J. S. Robertson

Current and current shear effects in the parabolic approximation for underwater sound channels

The effect of currents on the acoustic pressure field in an underwater sound channel is investigated. Based on fundamental fluid equations, model equations are formulated for sound pressure while including nonuniform currents in the source–receiver plane. Application of parabolic‐type approximations yields a collection of parabolic equations. Each of these is valid in a different domain determined by the magnitudes of current speed, current shear, and depth variation of sound speed. Under certain conditions, it is possible to interpret current effects in terms of an effective sound speed. Using this effective sound speed in an existing numerical code, we examine sound speed in a shallow water isospeed channel with a simple shear flow and a lossy bottom. It is found that even small currents can induce very substantial variations in relative intensity. The degree of variation depends upon current speed, source and receiver geometry, and acoustic frequency. Particular emphasis is placed on intensity‐differen...

Low‐frequency sound propagation modeling over a locally reacting boundary with the parabolic approximation

There is substantial interest in the analytical and numerical modeling of low‐frequency, long‐range atmospheric acoustic propagation. Ray‐based models, because of their frequency limitations, do not always give an adequate prediction of quantities such as sound pressure or intensity levels. However, the parabolic approximation method, widely used in ocean acoustics, and often more accurate than ray models for frequencies of interest, can be applied to acoustic propagation in the atmosphere. Modifications of an existing implicit finite‐difference implementation for computing solutions to the parabolic approximation are discussed. A locally reacting boundary is used together with a one‐parameter (the flow resistivity) ground impedance model. Intensity calculations are performed for a number of flow resistivity values in both quiescent and windy atmospheric sound channels. Variations in the value of this parameter are shown to have substantial effects on the spatial variation of the acoustic signal. [Work su...

Acoustical effects of ocean current shear structures in the parabolic approximation

In a previous article [Robertson et al., J. Acoust. Soc. Am. 77, 1768–1780 (1985)], the authors developed a family of parabolic equations that includes effects due to the presence of a time‐dependent, depth‐dependent current. Some of these equations contain a new term that explicitly depends on current gradient. In this article, certain effects of this new term are studied. By transforming the parabolic equations, it is possible to convert them to forms that can be directly solved numerically using existing IFD or FFT implementations. Ocean currents with vertical fine structure present situations that can require these new types of parabolic approximations. Propagation in a shallow isospeed channel is examined, with both rigid and lossy bottoms, and use is made of a shear flow with features of an actual ocean current. The vertical current variation can cause changes in relative intensity that are substantial and that depend on bottom loss, source and receiver locations, and acoustic frequency. Intensity d...

A treatment of three‐dimensional underwater acoustic propagation through a steady shear flow

The effect of a steady, depth‐dependent, horizontal shear current in an underwater sound channel is considered. Because the source–receiver direction and current direction need not lie in the same vertical plane, the propagation problem is inherently three dimensional. A three‐dimensional (3‐D) parabolic approximation for this channel is formulated by extending a two‐dimensional result obtained previously [Robertson et al., J. Acoust. Soc. Am. 77, 1768–1780 (1985)]. It is shown that, if the azimuthal derivatives are small enough to be neglected in the farfield, azimuthal effects appear only as coefficients in the parabolic equation. Therefore, an N×2‐D technique can be used to solve the parabolic equation. Numerical examples are used to examine cross‐current propagation. It is shown that substantial intensity variations can occur as the angle between the source–receiver direction and current varies from 0 to 180 deg.

An Efficient Enhancement of Finite-Difference Implementations for Solving Parabolic Equations

The parabolic approximation method is widely recognized as useful for accurately analyzing sound transmissions in diverse ocean environments. One reason for its attractiveness is because solutions are marched in range, thereby avoiding the massive internal storage required when using the full wave equation. Present implementations employ a range step size that is prescribed either by the user or by the code and remains fixed for the duration of the computation. An algorithm is presented in which the range step is adaptively selected by a procedure within the implicit finite‐difference (IFD) implementation of the parabolic approximation. An error indicator is computed at each range step, and its value is compared to a user‐specified error tolerance window. If the error indicator falls outside this window, a new range step size is computed and used until the error indicator again leaves the tolerance window. For a given tolerance, the algorithm generates a range step size that is optimal in a specified sens...

