Frederick D. Tappert

Ray chaos in underwater acoustics

Generically, in range‐dependent environments, the acoustic wave equation cannot be solved by techniques which require that variables be separated. Under such conditions, the acoustic ray equations, which have Hamiltonian form, are nonintegrable. At least some ray trajectories are expected to exhibit chaotic motion, i.e., extreme sensitivity to initial and environmental conditions. These ideas are illustrated numerically using simple models of the ocean sound channel with weak periodic range dependence. The use of Poincare sections, power spectra, and Lyapunov exponents to investigate and characterize ray chaos are discussed. The practical importance of chaotic ray trajectories—a limitation on one’s ability to make deterministic predictions using ray theory—is emphasized.

New full‐wave approximation for ocean acoustic travel time predictions

A new full‐wave parabolic approximation is introduced that is valid for a wide range of grazing angles. By Fourier synthesis it yields travel times of ocean acoustic multipaths that are insensitive to a reference speed of sound. After depths and sound speeds are transformed to new coordinates, the highly efficient ‘‘split‐step Fourier’’ algorithm is used to solve the new approximate wave equation for forward propagation. Accuracy of the new approximation has been tested by comparison to a broadband normal mode model in a range‐independent environment. At 1000 km range and with a pulse of resolution 20 ms at center frequency 75 Hz, computed travel times of 24 multipaths agreed with maximum difference 3.4 ms, mean difference 0.9 ms, and rms difference 1.5 ms. This approximation may prove to be an efficient method for accurate travel time predictions of multipaths over a wide range of acoustic frequencies and for basin scale distances.

Calculation of the effect of internal waves on oceanic sound transmission

The signal received by a hydrophone in the ocean many kilometers from a steady sound source fluctuates dramatically due to variations of the speed of sound in sea water. By inserting an empirical model of internal‐wave‐generated sound‐speed variations into an acoustic‐transmission computer code, we have shown that internal waves cause significant variations in sound transmission at 100 Hz, comparable in size and frequency to the variations observed in field experiments. We have also studied the usefulness of vertical hydrophone arrays.Subject Classification: 30.25, 30.82; 28.60.

Acoustic ray chaos induced by mesoscale ocean structure

It has previously been shown that some acoustic ray trajectories in ocean models with periodic range dependence exhibit chaotic behavior, thereby imposing a limit on one’s ability to make deterministic predictions using ray theory. The objective of the work reported here is to quantify the limitations of ray theory to make predictions of underwater sound fields in the presence of realistic mesoscale structure. This is done by numerically investigating sound ray propagation in an analytically prescribed sound speed model consisting of Munk’s canonical profile perturbed by a randomly phased superposition of several baroclinic modes of the linearized quasigeostrophic potential vorticity equation. The ray equations used are consistent with the parabolic wave equation. To investigate ray chaos, power spectra are calculated and Lyapunov exponents are estimated. For realistic strengths of the mesoscale field, near‐axial ray trajectories are found to be chaotic with characteristic e‐folding distances (inverse Lya...

KANEOHE ACOUSTIC THERMOMETER FURTHER VALIDATED WITH RAYS OVER 3700 KM AND THE DEMISE OF THE IDEA OF AXIALLY TRAPPED ENERGY

The Kaneohe acoustic source transmitted 133‐Hz, 60‐ms resolution signals over 3709 km from Oahu at 183‐m depth to a Naval receiver at 1433‐m depth near northern California. Ray theory successfully models the acoustic multipaths whose travel times are unambiguously tracked between 1983–89, despite the fact that the sound bounces one or more times from the Oahu slope before becoming trapped in the sound channel. The eigenrays are inclined at about 15° at the axis of the sound channel. The upper turning depths of the eigenrays are insensitive to realistic perturbations along the section. This supports the finding that the changes in delay of ∼±0.2 s between 1983–89 are due to temperature and not due to changes in the multipaths. Compared with transmission through a smoothed representation of the ocean’s acoustic waveguide, the mesoscale and submesoscale features vertically scatter axially trapped energy about 200 and 800 m, respectively. The submesoscale structure may be associated with internal waves. Scatt...

