Effects of front width on acoustic ducting by a continuous curved front over a sloping bottom.

The behavior of sound near an ocean front in a region with wedge bathymetry is examined. The front is parameterized as a zone of variation with inshore and offshore boundaries parallel to a straight coastline. The importance of frontal width and frontal sound speed on the ducting of acoustic energy is examined. Previous analytical studies of sound propagation and parameter sensitivity in an idealized wedge environment use an unphysical but convenient single interface front representation, which is here replaced by a continuous sound speed profile. The continuous profile selected is convenient for analytical investigation, but encourages the use of asymptotic approximation methods which are also described. The analytical solution method is outlined, and numerical results are produced with an emphasis on comparing to the single interface front. These comparisons are made to highlight the strengths and weaknesses of the idealized model for capturing the horizontal ducting effects.

Parameter dependence of acoustic mode quantities in an idealized model for shallow-water nonlinear internal wave ducts.

Nonlinear internal waves in shallow water have significant acoustic impacts and cause three-dimensional ducting effects, for example, energy trapping in a duct between curved wavefronts that propagates over long distances. A normal mode approach applied to a three-dimensional idealized parametric model [Lin, McMahon, Lynch, and Siegmann, J. Acoust. Soc. Am. 133(1), 37-49 (2013)] determines the dependence of such effects on parameters of the features. Specifically, an extension of mode number conservation leads to convenient analytical formulas for along-duct (angular) acoustic wavenumbers. The radial modes are classified into five types depending on geometric characteristics, resulting in five distinct formulas to obtain wavenumber approximations. Examples of their dependence on wavefront curvature and duct width, along with benchmark comparisons, demonstrate approximation accuracy over a broad range of physical values, even including situations where transitions in mode types occur with parameter changes. Horizontal-mode transmission loss contours found from approximate and numerically exact wavenumbers agree well in structure and location of intensity features. Cross-sectional plots show only small differences between pattern phases and amplitudes of the two calculations. The efficiency and accuracy of acoustic wavenumber and field approximations, in combination with the mode-type classifications, suggest their application to determining parameter sensitivity and also to other feature models.

Estimating the parameter sensitivity of acoustic mode quantities for an idealized shelf-slope front.

The acoustic modes of an idealized three-dimensional model for a curved shelf-slope ocean front [Lin and Lynch, J. Acoust. Soc. Am. 131, EL1-EL7 (2012)] is examined analytically and numerically. The goal is to quantify the influence of environmental and acoustic parameters on acoustic field metrics. This goal is achieved by using conserved quantities of the model, including the dispersion relation and a conservation of mode number. Analytic expressions for the horizontal wave numbers can be extracted by asymptotic approximations and perturbations, leading to accurate and convenient approximations for their parameter dependence. These equations provide the dependence on model parameter changes of both the real horizontal wavenumbers, leading to modal phase speeds and other metrics, and the imaginary parts, leading to modal attenuation coefficients. Further approximations for small parameter changes of these equations characterize the parameter sensitivities and produce assessments of environmental and acoustic influences.

Approximate formulas and physical interpretations for horizontal acoustic modes in a shelf-slope front model.

The structure and behavior of horizontal acoustic modes for a three-dimensional idealized model of a shelf-slope front are examined analytically. The Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) method is used to obtain convenient simple expressions and to provide physical insight into the structure and behavior of horizontal modes as trapped, leaky, or transition types. Validity regions for WKBJ expressions in terms of slope and frontal parameters are found, and outside the regions the asymptotic formulas for large order and large argument Hankel functions are used. These combined approximations have very good accuracy as shown by comparisons with numerical solutions for modal shapes and horizontal wavenumbers.

Treatment of a sloping fluid-solid interface and sediment layering with the seismo-acoustic parabolic equation.

