Isospeed Channel Research Articles

Effects of random-bottom topography on the parabolic-approximation determination of acoustic intensity

A model is presented for stochastic variations about a horizontal mean water-bottom interface below a shallow ocean channel. A pressure release surface and a rigid bottom are taken as boundary conditions for solutions of a parabolic approximation to the wave equation, when sound speed is depth dependent. A perturbation approach is used to generate a sequence of differential problems, the perturbation parameter being a measure of the deviation in depth of the random bottom from its mean. Green's functions are used to represent the solution to the first two differential problems. The mean and variance of received intensity are then obtained in terms of statistics of the random bottom. In an example, an isospeed channel is treated, and the ratio of the mean and standard deviation of intensity is expressed in terms of the autocorrelation lengths and variances of the random processes representing bottom height and slope variations. The behavior of this ratio is described as range and bottom statistics are varied. Further, the effect on intensity of bottom depth deviations and slope variations are compared. [Work supported by ONR.]

Sound propagation through random currents using parabolic approximations

The influence of random, horizontal, depth‐dependent ocean currents in the parabolic approximation is studied. The current is taken to have a depth‐dependent mean, upon which is superimposed a small fluctuating component. Emphasis is placed on received acoustic intensity. Recently derived parabolic equations including currents are examined using asymptotic and numerical methods. Expressions are derived for the mean and standard deviation of intensity. The fluctuating current component is taken as having an exponential‐cosine autocorrelation function with depth, although other functions could be used. Formulas obtained are sufficiently general to permit many types of depth‐dependent sound speeds and bottom interface models. Current fluctuation effects on intensity moments are illustrated for the specific case of an isospeed channel with a perfectly hard bottom. The acoustic consequences of changes in physical parameters directly related to current fluctuations, such as its standard deviation and correlation length, are discussed. Also exhibited are changes in random current effects on intensity that result from changes in noncurrent related parameters, such as acoustic frequency and channel depth.

Acoustical effects of ocean current shear structures in the parabolic approximation

In a previous article [Robertson et al., J. Acoust. Soc. Am. 77, 1768–1780 (1985)], the authors developed a family of parabolic equations that includes effects due to the presence of a time-dependent, depth-dependent current. Some of these equations contain a new term that explicitly depends on current gradient. In this article, certain effects of this new term are studied. By transforming the parabolic equations, it is possible to convert them to forms that can be directly solved numerically using existing IFD or FFT implementations. Ocean currents with vertical fine structure present situations that can require these new types of parabolic approximations. Propagation in a shallow isospeed channel is examined, with both rigid and lossy bottoms, and use is made of a shear flow with features of an actual ocean current. The vertical current variation can cause changes in relative intensity that are substantial and that depend on bottom loss, source and receiver locations, and acoustic frequency. Intensity differences are demonstrated and examined for reciprocal sound transmissions.

Effects of bottom attenuation on acoustic propagation with a modified ray theory

The influence on ray propagation of including both attenuation and beam displacement at the bottom of a shallow-water isospeed channel is determined. Attenuation is incorporated using the Mackenzie-bottom model, rather than the commonly used Rayleigh reflection theory. Properties of displacement are studied as a function of launch angle and parameters such as bottom-to-water density, channel depth in wavelengths, and channel aspect ratio. Differences in beam displacement for the two bottom models are shown to affect the number and geometry of rays between surfaced source and receiver. Formulas for per-ray amplitude and phase shift and for incoherent total-field intensity are developed. Comparisons are made between these quantities for both modified and classical ray theory with a Mackenzie bottom, and also for both Mackenzie- and Rayleigh-bottom models using modified rays. Numerical computations show significant phase and amplitude differences arising in both types of comparisons.

Sound propagation through random currents using parabolic approximations

The influence of random, horizontal, depth‐dependent ocean currents in the parabolic approximation is studied. Emphasis is placed on received acoustic intensity. Parabolic equations including currents, derived recently by our group, are examined using perturbation and asymptotic methods. Expressions are derived for the mean and standard deviation of intensity. The development assumes that the random current is wide‐sense stationary. For convenience, an exponential‐cosine autocorrelation function with depth is taken, although any such function can be used. Formulas obtained are sufficiently general to permit many types of depth‐dependent sound speeds and bottom interface models. We illustrate current fluctuation effects on intensity moments for the specific case of an isospeed channel with a perfectly hard bottom. The acoustic consequences of changes in physical parameters directly related to current fluctuations, such as its standard deviation and correlation length, are discussed. Also exhibited are changes in random current effects on intensity which result from changes in noncurrent related parameters, such as acoustic frequency and channel depth. [Work supported by ONR.]

