Application of the Foldy–Wouthuysen transformation to the reduced wave equation in range-dependent environments

Abstract

The Foldy-Wouthuysen transformation can be used to reduce the relativistic Klein-Gordon equation to the nonrelativistic Schrödinger equation. This technique is modified and applied to the problem of wave propagation through media with a range-dependent index of refraction. The forward and backward propagating components of the field are decoupled order-by-order to produce a perturbative expansion of the range-dependent parabolic equation. The result includes energy-conserving correction terms that can be associated with a rapid fluctuation of energy between forward and backward propagating solutions of the Helmholtz equation. The approach selects out physical processes which accumulate over the entire range of propagation, distinguishing them from effects which depend solely on the initial and final values of the index of refraction and its derivatives. It is also shown that the corresponding backscatter mechanism is fundamentally nonperturbative, so that the parabolic equation technique as applied to the problem of propagation through range-dependent media generates an asymptotic expansion of the exact solution. This procedure has been applied to long-distance low-frequency propagation through a sound channel with internal waves. For this application, the expansion parameters are typically very small, so the propagation distances must be very large for the effect to be detectable. ͓S0001-4966͑97͒02203-0͔

Zitterbewegungof Klein-Gordon particles and its simulation by classical systems

“…The Klein-Gordon equation appears in several classical systems, usually as a modification of the wave equation φ = 0. Under some conditions KGE is used to describe sound waves in ducts [18,19], electromagnetic waves in the ionosphere [20,21], transverse modes of wave guides [22] and oceanic waves [23]. Below we examine in more detail a model proposed by Morse and Feshbach in which one can simulate KGE with the use of a piano string and a thin rubber sheet [11].…”

A nonlocal effective operator for coupling forward and backward propagating modes in inhomogeneous media

“…1 The Foldy-Wouthysen transformation 2,3 applied to the Helmholtz Hamiltonian produces a first order correction term to the parabolic wave equation approximation. 1 Because the study in Ref.…”

“…1 The Foldy-Wouthysen transformation 2,3 applied to the Helmholtz Hamiltonian produces a first order correction term to the parabolic wave equation approximation. 1 Because the study in Ref. 1 did not examine the convergence characteristics of the perturbation series, one cannot gauge the accuracy of the first order correction term provided for the parabolic equation by the Foldy-Wouthuysen transformation.…”

The one-way wave equation and its invariance properties

“…The origin of the corrected one-way equations obtained is an analogy with the WKB solution in the one-dimensional case [11]. An alternative method [12][13][14] to derive improved one-way wave equations, still with claimed energy flux conservation, is to resort to the Foldy-Wouthuysen (FW) transformation, initially developed in order to derive relativistic corrections to the Schrödinger equation, from the Dirac or Klein-Gordon equations. A similar method has been developed for one-dimensional propagation [15], exhibiting WKB corrections at high orders.…”