The parabolic equation method is extended to handle problems in seismo-acoustics that have multiple fluid and solid layers, continuous depth dependence within layers, and sloping interfaces between layers. The medium is approximated in terms of a series of range-independent regions, and a single-scattering approximation is used to compute transmitted fields across the vertical interfaces between regions. The approach is implemented in terms of a set of dependent variables that is well suited to piecewise continuous depth dependence in the elastic parameters, but one of the fluid-solid interface conditions in that formulation involves a second derivative that complicates the treatment of sloping interfaces. This issue is resolved by using a non-centered, four-point difference formula for the second derivative. The approach is implemented using a matrix decomposition that is efficient when the parameters of the medium have a general dependence within the upper layers of the sediment but only depend on depth in the water column and deep within the sediment.

A scaled mapping parabolic equation for sloping range-dependent environments.

Parabolic equation solutions use various techniques for approximating range-dependent interfaces. One is a mapping approach [M. D. Collins et al., J. Acoust. Soc. Am. 107, 1937-1942 (2000)] where at each range the domain is vertically translated so that sloping bathymetry becomes horizontal, and range dependence is transferred to the upper surface. In this paper, a scaled mapping is suggested where the domain is vertically distorted so that both the bathymetry and upper surface are horizontal. Accuracy is demonstrated for problems involving fluid sediments. Generalizations of the approach should be useful for environments with layer thicknesses that vary with range.

Two parabolic equations for propagation in layered poro-elastic media.

Parabolic equation methods for fluid and elastic media are extended to layered poro-elastic media, including some shallow-water sediments. A previous parabolic equation solution for one model of range-independent poro-elastic media [Collins et al., J. Acoust. Soc. Am. 98, 1645-1656 (1995)] does not produce accurate solutions for environments with multiple poro-elastic layers. First, a dependent-variable formulation for parabolic equations used with elastic media is generalized to layered poro-elastic media. An improvement in accuracy is obtained using a second dependent-variable formulation that conserves dependent variables across interfaces between horizontally stratified layers. Furthermore, this formulation expresses conditions at interfaces using no depth derivatives higher than first order. This feature should aid in treating range dependence because convenient matching across interfaces is possible with discretized derivatives of first order in contrast to second order.

Horizontal ducting of sound by curved nonlinear internal gravity waves in the continental shelf areas.

The acoustic ducting effect by curved nonlinear gravity waves in shallow water is studied through idealized models in this paper. The internal wave ducts are three-dimensional, bounded vertically by the sea surface and bottom, and horizontally by aligned wavefronts. Both normal mode and parabolic equation methods are taken to analyze the ducted sound field. Two types of horizontal acoustic modes can be found in the curved internal wave duct. One is a whispering-gallery type formed by the sound energy trapped along the outer and concave boundary of the duct, and the other is a fully bouncing type due to continual reflections from boundaries in the duct. The ducting condition depends on both internal-wave and acoustic-source parameters, and a parametric study is conducted to derive a general pattern. The parabolic equation method provides full-field modeling of the sound field, so it includes other acoustic effects caused by internal waves, such as mode coupling/scattering and horizontal Lloyd's mirror interference. Two examples are provided to present internal wave ducts with constant curvature and meandering wavefronts.

Single-scattering parabolic equation solutions for elastic media propagation, including Rayleigh waves.

The parabolic equation method with a single-scattering correction allows for accurate modeling of range-dependent environments in elastic layered media. For problems with large contrasts, accuracy and efficiency are gained by subdividing vertical interfaces into a series of two or more single-scattering problems. This approach generates several computational parameters, such as the number of interface slices, an iteration convergence parameter τ, and the number of iterations n for convergence. Using a narrow-angle approximation, the choices of n=1 and τ=2 give accurate solutions. Analogous results from the narrow-angle approximation extend to environments with larger variations when slices are used as needed at vertical interfaces. The approach is applied to a generic ocean waveguide that includes the generation of a Rayleigh interface wave. Results are presented in both frequency and time domains.

