The scattering of waves by a large sphere for impedance boundary conditions

Abstract

A method is presented for obtaining the amplitude of waves scattered by a by a sphere whose radius is large compared with the wavelength. Impedance boundary conditions are imposed on the surface of the sphere. The solution is characterized by both a discrete and continuous spectrum with the discrete spectrum expressed in terms of the radial modes, or eigenfunctions. It is shown that the discrete spectrum is cancelled by the continous spectrum after the first few terms. Thus, a rapidly converging solution is obtained, consisting of only the first few radial modes and a definite integral. Numerical examples are given for the Dirichlet and Neumann boundary conditions in the forward direction. The frequency dependent correction terms obtained for the shadow region are in agreement with earlier results, obtained by more laborious methods.