The one-way wave equation and its invariance properties

Abstract

The harmonic wave equation in inhomogeneous media is exactly split into coupled first-order equations with respect to a principal direction of propagation according to the Bremmer scheme. The resulting one-way wave equation is shown not to conserve energy flux for dimensions two and three against the general belief in one-way wave propagation or parabolic equation literature. Conservation of energy flux is only ensured in the high frequency limit. On the other hand, a simple invariant is found that may be seen as a generalization of the Snell law to arbitrary, non-stratified, media. Similarly, the reciprocity property is not fully ensured in general and the time-reversal symmetry is ensured for propagating fields. Besides, in the one-way wave equation, the additional term to the standard parabolic equation is shown to strengthen mode coupling. The analysis encompasses the evanescent waves.