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.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


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.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+