groove

Wave scattering by a semi-circular groove in an otherwise flat surface is considered. A mode-matching technique is employed in which the inner region is circular, the lower half of which corresponds to the groove. A Galerkin approximation is then made in which the solution (or its derivative, depending on the boundary conditions) on the matching semi-circle is expanded in a set of basis functions involving ultraspherical Gegenbauer polynomials which incorporate the known singularity at the groove edges. In this way a highly accurate and robust scheme is developed which can be used as a benchmark against which to test more general solution techniques.