Waves in Unmagnetized Plasmas

where

(5.69) |

Here, we have neglected with respect to .

It is clear from Equation (5.68) that there are two types of waves. The first possesses the eigenvector , and has the dispersion relation

(5.70) |

The second possesses the eigenvector , and has the dispersion relation

Here, , , and are arbitrary non-zero quantities.

The former wave has
parallel to
, and is, thus, a
longitudinal (with respect to the electric field) wave. This wave is known as the *plasma wave*, and
possesses the fixed frequency
. Now, if
is parallel to
then it follows from Equation (5.5)
that
. In other words, the plasma wave is purely electrostatic
in nature. In fact, the plasma wave is an electrostatic oscillation of the type
discussed in Section 1.4.
Because
is independent of
, the so-called *group-velocity* (Fitzpatrick 2013),

associated with a plasma wave, is zero. As is demonstrated in Section 6.7, 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 non-zero, albeit very small, group velocity. (See Section 8.2.)

The latter 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 (5.71) can be rearranged to give

(5.73) |

showing that this wave is just the conventional electromagnetic wave, whose vacuum dispersion relation is , modified by the presence of the plasma. An important conclusion, which follows immediately from the previous expression, is that this wave can only propagate if . Because is proportional to the square root of the electron number density, it follows that electromagnetic radiation of a given frequency can only propagate through an unmagnetized plasma when the electron number density falls below some critical value.