Time reversal experiments for acoustic and electromagnetic waves promise to bring new useful developments [1] in material and communication sciences.
      A suitable object to study time reversed acoustic waves is aluminum acoustic cavity shown in figure 1. There are two piezoelectric transducers connected to it from the top and from the bottom, playing the role of the source and the receiver of acoustic waves.
 

The experiment works in general as follows. Short acoustic pulse is launched at the source location and due to complex wave scattering travels to the receiver location via multiple ‘delivery ways’. Figure 2 below shows a section of the reverberation response, caused by a short excitation pulse at time=0 ms. This cavity response is recorded at the receiver location. Different parts of the response, arriving with different time delays are shown in figure 2 with different colors.

After recording a part of the long slowly decaying response, like those shown in figure 2, this signal is replayed at the receiver location in the opposite direction of time. This implies that oscillations which reached the receiver first are to be released last. Figure 3 shows result of the measurement of the cavity response upon the time reversed section of the previously obtained (figure 2) response. Each response shown in figure 3 has the color corresponding to the previous figure, the same as the part of the signal that was generated backwards in time and used as input signal.

Signals shown in figure 3 have the following structure. The direct response to an incoming signal appears immediately, delayed no more than about 1 mks. However after the excited cavity oscillations last for quite a longer time it is possible to observe the spike of noticeably higher amplitude (figure 3), which is a reconstruction of the short incoming pulse, which originally caused the response in figure 1. The amplitude in figure 3 is normalized to the maximum value of the reconstructed peak for each signal.
Time marks in figures 2 and 3 demonstrate that the reconstruction pulse is delayed in figure 3 exactly as much as the time difference between the original incoming short pulse the end of the response part (figure 1) that has been reversed. 
The described experiment itself is already useful tool to test directly how reversible are ways of acoustic energy transfer in the studied system. However, this kind of experiment does not collect yet complete information about type and origin of time reversal symmetry breaking for wave propagation, what would enhance efficiency of all applications in science and technology. Additional analysis of the mentioned reverberation response due to random matrix theory (further named RMT) seems to offer this. Using RMT [2] one can compare statistical properties of eigenvalues of ensembles of random matrices, invariant under given type of transformation, to statistics of resonances in wave scattering experiments in physical systems.
The distribution of frequency intervals between the cavity resonances found in the measured response spectra of the studied aluminum cavity is shown in figure 4. It is comparable to those predicted by RMT for so called Gaussian Orthogonal Ensemble of matrices. This is the statistics expected for the case of waves that obey equations with time reversal symmetry. The correspondence was seen when accounting for certain amount of lost resonances.

The distribution expected for the random arrangement of resonances in the studied frequency band is shown as red dashed curve (exponential distribution). The distribution expected for Gaussian Orthogonal Ensemble is given by the blue dashed curve and the distribution shown in green is the distribution accounting for 25 % of resonances lost.
Statistics of cavity resonances in cavity response spectra can be used to study breaking of time reversal symmetry of the elastic wave dynamics in the cavity.

References
[1] M. Fink, Physics Today, Vol. 50, Iss. 3, p. 34, 1997.
[2] H.-J. Stöckmann, Quantum Chaos, Cambridge University Press, 1999.