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.
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.
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.[3] 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 Gordon, Webb and Wolpert.
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?
One-parameter family of isospectral drumsEigenmodes 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 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, Steve Zelditch proved that the answer to Kac's question is positive if one imposes restrictions to certain convex planar regions with 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∞ 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 surfaces 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 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 conjectureedit
For non-smooth boundaries, Michael Berry conjectured in 1979 that the correction should be of the order of
where D is the Hausdorff dimension of the boundary. This was disproved by J. Brossard and R. 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 1 (1993), but mostly disproved for higher dimensions (1996); both results are by Lapidus and Pomerance.
^Arrighetti, W.; Gerosa, G. (2005). Can you hear the fractal dimension of a drum?. pp. 65–75. arXiv:math.SP/0503748. doi:10.1142/9789812701817_0007. ISBN978-981-256-368-2. S2CID 119709456. {{cite book}}: |journal= ignored (help)
Referencesedit
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. Bibcode:1986CMaPh.104..103B. doi:10.1007/BF01210795. S2CID 121173871.
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, 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, doi:10.1007/BF01086550, S2CID 123935462 (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, Math. Sci. Res. Inst. Publ., vol. 17, New York: Springer, pp. 119–126, doi:10.1007/978-1-4613-9711-3_13, ISBN978-1-4613-9713-7
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 (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, S2CID 33567484
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 (4): 542ff, Bibcode:1964PNAS...51..542M, doi:10.1073/pnas.51.4.542, PMC300113, 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, arXiv:math/9901005, doi:10.1007/PL00001633, S2CID 16324240
External linksedit
Simulation showing solutions of the wave equation in two 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
3D rendering of the Buser-Conway-Doyle-Semmler homophonic drums
by Ivars Peterson at the Mathematical Association of America web site
hearing, shape, drum, this, article, includes, list, general, references, lacks, sufficient, corresponding, inline, citations, please, help, improve, this, article, introducing, more, precise, citations, march, 2022, learn, when, remove, this, template, messag. This article includes a list of general references but it lacks sufficient corresponding inline citations Please help to improve this article by introducing more precise citations March 2022 Learn how and when to remove this template message 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 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 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 1 For his paper Kac was given the Lester R Ford Award in 1967 and the Chauvenet Prize in 1968 2 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 3 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 Gordon Webb and Wolpert Contents 1 Formal statement 2 The answer 3 Weyl s formula 4 The Weyl Berry conjecture 5 See also 6 Notes 7 References 8 External linksFormal statement editMore 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 ln the Dirichlet eigenvalues for D that is the eigenvalues of the Dirichlet problem for the Laplacian D u l u 0 u D 0 displaystyle begin cases Delta u lambda u 0 u partial D 0 end cases nbsp 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 ln 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 edit nbsp One parameter family of isospectral drums nbsp Eigenmodes and corresponding eigenvalues of the Laplace operator on the GWW domainsIn 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 Steve Zelditch proved that the answer to Kac s question is positive if one imposes restrictions to certain convex planar regions with 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 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 surfaces of a given genus does not admit a continuous isospectral flow through any point and is compact in the Frechet Schwartz topology Weyl s formula editMain article Weyl law Weyl s formula states that one can infer the area A of the drum by counting how rapidly the ln grow We define N R to be the number of eigenvalues smaller than R and we get A w d 1 2 p d lim R N R R d 2 displaystyle A omega d 1 2 pi d lim R to infty frac N R R d 2 nbsp where d is the dimension and w d displaystyle omega d nbsp 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 N R 2 p d w d A R d 2 1 4 2 p d 1 w d 1 L R d 1 2 o R d 1 2 displaystyle 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 nbsp 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 editFor non smooth boundaries Michael Berry conjectured in 1979 that the correction should be of the order of R D 2 displaystyle R D 2 nbsp where D is the Hausdorff dimension of the boundary This was disproved by J Brossard and R 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 1 1993 but mostly disproved for higher dimensions 1996 both results are by Lapidus and Pomerance See also editGassmann triple Isospectral Spectral geometry Vibrations of a circular membrane an extension to iterated function system fractals 5 Notes edit 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 Kac Mark April 1966 Can One Hear the Shape of a Drum PDF American Mathematical Monthly 73 4 part 2 16 doi 10 2307 2313748 JSTOR 2313748 Buser et al 1994 Arrighetti W Gerosa G 2005 Can you hear the fractal dimension of a drum pp 65 75 arXiv math SP 0503748 doi 10 1142 9789812701817 0007 ISBN 978 981 256 368 2 S2CID 119709456 a href Template Cite book html title Template Cite book cite book a journal ignored help References editAbikoff William January 1995 Remembering Lipman Bers PDF Notices of the AMS 42 1 8 18 Brossard Jean Carmona Rene 1986 Can one hear the dimension of a fractal Comm Math Phys 104 1 103 122 Bibcode 1986CMaPh 104 103B doi 10 1007 BF01210795 S2CID 121173871 Buser Peter Conway John Doyle Peter Semmler Klaus Dieter 1994 Some planar isospectral domains International Mathematics Research Notices 1994 9 391 400 doi 10 1155 S1073792894000437 Chapman S J 1995 Drums that sound the same American Mathematical Monthly 102 February 124 138 doi 10 2307 2975346 JSTOR 2975346 Giraud Olivier Thas Koen 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 S2CID 119289493 Gordon Carolyn Webb David 1996 You can t hear the shape of a drum American Scientist 84 January February 46 55 Bibcode 1996AmSci 84 46G 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 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 doi 10 1007 BF01086550 S2CID 123935462 In Russian Kac Mark April 1966 Can One Hear the Shape of a Drum PDF American Mathematical Monthly 73 4 part 2 1 23 doi 10 2307 2313748 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 doi 10 1007 978 1 4613 9711 3 13 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 The Riemann zeta function and the one dimensional Weyl Berry conjecture for fractal drums Proc London Math Soc Series 3 66 1 41 69 CiteSeerX 10 1 1 526 854 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 S2CID 33567484 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 4 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 arXiv math 9901005 doi 10 1007 PL00001633 S2CID 16324240External links editSimulation 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 3D rendering of the Buser Conway Doyle Semmler homophonic drums 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 Retrieved from https en wikipedia org w index php title Hearing the shape of a drum amp oldid 1174290165 The Weyl Berry conjecture, wikipedia, wiki, book, books, library,
