Distribution of eigenfrequencies for vibrating plates

Acoustic spectra of free plates with a chaotic billiard shape have been mea sured, and all resonance frequencies in the range 0-500 kHz have been ident ified. The spectral fluctuations are analyzed and compared to predictions o f the Gaussian Orthogonal Ensemble (GOE) of random matrices. The best agree ment is found with a superposition of two independent GOE spectra with equa l density which indicates that two types of eigenmodes contribute to the sa me extent. To explain and predict these results a detailed theoretical anal ysis is carried out below the first cut-off frequency where only flexural a nd in-plane vibrations are possible. Using three-dimensional plate dispersi on relations and two-dimensional models for flexural and in-plane vibration s we obtained two first terms of the asymptotic expansion of the counting f unction of these eigenmodes. The contribution of edge modes is also discuss ed. The results are in a very good agreement with the experimentally measur ed number of modes. The analysis shows that the two types of modes have alm ost equal level density in the measured frequency interval, and this explai ns the observed spectral statistics. For a plate with broken symmetry in the up-down direction (where flexural a nd in-plane modes are strongly coupled) experimentally observed spectral fl uctuations correspond to a single GOE spectrum. Above the first cut-off fre quency a greater complexity of the spectral fluctuations is expected since a larger number of types of modes will contribute to the spectrum.