The Computation of Conical Diffraction Coefficients in High-Frequency Acoustic Wave Scattering

Abstract

When a high-frequency acoustic or electromagnetic wave is scattered by a surface with a conical point, the component of the asymptotics of the scattered wave corresponding to diffraction by the conical point can be represented as an asymptotic expansion, valid as the wave number $k \rightarrow \infty$. The diffraction coefficient is the coefficient of the principal term in this expansion and is of fundamental interest in high-frequency scattering. It can be computed by solving a family of homogeneous boundary value problems for the Laplace--Beltrami--Helmholtz equation (parametrized by a complex wave number--like parameter $\nu$) on a portion of the unit sphere bounded by a simple closed contour $\ell$, and then integrating the resulting solutions with respect to $\nu$. In this paper we give the numerical analysis of a method for carrying out this computation (in the case of acoustic waves) via the boundary integral method applied on $\ell$, emphasizing the practically important case when the conical scat...