Magnetosonic wave

A magnetosonic wave (also called magnetoacoustic wave) is a linear magnetohydrodynamic (MHD) wave that is driven by both pressure (thermal and magnetic) and magnetic tension. There are two types of magnetosonic waves, the "fast" magnetosonic wave and the "slow" magnetosonic wave. Both fast and slow magnetosonic waves have been recently discovered in the solar corona, which created an observational foundation for the novel technique for the coronal plasma diagnostics, coronal seismology.
Homogeneous plasma.
In an ideal homogeneous plasma of infinite extent, and in the absence of gravity, the magnetosonic waves form, together with the Alfvén wave, the three basic linear MHD waves. Under the assumption of normal modes, namely that the linear perturbations of the physical quantities are of the form
(with the constant amplitude), a dispersion relation of the magnetosonic waves can be derived from the system of ideal MHD equations:
where is the Alfvén speed, is the sound speed, is the magnitude of the wave vector and is the component of the wave vector along the background magnetic field (which is straight and constant, because the plasma is assumed homogeneous).
This equation can be solved for the frequency , yielding the frequencies of the fast and slow magnetosonic waves:
It can be shown that (with the Alfvén frequency), hence the name of "slow" and "fast" magnetosonic waves.
Limiting cases.
Absent magnetic field.
In the absence of a magnetic field, the whole MHD model reduces to the hydrodynamics (HD) model. In this case , and hence and . The slow wave thus disappears from the system, while the fast wave is just a sound wave, propagating isotropically.
Incompressible plasma.
In case the plasma is incompressible, the sound speed (this follows from the energy equation) and it can then be shown that and . The slow wave thus propagates with the Alfvén speed (although it remains different from an Alfvén wave in its nature), while the fast wave disappears from the system.
Cold plasma.
Under the assumption that the background temperature , it follows from the perfect gas law that the thermal pressure and thus that . In this case, and . Hence there are no slow waves in the system, and the fast waves propagate isotropically with the Alfvén speed.
Inhomogeneous plasma.
In the case of an inhomogeneous plasma (that is, a plasma where at least one of the background quantities is not constant) the MHD waves lose their defining nature and get mixed properties. In some setups, such as the axisymmetric waves in a straight cylinder with a circular basis (one of the simplest models for a coronal loop), the three MHD waves can still be clearly distinguished. But in general, the pure Alfvén, fast and slow magnetosonic waves don't exist and the waves in the plasma are coupled to each other in intricate ways.