Experimental testing of the variable rotated elastic parabolic equation.

A series of laboratory experiments was conducted to obtain high-quality data for acoustic propagation in shallow water waveguides with sloping elastic bottoms. Accurate modeling of transmission loss in these waveguides can be performed with the variable rotated parabolic equation method. Results from an earlier experiment with a flat or sloped slab of polyvinyl chloride (PVC) demonstrated the necessity of accounting for elasticity in the bottom and the ability of the model to produce benchmark-quality agreement with experimental data [J. M. Collis et al., J. Acoust. Soc. Am. 122, 1987-1993 (2007)]. This paper presents results of a second experiment, using two PVC slabs joined at an angle to create a waveguide with variable bottom slope. Acoustic transmissions over the 100-300 kHz band were received on synthetic horizontal arrays for two source positions. The PVC slabs were oriented to produce three different simulated waveguides: flat bottom followed by downslope, upslope followed by flat bottom, and upslope followed by downslope. Parabolic equation solutions for treating variable slopes are benchmarked against the data.

Parabolic equation solution of seismo-acoustics problems involving variations in bathymetry and sediment thickness.

Recent improvements in the parabolic equation method are combined to extend this approach to a larger class of seismo-acoustics problems. The variable rotated parabolic equation [J. Acoust. Soc. Am. 120, 3534-3538 (2006)] handles a sloping fluid-solid interface at the ocean bottom. The single-scattering solution [J. Acoust. Soc. Am. 121, 808-813 (2007)] handles range dependence within elastic sediment layers. When these methods are implemented together, the parabolic equation method can be applied to problems involving variations in bathymetry and the thickness of sediment layers. The accuracy of the approach is demonstrated by comparing with finite-element solutions. The approach is applied to a complex scenario in a realistic environment.

Statistical analysis of sound transmission results obtained on the New Jersey continental shelf.

Experiments have been conducted near the site of AMCOR Borehole 6010 on the New Jersey Shelf to evaluate propagation predictability in sandy shallow-water environments. The influence of a nonlinear frequency dependence of the sediment volume attenuation in the uppermost sediment layer at this location is examined. Previously it was determined that a frequency power-law exponent of 1.5 was required for the best modeling of experimental results over the band 50-1000 Hz. The approach here references the attenuation to an accepted value at 1 kHz and makes extensive comparisons between measurements and calculations, to determine a power-law exponent of 1.85+/-0.15.

Comparison of simulations and data from a seismo-acoustic tank experiment.

A tank experiment was carried out to investigate underwater sound propagation over an elastic bottom in flat and sloping configurations. The purpose of the experiment was to evaluate range-dependent propagation models with high-quality experimental data. The sea floor was modeled as an elastic medium by a polyvinyl chloride slab. The relatively high rigidity of the slab requires accounting for shear waves in this environment. Acoustic measurements were obtained along virtual arrays in the water column using a robotic apparatus. Elastic parabolic equation solutions are in excellent agreement with data.

A single-scattering correction for large contrasts in elastic layers.

The single-scattering solution is implemented in a formulation that makes it possible to accurately handle solid-solid interfaces with the parabolic equation method. Problems involving large contrasts across sloping stratigraphy can be handled by subdividing a vertical interface into a series of two or more scattering problems. The approach can handle complex layering and is applicable to a large class of seismic problems. The solution of the scattering problem is based on an iteration formula, which has improved convergence in the new formulation, and the transverse operator of the parabolic wave equation, which is implemented efficiently in terms of banded matrices. Accurate solutions can often be obtained by using only one iteration.

Generalization of the rotated parabolic equation to variable slopes.

