Sensitivity of ray travel times

Ray in a waveguide can be considered as a trajectory of the corresponding Hamiltonian system, which appears to be chaotic in a nonuniform environment. From the experimental and practical viewpoints, the ray travel time is an important characteristic that, in some way, involves an information about the waveguide condition. It is shown that the ray travel time as a function of the initial momentum and propagation range in the unperturbed waveguide displays a scaling law. Some properties of the ray travel time predicted by this law still persist in periodically nonuniform waveguides with chaotic ray trajectories. As examples we consider few models with special attention to the underwater acoustic waveguide. It is demonstrated for a deep ocean propagation model that even under conditions of ray chaos the ray travel time is determined, to a considerable extent, by the coordinates of the ray endpoints and the number of turning points, i.e., by a topology of the ray path. We show how the closeness of travel times for rays with equal numbers of turning points reveals itself in ray travel time dependencies on the starting momentum and on the depth of the observation point. It has been shown that the same effect is associated with the appearance of the gap between travel times of chaotic and regular rays. The manifestation of the stickiness (the presence of such parts in a chaotic trajectory where the latter exhibits an almost regular behavior) in ray travel times is discussed. (c) 2002 American Institute of Physics.     

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.     

Regions where transient signals are influenced between a source and receiver

At infinite frequency, the only place that a medium can influence the waveform of a received signal that is emitted from a source is along one or more infinitesimally thin ray paths. For any transient signal at finite frequencies, an exact method is developed to compute the regions in a medium that significantly influence the received signal for any specified window of signal travel time. This window is sometimes chosen to surround a peak. Results at finite frequencies differ from those at infinite frequency because of diffraction. The method has its foundation in the integral theorem of Helmholtz and Kirchhoff. Part of the method involves a filter that yields an imperfect but apparently useful picture of influential regions in the presence of interfering waves. The method is useful for quantifying differences between the region of influence and a ray, and for identifying regions in which medium fluctuations significantly influence signal aberrations at a receiver. Four principal results are found at low frequencies. They are: 1) For propagation in homogeneous media, a significant portion of the received signal is influenced by waves that traverse paths that are approximately integer and half-integer numbers of cycles greater than the straight path between source and receiver. Such paths are called 'constructive and destructive paths of influence', respectively. They correspond to edge-diffracted rays for the geometrical theory of diffraction. 2) For reflection from a flat interface in an otherwise homogeneous medium, the received signal is significantly influenced by constructive and destructive paths of influence whose angles of incidence and reflection differ (non-specular reflection). 3) For acoustic propagation centered at 100 Hz in an oceanic acoustic waveguide, the region of influence markedly departs from a ray path, particularly near the reflective ocean surface. The influential region is flat for O(10) km instead of O(1) km for a ray. 4) The first Fresnel zone is an inappropriate scale to characterize the region of influence for transient signals near a steep ray in inhomogeneous media, as assumed by at least one scattering theory. Modification of that theory may yield a better fit with data.     

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.     

High frequency sound propagation in a network of interconnecting streets

We propose a new model for the propagation of acoustic energy from a time-harmonic point source through a network of interconnecting streets in the high frequency regime, in which the wavelength is small compared to typical macro-lengthscales such as street widths/lengths and building heights. Our model, which is based on geometrical acoustics (ray theory), represents the acoustic power flow from the source along any pathway through the network as the integral of a power density over the launch angle of a ray emanating from the source, and takes into account the key phenomena involved in the propagation, namely energy loss by wall absorption, energy redistribution at junctions, and, in 3D, energy loss to the atmosphere. The model predicts strongly anisotropic decay away from the source, with the power flow decaying exponentially in the number of junctions from the source, except along the axial directions of the network, where the decay is algebraic.     

Internal-wave time evolution effect on ocean acoustic rays

A range-dependent field of sound speed in the ocean, c(x,z), caused by internal waves, can give rise to instabilities in acoustic ray paths. Past work has shown the importance of the background, range-independent, sound-speed profile; the ray initial conditions; the source-receiver geometry (depths and range); and the strength of the internal waves. However, in the past the time evolution of the internal waves has been ignored on the grounds that the speed of internal waves is much slower than the speed of the acoustic wave. It is shown here by numerical simulation that two rays with identical initial conditions, traveling through an ocean with the same background profile and the same random realization of internal waves, but with the internal waves frozen in one case and evolving in the other, travel significantly different trajectories. The dependence of this "frozen-unfrozen" difference on the initial ray launch angle, the background profile, and the strength of the internal-wave spectrum, is investigated. The launch-angle difference that generates similar arrival-depth differences to those induced by internal-wave time evolution is on the order of 100 microrad. The pattern of differences is measured here by the arrival depth at the final range of 1000 km. The observed pattern as a function of launch angle, change in the background profile, and change in internal-wave strength is found to be nearly the same for "frozen-unfrozen" change as for a slight change in launch angle.     

