Manifestation of scarring in a driven system with wave chaos

Abstract

We consider acoustic wave propagation in a model of a deep ocean acoustic waveguide with a periodic range-dependence. Formally, the wave field is described by the Schrödinger equation with a time-dependent Hamiltonian. Using methods borrowed from the quantum chaos theory it is shown that in the driven system under consideration there exists a "scarring" effect similar to that observed in autonomous quantum systems.It is well-known in the theory of quantum chaos that classical periodic orbits, to a significant extent, determine both the distribution of energy levels and structures of eigenfunctions [1,2,3]. In particular, for some eigenfunctions, the amplitudes are high in the vicinity of unstable periodic orbits and low elsewhere. This phenomenon, first discovered in a quantum billiard, is called the scarring [4]. It has been established that the scarring is a common property for many autonomous chaotic systems [3,5,6]. The description of scarring is related to the problem of construction of eigenfunctions on the basis of purely semiclassical calculations [7,8,9,10,11,12].It is also well-known that so different physical problems as quantum particle dynamics and wave propagation in nonuniform media in the narrow beam approximation are described by the same parabolic (or Shrödinger) equation. One of the most intriguing object is the underwater acoustic (UWA) and the long range sound propagation in the ocean [13,14,15]. Application of methods of quantum chaos to the wave chaos proves to be efficient in many publications [16,17,18,19,20]. The goal of this paper is to extend this analogy between quantum and wave chaos and to present simulations that demonstrate the phenomenon of scarring in a nonautonomous Hamiltonian system with 1.5 degrees of freedom related to the UWA. We consider a simplified case of wave propagation in a two dimensional range-dependent model of hydroacoustic waveguide that has been used to study wave and ray chaos at long range sound propagation in the ocean [21,22,23,24,25]. Formally, the simplified model of ray propagation is equivalent to a nonlinear oscillator perturbed by a periodic force, while the corresponding parabolic equation is equivalent to the Shrödinger equation for the oscillator.Consider a two-dimensional UWA waveguide with the sound speed c as a function of depth, z, and range, r. In the parabolic equation approximation (valid under assumptions that waves propagate at small grazing angles) the monochromatic wave field u at a carrier frequency f obeys the parabolic equation [26,27] i k ∂u ∂r =Ĥu,Ĥ = − 1 2k 2where U (r, z) = 1 −c 2 /c 2 (r, z) /2. Herec and k = 2πf /c are a reference sound speed and a reference wavenumber, respectively. Equation (1) formally coincides with the time-dependent Schrödinger equation. In this analogy r plays a role of time,Ĥ is the Hamiltonian operator, U (r, z) is the potential, and k −1 associates with the Planck constant. The ray paths are determined by the standard Hamilton equations dp/dt = −∂H/∂z and dz/dr = ∂H/∂p with the Hamiltonian H = p 2 /2...

Random Matrix Theory for Sound Propagation in a Shallow-Water Acoustic Waveguide with Sea Bottom Roughness

“…The main advantage of this approach is the possibility to take into account interference effects within a full-wave picture. We conducted early studies of a wavefield propagator for idealized range-periodic waveguides in the context of the wave chaos problem [22][23][24]. The propagator properties for waveguides with random inhomogeneities caused by linear internal waves were considered in [25][26][27].…”

Random matrix theory for an adiabatically-varying oceanic acoustic waveguide

“…This circumstance enables usage of well-developed methods of quantum chaos to the problem of long-range sound propagation. In particular, we can mention phase space analysis using the Wigner function or its smoothed versions [9,10,11,12,13], Lyapunov analysis [14], entropy calculation [15], periodic orbit theory [16,17,18], theory of nonlinear resonance [19,20,21], and action-angle formalism [22], to name a few. One of the most novel approaches is based on the unitary propagator governing wave evolution within the narrowangle approximation [7,23,24,25,26].…”

“…Under some assumptions this range dependence can be evaluated using the perturbation theory[34], i. e. without solving the Sturm-Liouville problem too frequently. However, even in this way, statistics of integrals(16) can be found only numerically, using Monte-Carlo sampling. It remarkably increases computational time needed to estimate variances σ 2 mn .…”