Spectral properties of a mixed system using an acoustical resonator

We experimentally study the spectral properties of a mixed system using the flexural modes of a clover shaped plate. The system is called mixed because the corresponding ray dynamics has both chaotic and integrable regions in its phase space. The eigenvalue statistics show intermediate properties between the universal statistics corresponding to chaotic geometries which show Gaussian orthogonal ensemble statistics and integrable geometries that show Poisson statistics. We further investigate the Fourier transform of the peaks to study the influence of the length scales of the plate on the properties of the acoustic resonances. We observe a weak signal of the periodic orbits in the experimental data. Although some of the peaks in the Fourier transform of the eigenvalue spectrum correspond to the shortest stable periodic orbits, other strong peaks are also observed. To understand the role of symmetries, we start with a clover shaped plate belonging to the C(4v) point symmetry group, and progressively reduce the symmetry by sanding one of the edges. A Shnirelman peak in P(s) is observed for the highly symmetric situation due to level clustering.