Abstract
If a Dirichlet or Neumann condition is imposed on the surface of the ellipsoid, the variables are separated in the scalar wave equation in ellipsoidal coordinates, and the problem in hand is reduced to a system of three identical ordinary differential equations, each being defined on a separate interval and subject to its own boundary conditions. Thus, the three-parameter self-adjoint Sturm-Liouville problem arises: the equations are coupled by two separation constants and the eigen frequency of the ellipsoid, i. e., the spectral parameters, which must be so chosen that all the equations of the system have simultaneously nontrivial solutions, each satisfying the corresponding boundary conditions. The effective globally converging numerical algorithm is proposed for calculating eigen frequencies and separation constants. When the modes of an ellipsoid are found, the caustic surfaces can be easily determined. The merit of the method is illustrated on the example of several calculations of the sound field and caustic surfaces in an ellipsoid.
A Singular Sturm-Liouville Problem with Limit Circle Endpoints and Eigenparameter Dependent Boundary Conditions
