Distribution of eigenfrequencies for vibrating plates

Acoustic spectra of free plates with a chaotic billiard shape have been measured, and all resonance frequencies in the range 0-500 kHz have been identified. The spectral fluctuations are analyzed and compared to predictions of the Gaussian Orthogonal Ensemble (GOE) of random matrices. The best agreement is found with a superposition of two independent GOE spectra with equal density which indicates that two types of eigenmodes contribute to the same extent. To explain and predict these results a detailed theoretical analysis is carried out below the first cut-off frequency where only flexural and in-plane vibrations are possible. Using three-dimensional plate dispersion relations and two-dimensional models for flexural and in-plane vibrations we obtained two first terms of the asymptotic expansion of the counting function of these eigenmodes. The contribution of edge modes is also discussed. The results are in a very good agreement with the experimentally measured number of modes. The analysis shows that the two types of modes have almost equal level density in the measured frequency interval, and this explains the observed spectral statistics. For a plate with broken symmetry in the up-down direction (where flexural and in-plane modes are strongly coupled) experimentally observed spectral fluctuations correspond to a single GOE spectrum. Above the first cut-off frequency a greater complexity of the spectral fluctuations is expected since a larger number of types of modes will contribute to the spectrum.