An investigation of sound ray dynamics in the ocean volume using an area preserving mapping

Abstract An area preserving mapping which describes sound ray propagation in a simple range-dependent model of the ocean sound channel is derived and studied. The unbounded ocean model has a bilinear sound speed profile in which the vertical sound speed gradient above the sound channel axis varies sinusoidally in range. It is assumed that the scale of the range-dependent perturbation is small compared to a typical upper loop length of a ray. The explicit mapping which results gives successive iterates of range and upgoing ray angle at the sound channel axis, ( r n , θ n ) → ( r n + 1 , θ n +1 . The degree of stochasticity of the mapping is governed by a single dimensionless parameter, e — the strength of the range dependent perturbation. Iterates of the mapping indicate that some ray trajectories are chaotic (i.e., exhibit estreme sensitivity to initial conditions) for perturbations comparable in strength to those produced by internal waves in the ocean. The chaotic nature of these rays is confirmed by the calculation of positive Lyapunov exponents.

A range refraction parabolic equation

Application of the standard parabolic wave equation to solve real problems requires a clever selection of the reference wavenumber k0. An extended parabolic equation, having range refraction capability, is reintroduced in such a manner so as to be totally independent of k0. An already existing Implicit Finite‐Difference (IFD) model was applied to test the range refraction parabolic equation. Results compare favorably with known solutions for weakly range‐dependent environments, but yield significant corrections for propagation through strong oceanic fronts.

Propagation and analysis issues in the prediction of long‐range reverberation

Data collected from the Office of Naval Research‐special research program (ONR‐SRP) bottom reverberation research cruises consist of various environmental measurements (sound‐speed profiles, bottom properties, etc.), extensive bathymetric mapping of the 300 km×150 km natural laboratory, and high‐quality acoustic reverberation data recorded in both monostatic and bistatic geometries.The analysis of the acoustic data and the effects of propagation are investigated. Accurate GPS tracking of the participating research vessels provided the precision necessary to attempt to correlate ‘‘spiky’’ reverberation events with bathymetric features. Understanding the limitations of our ability to resolve such features—due to imperfect signal processing, environmental variability, and complex multipath structures—is the main objective of this work. Using a simplified time‐to‐range conversion, the measured reverberation data can be displayed over the local bathymetry. This process shows good correlation between large‐scal...

Source localization using the PE method

The most important information for source localization is oceanographic knowledge supplied to a powerful machine that numerically computes acoustic Greens functions rapidly and accurately. By this means the distorting effects of the oceanic medium and its boundaries can be removed, thereby rendering the ocean transparent. Then targets can be detected, localized, and classified at low S/N ratios as though they were moving in free space, leading to the result there is “no place to hide.” As an illustration of this approach, we discuss in detail the solution of the problem of passive narrow‐band acoustic localization (an instance of the inverse source problem) following the pioneering work of Bucker [J. Acoust. Soc. Am. 59, 368–373 (1976)], in which beamforming was made obsolescent. Using the PESOGEN (Parabolic Equation SOlution GENerator) computer system [J. Acoust. Soc. Am. Suppl. 1 75, S26 (1984)] and an algorithm based on the generalized principle of reciprocity, we have demonstrated theoretically that targets can be accurately and reliably localized at long ranges and low S/N ratios using sparse configurations of sensors.

Weak chaos in an area-preserving mapping for sound ray propagation

Abstract A nonseparable underwater acoustic wave propagation problem is studied in the geometric limit. The combination of internal refraction and reflecting boundaries leads to a noncontinuously differentiable area-preserving mapping, to which the KAM theorem does not apply. The phenomenon of weak chaos, wherein an arbitrarily small perturbation to the separable problem causes observable chaotic behavior, is shown to occur.