In this paper, the propagation of a broadband sound pulse in three-dimensional ͑3D͒ shallow water waveguides is investigated numerically. Two cases are examined: ͑i͒ the 3D ASA benchmark wedge, and ͑ii͒ the 3D Gaussian canyon. The numerical method used to solve the four-dimensional acoustic problem is based on a Fourier synthesis technique. The frequency-domain calculations are carried out using the fully 3D parabolic equation based model 3DWAPE, recently modified to include a wide-angle paraxial approximation for the azimuthal component. A broadband sound pulse with a central frequency of 25 Hz and a bandwith of 40 Hz is considered. For both test cases, 3D results corresponding to a 25 Hz cw point source are first presented and compared with predictions from a 3D adiabatic modal model. Then, the acoustic problem is solved considering the broadband source pulse. The modal structure of the received signals is analyzed and exhibits multiple mode arrivals of the propagating signal.
In this paper, the issue of using higher-order finite difference schemes to handle the azimuthal derivative term in a three-dimensional parabolic equation based model is addressed. The three-dimensional penetrable wedge benchmark problem is chosen to illustrate the accuracy and efficiency of the proposed schemes. Both point source and modal initializations of the pressure field are considered. For each higher-order finite difference scheme used in azimuth, the convergence of the numerical solution with respect to the azimuth is investigated and the CPU times are given. Some comparisons with solutions obtained from another 3-D model [J. A. Fawcett, J. Acoust. Soc. Am. 93, 2627-2632 (1993)] are presented. The numerical simulations show that the use of a higher-order scheme in azimuth allows one to reduce the required number of points in the azimuthal direction while still obtaining accurate solutions. The higher-order schemes have approximately the same efficiency as a FFT-based approach (in fact, may outperform it slightly); however, the finite difference approach has the advantage that it may be more flexible than the FFT approach for various PE approximations.
SUMMARYWe consider the third-order Claerbout-type wide-angle parabolic equation (PE) of underwater acoustics in a cylindrically symmetric medium consisting of water over a soft bottom B of range-dependent topography. There is strong indication that the initial-boundary value problem for this equation with just a homogeneous Dirichlet boundary condition posed on B may not be well-posed, for example when B is downsloping. We impose, in addition to the above, another homogeneous, second-order boundary condition, derived by assuming that the standard (narrow-angle) PE holds on B, and establish a priori H 2 estimates for the solution of the resulting initial-boundary value problem for any bottom topography. After a change of the depth variable that makes B horizontal, we discretize the transformed problem by a second-order accurate finite difference scheme and show, in the case of upsloping and downsloping wedge-type domains, that the new model gives stable and accurate results. We also present an alternative set of boundary conditions that make the problem exactly energy conserving; one of these conditions may be viewed as a generalization of the Abrahamsson-Kreiss boundary condition in the wide-angle case.
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