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Plane Waves in a Homogeneous Plasma
The propagation of small amplitude waves is described by linearized equations.
These are obtained by expanding the equations of motion in powers
of the wave amplitude, and neglecting terms of order higher than unity.
In the following, we use the subscript 0 to distinguish equilibrium
quantities from perturbed quantities, for which we retain the previous
notation.
Consider a homogeneous, quasineutral plasma, consisting of equal
numbers of electrons and ions, in which both plasma species are at rest.
It follows that
, and
. In a homogeneous medium, the
general solution of a system of linear equations can be constructed as
a superposition of plane wave solutions:

(450) 
with similar expressions for and . The
surfaces of constant phase,

(451) 
are planes perpendicular to , traveling at the velocity

(452) 
where
, and is a unit vector
pointing in the direction of . Here,
is termed the phasevelocity.
Henceforth, we shall omit
the subscript from field variables, for ease of notation.
Substitution of the plane wave solution (450) into Maxwell's equations yields:
In linear theory, the current is related to the electric field via

(455) 
where the conductivity tensor
is a
function of both and . Note that the conductivity
tensor is anisotropic in the presence of a nonzero equilibrium
magnetic field. Furthermore,
completely specifies the
plasma response.
Substitution of Eq. (455) into Eq. (453) yields

(456) 
where we have introduced the dielectric permittivity tensor,

(457) 
Here, is the identity tensor. Eliminating the
magnetic field between Eqs. (454) and (456), we obtain

(458) 
where

(459) 
The solubility condition for Eq. (459),

(460) 
is called the dispersion relation. The dispersion relation
relates the frequency, , to the wavevector, .
Also, as the name
``dispersion relation'' indicates, it allows us to determine the rate at which the
different Fourier components in a wavetrain disperse due to
the variation of their phasevelocity with wavelength.
Next: ColdPlasma Dielectric Permittivity
Up: Waves in Cold Plasmas
Previous: Introduction
Richard Fitzpatrick
20110331