Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-24T09:43:21.830Z Has data issue: false hasContentIssue false
  • English
  • Français

Eigenvalues of the Laplacian with Neumann boundary conditions

Published online by Cambridge University Press:  17 February 2009

H. P. W. Gottlieb
H. P. W. Gottlieb
Affiliation:
School of Science, Griffith University, Nathan, Queensland 4111.
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.

Various grometrical properties of a domain may be elicited from the asymptotic expansion of a spectral function of the Laplacian operator for that region with apporpriate boundary conditions. Explicit calculations, using analytical formulae for the eigenvalues, are performed for the cases fo Neumann and mixed boundary conditions, extending earlier work involving Dirichet boundary conditions. Two- and three-dimensional cases are considered. Simply-connected regions dealt with are the rectangle, annular sector, and cuboid. Evaluations are carried out for doubly-connected regions, including the narrow annulus, annular cylinder, and thin concentric spherical cavity. The main summation tool is the Poission summation formula. The calculations utilize asymptotic expansions of the zeros of the eigenvalue equations involving Bessel and related functions, in the cases of curved boundaries with radius ratio near unity. Conjectures concerning the form of the contributions due to corners, edges and vertices in the case of Neumann and mixed boundary conditions are presented.

Type
Research Article
Information
The ANZIAM Journal , Volume 26 , Issue 3 , January 1985 , pp. 293 - 309
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Abramowitz, M. and Stegun, I. A., Handbook of mathematical functions (Dover, New York, 1972).Google Scholar
[2]Arfken, G., Methematical methods for physicists (Academic Press, New York, 2nd edition, 1970).Google Scholar
[3]Bauer, H. F., “Tables of zeros of cross product Bessel functions ”, Math. Comp., 18 (1964), 128–135.Google Scholar
[4]Bender, C. M. and Orszag, S. A., Advanced mathematical methods for scientists and engineers (McGraw-Hill, Auckland, 1978).Google Scholar
[5]Bogert, B. P., “Some roots of an equation involving Bessel functions”, J. Math. and Phys. 30 (1951), 102–105.CrossRefGoogle Scholar
[6]Bridge, J. F. and Angrist, S. W., “An extended table of roots of ”, Math. Comp. 16 (1962), 198–204.Google Scholar
[7]Buchholz, H., “Besondere Reihenentwicklungen für eine häufig vorkommende zweireihige Determinante mit Zylinderfunktionen und thre Nullstellen”, Z. Angew. Math. Mech. 29 (1949), 356–367.CrossRefGoogle Scholar
[8]Coulson, C. A. and Jeffery, A., Waves (Longman, London, 2nd edition, 1977).Google Scholar
[9]Dodziuk, J., “Eigenvalues of the Laplacian and the heat equation”, Amer. Math. Monthly 88 (1981), 686–695.CrossRefGoogle Scholar
[10]Dwight, H. B., “Table of roots for natural frequencies in coaxial type cavities”, J. Maht. and Phys. 27 (1948), 84–89.Google Scholar
[11]Dwight, H. B., Mathematical tables of elementary and some higher mathematical functions (Dover, New York, 3rd edition, 1961).Google Scholar
[12]Gottlieb, H. P.W., “Hearing the shape of an annular drum”, J. Austral. Math. Soc. Ser. B 24 (1983), 435–438.CrossRefGoogle Scholar
[13]Gottlieb, H. P. W., “Harmonic properties of the annular membrane”, J. Acoust. Soc. Amer. 66 (1979), 647–650.CrossRefGoogle Scholar
[14]Gottlieb, H. P. W., “On the exceptional zeros of cross-products of derivatives of Bessel functions”, preprint (1983).Google Scholar
[15]Gray, A., Mathews, G. B. and MacRobert, T. M., A treatise on Bessel functions and their applications to physics (Dover, New York, 2nd edition, 1966).Google Scholar
[16]Hildebrand, F. B., A dvanced calulus for applications (Prentice-Hall, Englewood Cliffs, 1962).Google Scholar
[17]Kac, M., “Can one hear the shape of a drum?”, Amer. Math. Monthly 73 (1966), 1–23.CrossRefGoogle Scholar
[18]Kennedy, G., “Boundary terms in the schwinger-De Witt expansion: flat space results”, J. Phys. A 11 (1978), L173–L178.Google Scholar
[19]King, R. W. P., Mimno, H. R. and Wing, A. H., Transmission lines, antennas and wave guides (Dover, New York, 1965).Google Scholar
[20]Kirkman, D., “Graphs and formulas for zeros of cross Product Bessel functions”, J. Math. and Phys. 36 (1957), 371–377.Google Scholar
[21]Lamb, H., The dynamical theory of sound (Dover, New York, 1960).Google Scholar
[22]Lamont, H. R. L., Wave guides (Methuen, New York, 3rd edition, 1969).Google Scholar
[23]McKean, H. P. and Singer, I. M., “Curvature and the eigenvalues of the Laplacian”, J. Differential Geom. 1 (1967), 43–69.CrossRefGoogle Scholar
[24]McMahon, J., “On the roots of the Bessel and certain related functions”, Ann. of Math. 9 (1894–1895), 23–30.CrossRefGoogle Scholar
[25]Morse, P. M. and Ingard, K. U., Theoretical acoustics (McGraw-Hill, New York, 1968).Google Scholar
[26]Pleijel, A., “A study of certain Green's functions with applications in the theory of vibrating membranes”, Ark. Mat. 2 (1953–1954), 553–569.CrossRefGoogle Scholar
[27]Rayleigh, J. W. S., A theory of sound, Vol. 2 (Dover, New York, 2nd edition, 1945).Google Scholar
[28]Sen, S., “A calculation of the Casimir force on a circular boundary”, J. Math. Phys. 22 (1981), 2968–2973.CrossRefGoogle Scholar
[29]Sleeman, B. D., “The inverse problem of acoustic scattering”, IMA J. Appl. Math. 29 (1982), 113–142.CrossRefGoogle Scholar
[30]Stewartson, K. and Waechter, R. T., “On hearing the shape of a drum: further results”, Proc. Cambridge Philos. Soc. 69 (1971), 353–363.CrossRefGoogle Scholar
[31]Stratton, J. A., Electromagnetic theory (McGraw-Hill, New York, 1941).Google Scholar
[32]Truell, R., “Concerning the roots of ”, J. Appl. Phys. 14 (1943), 350–352.CrossRefGoogle Scholar
[33]Waechter, R. T., “On hearing the shape of a drum: an extension to higher dimensions”, Proc. Cambridge Philos. Soc. 72 (1972), 439–447CrossRefGoogle Scholar
[34]Whittaker, E. T. and Watson, G. N., A course of modern analysis (Cambridge Univ. Press, 4th edition, 1965).Google Scholar