On a fine localization of the Mathieu azimuthal numbers by Cassini ovals

The study is devoted to numerical and analytical problems concerning generating periodic and antiperiodic solutions of the angular (circumferential) Mathieu equation obtained for the circumferential harmonics of an elliptic cylinder. The Mathieu eigenvalues localization problem and computations of elliptic azimuthal numbers are discussed. First, the Sturm–Liouville eigenvalue problem for the angular Mathieu equation is reformulated as an algebraic eigenvalue problem for an infinite linear self-adjoint pentadiagonal matrix operator acting in the complex bi-infinite sequence space $l_2$. The matrix operator is then represented as a sum of a diagonal matrix and an infinite symmetric doubly stochastic matrix, which is interpreted as a finite perturbation imposed on the diagonal matrix. Effective algorithms for computations of the Mathieu eigenvalues and associated circumferential harmonics are discussed. Azimuthal numbers notion is extended to the case of elastic and thermoelastic waves propagating in a long elliptic waveguide. Estimations of upper and low bounds and thus localizations of the angular Mathieu eigenvalues and elliptic azimuthal numbers are given. Those are obtained by algebraic methods employing the Gerschgorin theorems and Cassini ovals technique. The latter provides more accurate solution of the Mathieu eigenvalues localization problem.