Real group velocity in a medium with dissipation

Abstract

When a medium is dissipative, the classic expression for the group velocity, dω/dk, is complex with an imaginary part often being far from negligible. To clarify the role of this imaginary term, 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 familiar for the case of a nondissipative medium. However, the central wave number of the packet slowly changes because of a differential damping among the Fourier components: Im(dω/dk)=dγ/dk≠0, where ω≡ωr+iγ. The packet height can be computed self-consistently as integrated damping (or growth). The real group velocity becomes a time-dependent combination of Re(dω/dk) and Im(dω/dk). Only where the medium is both homogeneous and loss free, does the group velocity remain constant. 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 by two applications: the case of a medium with some resonant damping (or growth) and the propagation of whistler waves in a collisional plasma.