View source for Hearing the shape of a drum

{{Short description|Mathematical problem in spectral theory}}
{{more footnotes|date=March 2022}}
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[[Image:Isospectral drums.svg|frame|right|Mathematically ideal drums with membranes of these two different shapes (but otherwise identical) would sound the same, because the [[eigenfrequency|eigenfrequencies]] are all equal, so the [[Timbre#Spectra|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 [[mathematics|mathematical]] theory.

"Can One Hear the Shape of a Drum?" is the title of a 1966 article by [[Mark Kac]] in the ''[[American Mathematical Monthly]]'' which made the question famous, though this particular phrasing originates with [[Lipman Bers]]. Similar questions can be traced back all the way to physicist [[Arthur Schuster]] in 1882.<ref name="Crowell SciAm 2022">{{cite web | last=Crowell | first=Rachel | title=Mathematicians Are Trying to 'Hear' Shapes—And Reach Higher Dimensions | website=Scientific American | date=2022-06-28 | url=https://www.scientificamerican.com/article/mathematicians-are-trying-to-lsquo-hear-rsquo-shapes/ | access-date=2022-11-15}}</ref> For his paper, Kac was given the [[Lester R.&nbsp;Ford Award]] in 1967 and the [[Chauvenet Prize]] in 1968.<ref>{{Cite web | url=http://www.maa.org/programs/maa-awards/writing-awards/can-one-hear-the-shape-of-a-drum | title=Can One Hear the Shape of a Drum? &#124; Mathematical Association of America}}</ref>

The frequencies at which a drumhead can vibrate depend on its shape. The [[Helmholtz equation]] calculates the frequencies if the shape is known. These frequencies are the [[eigenvalues]] of the [[Laplacian]] in the space. A central question is whether the shape can be predicted if the frequencies are known; for example, whether a [[Reuleaux triangle]] can be recognized in this way.<ref>{{cite journal |first=Mark |last=Kac |author-link=Mark Kac |title=Can One Hear the Shape of a Drum? |url=http://old.maa.org/sites/default/files/pdf/upload_library/22/Ford/MarkKac.pdf |journal=[[American Mathematical Monthly]] |date=April 1966 |volume=73 |issue=4, part 2 |pages=16 |doi=10.2307/2313748 |jstor=2313748}}</ref> Kac admitted that he did not know whether it was possible for two different shapes to yield the same set of frequencies. The question of whether the frequencies determine the shape was finally answered in the negative in the early 1990s by [[Carolyn S. Gordon]], [[David Webb (mathematician)|David Webb]] and [[Scott A. Wolpert]].

==Formal statement==

More formally, the drum is conceived as an elastic membrane whose boundary is clamped. It is represented as a [[Domain (mathematical analysis)|domain]] ''D'' in the [[Plane (mathematics)|plane]]. Denote by ''λ''<sub>''n''</sub> the [[Dirichlet eigenvalue]]s for ''D'': that is, the [[eigenvalue]]s of the [[Dirichlet problem]] for the [[Laplacian]]:

:<math>
\begin{cases}
\Delta u + \lambda u = 0\\
u|_{\partial D} = 0
\end{cases}
</math>

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 series|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''<sub>''n''</sub>? 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 manifold]]s, as well as for other [[elliptic differential operator]]s such as the [[Cauchy–Riemann equations|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==
[[File:Барабаны.gif|thumb|One-parameter family of isospectral drums]]
[[File:Eigenmodes_of_GWW_Isospectral_Domains.png|thumb|306x306px|Eigenmodes and corresponding eigenvalues of the Laplace operator on the GWW domains]]

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 S. Gordon|Carolyn Gordon]], [[David Webb (mathematician)|David Webb]], and [[Scott A. Wolpert|Scott Wolpert]] constructed, based on the [[Toshikazu Sunada|Sunada method]], a pair of regions in the plane that have different shapes but identical eigenvalues. The regions are [[concave polygon]]s. 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{{sfn|Buser|Conway|Doyle|Semmler|1994}} 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, [[Steve Zelditch]] proved that the answer to Kac's question is positive if one imposes restrictions to certain [[convex set|convex]] planar regions with [[analytic function|analytic]] boundary. It is not known whether two non-convex analytic domains can have the same eigenvalues. It is known that the set of domains isospectral with a given one is compact in the C<sup>∞</sup> topology. Moreover, the sphere (for instance) is spectrally rigid, by [[Cheng's eigenvalue comparison theorem]]. It is also known, by a result of Osgood, Phillips, and Sarnak that the moduli space of [[Riemann surface]]s of a given genus does not admit a continuous isospectral flow through any point, and is compact in the Fréchet–Schwartz topology.

