CATS Acoustic Group
Quantum chaos and vibrations of plates
This page describes some of the experimental results described in my
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
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 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
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.
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.
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.
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
- 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)