A Spectral Split-Step Padé Method for Guided Wave Propagation

TL;DR

The paper addresses the challenge of efficiently simulating high-frequency guided wave propagation in ocean acoustics. It introduces a spectral split-step Padé (SSSP) method that uses a discrete sine transform to represent the vertical operator exactly in isovelocity regions and treats inhomogeneities with a Neumann-series expansion. Numerical experiments in range-independent and range-dependent scenarios show that SSSP achieves high accuracy on coarse depth grids, outperforming traditional finite-difference SSP and enabling scalable simulations. The work highlights the practical impact of spectral representations for large-scale ocean acoustic modeling and outlines future directions for absorbing boundaries and hybrid pseudospectral strategies.

Abstract

In this study, a Fourier-based, split-step Padé (SSP) method for solving the parabolic wave equation with applications in guided wave propagation in ocean acoustics is presented. Traditional SSP implementations rely in finite-difference discretizations of the depth-dependent differential operator. This approach limits accuracy in coarse discretizations as well as computational efficiency in dense discretizations since it does not significantly benefit from parallelization. In contrast, our proposed method replaces finite differences with a spectral representation using the discrete sine transform (DST). This enables an exact treatment of the vertical operator under homogeneous boundary conditions. For non-constant sound speed, we use a Neumann series expansion to treat inhomogeneities as perturbations. Numerical experiments demonstrate the method's accuracy in range-independent media and rage-dependent scenarios, including propagation in deep ocean with Munk profile and in the presence of a parametrized synoptic eddy. Compared to finite-difference SSP methods, the Fourier-based approach achieves higher accuracy with fewer depth discretization points and avoids the resolution bottleneck associated with sharp field features, making it well-suited for large-scale, high-frequency wave propagation problems in ocean environments.

Figures (4)

• Figure 1: Acoustic pressure (in dB re $1$ m) computed at $100~\mathrm{Hz}$ using the $[4/4]$ Padé approximation and a step size of $100~\mathrm{m}$ for a deep-water sound channel with the Munk profile \ref{['eq:Munk']}. The subplots (a-c) illustrate the convergence of the Neumann Series \ref{['eq:neumannseries']}. a) corresponds to $M=0$, equivalent to the isovelocity case. b) shows the field for $M=1$, c) for $M=2$. d) shows a reference field, generated by the method of normal modes while a slice (the acoustic pressure at a constant depth $z=1300$ m) is displayed in Figure \ref{['fig:slice']}.
• Figure 2: Acoustic field (in dB re 1 m) propagated through a range independent Munk profile using different correctors of order $M$ at the depth $z=1300~\mathrm{m}$. The reference is once again produced by the method of normal modes. The stepsize used is $h=100~\mathrm{m}$, the discretization constant in depth is $\Delta z = 0.5~\mathrm{m}$.
• Figure 3: Acoustic pressure (in dB re 1 m) due to a point source in the problem of sound propagation through a synoptic eddy. In all SSP-simulations, a $[6,6]$ Padé approximation, a step size of $h=100~\mathrm{m}$ and 2048 equidistant discretization points in depth are used. In a) the field without the eddy is displayed as a reference, computed using SSP. The field perturbed by a synoptic eddy computed using SSP is shown in b). The eddy center is indicated by a black "X", and its sphere of influence, measured by its variance, is indicated by the horizontal and vertical blue lines. c) shows the acoustic field propagated by SSSP through the eddy. d) shows the acoustic field at $z=900\mathrm{m}$, as the field enters the perturbation in sound speed caused by the eddy.
• Figure 4: Propagation of a modal starter through a Munk profile using different discretizations in $z$. Displayed is the acoustic pressure (in dB re 1 m). a) uses SSP with 128 points, b) uses SSSP with again 128 points. c) uses 256 points and SSP, while d) uses SSSP. e) uses 512 points and SSP, f) SSSP. A reference solution computed using normal modes is given in Figure \ref{['fig:order']}.