Newsgroups: sci.physics.plasma
From: fcrary@benji.Colorado.EDU (Frank Crary)
Organization: University of Colorado, Boulder
Subject: Re: complex group velocity
Approved: eastman@glue.umd.eduF

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.
In other situations, the problem is even messier: For example,
in a cylindirc geometry or particular types of non-uniform
media, the wave is described as a complex Bessel (i.e.
Hankel) function. Here, there isn't a well-defined
dispersion relation nor any wave number that can be
used to derive a group velocity: The solution to the
wave equation actually contains a superposition of
forward and backward propagating waves, as required
by the reflection or refraction of the media. And yet,
the wave does propagate, so it must have a group
velocity. It's just a group velocity where the
d \omega / d k approximation doesn't apply.

>As I solve D(k,\omega,z) = 0 for fixed \omega at several
>axial positions, wave relections seems to be taking place
>(as evidenced by a cusp in the real part of k and group
>velocity becoming very small (and then Vg changes sign)).

I'm not so sure this is a good way to look for reflection.
Sometimes, the group velocity goes to zero and then changes
sign without the wave being reflected. For example, in
sun spot work, when you model the sun spot as a vertical
cylinder with a continuous, rather than sharp, boundary,
the group velocity goes to zero in the boundary layer
but does not produce a reflection: The energy is, instead,
trapped. What you might want to do is compare the
energy fluxes of the continuous wave and the wave
packet: If the continuous wave has a constant flux
but the wave packet does not, then the system is not
absorbing energy; the loss of energy in the wave
packet must be going into reflection. If the continuous
wave also looses energy, then energy may be absorbed
rather than reflected where the group velocity goes
to zero.

>...I just wondered about the
>fact the Vg is complex and at some positions its imaginary
>part is larger than the real part.

I'm not sure about the group velocity, but if the
real part of the wave number isn't much larger than
the complex part, you may have even more problems.
I don't know if you are deriving the wave number from
the fluid or kinetic equations, but in either case,
the derivation hits problems when the wave number
is largely complex. In kinetic theory, the common
derivation of dispersion relations has a built-in
assumption that the complex part is small; if this
isn't true, then the derivation is not valid. In
the fluid treatment, a large, complex wave number
is also a problem: It means either very strong
damping (in which case, the wave doesn't propagate
so much as simply disappear) or strong growth
(in which case, the wave becomes nonlinear within
a wavelength and the linear approximation falls
apart completely.) If I were you, I'd take such a
large, imaginary component as a sign I should look
very carefully at the derivation of the dispersion
relation and make sure all the approximations
were self-consistent.

Frank Crary
CU Boulder