Ray dynamics of the propagation of sound in nonuniform moving media

This paper presents a new ray theory for the propagation of sound waves in nonuniformly moving media. It is found that the ray equations in weakly inhomogeneous and slowly moving media are analogous to the equations of motion of charged particles in nonuniform electric and magnetic fields. The adiabatic approximation is used to study the problem of the propagation of sound rays in a model of near-ocean-bottom waveguide with horizontal flow and slowly varying parameters along the direction of propagation of the wave. A general formula is derived that describes the transverse displacement of the trajectory of the ray relative to the direction of propagation of the wave.     

Wave chaos and mode-medium resonances at long-range sound propagation in the ocean

We study how the chaotic ray motion manifests itself at a finite wavelength at long-range sound propagation in the ocean. The problem is investigated using a model of an underwater acoustic waveguide with a periodic range dependence. It is assumed that the sound propagation is governed by the parabolic equation, similar to the Schrodinger equation. When investigating the sound energy distribution in the time-depth plane, it has been found that the coexistence of chaotic and regular rays can cause a "focusing" of acoustic energy within a small temporal interval. It has been shown that this effect is a manifestation of the so-called stickiness, that is, the presence of such parts of the chaotic trajectory where the latter exhibit an almost regular behavior. Another issue considered in this paper is the range variation of the modal structure of the wave field. In a numerical simulation, it has been shown that the energy distribution over normal modes exhibits surprising periodicity. This occurs even for a mode formed by contributions from predominantly chaotic rays. The phenomenon is interpreted from the viewpoint of mode-medium resonance. For some modes, the following effect has been observed. Although an initially excited mode due to scattering at the inhomogeneity breaks up into a group of modes its amplitude at some range points almost restores the starting value. At these ranges, almost all acoustic energy gathers again in the initial mode and the coarse-grained Wigner function concentrates within a comparatively small area of the phase plane.     

Classical chaos and nonlinear dynamics of rays in inhomogeneous media

An investigation is made of the classical nonlinear resonance and the classical stochastic dynamics of rays in waveguide media with irregular inhomogeneities. Analytic and numerical methods are used to study the characteristics of the ray trajectories, their confinement in a nonlinear resonance, and the development of chaotic behavior in waveguides with longitudinal periodic inhomogeneities. It is established that the localization of the rays has fractal properties; in particular, the cycle length of a ray and the time and velocity of propagation of a signal depend on the initial parameters of the ray in the form of a "devil's staircase." A waveguide with an inhomogeneous index of refraction and a periodically corrugated wall is considered.     

Restless rays, steady wave fronts

Observations of underwater acoustic fields with vertical line arrays and numerical simulations of long-range sound propagation in an ocean perturbed by internal gravity waves indicate that acoustic wave fronts are much more stable than the rays comprising these wave fronts. This paper provides a theoretical explanation of the phenomenon of wave front stability in a medium with weak sound-speed perturbations. It is shown analytically that at propagation ranges that are large compared to the correlation length of the sound-speed perturbations but smaller than ranges at which ray chaos develops, end points of rays launched from a point source and having a given travel time are scattered primarily along the wave front corresponding to the same travel time in the unperturbed environment. The ratio of root mean square displacements of the ray end points along and across the unperturbed wave front increases with range as the ratio of ray length to correlation length of environmental perturbations. An intuitive physical explanation of the theoretical results is proposed. The relative stability of wave fronts compared to rays is shown to follow from Fermat's principle and dimensional considerations.     

Ray escape from a range-dependent underwater sound channel

The problem of sound propagation in a spatially inhomogeneous underwater sound channel is considered. The effect of ray escape, i.e., the ray incidence on the absorbing bottom due to the chaotic swing of rays, is studied. With the use of the Poincaré map and maps of escape, the relation of ray escape to the properties of the phase space of the set of ray equations is demonstrated. It is found that the maximum escape occurs under the vertical resonance conditions, i.e., at the resonance of the ray oscillations in the waveguide with the vertical oscillations of the sound velocity perturbation. A qualitative theory of the vertical resonance is developed. It is shown that the ray escape considerably shortens the time spreading of the signal.     

