Plane Waves in Homogeneous Plasmas

The propagation of small amplitude plasma waves is described by linearized equations that are obtained by expanding the plasma equations of motion in powers of the wave amplitude, and then neglecting terms of order higher than unity.

Consider a homogeneous, magnetized, quasi-neutral plasma, consisting of equal numbers of electrons and ions, in which the mean velocities of both plasma species are zero. It follows that ${\bf E}_0 = {\bf0}$, and ${\bf j}_0 =\nabla\times {\bf B}_0 /\mu_0= {\bf0}$, where the subscript 0 denotes an equilibrium quantity. In a homogeneous medium, the general solution of a system of linear equations can be constructed as a superposition of plane wave solutions of the form (Fitzpatrick 2013)

$${\bf E} ({\bf r}, t) = {\bf E}_{\bf k} \,\exp[\,{\rm i}\,({\bf k} \cdot{\bf r} - \omega\, t)],$$ (5.1)

with analogous expressions for ${\bf B}({\bf r},t)$ and ${\bf V}({\bf r},t)$. Here, ${\bf E}$, ${\bf B}$, and ${\bf V}$ are the perturbed electric field, magnetic field, and plasma center-of-mass velocity, respectively. The surfaces of constant phase,

$${\bf k}\cdot{\bf r} - \omega\, t = {\rm constant},$$ (5.2)

are planes perpendicular to ${\bf k}$, traveling at the velocity

$${\bf v}_{\rm ph} = \frac{\omega}{k}\,\skew{-3}\hat{\bf k},$$ (5.3)

where $k\equiv \vert{\bf k}\vert$, and $\skew{-3}\hat{\bf k}$ is a unit vector pointing in the direction of ${\bf k}$. Here, ${\bf v}_{\rm ph}$ is termed the phase-velocity of the wave (Fitzpatrick 2013). Henceforth, for ease of notation, we shall omit the subscript ${\bf k}$ from field variables.

Substitution of the plane-wave solution (5.1) into Maxwell's equations yields

$${\bf k}\times{\bf B} = - {\rm i}\,\mu_0\,{\bf j} - \frac{\omega}{c^2}\,{\bf E},$$ (5.4)

$${\bf k}\times{\bf E} = \omega\,{\bf B},$$ (5.5)

where ${\bf j}({\bf r},t)$ is the perturbed current density. In linear theory, the current is related to the electric field via

$${\bf j} = \mbox{\boldmath$\sigma$} \cdot{\bf E},$$ (5.6)

where the electrical conductivity tensor, $\sigma$, is a function of both ${\bf k}$ and $\omega$. In the presence of a non-zero equilibrium magnetic field, this tensor is anisotropic in nature.

Substitution of Equation (5.6) into Equation (5.4) yields

$${\bf k}\times{\bf B} = -\frac{\omega}{c^2}\,{\bf K}\cdot{\bf E},$$ (5.7)

where

$${\bf K} = {\bf I} + \frac{{\rm i}\,{\mbox{\boldmath$\sigma$}}}{\epsilon_0\,\omega}$$ (5.8)

is termed the dielectric permittivity tensor. Here, ${\bf I}$ is the identity tensor. Eliminating the magnetic field between Equations (5.5) and (5.7), we obtain

$${\bf M}\cdot{\bf E} = {\bf0},$$ (5.9)

where

$${\bf M} = \left(\frac{c}{\omega}\right)^2{\bf k}{\bf k} - \left(\frac{c\,k}{\omega}\right)^2{\bf I} + {\bf K}.$$ (5.10)

The solubility condition for Equation (5.9),

$${\cal M} (\omega, {\bf k}) \equiv {\rm det}({\bf M}) = 0,$$ (5.11)

is called the dispersion relation, and relates the wave angular frequency, $\omega$, to the wavevector, ${\bf k}$. Also, as the name “dispersion relation” suggests, this relation allows us to determine the rate at which the different Fourier components of a wave pulse disperse due to the variation of their phase-velocity with frequency (Fitzpatrick 2013).