An intensity approximation for source motion in shallow water, with application to passive tracking

The behavior of incoherent total‐field intensity is considered for a moving cw source in shallow water. The ocean channel is assumed isospeed, with a planar perfectly reflecting surface and planar lossy bottom. The source moves on a linear path at constant depth, and the receiver is fixed on the bottom. An expression for incoherent total‐field intensity loss is derived in terms of source‐motion and environmental parameters. For a bottom with uniform loss per ray reflection, special functions are used to develop and analyze an approximation for intensity. The behavior of this novel approximation with respect to variations in source quantities, such as range, depth, and speed, as well as bottom‐loss variations, suggests its possible application in source‐motion prediction. Two properties of the approximation, peak curvature and peak width of intensity versus time, are selected as source‐motion descriptors. Both are shown to be relatively insensitive to bottom‐loss variations, but sensitive to changes in sou...

Parabolic approximations for coherent low‐frequency acoustic propagation through atmospheric turbulence.

A parabolic equation (PE) method for the prediction of coherent low‐frequency acoustic propagation through small‐scale atmospheric turbulence is presented. Frequency constraints on the applicability of stochastic parabolic approximations are avoided by first averaging the stochastic Helmholtz equation and then applying a parabolic approximation to the resulting deterministic equation. Turbulence effects are incorporated by means of spatially varying effective wave numbers. Comparison of exact solutions in the case of infinite‐space propagation demonstrates the advantages and limitations of this approach. A uniform asymptotic expression for the effective wave‐number profile in the case of isotropic turbulence is used to develop a half‐space PE formulation that is valid in the limit of low‐frequency, small‐scale inhomogeneity. For anisotropic turbulence that is correlated more strongly in range than height, a modified mean‐value theorem for the 2‐D Helmholtz operator is used to find the effective wave numbe...

The influence of an ideal wind profile on sound propagation over a ridge

The effects on low‐frequency acoustic propagation resulting from ideal atmospheric flow over a large ridge are investigated using the parabolic approximation. The ridge is taken to be triangularly shaped with a horizontal earth‐air interface on both sides. A Schwarz‐Christoffel transformation is employed to calculate the wind speeds that are then used to compute the effective sound‐speed profiles. These profiles are used by an implicit finite‐difference implementation of the parabolic approximation to estimate the intensity of the sound field. Several examples are examined to determine the effects of this wind‐modeling method on sound pressure levels over rigid earth‐air boundaries. [Work supported by NASA.]

An adaptive two‐dimensional algorithm for solving finite‐difference implementations of parabolic equations

In previous work algorithms have been developed for enhancing the efficient solution of finite‐difference implementations of the parabolic approximation that adaptively adjust range‐step size [J. Acoust. Soc. Am. 86, 252–260 (1989)] and depth‐step size [Proc. Second IMACS Symp. Comput. Acoust. (to appear)]. Here, an algorithm is presented that integrates previous work, i.e., both range‐ and depth‐step sizes are adaptively selected by a procedure within the implicit finite‐difference (IFD) implementation of the parabolic approximation. An error indicator that includes components resulting from both range and depth discretization is computed at each range step, and its value is compared to a user‐specified error tolerance window. If the error indicator falls outside this window, the range‐ and depth‐step sizes are adjusted so that the error is equidistributed between both components. Further adjustments are not made until the indicator leaves the tolerance window. In this sense, the algorithm generates a tw...

Three‐dimensional underwater acoustic propagation through a steady shear flow

Previous studies by the authors have incorporated current effects into parabolic approximations, when the current vector lay within the vertical source‐receiver plane. Here, the current and the source‐receiver plane are not constrained to be parallel, and thus a solution to a fully three‐dimensional problem is obtained. A class of sound‐speed profiles is examined that is advected in a steady, depth‐dependent ocean current. A cw source is submerged between a horizontal pressure‐release surface and a horizontal bottom. When certain asymptotic order conditions on derivatives of the current and sound speed are satisfied, parabolic approximations of canonical form result. Furthermore, if additional restrictions on azimuthal derivatives apply, the azimuthal modes are decoupled so that the azimuthal angle appears only as a parameter in the parabolic equations. Thus the pressure field may be calculated in decoupled vertical planes, i.e., in an N × 2‐D approximation. Several examples are discussed, and the acousti...