THE FREE VIBRATIONS OF MEMBRANES WITH ELLIPTICAL BOUNDARIES

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Abstract
The formal integration of partial differential equations defined over regions with elliptical boundaries may, in suitable circumstances, be achieved by expansion in terms of Mathieu functions. The numerical integration of such equations by this method, however, would be in general very difficult since numerical information in respect of the Mathieu functions, and more particularly their zeros, is extremely limited. The method described in this paper is intended as a practical alternative technique which, in addition, should be particularly suited to the solution of non-separable eigenvalue problems. To demonstrate it, investigations of the modes of vibration of stretched membranes with constant and variable densities respectively have been undertaken. In either case the problem is finally reduced to the solution of a simple set of simultaneous algebraic equations requiring relatively little labour to solve.

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