CATS Acoustic Group

Quantum chaos and vibrations of plates

This page describes some of the experimental results described in my M.Sci. thesis

The mean level density

From the Theory of Elasticity, E. Hugues (Orsay, France) have predicted the accumulated level density for the resonance spectrum of a plate. This is called the Weyl formula, and in the figure below, we compare it to the experimental spectrum for a Sinai stadium plate. The Weyl formula represents the experimental data well, only 6-7 levels are missing in the experimental curve.
Theoretical Weyl formula for the Sinai stadium plate Difference between theoretical and experimental staircase


A plate of any given shape will be symmetric with respect to the middle plane of the plate. The spectrum can therefore be divided into two symmetry classes, according to whether the deformation vector is symmetric or antisymmetric with respect to reflection around the middle plane. The contribution from these two classes is plotted explicit in the staircase plot above, where one can see that the two classes have almost equal density in the frequency range 200-500 kHz.
The two symmetry classes

The symmetric modes have horizontal deformation, and they are called flexural modes. The antisymmetric modes have vertical deformation, and they are called extensional modes.

The Sinai stadium plate Sinai stadium

Below, we present the spectral statistics for a Sinai stadium plate with no symmetries except the one mentioned above. Here, we use the unfolded spectrum, where we have divided the frequencies with the mean level spacing. The left plot is the distribution of the distance between neighbouring levels, and the plot to the right is the spectral rigidity, measuring the size of the fluctuations in the spectrum.
NND distribution for the Sinai stadium plate Delta_3 distribution for the Sinai stadium plate
The '1 GOE'-curve is the prediction from Random Matrix Theory for a chaotic spectrum. A regular non-chaotic spectrum usually follows the Poisson distribution (i.e. the distribution of random numbers). Due to the above mentioned symmetry, we have two independent (i.e. non-interacting) classes of modes (flexural and extensional) with almost equal density. Therefore, we do not see the '1 GOE' distribution (for fully chaotic systems), but '2 GOE' instead, which corresponds to a superposition of two non-interacting chaotic spectra with equal density.

Symmetry breaking

The two cuts in the plate

By making two cuts on one side of the plate, we break the symmetry. As shown above, an extensional mode (i.e. horizontal deformation) is now turned into a 'mixed' mode after passing the cut. The two classes of modes are therefore no longer separable. This effect changes the statistics significantly: Now we see only one GOE distribution.

NND distribution for the plate with cuts Delta3 distribution for the plate with cuts

Other shapes

If the plate has more symmetries, things get more complicated. Consider for example the rectangular plate, which has 3 symmetry planes, and therefore 8 different symmetry classes. Furthermore, these classes cannot be considered totally independent, due to the boundary conditions. This makes it more difficult to obtain theoretical predictions.

The circular and the square plate are integrable systems, and their spectra follows the Poisson distribution as expected. However, one must take into account that degenerate levels sometimes split up into two or more close-lying levels. The spectra of these two plates have been compared to numerical solutions of the elasticity equations and to simulations using the Finite Element Method. In both cases, the first 100 levels can be recognised in the experimental spectrum.

A detailed description of these results can be found in my M.Sci. thesis.


  • P. Bertelsen, C. Ellegaard, E. Hugues and M. Oxborrow, Distribution of eigenfrequencies for vibrating plates
  • P. Bertelsen, C. Ellegaard, M. Oxborrow and K. Schaadt Universality in spectra of vibrating plates (VERY preliminary version)

CATS Acoustic Group
Preben Bertelsen's homepage
September 10 , 1997
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