A transformation of the environment eliminates parabolic equation phase errors

Abstract

It is shown that phase errors associated with the standard parabolic wave equation can be eliminated in a range-independent environment by appropriately transforming the environment. Solutions to the standard parabolic equation ͑PE͒ in the transformed environment are close to solutions to the Helmholtz equation in the original environment in the sense that ray and asymptotic mode contributions to the PE wave fields have no phase errors relative to their Helmholtz equation counterparts. Although phase errors are eliminated in the PE wave fields, complete asymptotic equivalence to the Helmholtz equation wave fields is not attained. Ray-and mode-based wave-field expansions are used to illustrate similarities and differences between PE and Helmholtz equation wave fields. Numerical simulations show excellent agreement between PE wave fields in the transformed environment and mode-based solutions to the Helmholtz equation in the true environment.

Array design considerations for exploitation of stable weakly dispersive modal pulses in the deep ocean

“…The second profile, C1, is the same as C0 with a Gaussian disturbance added in the upper ocean [23]. C1 qualitatively resembles an environment constructed from the hydrographic data in the Eastern North Pacific ocean [24].…”

“…The first profile, called C0, is the canonical "Munk" mid-latitude ocean profile. The second profile, called C1 in this paper, is the same as in [23] and is a perturbed version of C0.…”

Developments of parabolic equation method in the period of 2000–2016

“…Compared with other sound propagation methods, the PE method cannot obtain an accurate phase results of sound field computation. To eliminate phase errors, in 2006, Rypina et al [69] applied the environmental transformation technique to the standard parabolic wave equation. The technique can eliminate PE phase errors for all propagation angles simultaneously in a range-independent environment.…”