Chaotic sound waves in a regular billiard
K. Schaadt,
1,2
A. P. B. Tufaile,
3
and C. Ellegaard
1
1
Complexity Lab, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark
2
CORE A/S, Lerso ” parkalle 42, 2100 Copenhagen Ø, Denmark
3
Instituto de Fisica, Universidade de Sa ˜ o Paulo, Caixa Postal 66318, 05315-970 Sa ˜ o Paulo, Brazil
Received 1 September 2002; published 24 February 2003
We present experimental results for the ultrasound transmission spectra and standing wave patterns of a
rectangular block of fused quartz. A comparison is made between our data and an approximation of the
theoretical staircase function for three-dimensional isotropic elasticity. The main emphasis of our study is on
the role of mode conversion in regular ray-splitting billiards. We present the fluctuation statistics and find that
these are described by the Gaussian orthogonal ensemble of random matrix theory, despite the fact that the
system is not classically chaotic, as demonstrated with numerical simulation. Using temperature perturbation,
we find that the vast majority of the resonances are mixtures of transverse and longitudinal wave motion, yet
a small number of special resonances remain pure. We further illuminate this by presenting standing wave
patterns measured on one face of the block.
DOI: 10.1103/PhysRevE.67.026213 PACS numbers: 05.45.Mt, 46.70.De, 62.30.+d
I. INTRODUCTION
Can you calculate the eigenfrequencies of a freely vibrat-
ing rectangular block of isotropic and homogeneous mate-
rial? Confronted with this question, many physicists wrongly
believe that the answer is positive. In fact, not even the exact
average resonance density Weyl formula is known, much
less the actual eigenfrequencies or the eigenfunctions. Re-
newed interest and insight in such classic problems of elas-
todynamics is now arising from the application of methods
used in the field of quantum chaos.
It has been established 1 that the fluctuation properties
of spectra from quantum billiards and from the flexing of
thin plates are identical. This result was confirmed experi-
mentally in Ref. 2, which also presented the theoretical
Weyl formula and found agreement with the experimental
result. Cavity scattering in elastodynamics has been investi-
gated 3, and an important breakthrough occurred very re-
cently when periodic orbits were used for the first time to
calculate the level density for the elastic disc 4. The rect-
angular plate was investigated experimentally 5 and an at-
tempt to calculate the level density for this system, using
periodic orbits, is under way 6.
The conjecture of Ref. 7 states that spectral fluctuations
of quantum chaotic systems obey random matrix theory
RMT, and Ref. 8 states that also the motion of the energy
levels of quantum chaotic systems, under a perturbation of
an external parameter, obeys RMT. Experimental results with
acoustic systems 9–13 strongly suggest the applicability of
these two conjectures to a wider range of systems than quan-
tum chaotic systems 14. In that capacity, these experiments
have served not just as analog systems of quantum billiards,
but more generally, to promote problems of elastodynamics
as interesting problems in their own right.
In this paper, we consider the free, resonant vibrations of
a rectangular block of fused quartz. We are thus studying the
three-dimensional version of the in-plane vibration modes of
the rectangular plate, treated in Ref. 5. For both of these
systems, mode conversion is important: On the boundary, a
purely transverse or purely longitudinal incoming wave is
converted into two outgoing waves, one of each type, ac-
cording to Snell’s law 24. Reference 15 found by numeri-
cal simulation of ray splitting in classical billiards that chao-
ticity is enhanced. Experimental studies of ray-splitting
billiards have been carried out with modified Sinai micro-
wave cavities, comparing results for the spectral fluctuations
and parametric correlators to numerical calculations for the
ray-splitting version of the annular billiard 16, and the tri-
angular step billiard 17. In Ref. 18 the spectra and wave
functions of such experiments are given a semiclassical in-
terpretation. These studies have all focused on ray-splitting
systems that are classically chaotic. Here, we are interested
in systems that are not classically chaotic, even when ray
splitting is present. The rectangular plate is an example of
such a system, and in Ref. 5 it was established experimen-
tally that mode conversion gives rise to chaotic spectral fluc-
tuations for this system. It is precisely this issue we now seek
to clarify for the rectangular block.
We first present a comparison of the measured staircase
function to an approximation including the two leading
terms, first calculated by Dupuis, Mazo, and Onsager 19,
then later by Safarov and Vassiliev 20. We then present the
fluctuation statistics of the resonances, in terms of the nearest
neighbor spacing distribution and the
3
( L ), and compare to
random matrix results. To investigate the character of the
resonances, we measure the distribution of normalized fre-
quency shifts due to a temperature perturbation. This result
can be directly compared to the corresponding result for the
rectangular plate 5, but also serves as a guide for selection
of resonances for which we measure the standing wave pat-
terns by scanning one face of the block. One of the interest-
ing aspects of our results is that we find both mixed states
and states that are ‘‘bouncing-ball’’-like 21.
II. EXPERIMENTAL SETUP
We measure ultrasound transmission spectra of a rectan-
gular block using piezoelectric transducers, see Fig. 1. There
PHYSICAL REVIEW E 67, 026213 2003
1063-651X/2003/672/0262137/$20.00 ©2003 The American Physical Society 67 026213-1