# Laplace-Dirichlet Eigenstates of an Ellipse

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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 .

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Contributed by: Braxton Osting (September 2012)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

For more detailed descriptions of Laplace–Dirichlet eigensolutions on an ellipse, see [1] or [2].

References

[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.

## Permanent Citation

"Laplace-Dirichlet Eigenstates of an Ellipse"

http://demonstrations.wolfram.com/LaplaceDirichletEigenstatesOfAnEllipse/

Wolfram Demonstrations Project

Published: September 10 2012