Applications of Mathematics

, Volume 63, Issue 1, pp 37–53 | Cite as

Estimation of Vibration Frequencies of Linear Elastic Membranes

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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.

Keywords

membrane Laplacian estimation of frequencies 

MSC 2010

74K15 53C20 53C21 58C40 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.Dipartimento di Scienze di Base e Applicate per l’IngegneriaUniversità degli studi di Roma La SapienzaRomaItaly

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