Periodic orbit analysis of an elastodynamic resonator using shape deformation

Abstract

We report the first definitive experimental observation of periodic orbits (POs) in the spectral properties of an elastodynamic system. The Fourier transform of the density of flexural modes show peaks that correspond to stable and unstable POs of a clover shaped quartz plate. We change the shape of the plate and find that the peaks corresponding to the POs that hit only the unperturbed sides are unchanged proving the correspondence. However, an exact match to the length of the main POs could be made only after a small rescaling of the experimental results. Statistical analysis of the level dynamics also shows the effect of the stable POs. PACS number(s): 05.45.+b,05.45.Mt,46.40.-f Typeset using REVT E X

Classical wave experiments on chaotic scattering

“…Again various sized octants of material were removed from the corner, thus allowing a study of the transition between a sub-Poissonian and a GOE case. Neicu and Kudrolli [156] also studied the evolution of eigenstatistics under shape deformations.…”

Wave chaos in elastodynamic cavity scattering

“…[3] attain Q values as high as 5 • 10 6 , making spectral measurements in elastodynamics competitive with measurements in microwave cavities at liquid-helium temperatures [4,5], and vastly superior to nuclear-physics and room temperature microwave experiments for which the Q values are orders of magnitude lower, typically ∼ 10 2 -10 3 . For elastodynamics there are only a few experimental demonstrations [6] of the existence of unstable periodic orbits, and no theory that would predict them. While Oxborrow et al [3] measure about 10 5 spectral lines, the current theory is barely adequate for computation of dozens of resonances.…”

Wave chaos in acoustics and elasticity

“…The perturbations applied in both experiments act globally and bouncing ball contributions do either not exist (Schaadt and Kudrolli 1999) or have been accounted for (Bertelsen et al 1999). This is in contrast to the experiments on a clover-shaped fused-quartz plate by Neicu and Kudrolli (2002), where deviations in c(τ ) for the flexural spectrum may be due to stable periodic orbits present in the system. Two further studies on ray-splitting billiards as discussed in section 2.3 tried to shed light on this issue.…”

“…Experimental verification for periodic orbit contributions in plate spectra has been obtained by Neicu et al (2001) and Neicu and Kudrolli (2002) using a clover-shaped fused-quartz plate. The validity of semiclassical trace formulae is not immediately evident given that the biharmonic equation is itself an approximation of the true plate dynamics.…”

“…The validity of semiclassical trace formulae is not immediately evident given that the biharmonic equation is itself an approximation of the true plate dynamics. Unambiguous result could only be obtained by Neicu and Kudrolli (2002) after including higher order corrections to the dispersion relation as given by Bertelsen et al (2000) and carefully tracing individual peaks in the Fourier transformed spectrum by making small changes to the shape of the plate. An overall scaling factor in the frequency remained unaccounted for.…”