Eigenvalues and eigenfunctions of a clover plate |
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Authors: | O Brodier T Neicu A Kudrolli |
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Institution: | (1) Department of Physics, Clark University, Worcester, MA 01610, USA, US |
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Abstract: | We report a numerical study of the flexural modes of a plate using semi-classical analysis developed in the context of quantum
systems. We first introduce the Clover billiard as a paradigm for a system inside which rays exhibit stable and chaotic trajectories.
The resulting phase space explored by the ray trajectories is illustrated using the Poincare surface of section, and shows
that it has both integrable and chaotic regions. Examples of the stable and the unstable periodic orbits in the geometry are
presented. We numerically solve the biharmonic equation for the flexural vibrations of the Clover shaped plate with clamped
boundary conditions. The first few hundred eigenvalues and the eigenfunctions are obtained using a boundary elements method.
The Fourier transform of the eigenvalues show strong peaks which correspond to ray periodic orbits. However, the peaks corresponding
to the shortest stable periodic orbits are not stronger than the peaks associated with unstable periodic orbits. We also perform
statistics on the obtained eigenvalues and the eigenfunctions. The eigenvalue spacing distribution P(s) shows a strong peak and therefore deviates from both the Poisson and the Wigner distribution of random matrix theory at
small spacings because of the C4v symmetry of the Clover geometry. The density distribution of the eigenfunctions is observed to agree with the Porter-Thomas
distribution of random matrix theory.
Received 12 February 2001 and Received in final form 17 April 2001 |
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Keywords: | PACS 05 45 Mt Semiclassical chaos (“ quantum chaos") – 46 40 -f Vibrations and mechanical waves |
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