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# Waves in Unmagnetized Plasmas

Let us now investigate the cold-plasma dispersion relation in detail. It is instructive to first consider the limit in which the equilibrium magnetic field is zero. In the absence of a 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 (5.42) reduces to

 (5.68)

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

 (5.71)

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),

 (5.72)

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.

Next: Low-Frequency Wave Propagation Up: Waves in Cold Plasmas Previous: Cutoff and Resonance
Richard Fitzpatrick 2016-01-23