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.