Elliptical membranes with smallest second eigenvalue

Author:
B. Andreas Troesch

Journal:
Math. Comp. **27** (1973), 767-772

MSC:
Primary 73.65

MathSciNet review:
0421277

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Abstract: The elliptic membranes with fixed boundary are determined, for which the second eigenfrequency is a minimum if the area or if the circumference is prescribed. The results are compared with those of some other shapes. A remark is made about the overtones of elliptic membranes.

**[1]**Milton Abramowitz and Irene A. Stegun,*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**0167642****[2]**P. R. Garabedian and M. Schiffer,*Variational problems in the theory of elliptic partial differential equations*, J. Rational Mech. Anal.**2**(1953), 137–171. MR**0054819****[3]**L. E. Payne,*Isoperimetric inequalities and their applications*, SIAM Rev.**9**(1967), 453–488. MR**0218975****[4]**G. Pólya,*On the characteristic frequencies of a symmetric membrane*, Math. Z.**63**(1955), 331–337. MR**0073047****[5]**John William Strutt Rayleigh Baron,*The Theory of Sound*, Dover Publications, New York, N. Y., 1945. 2d ed. MR**0016009****[6]**B. A. Troesch and H. R. Troesch,*Eigenfrequencies of an elliptic membrane*, Math. Comput.**27**(1973), 755–765. MR**0421276**, 10.1090/S0025-5718-1973-0421276-2

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1973-0421277-4

Keywords:
Linear membrane vibrations,
isoperimetric shape,
acoustical overtones

Article copyright:
© Copyright 1973
American Mathematical Society