Real group velocity in a medium with dissipation

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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.