Langmuir Wave edit

The Langmuir wave is a purely longitudinal wave, that is, the wave vector is in the same direction as the E-field. It is an electrostatic wave; as such, it doesn't have an oscillating magnetic field.

A plasma consists of charged particles which react to electric fields, in contrast with dielectric matter. When electrons in a uniform, homogeneous plasma are perturbed from their equilibrium position, a charge separation occurs creating an electric field which acts as restoring force on the electrons. Since electrons have inertia the system behaves as a harmonic oscillator, where the electrons oscillate at a frequency ω_pe, called electron plasma frequency. These oscillations do not propagate—the group velocity is 0.

When the thermal motion of the electrons is taken into account a shift in frequency from the electron plasma frequency ω_pe occurs. Now the electron pressure gradient acts as the restoring force, creating a propagating wave analogous to a sound wave in non-ionized gases. Combining these two restoring forces (from the electric field and electron pressure gradient) a type of wave, named Langmuir wave, is excited. The dispersion relation is:

${\displaystyle \omega ^{2}=\omega _{pe}^{2}+3C_{e}^{2}k^{2}}$ 

The first term on the right-hand side of the dispersion relation is the electron plasma oscillation related to the electric field force and the second term is related to the thermal motion of the electrons, where C_e is the electron thermal speed and k is the wave vector.^[1]

Electromagnetic wave edit

In an unmagnetized plasma, waves above the plasma frequency propagate through the plasma according to the dispersion relation:

${\displaystyle 1-{\frac {\omega _{pe}^{2}}{\omega ^{2}}}-{\frac {k^{2}c^{2}}{\omega ^{2}}}=0\rightarrow \omega ^{2}=c^{2}k^{2}+\omega _{pe}^{2}}$ 

In an unmagnetized plasma for the high frequency or low electron density limit, i.e. for ${\displaystyle \omega \gg \omega _{pe}=(n_{e}e^{2}/m_{e}\epsilon _{0})^{1/2}}$  or ${\displaystyle n_{e}\ll m_{e}\omega ^{2}\epsilon _{0}\,/\,e^{2}}$  where ω_pe is the plasma frequency, the wave speed is the speed of light in vacuum. As the electron density increases, the phase velocity increases and the group velocity decreases until the cut-off frequency where the light frequency is equal to ω_pe. This density is known as the critical density for the angular frequency ω of that wave and is given by ^[2]

${\displaystyle n_{c}={\frac {\varepsilon _{o}\,m_{e}}{e^{2}}}\,\omega ^{2}}$  (SI units)

If the critical density is exceeded, the plasma is called over-dense.

In a magnetized plasma, except for the O wave, the cut-off relationships are more complex.

The O wave is the "ordinary" wave in the sense that its dispersion relation is the same as that in an unmagnetized plasma, that is,

${\displaystyle 1-{\frac {\omega _{pe}^{2}}{\omega ^{2}}}-{\frac {k^{2}c^{2}}{\omega ^{2}}}=0\rightarrow \omega ^{2}=c^{2}k^{2}+\omega _{pe}^{2}}$  ^[3]

. It is plane polarized with E₁ || B₀. It has a cut-off at the plasma frequency.

The X wave is the "extraordinary" wave because it has a more complicated dispersion relation:^[4]

${\displaystyle n^{2}={\frac {[(\omega +\omega _{ci})(\omega -\omega _{ce})-\omega _{p}^{2}][(\omega -\omega _{ci})(\omega +\omega _{ce})-\omega _{p}^{2}]}{(\omega ^{2}-\omega _{ci}^{2})(\omega ^{2}-\omega _{ce}^{2})+\omega _{p}^{2}(\omega _{ce}\omega _{ci}-\omega ^{2})}}}$ 

Where ${\displaystyle \omega _{p}^{2}=\omega _{pe}^{2}+\omega _{pi}^{2}}$ .

It is partly transverse (with E₁⊥B₀) and partly longitudinal; the E-field is of the form

${\displaystyle (E_{x},-j{\frac {S}{D}}E_{x},0)}$ 

Where ${\displaystyle S,D}$  refer to the Stix notation.

As the density is increased, the phase velocity rises from c until the cut-off at ${\displaystyle \omega _{R}}$  is reached. As the density is further increased, the wave is evanescent until the resonance at the upper hybrid frequency ${\displaystyle \omega _{h}^{2}=\omega _{p}^{2}+\omega _{c}^{2}}$ . Then it can propagate again until the second cut-off at ${\displaystyle \omega _{L}}$ . The cut-off frequencies are given by ^[5]

${\displaystyle {\begin{aligned}\omega _{R}&={\frac {1}{2}}\left[\omega _{c}+\left(\omega _{c}^{2}+4\omega _{p}^{2}\right)^{\frac {1}{2}}\right]\\\omega _{L}&={\frac {1}{2}}\left[-\omega _{c}+\left(\omega _{c}^{2}+4\omega _{p}^{2}\right)^{\frac {1}{2}}\right]\end{aligned}}}$ 

where ${\displaystyle \omega _{c}}$  is the electron cyclotron resonance frequency, and ${\displaystyle \omega _{p}}$  is the electron plasma frequency.

The resonant frequencies for the X-wave are:

${\displaystyle \omega ^{2}={\frac {\omega _{e}^{2}+\omega _{i}^{2}}{2}}\pm {\sqrt {({\frac {\omega _{e}^{2}-\omega _{i}^{2}}{2}})^{2}+\omega _{pe}^{2}\omega _{pi}^{2}}}}$ 

where ${\displaystyle \omega _{a}^{2}=\omega _{pa}^{2}+\omega _{ca}^{2}}$  and ${\displaystyle a=e,i}$ .

The R wave and the L wave are right-hand and left-hand circularly polarized, respectively. The R wave has a cut-off at ω_R (hence the designation of this frequency) and a resonance at ω_c. The L wave has a cut-off at ω_L and no resonance. R waves at frequencies below ω_c/2 are also known as whistler modes.^[6]

The dispersion relation can be written as an expression for the frequency (squared), but it is also common to write it as an expression for the index of refraction ck/ω (squared).

• Appleton-Hartree equation