Two alternative expressions for the spherical wave expansion of the time domain scalar free-space Green's function and an application: Scattering by a soft sphere

The importance of expanding Green's functions, particularly free-space Green's functions, in terms of orthogonal wave functions is practically self-evident when frequency domain scattering problems are of interest. With the relatively recent and widespread interest in time domain scattering problems, similar expansions of Green's functions are expected to be useful in the time domain. In this paper, two alternative expressions, expanded in terms of orthogonal spherical wave functions, for the free-space time domain scalar Green's functions are presented. Although the two expressions are equivalent, one of them is seen to be more convenient for the calculation of the scattered field for a known equivalent source density, whereas the second expression is more suitable for setting up an integral equation for the equivalent source density. Such an integral equation may be setup, for example, by the application of a time domain equivalent of the T-matrix (extended boundary condition) method. (C) 1997 Acoustical Society of America