Estimation of Vibration Frequencies of Linear Elastic Membranes

Abstract

The free motion of a thin elastic linear membrane is described, in a simplyfied model, by a second order linear homogeneous hyperbolic system of partial differential equations whose spatial part is the Laplace Beltrami operator acting on a Riemannian 2- dimensional manifold with boundary. We adapt the estimates of the spectrum of the Laplacian obtained in the last years by several authors for compact closed Riemannian manifolds. To make so, we use the standard technique of the doubled manifold to transform a Riemannian manifold with nonempty boundary (M, ∂M, g) to a compact Riemannian manifold ($M\# M,\tilde g$ ) without boundary. An easy numerical investigation on a concrete semi-ellipsoidic membrane with clamped boundary tests the sharpness of the method.