Newsgroups:
sci.physics.plasma

From: ctd@mpe-garching.mpg.de (Christian T. Dum)

Organization:
MPE

Subject: Re: complex group velocity

Approved:
eastman@glue.umd.edu

>

>In article
<3atoep$nuh@mojo.eng.umd.edu> oliveria@down.engin.umich.edu

> (Roque Donizete de Oliveira) writes:

>From:
oliveria@down.engin.umich.edu (Roque Donizete de Oliveira)

>Subject:
Re: complex group velocity

>Date: 22 Nov 1994 21:39:37 GMT

>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

>>

>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 believe that you have come across an embarrassing problem which
is far

from having been resolved.
The usual assumption of negliglible imaginary

part probably is violated more often than it is
satisfied. Take cyclotron

waves,
for example. The frequency stays then relatively constant, near a

harmonic, while the growth rate generally will have a
strong wave number

dependence. Hence with the definition v_g=d\omega/dk
one can easily get

imaginary
parts of v_g comparable or even much larger than the real part, even

if
the growth rate remains small compared to the frequency.

I don't
see anything wrong with the derivation of such dispersion relations.

The restriction to small imaginary
parts mentioned in the comments above

applies only to approximate
solutions based on the separation of real and

imaginary parts,
\gamma=-Imag \epsilon/{d Real \epsilon/d\omega}, rather

than a rigorous
solution of \epsilon =0 for complex
\omega=\omega_r+i\gamma.

(or complex k).

The next step I think
is to examine the concept of group
velocity

itself. It is always
used in the context of some approximation to the problem

of wave
propagation in a dispersive medium.

The traditional way (Brillouin 1914) is to consider the
space-time evolution

of a wave packet. The group velocity describes then
the propagation speed

of the peak in the envelope of this packet. In the paper mentioned above (

Muschietti
and Dum, 1993) we have extended this approach to a dissipative

medium,
using saddle point methods to approximately evaluate the

corresponding
Fourier integral. The basic effect is
easily understood by

realizing that d\gamma/dk represents differential
growth, leading to

shifts in the
dominant wave number etc. with time.
The effective

propagation speed ( group velocity) thus is a
combination of the real and

imaginary
parts of d\omega/dk which depends on the width of the packet.

The
group velocity also appears in the ray tracing equations for an

inhomogenous
medium. Bernstein (1975) considered
first order corrections which

arise for weak dissipation. We have
numerically compared his approach with

ours in the limit of a homogenous
plasma. It would be importantant to extend

results to inhomogenous
plasmas. Another important extension is
needed in the

kinetic eqation for
the wave spectra of turbulent
plasmas.

C. T. Dum

Max Planck Institute f. Extraterr. Physik

and Center for Space Research,
M. I. T.