Ray transmissions over a sloping bottom in shallow water

The effects of a sloping bottom on acoustic transmissions, between a source and receiver at arbitrary but fixed locations, are investigated using ray theory. An isospeed channel is assumed, and bottom angles up to about 3° are considered. Sloping bottom influence on per-ray quantities, including travel time and transmission loss, are examined for cw transmissions. Significant variations are shown to occur, such as travel time changes of more than 200 ms over ranges of about 6 km. Per-ray transmission loss is found to be influenced strongly by bottom slope, the amount of influence depending upon source–receiver bearing and the bottom loss model used. Variations of more than 20 dB are demonstrated. Effects of a sloping bottom on the total acoustic field are examined also, and the results compared with those for a horizontal bottom. Finally, a simple model of a shallow water front is superposed over the sloping bottom, and travel time is investigated. The sloping bottom effect can induce travel-time changes more than 300% larger than the frontal effect for different source–receiver geometries and bottom inclinations.

Some new results for ray transmissions in a wedge-shaped ocean

The influence of a sloping bottom on cw ray transmissions in an isospeed channel is investigated. Bottom angles up to about 3° are considered, and regular perturbations off horizontal bottom results are employed. Accurate analytic approximations to ray-geometric quantities are derived. Then, sloping bottom influence on per-ray travel time and transmission loss is examined. Significant variations are shown to occur, such as travel time changes of more than 200 ms over ranges of about 6 km. Per-ray transmission loss is found to be strongly influenced by bottom slope. Variations of more than 20 dB are illustrated. Sloping bottom effects on the total acoustic field are examined also. A comparison of the influences on travel time of one model of a shallow-ocean front and the sloping bottom is made. The sloping bottom effect can induce travel time changes more than 300% larger than the frontal effect. [Work supported by ONR.]

Effects of random bottom structure and topography on transmissions in a shallow ocean

In a recent paper [C. E. Ashley, M. J. Jacobson, and W. L. Siegmann, J. Acoust. Soc. Am. 76, 1445–1455 (1984)], the authors studied the influences of stochastic horizontal bottom structure on underwater sound transmissions. Both bottom density and sound speed were taken to be random. Ray theory was used in an isospeed channel with horizontal boundaries. In this study, we add to random structural effects those of a lossy bottom interface consisting of large-scale random facets with curvature, on which may be superimposed small-scale roughness. Each facet is assumed to possess small random slope, depth deviation, and curvature. Initially, we take acoustic rays to be specularly reflected from a large-scale facet bottom, and derive formulas for the mean and variance of incoherent intensity at a point receiver for a transmitted cw signal. The results are sufficiently general to permit their use with different bottom-acoustic models. Relative effects of structure and topography are compared. Subsequently, small-scale roughness is added to the facets, and the consequences of scattering are considered. [Work supported by ONR.]

Acoustical effects of ocean current shear in the parabolic approximation

In a previous study [J. S. Robertson, W. L. Siegmann, and M. J. Jacobson, J. Acoust. Soc. Am. 77, 1768–1780 (1985)], the authors developed a family of parabolic equations which include effects due to the presence of a steady, depth-dependent current. Three of these equations contained a new term which explicitly depended on current gradient. In this work, we study the effects of this new term. By appropriately transforming and simplifying, the equations are transformed into parabolic equations which can be integrated numerically with existing implementations. Presence of ocean current fine structure is one mechanism which can require new terms. Propagation is examined in a shallow isospeed channel with a lossy bottom and a variety of shear flows, some of which model actual ocean flows. Current fine structure can induce variations in intensity which are substantial and which depend on shear structure, source and receiver locations, and frequency. Finally, intensity differences are examined in reciprocal sound transmissions. [Work supported by ONR.]

Current and current shear effects in the parabolic approximation for underwater sound channels

The effect of currents on the acoustic pressure field in an underwater sound channel is investigated. Based on fundamental fluid equations, model equations are formulated for sound pressure while including nonuniform currents in the source–receiver plane. Application of parabolic-type approximations yields a collection of parabolic equations. Each of these is valid in a different domain determined by the magnitudes of current speed, current shear, and depth variation of sound speed. Under certain conditions, it is possible to interpret current effects in terms of an effective sound speed. Using this effective sound speed in an existing numerical code, we examine sound speed in a shallow water isospeed channel with a simple shear flow and a lossy bottom. It is found that even small currents can induce very substantial variations in relative intensity. The degree of variation depends upon current speed, source and receiver geometry, and acoustic frequency. Particular emphasis is placed on intensity-difference predictions in reciprocal sound transmissions in the presence of an ocean current.

