Hearing the shape of a drum

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Mathematically ideal drums with membranes of these two different shapes (but otherwise identical) would sound the same, because the eigenfrequencies are all equal, so the timbral spectra would contain the same overtones. This example was constructed by Gordon, Webb and Wolpert. Notice that both polygons have the same area and perimeter.
Mathematically ideal drums with membranes of these two different shapes (but otherwise identical) would sound the same, because the eigenfrequencies are all equal, so the timbral spectra would contain the same overtones. This example was constructed by Gordon, Webb and Wolpert. Notice that both polygons have the same area and perimeter.

To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of basic harmonics, via the use of mathematical theory. "Can One Hear the Shape of a Drum?" was the witty title of an article by Mark Kac in the American Mathematical Monthly 1966 (see References below), but these questions can be traced back all the way to Hermann Weyl.

The frequencies at which a drumhead can vibrate depend on its shape. Known mathematical formulas tell us the frequencies if we know the shape. A central question is: can they tell us the shape if we know the frequencies? No other shape than a square vibrates at the same frequencies as a square. Is it possible for two different shapes to yield the same set of frequencies? Kac did not know the answer to that question.

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[edit] In the language of mathematicians

Somewhat more formally, we are given a domain D, typically in the plane but sometimes in higher dimension, and the eigenvalues of a Dirichlet problem for the Laplacian, which we will denote by λn. The question is: what can be inferred on D if one knows only the values of λn? Two domains are said to be isospectral (or homophonic) if they have the same eigenvalues. Another way to pose the question is: are there two distinct domains that are isospectral?

[edit] The answer

Almost immediately, Milnor produced a pair of 16-dimensional tori that have the same eigenvalues but different shapes. However, the problem in two dimensions remained open until 1992, when Gordon, Webb, and Wolpert constructed, based on the Sunada method, a pair of regions in the plane that have different shapes but identical eigenvalues. The regions are non-convex polygons (see picture). The proof that both regions have the same eigenvalues is rather elementary and uses the symmetries of the Laplacian. This idea has been generalized by Buser et al., who constructed numerous similar examples. So, the answer to Kac' question is: for many shapes, one cannot hear the shape of the drum completely. However, some information can be inferred.

On the other hand, Zelditch proved that the answer to Kac' question is positive if one restrict oneself to certain convex planar regions with analytic boundary. It is not known whether two non-convex analytic domains can have the same eigenvalues.

[edit] Weyl's formula

Weyl's formula states that one can infer the area V of the drum by counting how many of the λns are quite small. We define N(R) to be the number of eigenvalues smaller than R and we get

V=(2\pi)^d \lim_{R\to\infty}\frac{N(R)}{R^{d/2}}\,

where d is the dimension. Weyl also conjectured that the next term in the approximation below would give the perimeter of D. In other words, if A denotes the length of the perimeter (or the surface area in higher dimension), then one should have

\,N(R)=(2\pi)^{-d}VR^{d/2}+cAR^{(d-1)/2}+o(R^{(d-1)/2}).\,

where c is some constant that depends only on the dimension. For smooth boundary, this was proved by V. Ja. Ivrii in 1980.

[edit] The Weyl-Berry conjecture

For non-smooth boundaries, Michael Berry conjectured in 1979 that the correction should be of the order of

RD / 2

where D is the Hausdorff dimension of the boundary. This was disproved by J. Brossard and R. A. Carmona, who then suggested one should replace the Hausdorff dimension with the upper box dimension. In the plane, this was proved if the boundary has dimension 1 (1993), but mostly disproved for higher dimensions (1996). Both results are by Lapidus and Pomerance.

[edit] References

  • John Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proceedings of the National Academy of Sciences of the United States of America 51 (1964), 542ff.
  • Mark Kac, Can one hear the shape of a drum? American Mathematical Monthly 73:4 (1966), part II, 1-23.
  • V. Ja. Ivrii, The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary. Funktsional. Anal. i Prilozhen. 14:2 (1980), 25-34 (In Russian).
  • Jean Brossard and René Carmona, Can one hear the dimension of a fractal? Comm. Math. Phys. 104:1 (1986), 103-122.
  • Michel L. Lapidus, Can one hear the shape of a fractal drum? Partial resolution of the Weyl-Berry conjecture, Geometric analysis and computer graphics (Berkeley, CA, 1988), 119-126, Math. Sci. Res. Inst. Publ., 17, Springer, New York, 1991.
  • C. Gordon, D. Webb, and S. Wolpert, Isospectral plane domains and surfaces via Riemannian orbifolds, Inventiones mathematicae 110 (1992), 1-22.
  • Michel L. Lapidus, Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the Weyl-Berry conjecture, in: Ordinary and Partial Differential Equations (B. D. Sleeman and R. J. Jarvis, eds.), vol. IV, Proc. Twelfth Internat. Conf. (Dundee, Scotland,UK, June 1992), Pitman Research Notes in Math. Series, vol. 289, Longman and Technical, London, 1993, pp. 126-209.
  • Michel L. Lapidus and Carl Pomerance, The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums. Proc. London Math. Soc. (3) 66:1 (1993), 41-69.
  • Peter Buser, John Conway, Peter Doyle and Klaus-Dieter Semmler, Some planar isospectral domains, International Mathematics Research Notices, no. 9 (1994), 391ff.
  • Michel L. Lapidus and Carl Pomerance, Counterexamples to the modified Weyl-Berry conjecture on fractal drums, Math. Proc. Cambridge Philos. Soc. 119:1 (1996), 167-178.
  • M. L. Lapidus and M. van Frankenhuysen, Fractal Geometry and Number Theory: Complex dimensions of fractal strings and zeros of zeta functions, Birkhauser, Boston, 2000. (Revised and enlarged second edition to appear in 2005.)
  • T. Sunada, Riemannian coverings and isospectral manifolds", Ann. of Math. (2) '121 (1985), 169-186
  • S. Zelditch, Spectral determination of analytic bi-axisymmetric plane domains, Geometric and Functional Analysis 10:3 (2000), 628-677.
  • W. Arrighetti, G. Gerosa, Can you hear the fractal dimension of a drum?, arXiv:math.SP/0503748, in “Applied and Industrial Mathematics in Italy”, Series on Advances in Mathematics for Applied Sciences 69, 65–75, World Scientific, 2005. ISBN 978-981-256-368-2

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