On the retarded potentials of inhomogeneous ellipsoids and sources of arbitrary shapes in the three-dimensional infinite space

We analyze the infinite space solutions of the three-dimensional inhomogeneous wave equation (the retarded potentials or causal propagators ) for ellipsoidal sources and for sources of arbitrary shapes. The short-time characteristics of the retarded potential for a spatially inhomogeneous source density of d-shaped time profile is considered. It is found that, the short-time characteristics is governed by the spatial inhomogeneity of the source density in the immediate vicinity of a spacepoint. Surface integral representations are derived for spatial inhomogeneous source regions of ellipsoidal symmetry. For spherical sources these integral representations yield closed form solutions for the retarded potentials. We find that the wave field inside a spherical source consists of an incoming and outgoing spherical wave package, whereas the external wave field consists of an outgoing spherical wave package only. Characteristic runtime and superposition effects are discussed. Moreover, a numerical technique based on Gauss quadrature is applied to generate the wave field for a cubic source. The integral representations derived for the retarded potentials of inhomogeneous ellipsoidal sources are consistent with results previously derived by the authors for the Helmholtz potentials of homogeneous ellipsoids and ellipsoidal shells [Michelitsch, T.M., Gao, H., Levin, V.M., 2003. On the dynamic potentials of ellipsoidal shells. Q. J. Mech. Appl. Math. 56 (4), 629]. The derived solutions are crucial for many problems of wave propagation and diffraction theory as they may occur in materials science. As an example we give a formulation for the solution of