Complex group velocity and energy transport in absorbing media

Complex group velocity is common in absorbing and active media, yet its precise physical meaning is unclear. While in the case of a nondissipative medium the group velocity of propagating waves C_g=dω/dk is exactly equal to the observable energy velocity (defined as the ratio between the energy flux and the total energy density) C_g=F¯/E¯ , in a dissipative medium C_g=dω/dk is in general a complex quantity which cannot be associated with the velocity of energy transport. Nevertheless, we find that the complex group velocity may contain information about the energy transport as well as the energy dissipated in the medium. The presented analysis is intended to expound the connection between the complex group velocity and energy transport characteristics for a class of hyperbolic dissipative dynamical systems. Dissipation mechanisms considered herein include viscous and viscoelastic types of damping. Both cases of spatial and temporal decay are discussed. The presented approach stems from the Lagrangian formulation and is illustrated with identities that relate the complex group velocity and energy transport characteristics for the damped Klein-Gordon equation; Maxwell’s equations, governing electromagnetic waves in partially conducting media; and Biot’s theory, governing acoustic wave propagation in porous solids.