Parabolic Equation Development in Recent Decade

Abstract

Numerous contributions have been made in the enhancement of the Parabolic Equation (PE) approximation method, which has been shown to be a useful tool for solving realistic problems in many different scientific fields. Evidence of its usefulness is the application of PE to solve ocean acoustic propagation problems. In early years, when the PE was introduced to the field of underwater acoustics, its main purpose was to predict long-range, low-frequency acoustic propagations in range-dependent environments; thus, there were certain limitations. In the recent decade, these limitations have been relaxed a great deal due to many useful contributions. The time has come to survey and report these important contributions and to discuss how these contributions enhance the capability of the PE method. This paper gives a brief review of what had been done before 1984 and highlights some important PE developments from 1984 to 1994. Also, some applications of the PE to predict ocean acoustic propagation problems will be presented. We shall call attention to a few important issues related to the PE developments and applications. Looking ahead we will discuss what more a PE can do in order to stimulate future research, development, as well as applications.

Hybrid Ray-Fdtd Moving Coordinate Frame Approach for Long Range Tracking of Collimated Wavepackets

“…The first and second expressions in (27) correspond to the leading order terms in (19) and (22), respectively, showing, as expected, that the moving frame numerical dispersion relations for collimated PBs recover the continuous frame relations in the limit Ω, K ζ → 0 .…”

“…Moving frame concepts have been used extensively in various analytical and numerical techniques in the past, however mostly in the form of the parabolic equation (PE) method or its optical counterpart, the beam propagation method (BPM) [19,20 (Ch. 6), 22,23].…”

The Bremmer series for a multi-dimensional acoustic scattering problem

“…In multi-dimensions, wave decomposition have been used in a variety of large scale problems, e.g., exploration seismics, ocean acoustics and fiber optics [17,20]. In contrast to the one-dimensional case, it is difficult to use an exact wave decomposition in multi-dimensions.…”

“…Several papers consider the par-axial or beam approximation [10,13], where the one-way wave operator is approximated with a local operator, asymptotically correct in the preferred direction and in the high frequency limit. Here, we use a pseudo-differential approximation [5,10,15,17]. This gives a non-local one-way wave operator asymptotically correct in both the high frequency and high wave number limit.…”

Moving Coordinate Frame FDTD Analysis of Long Range Tracking of Pulsed Fields in Graded Index Waveguides

“…Moving frame concepts have been used extensively in various analytical and numerical techniques in the past, mostly in the form of the frequency domain parabolic equation (PE) method which is also termed in the optical context as the beam propagation method (BPM) [8,9 (Ch. 6), [10][11][12].…”