Recovery of ordered periodic orbits with increasing wavelength for sound propagation in a range-dependent waveguide

Abstract

We consider sound wave propagation in a range-periodic acoustic waveguide in the deep ocean. It is demonstrated that vertical oscillations of a sound-speed perturbation, induced by ocean internal waves, influence near-axial rays in a resonant way, producing ray chaos and forming a wide chaotic sea in the underlying phase space. We study interplay between chaotic ray dynamics and wave motion with signal frequencies of 50-100 Hz. The Floquet modes of the waveguide are calculated and visualized by means of the Husimi plots. Despite of irregular phase space distribution of periodic orbits, the Husimi plots display the presence of ordered peaks within the chaotic sea. These peaks, not being supported by certain periodic orbits, draw the specific "chainlike" pattern, reminiscent of KAM resonance. The link between the peaks and KAM resonance is confirmed by ray calculations with lower amplitude of the sound-speed perturbation, when the periodic orbits are well-ordered. We associate occurrence of the peaks with the recovery of ordered periodic orbits, corresponding to KAM resonance, due to suppressing of wave-field sensitivity to small-scale features of the sound-speed profile that take place with increasing wavelength.

Wave chaos in a randomly inhomogeneous waveguide: Spectral analysis of the finite-range evolution operator

“…Since eigenvalues of the FREO belong to the unit circle in the complex plane, the FREO corresponds to the circular ensemble [25]. The FREO has much in common with the Floquet operator governing wave propagation in a rangeperiodic waveguide [11,12] and quantum dynamics in time-periodic systems. For instance, quantity ǫ m is the analogue of quasienergy in quantum mechanics.…”

“…Only purely-water modes which propagate without reaching the bottom were taken into account. They were selected using the criterion E m < U (z = h), where E m is the m-th eigenvalue of the Sturm-Liouville problem (12). This criterion ensues from the WKB approximation for normal modes [9].…”

“…FREO can be represented as a matrix in the basis of normal modes φ j (z) satisfying the Sturm-Liouville problem (12,22). Matrix elements of the FREO are given by…”

“…During the last two decades ray chaos in ocean * makarov@poi.dvo.ru acoustics was an object of intense research, both theoretical and experimental [4][5][6][7][8]. Considerable attention was paid to wave chaos [3,[9][10][11][12][13][14]. The term wave chaos relates to wavefield manifestations of ray chaos.…”

Onset of quantum chaos in molecular systems and the zeros of the Husimi function

“…The regular or chaotic behavior of quantum states has been characterized through the distributions of the Husimi zeros in the spin-boson interaction model [5,6], in quantum billiards [7,8], in one-dimensional autonomous (then integrable) systems [9,10], in spin SternGerlach apparatus [11] (where it is possible to experimentally measure the zeros), in acoustic waveguides [12], and also in molecular systems [13][14][15][16][17].…”