On the dynamic potentials of ellipsoidal shells

Abstract

The solutions of many dynamical problems as wave propagation in electrodynamics, acoustics or elasticity very often require the solution of inhomogeneous Helmholtz equations (determination of dynamic potentials) for ellipsoidal source regions. As in the case of the static (Newtonian) potentials a compact representation of the dynamic potentials in terms of one-dimensional integrals is highly desirable. Due to the mathematical complexity of the problem for ellipsoids such a representation seems not to have been reported in the literature so far. In this paper we close this gap for the dynamic potential of an ellipsoidal shell for internal spacepoints. The derived solution of the inside region can easily be used to find the solution for the outside region by applying Ivory's theorem. In the static limit classical results of Ferrers and Dyson for the Newtonian potential of inhomogeneous ellipsoids are reproduced.