Interfering modes and an axial wave in an underwater sound channel

Experiments on long-range propagation of low-frequency sound that were conducted starting from the mid-1980s indicate a complex character of propagation in an underwater sound channel, in which a source and a receiver are located close to the channel axis. A burst of energy propagating along the axis follows early arrivals, which are well described by the formulas of geometrical acoustics, in plots of acoustic intensity as a function of propagation time and hydrophone depth. This energy burst cannot be described using geometrical acoustics because of caustics with caustic beaks located near the channel axis. Very complex interference processes occur near these caustics. As the distance from the source grows, the dimensions of the interference vicinity increase and start to overlap producing a peculiar “axial wave.” For an arbitrary two-dimensional underwater sound channel, the axial wave can be represented as a sum of the first normal modes and a residue. This conclusion is based on the use of two representations for an acoustic field. The first of them includes the sum of ray components and an axial wave. The second representation consists of ray addends, the sum of the first normal modes, and a residue. Numerical results are obtained for a canonical profile of sound velocity at the frequency of 200 Hz for the distances of 1600–1650 km.     

浅海周期起伏海底环境下的声传播

海底粗糙对水下声传播及水声探测等应用具有重要影响.利用黄海夏季典型海洋环境,分析了同时存在海底周期起伏和强温跃层条件下的声传播特性,结果表明:由于海底周期起伏的存在,对于低频(<1 kHz)、近程(10 km)的声信号,传播损失可增大5—30 dB.总结了声传播损失及脉冲到达结构随声源深度、海底起伏周期及起伏高度等因素变化的规律.当海底起伏周期不变时,起伏高度越大引起的异常声传播的影响随之变大;当起伏高度不变时,随着起伏周期变大,其对声传播的影响逐渐变小.用射线理论分析了其影响机理,由于海底周期起伏改变了声波与海底的入射和反射角度,使得原本小掠射角入射到海底的声线变为大掠射角,导致海底的反射损失增大;另一方面,声线反射角度的改变会使得原本可以到达接收点的声能量,由于与海底作用次数增加或变为反向传播而大幅度衰减.在浅海负跃层环境下,声源位于跃层上比位于跃层下对声传播影响更大.周期起伏海底对脉冲声传播的影响表现在引起不同角度的声线(或简正波号数)之间的能量发生转化,一些大角度声线能量衰减加大,多途结构变少.多途结构到达时间及相对幅度的变化进而影响声场的频谱,会使得基于匹配场定位的方法性能受到影响.所以,声呐在实际浅海环境中应用时,应对起伏海底的影响予以重视.此外,研究结果对海底地形测绘空间精度的提高也具有重要参考意义.     

Experimental investigation of the travel-time variance of an acoustic wave propagating through the grid-generated turbulence

An experimental technique for the investigation of the behaviour of acoustic wave propagation through a turbulent medium is discussed. The present study utilizes the ultrasonic travel-time technique to diagnose a grid-generated turbulence. Travel-time variance is studied versus mean flow velocity, travel distance and outer turbulence scale. The effect of thermal fluctuations, which result in fluctuations of sound speed, is studied using a heated-grid experiment. Experimental data obtained using ultrasonic technique confirm numerical and theoretical predictions of nonlinear increase of the travel-time variance with propagation distance, which could be connected to the occurrence of caustics. The effect of turbulent intensity on the travel-time variance and appearance of caustics is studied. It is demonstrated experimentally that the higher turbulence intensity leads to the shorter distance, at which the first caustic occurs. The probability density for caustics appearance is analysed against the measured wave amplitude fluctuations. The analysis reveals that the region of high-amplitude fluctuations corresponds to the region where the probability of formation of random caustics differs from zero. Experimental results are in very good agreement with theoretical and numerical predictions.     

The horizontal structure of the sound field in the ocean

Results of calculating the horizontal structure of acoustic field in the ocean with a canonical sound velocity profile are presented. The calculation is performed in the framework of geometrical acoustics by combining the fields of water-path rays at every point with allowance for their phases and amplitudes. The field distribution at a fixed depth is found to be not very informative: within a cycle length, it contains 1–3 maxima, which are mainly caused by the caustics present at the given depth. The width of caustics is ～0.1–1 km. Between the caustics, as well as in their absence, the field amplitude is much smaller than that in the regions of caustics and varies depending on the phases and amplitudes of rays arriving at a given point. The comparison of the calculated horizontal field structure with that obtained from full-scale measurements for the regions between the caustics is difficult because of the possible fluctuations of sound propagation conditions.     

Evaluation of the smoothed interference pattern under conditions of ray chaos

A ray-based approach has been considered for evaluation of the coarse-grained Wigner function. From the viewpoint of wave propagation theory this function represents the local spectrum of the wave field smoothed over some spatial and angular scales. A very simple formula has been considered which expresses the smoothed Wigner function through parameters of ray trajectories. Although the formula is ray-based, it nevertheless has no singularities at caustics and its numerical implementation does not require looking for eigenrays. These advantages are especially important under conditions of ray chaos when fast growing numbers of eigenrays and caustics are the important factors spoiling applicability of standard semiclassical approaches already at short ranges. Similar factors restrict applicability of some semiclassical predictions in quantum mechanics at times exceeding the so-called "logarithm break time." Numerical calculations have been carried out for a particular model of range-dependent waveguide where ray trajectories exhibit chaotic motion. These calculations have confirmed our conjecture that by choosing large enough smoothing scales, i.e., by sacrificing small details of the interference pattern, one can substantially enhance the validity region of ray theory. (c) 2000 American Institute of Physics.     