On the vertical structure of the sound field in a canonical waveguide at long ranges

The geometrical-acoustics approach is used to calculate the vertical structure of the sound field in an oceanic waveguide. The profile of the sound speed is specified to be canonical and range-independent along a 1000-km propagation path. A monochromatic sound source lies on the waveguide axis. It is shown that, at long distances from the source, the sound field formed by the water-path rays is mainly concentrated in the caustics, the number of which is determined by the number of the overlapping ray cycles at a given distance. A method for estimating the amplitude of the sound field produced by individual rays is proposed. The amplitudes obtained are used to calculate the total sound field along the vertical. A possible cause of the chaotic distribution of ray coordinates is considered. This cause may consist in the arbitrary choice of the number of rays and their departure angles without taking into account the discrete character of one of the variables. This mechanism of ray chaos formation furnishes an explanation for the fact that the chaos obtained in calculations is mainly associated with the flat rays.     

Classical chaotic dynamics of sound rays in 3-D inhomogeneous waveguide media

The chaotic dynamics of sound rays in a near-bottom waveguide channel is studied on the basis of the Hamiltonian dynamics of nonparaxial rays in inhomogeneous moving media. The bottom is assumed to have a two-dimensional roughness. The mapping of the coordinates of the rays upon reflection from the rough bottom is derived through a solution of the corresponding ray equations in an unperturbed waveguide with a horizontal bottom. A numerical analysis of the mapping reveals that a chaotic instability of rays which start out at small angles from the horizontal develops at short distances from the source. Because of this instability, the path segments of a ray along the horizontal coordinates and the signal passage time along a ray are random functions of the angle at which the ray emerges from the source. Upon a further reflection of rays from the rough bottom, there is a diffusion of rays in a stochastic ring which forms in the plane of horizontal ray directions as a result of the overlap and intersection of resonance curves. A qualitative analysis of this effect is carried out. This effect leads to a nearly isotropic distribution of ray directions.     

Turning point properties as a method for the characterization of the ergodic dynamics of one-dimensional iterative maps

Dynamical as well as statistical properties of the ergodic and fully developed chaotic dynamics of iterative maps are investigated by means of a turning point analysis. The turning points of a trajectory are hereby defined as the local maxima and minima of the trajectory. An examination of the turning point density directly provides us with the information of the position of the fixed point for the corresponding dynamical system. Dividing the ergodic dynamics into phases consisting of turning points and nonturning points, respectively, elucidates the understanding of the organization of the chaotic dynamics for maps. The turning point map contains information on any iteration of the dynamical law and is shown to possess an asymptotic scaling behaviour which is responsible for the assignment of dynamical structures to the environment of the two fixed points of the map. Universal statistical turning point properties are derived for doubly symmetric maps. Possible applications of the observed turning point properties for the analysis of time series are discussed in some detail. (c) 1997 American Institute of Physics.     

A circular system of coupled waveguides and an optical analogue of bloch oscillations

The process of light propagation in a nonuniform system of tunnel-coupled waveguides is studied. The nonuniform system of waveguides is realized by making a circular system of waveguides on the inner wall of a supporting tube with the help of the SPCVD technology, which is designed for the synthesis of preforms of optical fibers. It is shown that the waveguide light beam propagates in such a system along a wavelike trajectory and partially radiates outwards at the crests of this trajectory. The oscillation period of the trajectory is measured, and it is shown that it is possible to input radiation into the waveguide mode by the process that is inverse to the process of light emission from the structure.     

Chaotic transmission of waves and "cooling" of signals

Ray dynamics in waveguide media exhibits chaotic motion. For a finite length of propagation, the large distance asymptotics is not uniform and represents a complicated combination of bunches of rays with different intermediate asymptotics. The origin of the phenomena that we call "chaotic transmission," lies in the nonuniformity of the phase space with sticky domains near the boundary of islands. We demonstrate different fractal properties of ray propagation using underwater acoustics as an example. The phenomenon of the kind of Levy flights can occur and it can be used as a mechanism of cooling of signals when the width of spatial spectra dispersion is significantly reduced. (c) 1997 American Institute of Physics.     

Geometrical analysis of a lens-like medium with higher-order aberration

The propagation of light rays in a slab-waveguide whose dielectric constant varies radially with a quadratic law but has fourth and sixth order terms in it is described. Exact solutions consisting of Jacobi elliptic functions are shown. The exact ray trajectory and delay time of an optical pulse, therefore, can be obtained. The results show that the optimum fourth-order and sixth-order coefficients are near 2/3 and 17/45, respectively.     

