Next: Polarization
Up: Waves in Cold Plasmas
Previous: ColdPlasma Dielectric Permittivity
It is convenient to define a vector

(488) 
which points in the same direction as the wavevector, ,
and whose magnitude
is the refractive index (i.e., the ratio of the
velocity of light in vacuum to the phasevelocity). Note that should not be
confused with the particle density.
Equation (458) can be rewritten

(489) 
We may, without loss of generality, assume that the equilibrium
magnetic field is directed along the axis, and that the wavevector,
, lies in the plane. Let be the angle subtended between
and . The eigenmode equation (489) can be written

(490) 
The condition for a nontrivial solution is that the determinant of
the square matrix be zero. With the help of the identity

(491) 
we find that

(492) 
where
The dispersion relation (492) is evidently a quadratic in , with
two roots.
The solution can be written

(496) 
where

(497) 
Note that . It follows that is always real, which implies
that is either purely real or purely imaginary. In other words, the
coldplasma dispersion relation describes waves which either propagate
without evanescense, or decay without spatial oscillation.
The two roots
of opposite sign for , corresponding to a particular root for , simply describe
waves of the same type propagating, or decaying, in opposite directions.
The dispersion relation (492) can also be written

(498) 
For the special case of wave propagation parallel to the magnetic
field (i.e., ), the above expression reduces to
Likewise, for the special case of propagation perpendicular to the
field (i.e., ), Eq. (498) yields
Next: Polarization
Up: Waves in Cold Plasmas
Previous: ColdPlasma Dielectric Permittivity
Richard Fitzpatrick
20110331