Spheroidal Wave Functions: Their Use and Evaluation

Abstract

The Helmholtz or scaler wave equation (▿2 + k2) ψ = 0 is separable in both oblate and prolate spheroidal coordinates. Solutions to this equation constitute an essential element in the numerical calculation of the diffraction, radiation, and scattering of acoustic and electromagnetic waves by spheroids. They also find considerable application in other areas such as signal processing and nuclear models. FORTRAN computer programs have been developed to evaluate both efficiently and accurately the radial and angular parts of the solution in both oblate and prolate spheroidal coordinates. Extensive tables of oblate and prolate radial functions containing entries for values of h up to 40.0 and for values of the prolate radial coordinate as low as 1.00000001 are being published. (h = kd/2, where d is the interfocal distance.) These tables as well as the capabilities of the capabilities of the various computer programs are discussed. Examples illustrating the use of this coordinate system in acoustic radiation calculations are presented.