Shallow Water Wedge Research Articles

Mesh generation for underwater acoustic modeling with KRAKEN

Underwater sound modeling is applied in several environmental studies, from assessing underwater noise induced by offshore wind farms to the influence of climate change on ocean temperature. One of the most well-known underwater acoustics models is KRAKEN, an open-source code requiring a mesh to simulate 3D environments. This paper presents OMesh2KRAKEN, a software toolkit that uses OceanMesh2D to generate unstructured meshes for KRAKEN based on realistic environmental data. Two practical study cases in the North Atlantic are presented and discussed. In addition, an idealized 3D shallow water wedge was simulated. Practically, OMesh2KRAKEN can assist in implementing underwater acoustic modeling studies from deep waters to nearshore regions where, for instance, sound propagation maps are required. Indeed, higher mesh resolutions may be required for areas with complex bathymetric features. In these cases, the new tool presents several advantages as it automatically implements different mesh resolutions in areas of interest and prepares the respective environmental files for acoustic computations.

Underwater Sound Propagation Modeling in a Complex Shallow Water Environment

Three-dimensional (3D) effects can profoundly influence underwater sound propagation in shallow-water environments, hence, affecting the underwater soundscape. Various geological features and coastal oceanographic processes can cause horizontal reflection, refraction, and diffraction of underwater sound. In this work, the ability of a parabolic equation (PE) model to simulate sound propagation in the extremely complicated shallow water environment of Long Island Sound (United States east coast) is investigated. First, the 2D and 3D versions of the PE model are compared with state-of-the-art normal mode and beam tracing models for two idealized cases representing the local environment in the Sound: (i) a 2D 50-m flat bottom and (ii) a 3D shallow water wedge. After that, the PE model is utilized to model sound propagation in three realistic local scenarios in the Sound. Frequencies of 500 and 1500 Hz are considered in all the simulations. In general, transmission loss (TL) results provided by the PE, normal mode and beam tracing models tend to agree with each other. Differences found emerge with (1) increasing the bathymetry complexity, (2) expanding the propagation range, and (3) approaching the limits of model applicability. The TL results from 3D PE simulations indicate that sound propagating along sand bars can experience significant 3D effects. Indeed, for the complex shallow bathymetry found in some areas of Long Island Sound, it is challenging for the models to track the interference effects in the sound pattern. Results emphasize that when choosing an underwater sound propagation model for practical applications in a complex shallow-water environment, a compromise will be made between the numerical model accuracy, computational time, and validity.

Three-dimensional sound scattering from transversely symmetric surface waves in deep and shallow water using the equivalent source method.

This paper proposes a propagation model to calculate the three-dimensional (3-D) sound scattering from transversely symmetric sea surface waves in both deep and shallow water using the equivalent source method (ESM). The 3-D sound field is calculated by integrating an assembly of two-dimensional (2-D) transformed fields with different out-of-plane wavenumbers through a cosine transform. Each 2-D solution is calculated using the ESM incorporating a complex image method that can efficiently and accurately solve the 2-D water/seabed Green's function. The oscillatory cosine integral is accurately calculated using a segmented integral scheme requiring relatively few 2-D solutions, which can be further improved through the use of parallel computation. The model is validated by comparison with a 3-D Helmholtz-Kirchhoff method for deep water and a finite element method for a shallow water wedge with both a fluid and an elastic seabed. The model is as accurate as the finite element approach but more numerically efficient, which enables Monte Carlo simulations to be performed for random rough surfaces in order to study the scattering effects at a reasonable computational cost. Also, 3-D pulse propagation in the shallow water wedge is demonstrated to understand the out-of-plane scattering effects further.

Slope inversion in a single-receiver context for three-dimensional wedge-like environments

In a single-receiver context, time-frequency (TF) analysis can be used to analyze modal dispersion of low-frequency broadband sound pulses in shallow-water oceanic environments. In a previous work, TF analysis was used to study the propagation of low-frequency broadband pulses in three-dimensional (3-D) shallow-water wedge waveguides. Of particular interest is that TF analysis turns out to be a suitable tool to better understand, illustrate and visualize 3-D propagation effects for such wedge-like environments. In the present work, it is shown that TF analysis can also be used at the core of an inversion scheme to estimate the slope of the seabed in a same single hydrophone receiving configuration and for similar 3-D wedge-shaped waveguides. The inversion algorithm proposed, based on a masking process, focuses on specific parts of the TF domain where modal energy is concentrated. The criterion used to quantify the match between the received signal and the replicas by a fully 3-D parabolic equation code, is defined as the amount of measured time-frequency energy integrated inside the masks. Its maximization is obtained using an exhaustive search. The method is first benchmarked on numerical simulations and then successfully applied on experimental small-scale data.

