Newsgroups: sci.physics.plasma
From: lawson@pax.llnl.gov (William S. Lawson)
Organization: Lawrence Livermore National Laboratory
Subject: Re: complex group velocity

In article <3astj4$9b1@mojo.eng.umd.edu>, fcrary@benji.Colorado.EDU (Frank Crary) writes:
|> In article <3aiqbe$m3n@mojo.eng.umd.edu>,
|> Roque Donizete de Oliveira <oliveria@hadron.engin.umich.edu> wrote:
|> >I'm looking for recent references that discuss
|> >the problem of whether or not complex group
|> >velocity makes sense
|>
|> Well, no. More or less by definition, a complex group
|> velocity doesn't makes sense. (The distance a
|> wave packet travels in a given, real, time has
|> to be real.)
|>
|> > d \omega dD/dk
|> > Vg = --- = - ---------
|> > dk dD/d\omega
|>
|> >I'm solving the plasma dispersion equation D(k,\omega,z) = 0
|> >for parallel propagation (whistler waves, k parallel
|> >to static magnetic field along the z-axis).
|> >I'm solving it for the boundary value problem, i.e.,
|> >complex wavenumber k and real frequency \omega.
|>
|> I think the problem is that you are taking this definition
|> of group velocity a bit to seriously. Jackson's text has
|> an interesting derivation of group velocity that implies
|> Vg = d \omega / d k is just an approximation. For a
|> linear wave in a plane-parallel coordinate system, it
|> is an extremely good approximation, but sometimes you
|> have to be a bit more careful. In your case, I think
|> you want to rewrite the dependence on k, separating
|> the real and complex components into a real wave number
|> and an exponential damping (or growth) term. Then derive
|> the group velocity from the real part of the wave number.

Even this doesn't really work. In a uniform, resistive medium, the damped light
waves have a group velocity that exceeds c. The conditions under which the
group velocity is the signal propagation velocity are quite restrictive (but
often satisfied).

-- Bill Lawson