Isovelocity Ocean Research Articles

Ambient noise modeling using sound field reciprocity

An efficient numerical calculation of the vertical and horizontal coherence of the noise field in the ocean due to a distributed sheet of sources can be carried out using a parabolic equation (PE) propagation model and the reciprocity property of the complex pressure field. In the case of an infinite sheet of sources near the surface in an infinitely deep, isovelocity ocean, the calculation of vertical coherence using this method is in agreement with the Cron and Sherman model for wind-driven surface noise, as well as with measurements made of vertical coherence in the Philippine Sea. This numerical model also gives the effective surface-listening radius for a single hydrophone by calculating the contribution to the noise field at the receiver from a small noise patch at the surface, as a function of range. The model is capable of calculating the vertical and horizontal coherence (directionality) of the noise field and the depth-dependent surface-listening radius of a hydrophone in a shallow water waveguide and over range-dependent environments, such as a wedge, canyon, or shelf break. Additionally, this technique can be used to calculate the spatial properties of the noise field due a finite patch of surface sources in motion, such as a rainstorm.

Theory of acoustic radiation over a 3-D hyperbolic ocean ridge

An analytic solution to a 3-D problem of ocean acoustics involving horizontal refraction is presented. An expression for the acoustic field due to a harmonic point source in an isovelocity ocean overlying a hyperbolic ridge with perfectly reflecting boundaries is given as a finite modal sum of integrals. The integrals are in terms of Mathieu functions, the eigenfunctions of the reduced wave equation in the elliptical coordinate system. These eigenfunctions are approximated away from the low-frequency limit by standard WKB techniques and the resulting integrals are then estimated by first- and second-order stationary-phase asymptotics. This gives an approximation in terms of elliptical functions for which standard numerical routines are available. The approximation is very accurate in the shallow depth and gentle slope limits appropriate to the physical situation of ridges in an isovelocity ocean. The solution includes the locations of the caustics and shadow zones and predicts a complicated intramodal interference pattern resulting from the intersection of up to three rays in a given mode. The intermodal interference from the finite modal sum is also evident in the full solution. The explicit representation of the features arising from horizontal refraction makes this model useful as a 3-D benchmark solution.

Theory of acoustic radiation near a hyperbolic ridge

A solution to the acoustic field due to a harmonic point source near a hyperbolic ridge with perfectly reflecting boundaries in an isovelocity ocean is derived and developed. An exact expression for the field is given as a modal sum of integrals. The integrals are in terms of eigenfunctions of the reduced wave equation in the three-dimensional elliptical coordinate system. The eigenfunctions are approximated away from the low-frequency limit by standard WKB techniques. The resulting integrals are estimated by first and second order stationary phase asymptotics, which are matched in the vicinity of the caustics, yielding a complete representation of the field. The field is given in terms of standard elliptic functions for which fast numerical routines are available. The solution includes the locations of the caustics and shadow zones, as well as predicting a complicated intramodal interference pattern resulting from the intersection of up to three rays in a given mode. The explicit representation of these features arising from horizontal refraction makes this theoretical model useful as a new fully three-dimensional benchmark solution.

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Stationary phase evaluation of the bottom interacting field in isovelocity water

The near surface sound pressure field in an isovelocity ocean is calculated analytically using ray theory and then normal mode theory for two different bottom types: first, a simple reflecting bottom with constant density and sound speed for which the two methods give identical solutions; then, a refracting bottom with constant sound-speed gradient. In the latter case the two solutions differ only (a) by a cumulative π/2 phase difference in the transition through each caustic, and (b) in the vicinity of the caustics themselves where the ray theory is inapplicable. The normal mode sum is evaluated by means of a Poisson sum, each term of which is then integrated using the stationary phase approximation, giving closed form expressions for the contribution from each ray path (or group of ray paths) that are identical to the ray approximation (apart from the phase) away from caustics, and provide a uniformly valid solution through the caustics. The resulting formulas are used to simplify the interpretation of complicated interference effects often predicted by numerical models. Specifically, comparisons are made with numerical predictions using the fast field method at 25 and 250 Hz, showing close agreement through caustics as well as in shadows for a 1000-m water depth.

Normal mode sound propagation in an ocean with random narrow-band surface waves

Normal mode sound propagation in an isovelocity ocean with random narrow-band surface waves is considered, assuming the root-mean-square wave height to be small compared to the acoustic wavelength. Nonresonant interaction among the normal modes is studied using a straightforward perturbation technique. The more interesting case of resonant interaction is investigated using the method of multiple scales to obtain a pair of stochastic coupled amplitude equations which are solved using the Peano–Baker expansion technique. Equations for the spatial evolution of the first and second moments of the mode amplitudes are also derived and solved. It is shown that, irrespective of the initial conditions, the mean values of the mode amplitudes tend to zero asymptotically with increasing range, the mean-square amplitudes tend towards a state of equipartition of energy, and the total energy of the modes is conserved.