An exact representation of the image field in a perfect wedge

A solution is presented for the image component of the three-dimensional (3-D) acoustic wave-field from an impulsive point source in a liquid wedge with impenetrable boundaries (mixed Dirichlet-Neumann) modeling shallow water over sloping bottom (shallow-water wedge). This exact and complete solution is in the form of a sum of partial waves including the pulse emitted from the source plus a finite number of pulses reflected off the wedge boundaries, and it is generally valid for all ranges and depths. The 3-D acoustic propagation effects in an impenetrable wedge are examined for range propagation when the receivers are located directly up-slope and down-slope of the source and at the source-point itself, and for cross-range propagation when the receivers are located directly cross-slope of the source. When the receiver recedes up-slope from the source, the duration of the pressure response is shortened, and the number of pulses arriving along the indirect ray paths (the backscattered pulses) decreases. When the receiver recedes down-slope from the source, the response duration is constant, and the number of backscattered pulses also decreases. As the source and the receiver separate in the cross-range-direction, the response duration is so shortened that the arrival of the last-arriving pulse approaches that of the first-arriving pulse in the limit (i.e., when the separation between the source and the receiver is “large” in the cross-range-direction), and the number of backscattered pulses remains constant.

Coupled perturbed modes over a sloping penetrable bottom.

The development of coupled perturbed mode theory--the combination of conventional coupled modes and perturbation theory for problems of sound speed range dependence--is extended using a transformation of the normal modes to include a range-dependent penetrable bottom. The result is a fast, direct, and accurate range-dependent modal solution of the acoustic field in the water column. The method is applied to upslope propagation in shallow water wedge waveguides.

A two-way coupled mode simulation of sound propagation in a shallow water wedge

Near a shoreline an acoustic waveguide can be modeled by a wedge-shaped geometry with a penetrable bottom. The propagation of sound in such a waveguide was studied in a tank experiment conducted by Tindle et al. [J. Acoust. Soc. Am. 81, 275–286 (1987); 81, 287–294 (1987)]. An analysis of the time series data measured in this experiment was carried out by Royal et al. (unpublished), in which time series were simulated using a broadband, one-way, coupled normal mode approach. This analysis is extended using a two-way propagation approach that solves the integral equation form of the coupled mode equation. Also, the modal continuum is treated more realistically through the employment of a leaky mode decomposition. In particular, the importance of backscattering and coupling to the continuum is gauged by comparing the experimental data with the results of the two approaches.

Broadband source imaging in a shallow water wedge by an array of receivers

The combination of time domain source imaging techniques and arrays to consider the localization of an impulsive source in a wedge waveguide, with a free surface and rigid bottom, is investigated. An impulsive signal is transmitted from an unknown location, received by an array of hydrophones, and stored. The signals are time reversed and transmitted into the model wedge by a transducer at each receiver location. Model transmission functions are used to compute the pressures at a set of locations. The field calculations use an image construction from D. Chu’s exact solution for a density contrast wedge [J. Acoust. Soc. Am. 86, 1883–1896 (1989)]. A map of the peak pressures gives an image of the source location. Particular attention is given to the peak amplitudes, sidelobe amplitudes, and spatial resolution as a function of the number and placement of receivers. Source localization is enhanced by the range dependency of the environment, which eliminates ‘‘range’’ sidelobes, even for a single hydrophone. Adding receivers along an arc eliminates ‘‘angle’’ sidelobes; but adding receivers along an arc or radial gives little reduction in overall background level. Adding receivers parallel to the wedge axis gives sharp, unambiguous azimuthal imaging.

