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P. Amodio, A. Arnold, T. Levitina, G. Settanni, E. Weinmüller:
"On the Abramov approach for the approximation of whispering gallery modes in prolate spheroids";
Applied Mathematics and Computation, 1 (2021), 125599; 14 pages.

A whispering gallery is usually a circular, hemispherical, elliptical or ellipsoidal enclosure, often beneath a dome or a vault, in which whispers can be heard clearly in other parts of the gallery.´ Famous examples are St. Paul´s Cathedral in London and the Temple of Heaven in Beijing, for more details see [30]. The involved oscillations are strongly localized within a narrow ring on the `equator´ of the resonator surface and show an extremely high quality factor (QF) [16,24]. Especially, the high QF explains growing interest in whispering gallery modes (WGMs). This phenomenon is related to standing waves, typically described by the Helmholtz equation, which has the advantage to be separable in spheroidal coordinates [25]. It also arises in many important scientific and industrial applications in the context of optics and photonics [14,17,22-24,31].
In our earlier work, [5,6], we discussed the numerical simulation of scalar (acoustic) eigen-oscillations inside a pro- late and an oblate spheroid with a focus on WGMs. In order to compute the eigenfunctions of the Helmholtz equation in spheroidal geometries, separation of variables simplifies the matter. There, one uses a product ansatz for the Helmholtz solu- tion consisting of three factors. While the azimuthal contribution is explicitly known (trigonometric with m oscillations), the angular or radial solution components are described by a system of two ODEs called prolate spheroidal wave equations.

http://dx.doi.org/10.1016/j.amc.2020.125599