Normal-mode sound propagation in an ocean with sinusoidal surface waves

The normal-mode solution to the problem of acoustic wave propagation in an isovelocity ocean with a wavy surface is considered. The surface wave amplitude is assumed to be small compared to the acoustic wavelength, and the method of multiple scales is employed to study the interaction between normal-mode acoustic waves and the surface waves. A nonresonant interaction causes small fluctuations of the amplitude and phase of the acoustic wave at a rate dependent on the frequency of the surface wave. Backscatter occurs if the wavenumber of the surface wave is larger than that of the acoustic wave. The interaction becomes resonant if appropriate phase-matching conditions are satisfied. In this case, two acoustic normal modes get coupled, resulting in a large-scale periodic exchange of energy from one mode to another.

REVSPEC—A reverberation spectrum numerical model

The REVSPEC model provides a numerical evaluation of the reverberation spectrum for a general set of sonar system parameters and environmental conditions. The model is a numerical evaluation scheme based on the discretization approach of Ackerman for a general formulation of reverberation as developed by Fauer, Ol'shevskii, and Middleton. The model produces surface, volume, and/or bottom reverberation spectra and power levels as functions of the sonar system (transmit signal waveform and power level, sampling rate, transmit/receive and horizontal/vertical beampatterns), environment (spreading and absorption propagation losses, backscattering strengths, wind speed and direction, scatterer motion), and sonar platform (motion, depth, direction). The reverberation spectra at the receiver input and at the receiver output for a range‐gated and waveform‐weighted receiver are generated. The reverberation is nonstationary in that the spectra vary with time (and range) both in their spectral shape and power levels. Intrinsic assumptions include: primary scattering only (no secondary scattering), isovelocity ocean (straight line propagation paths), direct path (no boundary forward reflections), quasistationary narrowband transmit signals, backscattering from a large number of randomly distributed, weak, discrete scatterers, and radial velocity distributions of boundary and volume scatterers are uniform in space. The model formulation is presented and examples which illustrate its applicability for sonar systems analysis are given.

Comparison of coupled-mode theory with the small-waveheight approximation for sea-surface scattering

The coupled-mode theory of sea-surface scattering is shown to be identical to the small-waveheight theory for the case of a deep isovelocity ocean where the modal spectrum forms a continuum. Numerical calculations demonstrate that the same results are also obtained for a shallow-water example where the modal spectrum is discrete.

A technique for measuring the plane-wave reflection coefficient of the ocean bottom

A new technique for the measurement of the plane-wave reflection coefficient of a horizontally stratified ocean bottom is described. It is based on the exact Hankel transform relationship between the reflection coefficient and the bottom reflected field due to a point source. The method employs a new algorithm for the numerical evaluation of the Hankel transform which is based on the ’’projection-slice’’ theorem for the two-dimensional Fourier transform. The details of the algorithm are described in the companion paper. Although the algorithm is applied to the case of an isovelocity ocean, the general theory for measuring the plane-wave reflection coefficient in a refracting ocean is developed. The technique provides information about the reflection coefficient, not only for real incident angles, but also for complex angles, thus potentially providing substantial additional structural information about the bottom. The method is shown to yield excellent results with synthetically generated data for the cases of a hard bottom and slow isovelocity bottom.

Mode interactions in an isovelocity ocean of uniformly varying depth

Pierce’s treatment of normal-mode propagation for an ocean in which physical parameters vary slowly with range is used to calculate mode interactions for isovelocity water of depth varying linearly with range. Simplifying assumptions are (a) two dimensions only, depth and range, and (b) a pressure-release bottom interface (reasonably valid for all but the highest modes). After general procedures are established, an explicit result is derived for the interaction contribution of mode 2 to mode 1 (the lowest mode), for down-slope propagation with several allowed modes. Changes, mainly only of signs and phases, are derived to cover up-slope propagation as well as the contributions of mode 1 to mode 2. Each interaction yields a forward-scattered component and a considerably weaker back-scattered one. Each is proportional to the bottom slope. The forward component is also proportional to the ratio depth to acoustic wavelength; the back component follows the inverse ratio. When normalized to the unperturbed outgoing mode functions, the interaction contributions are essentially of equal magnitudes, as would be expected in the absence of loss mechanisms.

