© 2003 by Institute of Mathematics and its Applications
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Fundamental frequency of a doubly connected membrane: a modified perturbation method
1 Department of Mathematics, National Chung Cheng University, Minghsiung, Chiayi 621, Taiwan, Republic of China
The fundamental frequency of a membrane is the square root of the lowest eigenvalue of the negative Laplace operator with Dirichlet boundary conditions. A doubly connected membrane with the inner region of vanishing maximal dimension 2c is considered in this paper. A modified perturbation method is developed to provide an asymptotic expansion (c 
0) for the fundamental frequency of the membrane. The first three order terms of the asymptotic expansion for the fundamental frequency of a doubly connected membrane with the circular inner region are derived explicitly. The results are compared with the exact solutions and the approximations determined by other investigators. The error of the perturbation calculations compared with the exact values is less than 1% as c is less than or equal to 0·25 and is less than 4% as c is less than or equal to 0·35.
Keywords: asymptotic expansions; eigenvalues; frequency; Helmholtz equation; Laplace operator; membrane.
Received 29 October 2001. Revised 20 June 2002.