Determination of Spheroidal Normal Modes: Mathematical Aspects

Abstract

AbstractConvenient fourth-order systems of differential equations in the radial coordinate are derived. Next, a mathematical method for the determination of radial and non-radial spheroidal normal modes is presented as the resolution of an eigenvalue problem with two end points, at which the governing differential equations display singularities. The standard procedure rests on an application of initial-value techniques by which admissible trial solutions are constructed from each singular end point. In the case of the radial normal modes, the basic equation is a linear, homogeneous second-order differential equation for the relative radial displacement. From both end points, the equation admits of a particular solution that satisfies the boundary conditions. The equation for the eigenvalues results from the requirement that the logarithmic derivative of the relative radial displacement must be continuous at any point of the common domain of validity of the admissible solutions. In the case of the non-radial spheroidal normal modes, the basic equations consist of a fourth-order linear, homogeneous system of ordinary differential equations, and an appropriate extension of the method that applies to the determination of the radial normal modes is applied.KeywordsPower SeriesBoundary PointRadial DisplacementSolution VectorAdmissible SolutionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.