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On hearing the shape of a drum: further results

Published online by Cambridge University Press:  24 October 2008

K. Stewartson  and
R. T. Waechter
K. Stewartson
Affiliation:
Department of Mathematics, University College London
R. T. Waechter
Affiliation:
Department of Mathematics, University College London
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Extract

1. Introduction. The underlying problem is to deduce the shape of a drum or plane uniform membrane from the knowledge of its spectrum of eigenvalues ωn = i√λn. It has been shown by Kac(3) that some progress is possible on establishing the leading terms of the asymptotic expansion of the trace function for small positive t. In particular, for a simply connected membrane Ω bounded by a smooth convex curve Γ for which the displacement satisfies the wave equation ∇2ø = ∂2ø/∂t2 and Dirichlet conditions on Γ

where |Ω| = area of Ω, L = length of Γ, and the constant ⅙ is determined (apart from a factor) on integrating the curvature of the boundary; moreover, if Ω is permitted to have a finite number of smooth convex holes then the constant becomes ⅙(1–r), where r = number of holes.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 69 , Issue 2 , March 1971 , pp. 353 - 363
Copyright
Copyright © Cambridge Philosophical Society 1971

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References

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