A higher order asymptotic approximation for the fundamental frequency of a multiply connected membrane

The fundamental frequency of a fixed membrane is the square root of the lowest eigenvalue of negative Laplace operator with Dirichlet boundary conditions. A multiply connected membrane with inner cores of vanishing maximal dimensions 2 c j is considered in the present article. The modified perturbation method developed for a doubly connected membrane is extended to provide a general formula for the fundamental frequency of the multiply connected membrane. A higher order asymptotic approximation (as c j → 0 ) for the fundamental frequency of a membrane with inner circular cores of radius c j is specified. It is an excellent extension of the results in the literature. Moreover, a second-order asymptotic approximation (as c → 0 ) for the fundamental frequency of a circular membrane of radius 1 with finitely many inner circular cores of small radius c is found and computed explicitly. The effects of the positions of the inner cores on the second-order asymptotic approximation are investigated. The accuracy of the second-order asymptotic approximation is also shown by the comparisons among the asymptotic approximations and the numerical values computed by other investigators.