Editing Magnetohydrodynamics (section)

== Waves ==
{{See also|Waves in plasmas}}
The wave modes derived using the MHD equations are called '''magnetohydrodynamic waves''' or '''MHD waves'''. There are three MHD wave modes that can be derived from the linearized ideal-MHD equations for a fluid with a uniform and constant magnetic field:
* Alfvén waves
* Slow magnetosonic waves
* Fast magnetosonic waves
{{multiple image
 | width=250
 | direction=vertical
 | align=right
 | header=Phase velocity plotted with respect to {{mvar|θ}}
 | image1=MHD wave mode 1.svg
 | alt1=<math>v_A>v_s</math>
 | caption1={{math|''v<sub>A</sub>'' > ''v<sub>s</sub>''}}
 | image2=MHD wave mode 2.svg
 | alt2=<math>v_A<v_s</math>
 | caption2={{math|''v<sub>A</sub>'' < ''v<sub>s</sub>''}}
}}
These modes have phase velocities that are independent of the magnitude of the wavevector, so they experience no dispersion. The phase velocity depends on the angle between the wave vector {{math|'''k'''}} and the magnetic field {{math|'''B'''}}.  An MHD wave propagating at an arbitrary angle {{mvar|θ}} with respect to the time independent or bulk field {{math|'''B'''<sub>0</sub>}} will satisfy the dispersion relation
:<math>\frac{\omega}{k} = v_A \cos\theta</math>
where
:<math>v_A = \frac{B_0}{\sqrt{\mu_0\rho}}</math>
is the Alfvén speed.  This branch corresponds to the shear Alfvén mode.  Additionally  the dispersion equation gives
:<math>\frac{\omega}{k} = \left( \tfrac12\left(v_A^2+v_s^2\right) \pm \tfrac12\sqrt{\left(v_A^2+v_s^2\right)^2 - 4v_s^2v_A^2\cos^2\theta}\right)^\frac12</math>
where
:<math>v_s = \sqrt{\frac{\gamma p}{\rho}}</math>
is the ideal gas speed of sound.  The plus branch corresponds to the fast-MHD wave mode and the minus branch corresponds to the slow-MHD wave mode. A summary of the properties of these waves is provided:
{|class="wikitable"
! rowspan=2 | Mode || rowspan=2 | Type || colspan=2 | Limiting phase speeds || rowspan=2 | Group velocity || rowspan=2 | Direction of energy flow
|-
! <math>\mathbf{k} \parallel \mathbf{B}</math>
! <math>\mathbf{k} \perp \mathbf{B}</math>
|-
| Alfvén wave || transversal; incompressible || <math>v_A</math> || <math>0</math> || <math>\frac{\mathbf{B}}{\sqrt{\mu_0 \rho}}</math> || <math>\mathbf{Q} \parallel \mathbf{B}</math>
|-
| Fast magnetosonic wave || rowspan=2 | neither transversal nor longitudinal; compressional ||<math>\max (v_A, v_s)</math>|| <math>\sqrt{v_A^2 + v_s^2}</math> || rowspan=2 | equal to phase velocity || approx. <math>\mathbf{Q} \parallel \mathbf{k}</math>
|-
| Slow magnetosonic wave || <math>\min (v_A, v_s)</math> || <math>0</math> || approx. <math>\mathbf{Q} \parallel \mathbf{B}</math>
|}

The MHD oscillations will be damped if the fluid is not perfectly conducting but has a finite conductivity, or if viscous effects are present.

MHD waves and oscillations are a popular tool for the remote diagnostics of laboratory and astrophysical plasmas, for example, the [[solar corona|corona]] of the Sun ([[Coronal seismology]]).