==Weyl's formula==
{{main|Weyl law}}
Weyl's formula states that one can infer the area ''A'' of the drum by counting how rapidly the ''λ''<sub>''n''</sub> grow. We define ''N''(''R'') to be the number of eigenvalues smaller than ''R'' and we get

: <math>A = \omega_d^{-1}(2\pi)^d \lim_{R\to\infty}\frac{N(R)}{R^{d/2}},</math>

where ''d'' is the dimension, and <math>\omega_d</math> 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

: <math>N(R) = (2\pi)^{-d}\omega_d AR^{d/2} \mp \frac{1}{4}(2\pi)^{-d+1}\omega_{d-1} LR^{(d-1)/2} + o(R^{(d-1)/2}).</math>

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 (physicist)|Michael Berry]] conjectured in 1979 that the correction should be of the order of

: <math>R^{D/2},</math>

where ''D'' is the [[Hausdorff dimension]] of the boundary. This was disproved by J.&nbsp;Brossard and R.&nbsp;A. Carmona, who then suggested that one should replace the Hausdorff dimension with the [[upper box dimension]]. In the plane, this was proved if the boundary has dimension&nbsp;1 (1993), but mostly disproved for higher dimensions (1996); both results are by [[:fr:Michel Lapidus|Lapidus]] and [[Carl Pomerance|Pomerance]].

==See also==
* [[Gassmann triple]]
* [[Isospectral]]
* [[Spectral geometry]]
* [[Vibrations of a circular membrane]]

