Assessment of the surface radiation condition by reference to acoustic scattering by a hard sphere

Abstract

Two forms of the surface radiation condition (SRC) introduced by different approaches are considered and their relative merits are examined via a special problem, namely acoustic scattering by a hard spherical object. Each of the methods offers a differential equation on the scatterer to determine the surface field and eventually die calculation of the scattered field is reduced to quadra-tures. These equations are first solved exactly by a series expansion method for an incident plane wave and the far field is obtained analytically. A comparison between the exact series solutions constructed by the SRC techniques and the exact answer of the problem shows that both approaches, almost equivalently, provide the scattered field very accurately in the low and middle frequency ranges, i.e. for ka ≤5, but as the frequency increases, although the phase remains remarkably accurate, the relative error in the far-field amplitude grows in the forward region; nevertheless, the results are still qualitatively quite satisfactory and the accuracy increases with increasing frequency in the backward direction. Since the series expansion method is limited to the geometries where the Helmholtz operator is separable, some supplementary techniques are needed to apply the SRC concept for arbitrary convex objects. For this purpose, the effects of introducing a high-frequency asymptotic expansion and then an iterative technique for the solution of the SRC equations are investigated. The first-order approximations of both techniques also yield sufficiently accurate results for the far field. Thus they appear to be reliable supplementary methods for a general obstacle.