Currents and the parabolic approximation in underwater sound channels

The effect of currents on the acoustic pressure field in an underwater channel is investigated. Based on the fundamental fluid equations, model equations are formulated which predict sound pressure while including nonuniform currents in the source-receiver plane. Application of parabolic-type approximations yields a collection of parabolic equations. Each of these is valid in a different domain determined by the magnitudes of current speed, current shear, and depth variation of sound speed. Under suitable conditions, it is useful to interpret current effects in terms of an “effective” sound speed. Using this effective sound speed in an existing numerical code, we examine propagation in a shallow isospeed channel with a simple shear flow and a lussy bottom. It is found that even small currents can induce very subtantial variations in relative intensity. The degree of variation depends upon current speed, source and receiver geometry, and acoustic frequency. Particular emphasis is placed on intensity-difference predictions in reciprocal sound transmissions in the presence of an ocean current. [Work supported by ONR.]

Effect of a random ocean current on acoustic transmission in an isospeed channel

The effects of a random current on the fluctuations of underwater cw sound transmissions are considered for a horizontal isospeed channel. A statistical ensemble of currents is employed, whose members are depth dependent only, and current influence on ray geometry is investigated. Approximations to the total acoustic field at a receiving point are obtained, and it is shown that a current ensemble member has a significant effect on phase. A mean current is taken, on which are superimposed small depth-dependent current changes in speed and direction. These changes are members of statistical ensembles, which lead to an ensemble of phase functions. Expressions for the mean and standard deviation of the phase, in terms of statistics of the depth-averaged current fluctuations, are determined and analyzed. The inverse problem of determining second-order moments of the random current, given the standard deviation of phase, is examined also. If the mean current is known, phase information at three receivers is sufficient to specify the second moments of the current fluctuations.

Effect of a random ocean current on acoustic propagation

The effects of a random current on the fluctuations of an underwater sound transmission are considered for a horizontal isospeed channel. The current is assumed to have a constant unperturbed part, plus small depth-dependent random variations in magnitude and direction. Using ray theory the current influence on ray geometry, travel time, spreading loss, and boundary effects is investigated. Then consistent approximations to total-field phase and amplitude at a fixed receiver are obtained. Acoustically pertinent statistics, including the mean and variance of total-field phase, are found as functions of the statistics of the current fluctuations. Conversely, it is shown how statistics of the assumed current structure can be described in terms of the total-field phase statistics. This initial study provides a basis for future investigations which will utilize more elaborate current and sound-speed models. [Work supported by ONR.]

General treatment of source motion on the total acoustic field with application to an isospeed channel

A treatment of the effects of arbitrary motion of a cw source and depth-dependent sound speed on the total acoustic field at a fixed receiving point is considered for an ocean with horizontal boundaries. Application of our general method is made to a constant sound-speed channel in which the range-to-depth ratio is large, when the source follows a short straight-line path with constant velocity. Total-field phase is investigated as a function of receiver time for various source trajectories and phase rate is examined in terms of an arbitrary, but fixed, acoustic frequency. It is shown that source motion may be accounted for by assuming the sound source to be stationary, and by replacing its frequency by approximate Doppler frequency. For long source trajectories, cumulative phase can be approximated as a hyperbolic function of time. The outputs of two uniform colinear arrays, together with power spectra there, are employed to illustrate one method for determing source speed, location, bearing, and frequency. Subject Classification: [43]30.20; [43]20.20..

Estimates of spatial coherence in multipath channels

Pure‐tone response in a multipath channel, wherein by virtue of reflection or refraction many paths connect the source to any receiver point, is highly variable due to interference between the components transmitted along different paths. A statistical measure of these fluctuations is the spatial coherence—the averaged product of the complex response field evaluated at two spatially lagged points. Estimates of the coherence are readily formed by combining (i) a field description in which the average response “energy” is expressed as a density in wavenumber or propagation direction, and (ii) the hypothesis that the local properties of each component are those of a free plane wave. This procedure models the field as being locally homogeneous, but with statistical parameters that vary slowly as a consequence of refraction and attenuation. Analytical estimates of coherence have been made for several canonical problems of ocean acoustics: the isospeed channel with flat bottom; the bigradient duct with a limiting ray.