Isospectral drumsIn 1991, mathematicians Gordon, Webb, and Wolpert (Invent. Math. 110, pp 1--22) solved a famous problem posed by M. Kac: "Can one hear the shape of a drum?" That is, do the Dirichlet eigenvalues of a membrane determine the shape of the membrane? Their answer was "No!", and they used a powerful mathematical technique to produce a counterexample, which in its simplest form is a pair of eight-sided nonconvex polygons.
Gordon, Webb, and Wolpert also noted that a more elementary technique known as transplantation can be used to show that the spectrum of the Laplacian with Dirichlet conditions is the same for both regions. However, actually finding what the eigenvalues are is far more difficult; in fact, it's essentially impossible to do analytically.
Determining the spectrum is also difficult to do numerically, at least to very high precision. Standard finite-element approaches, as implemented in PLTMG and the PDE Toolbox for MATLAB, are unable to produce more than a few accurate digits using reasonable time and storage. Surprisingly, the method of fundamental solutions, a well-known approach that was used successfully to compute the eigenvalues of the L-shaped membrane thirty years ago, fails to produce reliable results for these drums. Some progress can be made with Schwarz-Christoffel mapping techniques to transplant the polygons to squares, but accurate estimates again would require unreasonably large machine resources.
Wu, Sprung, and Martorell (Physical Review E , Jan 1995) present the results of computations with a "mode matching" method and finite differences. Their work appears to improve upon the accuracy of the first 25 eigenvalues, but there is not much indicating how accurate their numbers are. Furthermore, their mode matching method can't be used on general polygons.
A little-known method due to Descloux and Tolley performs far better than all of these. The underlying principle is to exploit the well-known expansions of an eigenfunction near the corners. The criterion that selects eigenvalues is the matching of values and normal derivatives of the local expansions at interfaces within the polygons. Using this method in MATLAB, and incorporating a crucial improvement, I have computed 25 eigenvalues and modes for each polygon to about 12 digits of accuracy. Each eigenpair takes just a few minutes on a Sun workstation.
Here are some side-by-side comparisons of the first twelve modes, with estimates of the eigenvalues. Exercise: Find an analytic expression for the ninth eigenvalue!
Physicists Sridhar and Kudrolli at Northeastern University have actually built microwave cavities in the shapes of the two polygons and determined 54 eigenvalues experimentally. Their results, however, are accurate only to about 0.1%. Take another look at modes 1,3, and 6, to compare directly with Sridhar and Kudrolli's experimental results. For more details, see Physical Review Letters, April 4, 1994, or Science News, September 17, 1994.
One of the most basic uses of eigenmodes is to represent vibrations governed by the wave equation. Below are a few animations of such vibrations, based on combinations of the first sixteen modes. Each movie runs for three periods of the first mode, and is about 200K-300K in MPEG format.
You can obtain a copy of "Eigenmodes of isospectral drums," which appeared in the March 1997 SIAM Review. For hard copies with color figures, send me mail. You can see more examples of isospectral regions in this paper by Buser, Conway, Doyle, and Semmler. They also produce a pair of "homophonic" drums: each has a special point at which the corresponding normalized eigenfunctions have identical values. These drums therefore sound alike when struck at the special points (the pair above would sound different if played in the conventional manner).