Details

Consider the two-dimensional Helmholtz equation

and use elliptic coordinates

,

with ( is the semiminor axis of the ellipse). Solutions of the type lead to separation of variables, with the pair of differential equations

,

.

Now, consider the wave equation

.

Here , where is the tension and is the surface mass density, and we assume the harmonic dependence . The wave equation can be rewritten as the Helmholtz equation, with solutions with Dirichlet boundary conditions . Thus

where and are even and odd angular solutions. These functions are represented in Mathematica as MathieuC[a,q,z] and MathieuS[a,q,z], respectively; is the characteristic (MathieuCharacteristicA[n,q] and MathieuCharacteristicB[n,q]), a value for which the Mathieu functions are periodic. The radial functions and correspond to the same functions with . The solutions have angular nodes and radial nodes, with for even modes and for the odd ones.

Reference

[1] J. C. Gutiérrez-Vega, R. M. Rodrı́guez-Dagnino, M. A. Meneses-Nava, and S. Chávez-Cerda, "Mathieu Functions, a Visual Approach," American Journal of Physics, 71(3), 2003 pp. 233–242. doi:10.1119/1.1522698.