Computation of Eigenfrequencies of an Acoustic Medium in a Prolate Spheroid by a Modified Abramov Method
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RY DIFFERENTIAL EQUATIONS
Computation of Eigenfrequencies of an Acoustic Medium in a Prolate Spheroid by a Modified Abramov Method T. V. Levitina* Max Planck Institute for Solar System Research, Göttingen, 37077 Germany *e-mail: [email protected] Received January 21, 2020; revised April 15, 2020; accepted June 3, 2020
Abstract—The method presented and studied in [1, 2] for solving self-adjoint multiparameter spectral problems for weakly coupled systems of ordinary differential equations is based on marching with respect to a parameter introduced into the problem. Although the method is formally applicable to systems of ordinary differential equations with singularities, its direct use for the numerical solution of the problem indicated in this paper’s title is limited. A modification of the method is proposed that applies to the computation of various, including high-frequency, acoustic oscillations in both nearly spherical and strongly prolate spheroids. Keywords: three-dimensional Helmholtz equation, separation of variables in a prolate spheroidal system of coordinates, two-parameter singular self-adjoint spectral problem, evaluation of spectral points, parameter marching, Newton’s method, prolate spheroidal wave functions, whispering gallery modes DOI: 10.1134/S0965542520100103
INTRODUCTION In the case of acoustic oscillations in a prolate spheroid, separation of variables in the three-dimensional Helmholtz equation in spheroidal coordinates leads to a two-parameter Sturm–Liouville problem: the separated angular and radial spheroidal wave equations are related by two spectral parameters: the separation constant λ and the dimensionless wave number c, which determines the frequency of the oscillation mode. Each mode is characterized by a multi-index (l, n, m), l, n, m = 0, 1, 2, ..., i.e., where m is the azimuthal index, while l and n are the numbers of internal zeros of solutions to the angular and radial equations, respectively. The solvability of the problem, despite its singularity, has been proved for any multiindex, but its numerical solution faces certain difficulties. The method presented in [1, 2] for solving multiparameter spectral problems combines Newton’s iterations with parameter marching that connects the original problem with a simpler one with frozen coefficients multiplying the spectral parameters. In [1, 2] the solvability of all intermediate problems was proved and the possibility of computing Newton’s iterations at every step in the transition from the frozen problem to the original one was shown, including the proof of the invertibility of the corresponding Jacobian matrices. The residual arising in the computations was estimated depending on the maximum size of the intermediate steps. Nevertheless, it turned out that the method of [1, 2] in its original form cannot be used to compute the frequencies of whispering gallery modes (WGMs), which are important for numerous applications [3–5]. The primary cause is that the equations resulting from the separation of variables are singular. The tr
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