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Waves in plasmas

Waves can be further classified by the oscillating species. In most plasmas of interest, the electron temperature is comparable to or larger than the ion temperature. This fact, coupled with the much smaller mass of the electron, implies that the electrons move much faster than the ions. An electron mode depends on the mass of the electrons, but the ions may be assumed to be infinitely massive, i.e. stationary. An ion mode depends on the ion mass, but the electrons are assumed to be massless and to redistribute themselves instantaneously according to the Boltzmann relation. Only rarely, e.g. in the lower hybrid oscillation, will a mode depend on both the electron and the ion mass.

The various modes can also be classified according to whether they propagate in an unmagnetized plasma or parallel, perpendicular, or oblique to the stationary magnetic field. Finally, for perpendicular electromagnetic electron waves, the perturbed electric field can be parallel or perpendicular to the stationary magnetic field.

Summary of elementary plasma waves

EM character	oscillating species	conditions	dispersion relation	name
electrostatic	electrons	${\displaystyle {\vec {B}}_{0}=0\ {\rm {or}}\ {\vec {k}}\|{\vec {B}}_{0}}$	${\displaystyle \omega ^{2}=\omega _{p}^{2}+3k^{2}v_{th}^{2}}$	plasma oscillation (or Langmuir wave)
		${\displaystyle {\vec {k}}\perp {\vec {B}}_{0}}$	${\displaystyle \omega ^{2}=\omega _{p}^{2}+\omega _{c}^{2}=\omega _{h}^{2}}$	upper hybrid oscillation
	ions	${\displaystyle {\vec {B}}_{0}=0\ {\rm {or}}\ {\vec {k}}\|{\vec {B}}_{0}}$	${\displaystyle \omega ^{2}=k^{2}v_{s}^{2}=k^{2}{\frac {\gamma _{e}KT_{e}+\gamma _{i}KT_{i}}{M}}}$	ion acoustic wave
		${\displaystyle {\vec {k}}\perp {\vec {B}}_{0}}$ (nearly)	${\displaystyle \omega ^{2}=\Omega _{c}^{2}+k^{2}v_{s}^{2}}$	electrostatic ion cyclotron wave
		${\displaystyle {\vec {k}}\perp {\vec {B}}_{0}}$ (exactly)	${\displaystyle \omega ^{2}=[(\Omega _{c}\omega _{c})^{-1}+\omega _{i}^{-2}]^{-1}}$	lower hybrid oscillation
electromagnetic	electrons	${\displaystyle {\vec {B}}_{0}=0}$	${\displaystyle \omega ^{2}=\omega _{p}^{2}+k^{2}c^{2}}$	light wave
		${\displaystyle {\vec {k}}\perp {\vec {B}}_{0},\ {\vec {E}}_{1}\|{\vec {B}}_{0}}$	${\displaystyle {\frac {c^{2}k^{2}}{\omega ^{2}}}=1-{\frac {\omega _{p}^{2}}{\omega ^{2}}}}$	O wave
		${\displaystyle {\vec {k}}\perp {\vec {B}}_{0},\ {\vec {E}}_{1}\perp {\vec {B}}_{0}}$	${\displaystyle {\frac {c^{2}k^{2}}{\omega ^{2}}}=1-{\frac {\omega _{p}^{2}}{\omega ^{2}}}\,{\frac {\omega ^{2}-\omega _{p}^{2}}{\omega ^{2}-\omega _{h}^{2}}}}$	X wave
		${\displaystyle {\vec {k}}\|{\vec {B}}_{0}}$ (right circ. pol.)	${\displaystyle {\frac {c^{2}k^{2}}{\omega ^{2}}}=1-{\frac {\omega _{p}^{2}/\omega ^{2}}{1-(\omega _{c}/\omega )}}}$	R wave (whistler mode)
		${\displaystyle {\vec {k}}\|{\vec {B}}_{0}}$ (left circ. pol.)	${\displaystyle {\frac {c^{2}k^{2}}{\omega ^{2}}}=1-{\frac {\omega _{p}^{2}/\omega ^{2}}{1+(\omega _{c}/\omega )}}}$	L wave
	ions	${\displaystyle {\vec {B}}_{0}=0}$		none
		${\displaystyle {\vec {k}}\|{\vec {B}}_{0}}$	${\displaystyle \omega ^{2}=k^{2}v_{A}^{2}}$	Alfvén wave
		${\displaystyle {\vec {k}}\perp {\vec {B}}_{0}}$	${\displaystyle {\frac {\omega ^{2}}{k^{2}}}=c^{2}\,{\frac {v_{s}^{2}+v_{A}^{2}}{c^{2}+v_{A}^{2}}}}$	magnetosonic wave

${\displaystyle \omega }$ - wave frequency, ${\displaystyle k}$ - wave number, ${\displaystyle c}$ - speed of light, ${\displaystyle \omega _{p}}$ - plasma frequency, ${\displaystyle \omega _{i}}$ - ion plasma frequency, ${\displaystyle \omega _{c}}$ - electron gyrofrequency, ${\displaystyle \Omega _{c}}$ - ion gyrofrequency, ${\displaystyle \omega _{h}}$ - upper hybrid frequency, ${\displaystyle v_{s}}$ - plasma "sound" speed, ${\displaystyle v_{A}}$ - plasma Alfvén speed

(The subscript 0 denotes the static part of the electric or magnetic field, and the subscript 1 denotes the oscillating part.)

Bibliography[edit]

• Swanson, D.G. Plasma Waves (2003). 2nd edition.
• Stix, Thomas Howard. Waves in Plasmas (1992).
• Chen, Francis F. Introduction to Plasma Physics and Controlled Fusion, 2nd edition (1984).

See also[edit]