Eigenvalues of a class of spherical wave functions

An analytically simple and sufficiently accurate solution for the eigenvalues of a class of spherical wave functions is presented. The class of spherical wave functions considered are modes in conical and quasipyramidal waveguides with perfectly conducting walls and hybrid modes in corrugated conical and quasipyramidal horns with an impedance boundary. The indicated solution has been first used to obtain in closed form eigenvalues of the class of spherical wave functions considered which are subsequently used as the starting values for evaluating the exact eigenvalues with a simple digital-computer based iterative algorithm. The digital-computer evaluation of the eigenvalues has been found to be very fast, since the starting values are close to the exact solution, irrespective of the flare angle of the radial waveguides considered. Further, some mathematical insight has been provided in order to explain why the asymptotic solution, which appears to be valid only for small flare angles, yields eigenvalues close to the exact one even for wide flare angle(s) of the radial waveguides.