Energy propagation and group speed for complex exponential waves

Abstract eng:
In classical dispersive wave equations, the solution is specified by a superposition of plane waves of the form exp(i(k · x − ωt)) (with k and ω real valued) along with a dispersion relation G(ω, k) = 0. However, in many problems one needs to consider solutions of the form exp(d · x + st), where d and s are complex and correspondingly related by G(is, −id) = 0. In these cases the classical theory providing the speed and direction of the energy propagation no longer applies. In this paper we derive general energy propagation formulas for generic complex exponential wave solutions. We show that: parallel to the direction of the exponential spatial envelope, the energy propagation speed is ce∥ = −Re(s)/Re(d∥ ), while in the orthogonal directions it is cei⊥ = −∂Im(s)/∂Im(di⊥ ). In the classical limit this is consistent with the standard group velocity.

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