The rotated parabolic equation [J. Acoust. Soc. Am. 87, 1035-1037 (1990)] is generalized to problems involving ocean-sediment interfaces of variable slope. The approach is based on approximating a variable slope in terms of a series of constant slope regions. The original rotated parabolic equation algorithm is used to march the field through each region. An interpolation-extrapolation approach is used to generate a starting field at the beginning of each region beyond the one containing the source. For the elastic case, a series of operators is applied to rotate the dependent variable vector along with the coordinate system. The variable rotated parabolic equation should provide accurate solutions to a large class of range-dependent seismo-acoustics problems. For the fluid case, the accuracy of the approach is confirmed through comparisons with reference solutions. For the elastic case, variable rotated parabolic equation solutions are compared with energy-conserving and mapping solutions.

Modeling Rayleigh and Stoneley waves and other interface and boundary effects with the parabolic equation.

An improved approach for handling boundaries, interfaces, and continuous depth dependence with the elastic parabolic equation is derived and benchmarked. The approach is applied to model the propagation of Rayleigh and Stoneley waves. Depending on the choice of dependent variables, the operator in the elastic wave equation may not factor or the treatment of interfaces may be difficult. These problems are resolved by using a formulation in terms of the vertical displacement and the range derivative of the horizontal displacement. These quantities are continuous across horizontal interfaces, which permits the use of Galerkin's method to discretize in depth. This implementation extends the capability of the elastic parabolic equation to handle arbitrary depth dependence and should lead to improvements for range-dependent problems.

Measurement and modeling of three-dimensional sound intensity variations due to shallow-water internal waves.

Broadband acoustic data (30-160 Hz) from the SWARM'95 experiment are analyzed to investigate acoustic signal variability in the presence of ocean internal waves. Temporal variations in the intensity of the received signals were observed over periods of 10 to 15 min. These fluctuations are synchronous in depth and are dependent upon the water column variability. They can be explained by significant horizontal refraction taking place when the orientation of the acoustic track is nearly parallel to the fronts of the internal waves. Analyses based on the equations of vertical modes and horizontal rays and on a parabolic equation in the horizontal plane are carried out and show interesting frequency-dependent behavior of the intensity. Good agreement is obtained between theoretical calculations and experimental data.

Analysis and modeling of broadband airgun data influenced by nonlinear internal waves.

To investigate acoustic effects of nonlinear internal waves, the two southwest tracks of the SWARM 95 experiment are considered. An airgun source produced broadband acoustic signals while a packet of large nonlinear internal waves passed between the source and two vertical linear arrays. The broadband data and its frequency range (10-180 Hz) distinguish this study from previous work. Models are developed for the internal wave environment, the geoacoustic parameters, and the airgun source signature. Parabolic equation simulations demonstrate that observed variations in intensity and wavelet time-frequency plots can be attributed to nonlinear internal waves. Empirical tests are provided of the internal wave-acoustic resonance condition that is the apparent theoretical mechanism responsible for the variations. Peaks of the effective internal wave spectrum are shown to coincide with differences in dominant acoustic wavenumbers comprising the airgun signal. The robustness of these relationships is investigated by simulations for a variety of geoacoustic and nonlinear internal wave model parameters.

A wide angle and high Mach number parabolic equation.

Various parabolic equations for advected acoustic waves have been derived based on the assumptions of small Mach number and narrow propagation angles, which are of limited validity in atmospheric acoustics. A parabolic equation solution that does not require these assumptions is derived in the weak shear limit, which is appropriate for frequencies of about 0.1 Hz and above for atmospheric acoustics. When the variables are scaled appropriately in this limit, terms involving derivatives of the sound speed, density, and wind speed are small but can have significant cumulative effects. To obtain a solution that is valid at large distances from the source, it is necessary to account for linear terms in the first derivatives of these quantities [A. D. Pierce, J. Acoust. Soc. Am. 87, 2292-2299 (1990)]. This approach is used to obtain a scalar wave equation for advected waves. Since this equation contains two depth operators that do not commute with each other, it does not readily factor into outgoing and incoming solutions. An approximate factorization is obtained that is correct to first order in the commutator of the depth operators.