Newsgroups: sci.physics.plasma
From: oliveria@down.engin.umich.edu (Roque Donizete de Oliveira)
Organization: University of Michigan Engineering, Ann Arbor
Subject: Re: complex group velocity

In article <aiqbe$m3n@mojo.eng.umd.edu>,
Frank Crary <fcrary@benji.Colorado.EDU> wrote:

>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.

I found a paper that promises to be interesting in this subject:

\bibitem{Muschietti} L. Muschietti and C. T. Dum,
{\em Real Group Velocity in a Medium with Dissipation},
Phys. Fluids B {\bf 5}, 1383 (1993).

>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.
>
I start with the Vlasov-Maxwell equations (kinetic theory)
for parallel propagating waves and a nonrelativistic electron
distribution fucntion of the form
f(v,\theta,z) = C exp(-v^2/V^2) G(\theta,z)
where \theta is the velocity pitch angle and
G(\theta,z) is the local form of the pitch angle distribution
obtained by applying Liouville's theorem to the midplane
pitch angle distribution G(\theta) below

( \theta - a)^2
G(\theta) = exp( - ----------------- )
b^2

where V, C, a and b are constants.
It is this pitch angle distribution that makes the
dispersion equation very messy (it has an integral term).
This is another problem though.

I'm trying to recall if there is any assumption in the derivation
(in the WKB approximation) of the local dispersion equation
equation, D(k,\omega,z) = 0, that says that the imaginary
part of index of refraction has to be smaller than its real
part, as Frank Crary mentioned in his reply. I thought as long
as the WBK condition is satisfied, i.e.,

| 1 dk(z) |
| ------ ------ | << 1
| k^2(z) dz |

the results would still be valid. I thought that assumptions
about the Im(k) << Re(k) are only made when one tries to get
nice analytical expressions for k as a function of \omega.
In my case, I do get (in some regions of axial z axis) solutions
in which imaginary part of the index of refraction is comparable
(but still smaller) to its real part. The WKB condition above
was marginally satisfied (I would get for the worst case 0.95 << 1).

Roque