ON THE SOLUTIONS OF THE INHOMOGENEOUS HELMHOLTZ WAVE EQUATION FOR ELLIPSOIDAL SOURCES

摘要

The solution of the inhomogeneous Helmholtz equation (the 'dynamic' or 'Helm-holtz potential') and its time domain representation (the retarded potentials) for an ellipsoidal source region is analyzed. They occur in many dynamic problems of mathematical physics such as wave propagation and scattering phenomena. From an aesthetic and practical point of view 1D-integral representations for dynamic potentials are highly desirable. So far such representations seem to be absent in the literature. Here we close this gap for the internal dynamic potential of an ellipsoidal shell. The solution of the external space can be constructed by applying Ivory's theorem. Moreover we construct surface integral representations for inhomogeneous ellipsoidal sources for source densities of the form ρ = Θ(1 - P)f(P~2), (P~2 =x~2/a_1~2 + y~2/a_2~2 + z~2/a_3~2). Closed form solutions are found for the retarded potentials of inhomogeneous spherical sources. In the static limit the dynamic potentials coincide with well known classical results for the Newtonian potentials of ellipsoids of Dyson and Ferrers.