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Waves in an Unmagnetized Plasma
Let us now investigate the coldplasma dispersion relation in detail. It is
instructive to first consider the limit in which the equilibrium magnetic
field goes to zero. In the absence of the magnetic field, there is
no preferred direction, so we can, without loss of generality,
assume that is directed along the axis (i.e., ).
In the zero magnetic field limit (i.e.,
), the eigenmode equation (490) reduces to

(516) 
where

(517) 
Here, we have neglected with respect to .
It is clear from Eq. (516) that there are two types of wave.
The first possesses
the eigenvector , and has the dispersion relation

(518) 
The second possesses the eigenvector , and has the dispersion
relation

(519) 
Here, , , and are arbitrary nonzero quantities.
The first wave has parallel to , and is, thus, a
longitudinal wave. This wave is know as the plasma wave, and
possesses the fixed frequency
. Note that if
is parallel to then it follows from Eq. (454)
that
. In other words, the wave is purely electrostatic
in nature. In fact, a plasma wave is an electrostatic oscillation of the type
discussed in Sect. 1.5.
Since is independent of , the group
velocity,

(520) 
associated with a plasma wave,
is zero. As we shall demonstrate later on, the group velocity is the propagation
velocity of localized wave packets. It is clear that the plasma wave
is not a propagating wave, but instead has the property than an oscillation
set up in one region of the plasma remains localized in that region. It
should be noted, however, that in a ``warm'' plasma (i.e., a plasma with a finite
thermal velocity) the plasma wave acquires a nonzero,
albeit very small, group velocity (see Sect. 6.2).
The second wave is a transverse wave, with perpendicular to
. There are two independent linear polarizations of this wave,
which propagate at identical velocities,
just like a vacuum electromagnetic wave. The dispersion relation
(519) can be rearranged to give

(521) 
showing that this wave is just the conventional electromagnetic wave,
whose vacuum dispersion relation is
, modified by
the presence of the plasma. An important property, which follows
immediately from the above expression, is that for the propagation of
this wave we need
. Since is
proportional to the square root of the plasma density, it follows that
electromagnetic radiation of a given frequency will only propagate through
a plasma when the plasma density falls below a critical value.
Next: LowFrequency Wave Propagation
Up: Waves in Cold Plasmas
Previous: Cutoff and Resonance
Richard Fitzpatrick
20110331