Ellipsoidal functions and their application to some wave problems

Abstract

The analysis of wave motion connected with the circular disc and circular aperture is of such great analytical and physical interest, that no excuse is required for a somewhat extended treatment. The theory of wave motion has been very fully developed when the surfaces bounding the medium in which the motion takes place are spheres and circular cones, or cylinders and planes. The functions involved in the analysis are of the hypergeometric type. Next in order of importance is the theory required for the treatment of problems in which the surfaces bounding the medium are :— A. Ellipsoids of revolution and hyperboloids of revolution of one and two sheets ; B. Elliptic and hyperbolic cylinders. The functions involved in case A are functions associated with the ellipsoid of revolution, or spheroid. The functions involved in case B are commonly called MATHIEU functions. The MATHIEU functions and the functions associated with the ellipsoid of revolution are of considerable importance but, unfortunately, of great complexity.