Eigenvalues and eigenfunctions of a clover plate

We report a numerical study of the flexural modes of a plate using semi-cla ssical analysis developed in the context of quantum systems. We first intro duce the Clover billiard as a paradigm for a system inside which rays exhib it stable and chaotic trajectories. The resulting phase space explored by t he ray trajectories is illustrated using the Poincare surface of section, a nd shows that it has both integrable and chaotic regions. Examples of the s table and the unstable periodic orbits in the geometry are presented. We nu merically 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 co rrespond to ray periodic orbits. However, the peaks corresponding to the sh ortest stable periodic orbits are not stronger than the peaks associated wi th unstable periodic orbits. We also perform statistics on the obtained eig envalues and the eigenfunctions. The eigenvalue spacing distribution P(s) s hows a strong peak and therefore deviates from both the Poisson and the Wig ner distribution of random matrix theory at small spacings because of the C -4v symmetry of the Clover geometry. The density distribution of the eigenf unctions is observed to agree with the Porter-Thomas distribution of random matrix theory.