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.     

Complex Ray Analysis for Plasmas

An extension of ray tracing techniques is considered for a variety of cases in which the dispersion relation of the plasma medium is complex. The ray trajectories are permitted to begin and/or at least travel through complex space-time; the wave propagation process so characterized becomes significant only where the rays intersect real space-time. It is found that rules and guidelines can be established for limited application of this idea.     

Shallow water sound propagation with surface waves

The theory of wavefront modeling in underwater acoustics is extended to allow rapid range dependence of the boundaries such as occurs in shallow water with surface waves. The theory allows for multiple reflections at surface and bottom as well as focusing and defocusing due to reflection from surface waves. The phase and amplitude of the field are calculated directly and used to model pulse propagation in the time domain. Pulse waveforms are obtained directly for all wavefront arrivals including both insonified and shadow regions near caustics. Calculated waveforms agree well with a reference solution and data obtained in a near-shore shallow water experiment with surface waves over a sloping bottom.     

Caustics method in dynamic fracture problem of orthotropic materials

Caustics method is a powerful optical technique in fracture mechanics because of its high sensitivity to stress gradients. In this paper, it is applied to resolve dynamic fracture problems in orthotropic composites. Considering most orthotropic materials are opaque, reflective caustics method is derived here by combining the fundamental principle of caustics method with the mechanical properties of orthotropic materials. Meanwhile, corresponding experiments are carried out for typical glass fiber-reinforced composites, where mode I and mixed-mode fracture states are taken into account. By recording and analyzing shadow spot patterns during the crack propagation process carefully, crack onset time, dynamic fracture toughness and crack growth velocity of orthotropic composite are determined. These results will be useful to evaluate the dynamic fracture properties of composites and further to optimize their designs.     

Statistical properties of nonlinear diffracting <Emphasis Type="Italic">N</Emphasis>-wave behind a random phase screen

Propagation of high amplitude N-wave behind a random phase screen is modeled based on the Khokhlov-Zabolotskaya-Kuznetsov equation. One-dimensional random phase screens with Gaussian power spectrum density are considered. The effects of nonlinear propagation, random focusing, and diffraction on the statistical properties of the acoustic field behind the screen, including propagation through caustics and beyond caustics, are analyzed. Statistical distributions and mean values of the acoustic field parameters obtained within the developed diffraction model and using nonlinear geometrical acoustics approach are compared.     

A ray model of sound focusing with a balloon lens: an experiment for high school students

A weather balloon filled with carbon dioxide gas is used as a positive spherical acoustic lens. High frequency but audible sound from a circular loudspeaker ensonifies the balloon and produces increased sound pressure levels in a region along the principal axis according to a ray acoustics model. This enhancement was measured experimentally and was found to agree with theory. The possibility that interference from reflected sound off walls or the floor could mask or mimic the expected focusing was countered by calculating and measuring within a "shadow zone" in which only direct rays or rays refracted by the balloon exist by the method of Fresnel volumes. The experiment described in this paper would be a suitable learning experience for junior high and high school students showing how rays and Snell's law apply to sound as well as light and giving them a measurable predicted focal region for enhanced sound pressure levels.     

Comparison between ocean-acoustic fluctuations in parabolic-equation simulations and estimates from integral approximations

Line-integral approximations to the acoustic path integral have been used to estimate the magnitude of the fluctuations in an acoustic signal traveling through an ocean filled with internal waves. These approximations for the root-mean-square (rms) fluctuation and the bias of travel time, rms fluctuation in a vertical arrival angle, and the spreading of the acoustic pulse are compared here to estimates from simulations that use the parabolic equation (PE). PE propagations at 250 Hz with a maximum range of 1000 km were performed. The model environment consisted of one of two sound-speed profiles perturbed by internal waves conforming to the Garrett-Munk (GM) spectral model with strengths of 0.5, 1, and 2 times the GM reference energy level. Integral-approximation (IA) estimates of rms travel-time fluctuations were within statistical uncertainty at 1000 km for the SLICE89 profile, and in disagreement by between 20% and 60% for the Canonical profile. Bias estimates were accurate for the first few hundred kilometers of propagation, but became a strong function of time front ID beyond, with some agreeing with the PE results and others very much larger. The IA structure functions of travel time with depth are predicted to be quadratic with the form theta(2)vc0(-2)deltaz(2), where deltaz is vertical separation, c0 is a reference sound speed, and thetav is the rms fluctuation in an arrival angle. At 1000 km, the PE results were close to quadratic at small deltaz, with values of thetav in disagreement with those of the integral approximation by factors of order 2. Pulse spreads in the PE results were much smaller than predicted by the IA estimates. Results imply that acoustic tomography of internal waves at ranges up to 1000 km can use the IA estimate of travel-time variance with reasonable reliability.