Acoustic `Wave Functions'
InspirationSolving the Shrödinger equation for a quantum particle trapped inside a two dimensional well of some geometry, can be a non-trivial task if this geometry is complicated. The solutions, eigenvalues and eigenfunctions, are of great interest within the study of Quantum Chaos. The fluctuation properties of the eigenvalue spectra have been conjectured to possess universal features, described by Random Matrix Theory, when the system is chaotic in the classical limit. This has been confirmed in numerous experimental and numerical studies.
However, little is known about the eigenfunctions, stemming from the fact that they are inaccessible to measurement in quantum systems. One must then turn to numerical methods or - as we do - experimental study of systems that model (qualitatively) the behaviour, one wishes to study. One such model system is a thin elastic plate, subjected to mechanical excitation. It will display a spectrum of eigenfrequencies and the corresponding stationary waves are the acoustic analogues of the quantum wave functions.
Preliminary resultsStarting September, 1995, an experimental study of the properties of stationary waves was planned within the acoustic group at CATS, NBI.
Low frequencyBelow, we present the very first results, obtained for an aluminium plate with the shape of a quarter Sinai stadium. Billiard systems of such geometry are known to be non-integrable. This work was done around October, 1996.
Dark areas correspond to large amplitude, bright areas to low amplitude. Note the complicated pattern of nodal lines (connected regions of zero amplitude), characteristic for non integrable systems. Note also that although the four modes presented here are all around the same frequency, one of them (lower right plot) has significantly fewer nodes. This is evidence of the unique acoustic phenomenon that two separate classes of modes exist for thin plates, flexural modes and extensional modes, and these have different dispersion relations.
High frequencyAt higher frequency the nodal line pattern looks increasingly complicated, see below. Various statistical quantities can be investigated, for which Random Matrix Theory predicts universality in the case of a chaotic geometry. In the Acoustic Group, we have proven for the first time that Random Matrix Theory indeed describes not only the spectral fluctuations of chaotic acoustic systems, but also properties of the acoustic `wave functions'. To learn about the details, you have to read my thesis, but one strong indication that we are not kidding can be seen in the plots below.
The two plots show the probability density for normalized squared amplitude (stepfunction) and the Porter-Thomas distibution (smooth curve) which is a Random Matrix Theory prediction. Note the nice agreement.
Integrable systemsIntegrable systems are expected to display non-universal behaviour. In particular, the amplitude distribution for a wavefunction of an integrable system is expected to show deviatios from the Porter-Thomas distribution. Early in 1997, we started studying a rectangular plate, which is expected to be integrable, at least for the flexural modes.
Low frequencyLet us start out looking at low frequncy modes. Numerous studies - theoretical as well as experimental - have shown that the flexural modes can be classified simply by the number of nodal lines along the axes.
Thus, the 116.0kHz mode is labelled (4,8) and the 117.5kHz mode (13,2). These are acoustic analouges of `good quantum numbers'. Considering only the flexural modes, the rectangle is integrable, non-universal and boring. The two lower plots show the reason why the rectangle is interesting anyway: There is a large class of non-integrable modes, namely the extensional modes mentioned above.
High frequencyJust to make sure that one can really see a difference in the amplitude statistics, let us examine some high frequency modes.
Exercise: Find the `quantum numbers' for the three flexural modes plotted here. For the two highest frequency flexural modes, we plot the amplitude distribution, see below.
The stepfunction represents the experiments while the smooth curve is the Porter-Thomas distribution. Need we say more?
Papers in preparation
Experimental investigation of amplitude correlations for acoustic waves in thin plates, K.Schaadt, M. Oxborrow and C. Ellegaard In preparation