Hearing the shape of a drum

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 overtones, 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 , 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. The Helmholtz equation tells us the frequencies if we know the shape. These frequencies are the eigenvalues of the Laplacian in the region. 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.

Formal statement

More formally, the drum is conceived as an elastic membrane whose boundary is clamped. It is represented as a domain D in the plane. Denote by λ_n the Dirichlet eigenvalues for D: that is, the eigenvalues of the Dirichlet problem for the Laplacian:

${\begin{cases}\Delta u+\lambda u=0\\u|_{\partial D}=0\end{cases}}$

Two domains are said to be isospectral (or homophonic) if they have the same eigenvalues. The term "homophonic" is justified because the Dirichlet eigenvalues are precisely the fundamental tones that the drum is capable of producing: they appear naturally as Fourier coefficients in the solution wave equation with clamped boundary.

Therefore the question may be reformulated as: what can be inferred on D if one knows only the values of λ_n? Or, more specifically: are there two distinct domains that are isospectral?

Related problems can be formulated for the Dirichlet problem for the Laplacian on domains in higher dimensions or on Riemannian manifolds, as well as for other elliptic differential operators such as the Cauchy–Riemann operator or Dirac operator. Other boundary conditions besides the Dirichlet condition, such as the Neumann boundary condition, can be imposed. See spectral geometry and isospectral as related articles.

The answer

Almost immediately, John Milnor observed that a theorem due to Ernst Witt implied the existence of 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

• <ref>Brossard, Jean; Carmona, René (1986). "Can one hear the dimension of a fractal?". Comm. Math. Phys. 104 (1): 103–122. Bibcode:1986CMaPh.104..103B. doi:10.1007/BF01210795.<ref>

References

• Buser, Peter; Conway, John; Doyle, Peter; Semmler, Klaus-Dieter (1994), "Some planar isospectral domains", International Mathematics Research Notices, 9: 391ff
• Chapman, S.J. (1995), "Drums that sound the same", American Mathematical Monthly (February): 124–138
• Giraud, Olivier (2010). "Hearing shapes of drums - mathematical and physical aspects of isospectrality". Reviews of Modern Physics. 82 (3): 2213–2255. arXiv:1101.1239. Bibcode:2010RvMP...82.2213G. doi:10.1103/RevModPhys.82.2213. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Gordon, Carolyn; Webb, David, "You can't hear the shape of a drum", American Scientist, 84 (January–February): 46–55
• Gordon, C.; Webb, D.; Wolpert, S. (1992), "Isospectral plane domains and surfaces via Riemannian orbifolds", Inventiones Mathematicae, 110 (1): 1–22, Bibcode:1992InMat.110....1G, doi:10.1007/BF01231320
• Ivrii, V. Ja. (1980), "The second term of the spectral asymptotics for a Laplace–Beltrami operator on manifolds with boundary", Funktsional. Anal. i Prilozhen, 14 (2): 25–34 (In Russian).
• Lapidus, Michel L. (1991), "Can one hear the shape of a fractal drum? Partial resolution of the Weyl–Berry conjecture", Geometric analysis and computer graphics (Berkeley, CA, 1988), Math. Sci. Res. Inst. Publ. (17), New York: Springer: 119–126
• Lapidus, Michel L. (1993), "Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the Weyl–Berry conjecture", Ordinary and Partial Differential Equations, Vol IV, Proc. Twelfth Internat. Conf. (Dundee, Scotland,UK, June 1992), Pitman Research Notes in Math. Series, vol. 289, London: Longman and Technical, pp. 126–209 {{citation}}: Unknown parameter |editors= ignored (|editor= suggested) (help)
• Lapidus, Michel L.; Pomerance, Carl (1993), "The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums", Proc. London Math. Soc. (3), 66 (1): 41–69, doi:10.1112/plms/s3-66.1.41
• Lapidus, Michel L.; Pomerance, Carl (1996), "Counterexamples to the modified Weyl–Berry conjecture on fractal drums", Math. Proc. Cambridge Philos. Soc., 119 (1): 167–178, Bibcode:1996MPCPS.119..167L, doi:10.1017/S0305004100074053
• Lapidus, M. L.; van Frankenhuysen, M. (2000), Fractal Geometry and Number Theory: Complex dimensions of fractal strings and zeros of zeta functions, Boston: Birkhauser. (Revised and enlarged second edition to appear in 2005.)
• Milnor, John (1964), "Eigenvalues of the Laplace operator on certain manifolds", Proceedings of the National Academy of Sciences of the United States of America, 51: 542ff, Bibcode:1964PNAS...51..542M, doi:10.1073/pnas.51.4.542, PMC 300113, PMID 16591156
• Sunada, T. (1985), "Riemannian coverings and isospectral manifolds", Ann. Of Math. (2), 121 (1): 169–186, doi:10.2307/1971195, JSTOR 1971195
• Zelditch, S. (2000), "Spectral determination of analytic bi-axisymmetric plane domains", Geometric and Functional Analysis, 10 (3): 628–677, doi:10.1007/PL00001633

External links

• Isospectral Drums by Toby Driscoll at the University of Delaware
• Some planar isospectral domains by Peter Buser, John Horton Conway, Peter Doyle, and Klaus-Dieter Semmler
• Drums That Sound Alike by Ivars Peterson at the Mathematical Association of America web site
• Weisstein, Eric W. "Isospectral Manifolds". MathWorld.
• Benguria, Rafael D. (2001) [1994], "Dirichlet eigenvalue", Encyclopedia of Mathematics, EMS Press