Hearing the shape of a drum
This article includes a list of general references, but it lacks sufficient corresponding inline citations. (March 2022) |
To hear the shape of a drum is to infer information about the shape of the
"Can One Hear the Shape of a Drum?" is the title of a 1966 article by
The frequencies at which a drumhead can vibrate depend on its shape. The
Formal statement
More formally, the drum is conceived as an elastic membrane whose boundary is clamped. It is represented as a
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 λn? 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
The answer
In 1964, John Milnor observed that a theorem on lattices due to Ernst Witt implied the existence of a pair of 16-dimensional flat tori that have the same eigenvalues but different shapes. However, the problem in two dimensions remained open until 1992, when Carolyn Gordon, David Webb, and Scott Wolpert constructed, based on the Sunada method, a pair of regions in the plane that have different shapes but identical eigenvalues. The regions are concave polygons. The proof that both regions have the same eigenvalues uses the symmetries of the Laplacian. This idea has been generalized by Buser, Conway, Doyle, and Semmler[4] who constructed numerous similar examples. So, the answer to Kac's question is: for many shapes, one cannot hear the shape of the drum completely. However, some information can be inferred.
On the other hand,
Weyl's formula
Weyl's formula states that one can infer the area A of the drum by counting how rapidly the λn grow. We define N(R) to be the number of eigenvalues smaller than R and we get
where d is the dimension, and is the volume of the d-dimensional unit ball. Weyl also conjectured that the next term in the approximation below would give the perimeter of D. In other words, if L denotes the length of the perimeter (or the surface area in higher dimension), then one should have
For a smooth boundary, this was proved by Victor Ivrii in 1980. The manifold is also not allowed to have a two-parameter family of periodic geodesics, such as a sphere would have.
The Weyl–Berry conjecture
For non-smooth boundaries, Michael Berry conjectured in 1979 that the correction should be of the order of
where D is the
See also
- Gassmann triple
- Isospectral
- Spectral geometry
- Vibrations of a circular membrane
- an extension to iterated function system fractals[5]
Notes
- ^ Crowell, Rachel (2022-06-28). "Mathematicians Are Trying to 'Hear' Shapes—And Reach Higher Dimensions". Scientific American. Retrieved 2022-11-15.
- ^ "Can One Hear the Shape of a Drum? | Mathematical Association of America".
- JSTOR 2313748.
- ^ Buser et al. 1994.
- )
References
- Abikoff, William (January 1995), "Remembering Lipman Bers" (PDF), Notices of the AMS, 42 (1): 8–18
- Brossard, Jean; Carmona, René (1986). "Can one hear the dimension of a fractal?". Comm. Math. Phys. 104 (1): 103–122. S2CID 121173871.
- Chapman, S.J. (1995). "Drums that sound the same". JSTOR 2975346.
- Giraud, Olivier; S2CID 119289493.
- Bibcode:1996AmSci..84...46G
- S2CID 122258115
- 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, ).
- JSTOR 2313748.
- 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, Math. Sci. Res. Inst. Publ., vol. 17, New York: Springer, pp. 119–126, ISBN 978-1-4613-9713-7
- Lapidus, Michel L. (1993), "Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the Weyl–Berry conjecture", in B. D. Sleeman; R. J. Jarvis (eds.), 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
- 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.)
- Lapidus, Michel L.; Pomerance, Carl (1993),
- 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, S2CID 33567484
- PMID 16591156
- JSTOR 1971195
- Zelditch, S. (2000), "Spectral determination of analytic bi-axisymmetric plane domains", Geometric and Functional Analysis, 10 (3): 628–677, S2CID 16324240
External links
- Simulation showing solutions of the wave equation in two isospectral drums
- 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