Laplace-Dirichlet Eigenstates of an Ellipse

The fundamental modes of vibration for an idealized drum of given shape satisfy the Laplace–Dirichlet eigenproblem. This Demonstration computes solutions to the Laplace–Dirichlet eigenproblem on an ellipse with unit area and eccentricity . For an ellipse with semiaxes , the eccentricity is and the area is . The Laplace–Dirichlet eigenvalues and eigenfunctions satisfy in the interior of , and the Dirichlet boundary condition on the boundary .
In elliptical coordinates, the Laplace–Dirichlet equations on an ellipse are separable. In the "angular" coordinate (parameterizing confocal ellipses), the solution satisfies the Mathieu equation, and in the "radial" coordinate (parameterizing confocal hyperbolas), the solution satisfies the modified Mathieu equation. The eigenvalues are such that the solutions to the Mathieu equation are periodic, and the solutions to the modified Mathieu equation vanish on the boundary of the ellipse.


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For more detailed descriptions of Laplace–Dirichlet eigensolutions on an ellipse, see [1] or [2].
[1] B. A. Troesch and H. R. Troesch, "Eigenfrequencies of an Elliptic Membrane," Mathematics of Computation, 27(124), 1973 pp. 755–765. doi:10.1090/S0025-5718-1973-0421276-2.
[2] B. A. Troesch, "Elliptical Membranes with Smallest Second Eigenvalue," Mathematics of Computation, 27(124), 1973 pp. 767–772. doi:10.1090/S0025-5718-1973-0421277-4.
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