The diffraction of a whispering-gallery mode by a spherical soft-hard cap

Abstract

Abstract An asymptotic solution is obtained for the problem of the diffraction of a rotationally symmetric high-frequency scalar whispering-gallery mode by a spherical cap. The convex side of the cap is assumed to have a soft (pressure release) boundary condition while the concave side is a rigid boundary. By considering the fact that the diffraction of a high frequency wave is a local phenomenon, the problem is formulated in an extended abstract space for which the polar angle θ varies in (−∞, ∞). This extension permits one to reduce the problem into a matrix Hilbert equation which is solved by using the Daniele-Khrapkov method of factorization. Various ray contributions are isolated and fairly simple non-uniform expressions for certain diffraction coefficients are obtained.