Nonseparable Solutions of the Helmholtz Wave Equation Used in Approximating Natural Frequencies for Membranes of Arbitrary Shape.

摘要

A new method for approximating the natural frequencies of vibrating membranes was developed. The method employs the nonseparable solutions of the Helmholtz wave equation discovered by D. S. Moseley as the elements of a series expansion for membrane displacement. The boundary conditions for a membrane of any shape are then approximated by selecting the coefficients for the expansion so as to minimize the mean square error over a number of points on the boundary greater than or equal to the number of terms in the series representation. This procedure leads to an eigenvalue problem,and the minima in the eigenvalues lead to approximations of the natural frequencies. The resulting approximate solution satisfies the differential equation exactly and approximates the boundary condition. One advantage of the new method is that it furnishes approximations to frequencies of higher modes,as well as the first. The method was applied to shapes for which the exact solutions are known;the results showed good agreement. (Author)