Exact mode representation of sound propagation in a shallow-water wedge with a penetrable bottom

A new solution for the acoustic wave propagating in a wedge-shaped shallow-water waveguide with a penetrable bottom (ρ1≠ρ2, c1≠c2) is presented. By using a special transformation of range function (Bessel function), the inherent nonseparable problem can be converted to a pseudoseparable one. The solution is exact and is expressed as a summation of wedge mode sets, i.e., each eigenvalue corresponds to a set of eigenfunctions. There are several advantages of the present method over the conventional coupling mode method: (1) since the cylindrical coordinates corresponding to wedge modes are used, the media are bounded in angular direction 0←θ←π; therefore, the mode spectrum of the exact solution only contains discrete modes. (2) The diffracted wave from wedge apex is included automatically. The contribution of the diffracted wave to the total field is important when kr≪1, where k is the wave number and r is the range from the wedge apex. (3) The mode coefficients are obtained through recursive formulas instead of solving M linear equations simultaneously, where M is the number of modes. When c1=c2, the problem becomes separable and for each eigenvalue, only the primary eigenfunction corresponding to a conventional wedge mode remains nonzero. Hence, the solution smoothly reduces to that for an isovelocity/density contrast wedge [D. Chu, J. Acoust. Soc. Am. 87, 2442–2450 (1990)]. Numerical computations are presented to show the effect of mode coupling in terms of different geometrical and physical parameters.

Numerical modeling results for mode propagation in a wedge

A PE (parabolic equation) code is used to study propagation in a shallow water wedge with a penetrable bottom. Particular attention is given to ‘‘wedge modes’’ that have wave fronts that are arcs of circles centered on the wedge apex. For downslope propagation and for upslope propagation before cutoff, the wedge modes propagate independently without coupling. For upslope propagation through cutoff, there is a small amount of coupling to the next lower mode and a trace of coupling to the next higher mode. The wedge modes can be considered the natural modes of the wedge.

Normal mode filtering for downslope propagation in a shallow water wedge

Experiments to measure downslope sound propagation in a shallow water wedge with a penetrable bottom have been conducted in an indoor tank. Mode filtering has been applied to the signals received on a vertical array to extract the waveforms of individual normal modes. Results show that the normal modes of the wedge propagate with wave fronts which are curved into arcs of circles centered on the wedge apex. These wedge modes propagate downslope without coupling and are therefore the true normal modes of the wedge. Conventional normal modes with vertical wave fronts show coupling effects for slopes of 2° and higher. The group velocities of the wedge modes are strongly dependent on the slope and water depth and are in agreement with adiabatic normal mode theory. Modes which are not trapped in the water depth at the source can become trapped when the water depth is sufficient to support them.

Downslope propagation of normal modes in a shallow water wedge

Experimental results are presented for the downslope propagation of acoustic normal modes in a shallow water wedge with a penetrable bottom and a slope of up to 9°. A 7-element line source was used to generate individual low-order normal modes. To obtain single modes in a wedge it was necessary to align the line source to form an arc of a circle centered on the wedge apex. The resulting normal modes propagate with wave fronts which are also curved into arcs of circles centered on the wedge apex. These ‘‘wedge modes’’ propagate without coupling. The amplitudes, mode functions, and group velocities of the wedge modes are described by simple adiabatic normal mode theory, but the wave front curvature is not.

Parabolic equation modeling of normal mode propagation in a wedge

The parabolic equation model has been used to investigate the behavior of the “wedge modes” described in the experimental results of Hobaek, Tindle, and Muir [J. Acoust. Soc. Am. Suppl. 1 78, S70 (1985)] for propagation in a shallow water wedge. The wavefronts of wedge modes are not vertical, but are curved into arcs of circles centered on the wedge apex. Individual wedge modes can be examined by simulating a curved line source. A wedge mode propagates without coupling to other modes. By contrast, if the line source curvature is omitted it is not possible to excite a pure mode, and strong mode interference is observed. As wedge modes propagate upslope through their cutoff depth, their energy is dumped into the bottom as a beam. Any coupling to other modes as a mode passes through cutoff is very small and may be ignored.

Model experiments on mode propagation in a shallow water wedge

Guided mode propagation in a wedge bounded by the ocean surface and a sloped bottom is modeled in a 80‐kHz experiment conducted in an indoor tank, under controlled conditions. The bottom is simulated by a sand filled tray, 10 m long, 1 m wide, and 20 cm deep. The tray is suspended beneath the water surface, with one end pivoted so as to vary the angle of the wedge from 0°–10°. A vertical line array of seven elements is used as a source to preferentially excite low ordered modes in the waveguide. The field of propagating modes is studied with a probe hydrophone as a function of range and depth in both the water column and the sand sediment. Measurements on mode conversion in propagation out of the wedge are discussed and compared to theory. [Work supported by the Office of Naval Research.]