## 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 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 ( 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 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. -
Mode
`n`contributes (1/`n`)^2 to the solution. -
Mode
`n`contributes 1/`n`to the solution. -
Odd mode
`n`contributes 1/`n`to the solution with an alternating sign. - Approximation to the drums being "plucked" at their centers.
You can obtain a copy of "Eigenmodes of isospectral
drums," which appeared in the March 1997 |