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Solutions of the Elliptic Membrane Problem

The problem of the free oscillations of an elliptic membrane was studied by Mathieu in 1868. The membrane is anchored to the perimeter of the ellipse. Mathieu functions can appear in problems with elliptical geometry and in certain periodic phenomena.

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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.
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