Wave packets, rays, and the role of real group velocity in absorbing media

Abstract

In an absorbing medium, where the vector $\mathbf{W}=\ensuremath{\partial}\ensuremath{\omega}/\ensuremath{\partial}\mathbf{k}$ usually is complex for real values of the wave vector k, the group velocity W may become real for some complex values of k. The role of real group velocity in the propagation of one-dimensional wave packets in homogeneous absorbing media is examined. Applying the saddle point method to an analysis of the asymptotic behavior of the Gaussian wave packets shows that for absorbing media, at large times and distances, the real group velocity appears as a local characteristic of any small section of a wave packet. For each section we can find the complex values of the local wave number and the local frequency defining a real group velocity. Thus, the real group velocity concepts in absorbing media do not have to be based on the signals having real wave vectors or real frequencies. The analysis of the exact solution for a Gaussian wave packet in a medium with a complex law of dispersion describing whistler waves in a collisional plasma is performed. It is shown that at all times the initial carrier wave number exists as a real part of the local complex wave number at some point of the Gaussian envelope and this point moves with a constant real group velocity. For large times the local wave group with the initial carrier wave number can be found far away from the envelope center.