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P. Amodio, T. Levitina, G. Settanni, E. Weinmüller:
"Numerical simulation of the whispering gallery modes in prolate spheroids";
in: "ASC Report 34/2013", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2013, ISBN: 978-3-902627-06-3, 1 - 20.

In this paper, we discuss the progress in the numerical simulation of the socalled `whispering gallery´ modes (WGMs) occurring inside a prolate spheroidal cavity. These modes are mainly concentrated in a narrow domain along the equatorial line of a spheroid and they are famous because of their extremely high quality factor. The scalar Helmholtz equation provides a sufficient accuracy for WGM simulation and (in a contrary to its vector version)is separable in spheroidal coordinates. However, the numerical simulation of `whispering gallery´ phenomena is not straightforward. The separation of
variables yields two spheroidal wave ordinary differential equations (ODEs), first only depending on the angular, second on the radial coordinate. Though separated, these equations remain coupled through the separation constant and the eigenfrequency, so that together with the boundary conditions they form a singular self-adjoint two-parameter Sturm-Liouville problem.
We discuss an efficient and reliable technique for the numerical solution of this problem which enables calculation of highly localized WGMs inside a spheroid. The presented approach is also applicable to other separable geometries.
We illustrate the performance of the method by means of numerical
experiments.

Morphology dependent resonances, `Whispering gallery´ mode, Multiparameter spectral problems, Pr¨ufer angle, High order finite difference schemes

http://www.asc.tuwien.ac.at/preprint/2013/asc34x2013.pdf