Wave chaos in elastodynamic cavity scattering

– The exact elastodynamic scattering theory is constructed to describe the spectral properties of twoand morecylindrical cavity systems, and compared to an elastodynamic generalization of the semi-classical Gutzwiller unstable periodic orbits formulas. In contrast to quantum mechanics, complex periodic orbits associated with the surface Rayleigh waves dominate the low-frequency spectrum, and already the two-cavity system displays chaotic features. Introduction. – The Gutzwiller semi-classical quantization of classically chaotic systems relates quantum observables such as spectral densities to sums over classical unstable periodic orbits [1, 2]. The work presented here is a step toward a formulation of such approximate short-wavelength theory of wave chaos for the case of linear elastodynamics. Why elastodynamics? The experiments initiated in ref. [3] attain Q values as high as 5 · 10, 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 ∼ 10–10. 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 spectral lines, the current theory is barely adequate for computation of dozens of resonances. A more effective theory would find many applications such as in the frequency domain quality testing for small devices built from high-Q materials. This unsatisfactory state of affairs is the raison d’être for the theoretical effort undertaken here. While current experiments excel in measurements of eigenspectra of compact resonators, the periodic orbit theory computations of such bound system spectra are rendered difficult by the presence of non-hyperbolic phase space regions. As our primary goal is to derive and test rules for replacing wave mechanics by the short-wavelength ray-dynamic trajectories, we concentrate here instead on the problem of scattering off cylindrical cavities, for which the classical dynamics is fully under control. In the case of one cavity the exact scattering spectrum is known [7]. For the multiple-cavities case we generalize the quantum-mechanical (QM) Smatrix formalism for N -disk scattering [8–11], and compute the exact resonances and the Wigner time delays from the full elastodynamic wave-mechanical scattering matrix. We then c © EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx.doi.org/10.1209/epl/i2005-10286-8 2 EUROPHYSICS LETTERS compare the exact results with the corresponding quantities calculated in the short-wavelength approximation (SWA), and discover that the QM intuition fails us: the Rayleigh surface waves (which have no analog in the QM scattering problem) dominate the low-frequency spectrum because of their weights and number, such that already the two-disk elastodynamic scattering problem displays chaotic features in this regime in contrast to its QM counterpart. Elastodynamics. – Consider an infinite slab of an isotropic and homogeneous elastic material (e.g., polyethylene or isotropic quartz) with parallel top and bottom plane boundaries, and an in-phase stimulus such that the system behaves quasi–two-dimensionally along the slab, with no excitation of or coupling to waves propagating perpendicular to the slab. The propagating waves are either the pressure or the shear solutions of the Navier-Cauchy equation [12] μ∇2u + (λ + μ)∇(∇ · u) + ρωu = 0, where u is a vectorial displacement field, λ and μ are the Lamé constants, ρ the mass density and ω the frequency. The experiments dictate free boundary conditions, with vanishing traction t(u) = 0, where