Manifestation of ray stochastic behavior in a modal structure of the wave field

A ray-based mathematical formalism is described to analyze modal structure variations in a range-dependent wave guide. In the scope of this formalism mode amplitudes are expressed through parameters of ray trajectories. Therefore, the approach under consideration provides a convenient tool to study how chaotic ray motion manifests itself in an irregular range dependence of the modal structure. The phenomenon of nonlinear ray-medium resonance playing a crucial role in the emergence of ray chaos has been interpreted from the viewpoint of normal modes. It has been shown that in terms of modes the coexistence of regular and chaotic rays means the presence of regular and irregular constituents of mode amplitudes. An analog to incoherent summation of rays has been proposed to evaluate mode intensities (squared mode amplitudes) smoothed over the mode number. Numerical calculations have shown that it gives correct results for smoothed mode intensities at surprisingly long ranges.     

The Ray-Wave correspondence and the suppression of chaos in long-range sound propagation in the ocean

The long-range sound propagation in a horizontally inhomogeneous underwater sound channel is considered. It is shown that the vertical oscillations of inhomogeneity affect the near-axis rays in a resonant way and destroy their stability. As a consequence, the near-axis rays exhibit a chaotic behavior. The ray chaos manifests itself as intensification of interaction between adjacent low-order modes. In this case, a great number of strongly coupled modes appear, and the field structure near the channel axis becomes diffusive. However, as the signal frequency decreases, the inhomogeneity oscillations in depth lead to decoupling of adjacent modes and, hence, to suppression of chaos. As a result, the field structure near the channel axis becomes regular, which is confirmed by numerical simulation.     

Ray chaos and ray clustering in an ocean waveguide

We consider ray propagation in a waveguide with a designed sound-speed profile perturbed by a range-dependent perturbation caused by internal waves in deep ocean environments. The Hamiltonian formalism in terms of the action and angle variables is applied to study nonlinear ray dynamics with two sound-channel models and three perturbation models: a single-mode perturbation, a randomlike sound-speed fluctuations, and a mixed perturbation. In the integrable limit without any perturbation, we derive analytical expressions for ray arrival times and timefronts at a given range, the main measurable characteristics in field experiments in the ocean. In the presence of a single-mode perturbation, ray chaos is shown to arise as a result of overlapping nonlinear ray-medium resonances. Poincare maps, plots of variations of the action per ray cycle length, and plots with rays escaping the channel reveal inhomogeneous structure of the underlying phase space with remarkable zones of stability where stable coherent ray clusters may be formed. We demonstrate the possibility of determining the wavelength of the perturbation mode from the arrival time distribution under conditions of ray chaos. It is surprising that coherent ray clusters, consisting of fans of rays which propagate over long ranges with close dynamical characteristics, can survive under a randomlike multiplicative perturbation modelling sound-speed fluctuations caused by a wide spectrum of internal waves.     

Determining tomographic arrival times based on matched filter processing: considering the impact of ocean waves

Reflection of high-frequency acoustic signals from an air-sea interface with waves is considered in terms of determining travel times for acoustic tomography. Wave-induced, multi-path rays are investigated to determine how they influence the assumption that the time of the largest matched filter magnitude between the source and receiver signals is the best estimate of the arrival time of the flat-surface specular ray path. A simple reflection model is developed to consider the impact of in-plane, multi-path arrivals on the signal detected by a receiver. It is found that the number of multi-path rays between a source and receiver increases significantly with the number of times the ray paths strike the ocean surface. In test cases, there was always one of the multi-path rays that closely followed the flat-surface specular ray path. But all the multi-path rays arrive at the receiver almost simultaneously, resulting in interference with the signal from the flat-surface specular ray path. As a result, multi-path arrivals due to open ocean surface waves often distort the received signal such that maxima of matched filtering magnitudes will not always be a reliable indicator of the arrival time of flat-surface specular ray paths.     

Fresnel zones for modes and analysis of field fluctuations in random multimode waveguides

The notion of Fresnel zones for modes is introduced, which is analogous to the usual Fresnel zones introduced for rays. It is shown that using Fresnel zones for modes one can simplify the analysis of mode scattering at large-scale and random inhomogeneities of a medium in waveguides. Simple formulae to calculate fluctuations of mode amplitudes are obtained. They are similar to well-known formulae of geometrical optics and to those of the Rytov method used to calculate fluctuations of ray complex amplitudes. Relations deduced can be used for calculating field fluctuations both at regular waveguide points and at caustics.