Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-13T06:52:36.904Z Has data issue: false hasContentIssue false

Hearing the shape of an annular drum

Published online by Cambridge University Press:  17 February 2009

Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The asymptotic expansion for a spectral function of the Laplacian operator, involving geometrical properties of the domain, is demonstrated by direct calculation for the case of a doubly-connected region in the form of a narrow annular membrane. By utilizing a known formula for the zeros of the eigenvalue equation containing Bessel functions, the area, total perimeter and connectivity are all extracted explicitly.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Abramowitz, M. and Stegun, I. A. (eds.), Handbook of maihematical functions (Dover, New York, 1972).Google Scholar
[2]Balian, R. and Bloch, C., “Distribution of eigenfrequencies for the wave equation in a finite domain I”, Ann. Physics 60 (1970), 401447.CrossRefGoogle Scholar
[3]De, S., “Vibrations of a non-homogeneous annular membrane”, J. Indian Math. Soc. 36 (1972), 297305.Google Scholar
[4]Fisher, M. E., “On hearing the shape of a drum”, J. Combin. Theory 1 (1966), 105125.CrossRefGoogle Scholar
[5]Gottlieb, H. P. W., “Harmonic properties of the annular membrane”, J. Acoust. Soc. Amer. 66 (1979), 647650.CrossRefGoogle Scholar
[6]Kac, M., “Can one hear the shape of a drum?”, Amer. Math. Monthly 73 (1966), 123.CrossRefGoogle Scholar
[7]McKean, H. P. and Singer, I. M., “Curvature and the eigenvalues of the Laplacian”, J. Differential Geom. 1 (1967), 4369.CrossRefGoogle Scholar
[8]Reed, M. and Simon, B., Methods of modern mathenwtical physics, Vol. 4 (Academic, 1978), Chapter 13.Google Scholar
[9]Sommerfeld, A., Partial differential equations in physics (Academic, 1964).Google Scholar
[10]Ziff, R. M., Uhlenbeck, G. E. and Kac, M., “The ideal Bose-Einstein gas, revisited”, Phys. Rep. 32 (1977), 169248.CrossRefGoogle Scholar