==Notes==
{{Reflist}}

==References==
* {{citation|url=https://www.ams.org/notices/199501/bers.pdf|first=William|last=Abikoff|title=Remembering Lipman Bers|journal=[[Notices of the AMS]]|date=January 1995|pages=8–18|volume=42|issue=1}}
* {{cite journal|doi=10.1007/BF01210795|first1=Jean|last1=Brossard|first2=René|last2=Carmona|title=Can one hear the dimension of a fractal?|journal=Communications in Mathematical Physics|volume=104|issue=1|year=1986|pages=103–122|bibcode=1986CMaPh.104..103B|s2cid=121173871|url=http://projecteuclid.org/euclid.cmp/1104114935}}
* {{citation |first1=Peter |last1=Buser |author-link=Jürg Peter Buser |first2=John |last2=Conway |authorlink2=John Horton Conway |first3=Peter |last3=Doyle |first4=Klaus-Dieter |last4=Semmler |title=Some planar isospectral domains |journal=[[International Mathematics Research Notices]] |volume=1994 |year=1994 |issue=9 |pages=391–400 |arxiv=1005.1839 |doi=10.1155/S1073792894000437 |doi-access=free}}
* {{cite journal|last=Chapman|first=S.J.|year=1995|title=Drums that sound the same|journal=[[American Mathematical Monthly]]|volume=102|issue=February|pages=124–138|doi=10.2307/2975346|jstor=2975346}}
* {{cite journal|last=Giraud|first=Olivier|author2=Thas, Koen|author2-link=Thas, Koen|title=Hearing shapes of drums – mathematical and physical aspects of isospectrality|journal=Reviews of Modern Physics|year=2010|volume=82|issue=3|pages=2213–2255|doi=10.1103/RevModPhys.82.2213|arxiv=1101.1239|bibcode=2010RvMP...82.2213G|s2cid=119289493}}
* {{citation|first1=Carolyn|last1=Gordon|author1-link=Carolyn S. Gordon|first2=David|last2=Webb|author2-link=David Webb (mathematician)|title=You can't hear the shape of a drum|journal=[[American Scientist]] |year=1996|volume=84|issue=January–February|pages=46–55|bibcode=1996AmSci..84...46G}}
* {{citation|doi=10.1007/BF01231320|first1=C.|last1=Gordon|author1-link=Carolyn S. Gordon|first2=D.|last2=Webb|author2-link=David Webb (mathematician)|first3=S.|last3=Wolpert|title=Isospectral plane domains and surfaces via Riemannian orbifolds|journal=Inventiones Mathematicae|volume=110|year=1992|issue=1|pages=1–22|bibcode = 1992InMat.110....1G |s2cid=122258115}}
* {{citation|first=V. Ja.|last=Ivrii|title=The second term of the spectral asymptotics for a Laplace–Beltrami operator on manifolds with boundary|journal=Funktsional. Anal. I Prilozhen|volume=14|issue=2|year=1980|pages=25–34|doi=10.1007/BF01086550|s2cid=123935462}} (In [[Russian language|Russian]]).
* {{cite journal|last=Kac|author-link=Mark Kac|first=Mark|title=Can One Hear the Shape of a Drum?|url=http://old.maa.org/sites/default/files/pdf/upload_library/22/Ford/MarkKac.pdf|journal=[[American Mathematical Monthly]]|date=April 1966|volume=73|issue=4, part 2|pages=1–23|doi=10.2307/2313748|jstor=2313748}}
* {{citation|first=Michel L.|last=Lapidus|chapter=Can One Hear the Shape of a Fractal Drum? Partial Resolution of the Weyl-Berry Conjecture |title=Geometric Analysis and Computer Graphics |volume=17|pages=119–126|series=Math. Sci. Res. Inst. Publ.|issue=17|publisher=Springer|location=New York|year=1991|doi=10.1007/978-1-4613-9711-3_13|isbn=978-1-4613-9713-7}}
* {{citation|first=Michel L.|last=Lapidus|contribution=Vibrations of fractal drums, the [[Riemann hypothesis]], waves in fractal media, and the Weyl–Berry conjecture|title=Ordinary and Partial Differential Equations, Vol IV, Proc. Twelfth Internat. Conf. (Dundee, Scotland, UK, June 1992)|editor1=B. D. Sleeman |editor2=R. J. Jarvis|series=Pitman Research Notes in Math. Series|volume=289|publisher=Longman and Technical|location=London|year=1993|pages=126–209}}
* {{citation|first1=M. L.|last1=Lapidus|first2=M.|last2=van Frankenhuysen|title=Fractal Geometry and Number Theory: Complex dimensions of fractal strings and zeros of zeta functions|publisher=Birkhauser|location=Boston|year=2000}}. (Revised and enlarged second edition to appear in 2005.)
* {{citation|doi=10.1112/plms/s3-66.1.41|first1=Michel L.|last1=Lapidus|first2=Carl|last2=Pomerance|title=The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums|journal=Proceedings of the London Mathematical Society |series=Series 3|volume=66|issue=1|year=1993|pages=41–69|title-link=Riemann zeta-function|citeseerx=10.1.1.526.854}}
* {{citation|doi=10.1017/S0305004100074053|first1=Michel L.|last1=Lapidus|first2=Carl|last2=Pomerance|authorlink2=Carl Pomerance|title=Counterexamples to the modified Weyl–Berry conjecture on fractal drums|journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]|volume=119|issue=1|year=1996|pages=167–178|bibcode = 1996MPCPS.119..167L |s2cid=33567484 }}
* {{citation|first=John|last=Milnor|author-link=John Milnor|title=Eigenvalues of the Laplace operator on certain manifolds|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=51|issue=4|year=1964|pages=542ff|bibcode = 1964PNAS...51..542M |doi = 10.1073/pnas.51.4.542|pmid=16591156|pmc=300113|doi-access=free}}
* {{citation|doi=10.2307/1971195|first=T.|last=Sunada|author-link=Toshikazu Sunada|title=Riemannian coverings and isospectral manifolds|journal=Ann. of Math. |series= 2|volume=121|issue=1|year=1985|pages=169–186|jstor=1971195}}
* {{citation|doi=10.1007/PL00001633|first=S.|last=Zelditch|title=Spectral determination of analytic bi-axisymmetric plane domains|journal=Geometric and Functional Analysis|volume=10|issue=3|year=2000|pages=628–677|arxiv=math/9901005|s2cid=16324240}}

==External links==
* [https://www.youtube.com/watch?v=RaQZ11ydBZo Simulation] showing solutions of the wave equation in two isospectral drums
* [https://web.archive.org/web/20210225100031/http://www.math.udel.edu/~driscoll/research/drums.html Isospectral Drums] by Toby Driscoll at the University of Delaware
* [http://math.dartmouth.edu/~doyle/docs/drum/drum.pdf Some planar isospectral domains] by Peter Buser, [[John Horton Conway]], Peter Doyle, and Klaus-Dieter Semmler
** [https://www.youtube.com/watch?v=1nOQF8MRK3w 3D rendering of the Buser-Conway-Doyle-Semmler homophonic drums]
* [https://web.archive.org/web/20110727051224/http://enterprise.maa.org/mathland/mathland_4_14.html Drums That Sound Alike] by Ivars Peterson at the Mathematical Association of America web site
* {{MathWorld | title=Isospectral Manifolds | urlname=IsospectralManifolds}}
* {{springer|title=Dirichlet eigenvalue|id=d/d130170|first=Rafael D.|last=Benguria}}


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