Sideband structure of sound from a harmonic point source scattered by a rough surface moving over an upward-refracting ocean

A normal-mode theory, for scattering of sound by the ocean surface in the presence of refraction, is reviewed. The theory is based on a perturbation expansion for small surface-wave heights. New results, based on this theory, are presented for the Doppler-shifted signal sidebands. These sidebands are the solution of the first-order perturbation problem and are represented exactly by a double normal-mode expansion with coefficients given by certain integral representations. The new results are derived by a saddle-point analysis of the integral representations, due account being taken of the presence of singularities. Specific numerical results for the case of upward refraction are shown to display considerably more structure than in the case of an isovelocity ocean. The increased structure is explained in terms of the effects of backscatter and focusing through the use of wave-number diagrams and ray-trajectory constructions.

Erratum: ’’Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean’’

Erratum: ’’Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean’’

Range-dependent normal modes in underwater sound propagation: Application to the wedge-shaped ocean

We use the adiabatic range variation method of Pierce and Milder to perform an approximate separation into normal modes of the wave equation for the problem of underwater sound propagation in an almost-stratified under-ocean channel with gradual range dependence of medium and boundaries. Our solution is illustrated by an application to the isovelocity ocean wedge with rigid ocean floor, and compared with the exact solution of Bradley and Hudimac. Even when mode coupling is neglected, good agreement is obtained for moderately large wedge angles. Subject Classification: 30.20; 20.15, 20.40.

Scattering of sound from a point source by a rough surface progressing over a refracting ocean

A normal-mode theory for scattering in the presence of refraction is reviewed. The theory is based on a perturbation expansion for small-surface wave heights. New results, based on this theory, are presented for the Doppler-shifted signal sidebands. The results are derived by a saddle-point analysis of the exact integral solution of the first-order perturbation problem. Specific numerical results for the case of upward refraction display considerably more structure than in the case of an isovelocity ocean, due to the effects of backscatter and focusing. [Research supported by ONR, Code 480.]

Asymptotic theory of scattering by a rough surface progressing over an inhomogeneous ocean

An asymptotic theory of scattering by a rough surface is presented which is appropriate to small ocean surface wave heights and valid for a slowly varying refractive index. The ocean surface is described as a sum of gravity waves of varying frequency progressing in different directions. It is demonstrated that, to a given order of approximation in the asymptotic theory, the average energy associated with the sideband rays is precisely accounted for by a reduction in intensity of the reflected carrier. The theory is shown to be in complete agreement with results obtained by asymptotic analysis of exact integral representations for the case of an isovelocity ocean. Wavenumber diagrams are introduced which are extremely useful for depicting the directions of the incident and scattered rays at the surface, demonstrating the effect of varying surface wave and acoustic frequency and demonstrating the frequency cutoff of the sideband power spectrum. It is shown how the theory can accommodate the effects of energy focusing, both in the carrier and sidebands, through the use of known uniformly valid asymptotic expansions of the field in the presence of caustics. [Research supported by the Office of Naval Research, Code 480.]

Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean

A perturbation procedure is applied, through the second order of approximation, to the three-dimensional problem of scattering of sound from a point source by a rough surface progressing in the wind direction over an isovelocity ocean. The results satisfy conservation of energy, in a ray-theoretic sense, and are uniformly valid throughout the field. At points in the region of surface-image interference (Lloyd’s-mirror region), the signal is shown to be strongly modulated. Results are presented which depict the power in the carrier and signal sidebands. The latter are not, in general, symmetric in amplitude about the carrier. However, it is demonstrated analytically that, on two specific planes in the field, the signal power spectrum is symmetric about the carrier. Subject Classification: 30.40, 30.20, 30.25.

New approach to scattering by random media and interfaces. II. Active underwater acoustic channels

The general method of paper I above is systematically applied to the characterization of active underwater acoustic channels, in an isovelocity ocean. Expressions for the second-order statistics of the received scattered field are obtained, which, apart from a possible coherent term (k=0), exhibit a combination of the Poisson [or Faure-Olshevskii-Middleton (FOM) model] contribution (k=1), and the earlier continuum, or “classical” component (k=2). Unlike all earlier “classical” scatter theories, the dynamical effects of Doppler are explicitly obtained, from the ensemble of appropriate “Dopplerized” wave equations, e.g., Langevin equation. Random-path delays produce the expected random phase modulation, as do the various Dopplers, but unlike the former the latter have variances quadratic in time. Because of this, the scattering medium, even when stationary, progressively destroys the coherence of the injected signal. The continuum component (k=2) of the medium itself may be described by delay-spread and Doppler-spread “cells” which are now correlated (i.e., non WSSUS), unlike the Poisson term (k=1), which is WSSUS. Other channel features are considered, including conditions for validity of the FOM models, and the role of surface interactions. [Work supported by Naval Sea Systems Command.]