Random matrix theory and acoustic resonances in plates with an approximate symmetry

Abstract

We discuss a random matrix model of systems with an approximate symmetry and present the spectral fluctuation statistics and eigenvector characteristics for the model. An acoustic resonator like, e.g., an aluminum plate may have an approximate symmetry. We have measured the frequency spectrum and the widths for acoustic resonances in thin aluminum plates, cut in the shape of the so-called three-leaf clover. Due to the mirror symmetry through the middle plane of the plate, each resonance of the plate belongs to one of two mode classes and we show how to separate the modes into these two classes using their measured widths. We compare the spectral statistics of each mode class with results for the Gaussian orthogonal ensemble. By cutting a slit of increasing depth on one face of the plate, we gradually break the mirror symmetry and study the transition that takes place as the two classes are mixed. Presenting the spectral fluctuation statistics and the distribution of widths for the resonances, we find that this transition is well described by the random matrix model.

Phenomenological model for symmetry breaking in a chaotic system

“…This relation is tested in Section IV by a comparison of the NNS distributions of the composite spectrum with the ones obtained by Leitner [6] through a diagonalization of the Rosenzweig-Porter Hamiltonian. Section V shows that the model under investigation is consistent with the results of the acousticresonance experiment by Andersen et al [13]. The conclusion and summary of this work are given in Section VI.…”

“…( 14), with f 1 calculated by inserting these values of Λ into Eqs. ( 11) and (13). We see that the proposed model presents a satisfactory agreement with the numerical results while using the same values for the parameter Λ.…”

Spatial, temporal, and thermal contributions to focusing contrast by time reversal in a cavity

“…12(a,b). The contrast was not affected by small temperature variations but decreased after a threshold that was independent from the number of actuators employed, as predicted by (19). The contrast increased when T decreased and decreased faster for small number of actuators as shown in Fig.…”

“…This result follows from the description of a wave field as the superposition of plane waves [18]. It is known that breaking symmetries in the placements of transducers, deviating from strict right angles, or locally providing smoothly changing normals in the cavity boundaries are sufficient measures to avoid pathological cases [19], especially when dealing with large numbers of modes [20]. A specific example of a pathological case is discussed in Section 3.3.…”

Deviations from Poisson statistics in the spectra of free rectangular thin plates

“…There is a plenitude of studies on vibrating plates, especially in the engineering literature [23][24][25]. There are some works in a wave chaos context [21,[26][27][28][29][30][31]. Despite all efforts made so far, the precise nature of wave chaos in rectangular plates has remained an open question for many years.…”