We have reported previously [Ralph N. Baer, J. Acoust. Soc. Am. 69, 70–75 (1981)] on an algorithm, based on the parabolic approximation to the reduced wave equation, for the propagation of sound in three dimensions in the ocean. We present here a simpler algorithm: solve N two-dimensional problems and combine the results to form an approximate three-dimensional solution. Analytic and numerical results show that this N×2D approach is an excellent approximation to the original algorithm for realistic ocean environments, even those where fronts and eddies are present, provided redirection of energy in azimuth due to boundary interaction is not important. We compare the two computer models based on these algorithms for several test cases by considering how they distribute energy spatially, and by simulating the performance of a hypothetical horizontal array of hydrophones placed in the calculated complex-valued acoustic fields. In an example of propagation through a large and strong ocean eddy, the N×2D method predicts intensities and array performance parameters (such as array signal gain) which are typically within 8% of the values from the full 3D algorithm. In another example, with a large linear cross-range gradient on the index of refraction, the N×2D and 3D solutions are essentially identical out to a range of 120 km. The new method has made it practical to consider higher source frequencies and has reduced the substantial computer time of the three-dimensional program.

By means of a corrected split-step parabolic-equation numerical algorithm, acoustic propagation through an ocean region characterized by a sound-speed distribution produced by an analytic model of an eddy is investigated. Parameter values for a moderate sized, cyclonic, Gulf Stream eddy are used. It is found that the presence of an eddy causes significant changes, both in nature and level, in the received acoustic field produced by an omnidirectional cw source. The eddy causes major changes in the arrival structure seen by a vertical array. The percentage of energy arriving at angles less than 5° from broadside increases from 5% to 50% in the presence of the eddy under consideration at 100 Hz. Greatly increased energy corresponding to horizontal arrival is noted. In addition, the effects of an eddy progressing through the region between an acoustic source and receiver is studied. The position of the eddy relative to the source causes changes in transmission loss of as much as 20 dB.

The possibility of determining the location of an acoustic source in the presence of gross sediment uncertainties is investigated. Promising results are obtained using focalization, which involves constructing ambiguity surfaces corresponding to randomly selected realizations of the sediment parameters. Due to a parameter hierarchy in which the source location is more important than environmental parameters, it is often possible to reliably determine the source position without determining the correct sediment parameters. The examples involve multiple sediment layers, with sound speeds and range-dependent thicknesses that are unknown. An example that includes both sediment uncertainties and internal waves is also included.

Propagation through a three-dimensional model of the sound–speed structure of an eddy is studied by means of a three-dimensional computer model based on the parabolic approximation to the reduced wave equation. Illustrations of the effect of the eddy on transmission loss in both vertical and horizontal planes are provided. Refractive variations cause a 20-km difference in the range of a given acoustic feature (such as a convergence zone) between the vertical plane through the eddy center and one exterior to the eddy. Eddy-induced variations on acoustic array performance are also considered. It is shown that an eddy can produce over one-half degree deviation in apparent source bearing angle. The influence on the acoustic 3-dB angular width and array signal gain are noted.

Wave propagation in heterogeneous media can be modeled efficiently with the parabolic equation method, which has been extended to problems with homogeneous anisotropic layers [A. J. Fredricks et al., Wave Motion 31, 139–146 (2000)]. This approach is currently being extended to the heterogeneous case, including piecewise continuous vertical dependence and horizontal dependence that is relatively gradual but which may be large over sufficient distances. Vertical dependence is included by applying appropriate heterogeneous depth operators in the equations of motion. Horizontal dependence is included by extending the single-scattering solution [E. T. Kusel et al., J. Acoust. Soc. Am. 121, 808–813 (2007)] to the anisotropic case. [Work supported by the Office of Naval Research.]

The effects in a bounded channel of a bilinear sound-speed profile on ray geometry, travel time, and spreading loss are examined. General equations are derived. The case of bottomed source and receiver is considered in detail. Appropriate approximations are made and properties of the ray geometry, travel time, spreading loss, and the total acoustic field at the receiver are deduced. It is shown that the gross characteristics of observed phase variations off Bermuda may be generated by changes in sound speed associated with Rossby waves and tidal effects. The addition to the model of tidal depth variation leads to intensity variation which accounts for 80% of the range of intensity variations observed near Bermuda.

A one-way wave equation is presented with the following properties. (1) For low angles and small sound-speed variations, it reduces to the standard parabolic approximation. (2) It allows a split-step solution. (3) The rays associated with this equation are exactly the rays of the Helmholtz equation in a range-independent environment. It is in the last sense an optimal one-way wave equation. Results of the split-step solution of this equation are presented and compared to normal-mode calculations and results of another modification of the standard parabolic equation, which was given by Thomson and Chapman [D. J. Thomson and N. R. Chapman, J. Acoust. Soc. Am. 74, 1848–1854 (1983)].

The possibility of determining the location of an acoustic source in the presence of internal waves is investigated. Source localization problems require environmental information as inputs. Internal waves cause uncertainties in the sound speed field. In previous work ͓J. Acoust. Soc. Am. 90, 1410-1422 ͑1991͔͒, it was found that a source can often be localized in an uncertain environment by including environmental parameters in the search space and tweaking them, but usually not determining their true values. This is possible due to a parameter hierarchy in which the source position is more important than the environmental parameters. The parameter hierarchy is shown to also apply to uncertainties associated with internal waves. Due to differences in the nature of the parameter space, this problem is solved with a statistical approach rather than a parameter search technique such as simulated annealing.