Real group velocity in a medium with dissipation

Abstract

In order to clarify the role of the imaginary term Im(d(omega)/dk), the motion of a wave packet in a dissipative, homogeneous medium is examined. The integral representation of the packet is analyzed by means of a saddle-point method. It is shown that in a moving frame attached to its maximum the packet looks self-similar. A Gaussian packet keeps its Gaussian identity, as is typical for the case of a nondissipative medium. The central wave number of the packet slowly changes because of a different damping among the Fourier components. Simple 'ray-tracing equations' are derived to follow the packet centers in coordinate and Fourier spaces. The analysis is illustrated with a comparison to geometric optics, and with two applications: the case of a medium with some resonant damping (or growth) and the propagation of whistler waves in a collisional plasma.