Comment on spectral statistics in elastodynamics

Recently, a set of high eigenfrequencies of small aluminum blocks have been measured and analyzed quantitatively by using methods initially developed in nuclear physics on the basis of an analogy with random matrix theory [R. L. Weaver, J. Acoust. Soc. Am. 85, 1001 (1989)]. At the foundation of the application of random matrix theory is the (classical motion) ⇔ (finite frequency vibration) correspondence according to which the nature of classical geometrical acoustic trajectories determine the correspondence with random matrix theory and thereby the structure of the spectrum (classical chaos→GOE spectrum; regular motion→Poissonian spectrum). Although the classical geometrical acoustics trajectories are not chaotic in the system studied by Weaver, he finds a good agreement with the GOE prediction, in contradiction to the usual wisdom concerning the classical-vibration correspondence. It is suggested that this paradox stems from finite wavelength effects that introduce a coarse graining in the relevant classical dynamics. Interpreted in this way, Weaver’s results seem to confirm that the GOE spectrum initially studied for simple scalar Helmholtz equation may also be valid in the more complicated case of elastodynamics.