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An Alfvén wave is a magnetohydrodynamic wave in which magnetic tension restores a sideways disturbance of a conducting fluid. In the small-amplitude, uniform-background ideal magnetohydrodynamics model, its shear polarization of waves has displacement perpendicular to both the background magnetic field and the wavevector. It is an incompressible flow disturbance: the mass density and gas pressure have no first-order fluctuation. Its dispersion relation is ${\omega }^2=(\mathbf{k}\cdot {\mathbf{v}}_A{)}^2$, where ${\mathbf{v}}_A$ is the background Alfvén velocity.

The mechanism resembles a plucked taut string: magnetic flux freezing makes sideways fluid flow bend magnetic field lines, and their magnetic tension accelerates neighboring fluid. The fluid supplies inertia and the bent magnetic field supplies the restoring stress. This permits energy transport without a first-order compression, unlike an acoustic wave. Hannes Alfvén's 1942 paper, Existence of Electromagnetic-Hydrodynamic Waves, considered precisely a uniform axial magnetic field, infinite electrical conductivity and sideways velocity depending on axial position and time; it proposed a solar application. This was a theoretical prediction, followed by the distinct Lundquist liquid-metal Alfvén wave experiment.^[ref]

For oblique propagation, the shear Alfvén wave has velocity normal to the plane containing the wavevector and background magnetic field. The fast magnetosonic wave and slow magnetosonic wave instead have velocity in that plane and generally involve compression, with gas pressure and magnetic pressure contributing to their restoring forces. For exactly parallel propagation the plane is not unique: either transverse polarization of waves is allowed, and some modal speeds coincide. For $k_{\parallel }=0$ the shear branch has $\omega =0$, while the fast magnetosonic wave can still propagate perpendicular to the magnetic field. One should identify the polarization of waves as well as the phase velocity. The statement that the shear mode is incompressible concerns this linear mode; a finite linearly polarized disturbance can drive a quadratic compressive response, as in the magnetic-pressure obstruction to a linearly polarized Alfvén wave.

If the cylindrical components of a vector field have dependence $e^{im\varphi }$, differentiation of the rotating basis adds the displayed cross product term. Scalar-component differentiation alone misses this contribution. For $\mathbf{B}=(J/2)s\hat{\mathbf{\varphi }}$, one also has $(\mathbf{v}\cdot \nabla )\mathbf{B}=(J/2)\hat{\mathbf{z}}\times \mathbf{v}$. These identities are useful in cylindrical magnetohydrodynamic waves.

A magnetohydrodynamic wave is a linear disturbance of a conducting fluid and its magnetic field. A uniform ideal plasma supports an Alfvén mode and fast and slow magnetosonic modes.

A polarization specifies the direction or trajectory of the oscillating velocity and magnetic field in a magnetohydrodynamic wave. The linear Alfvén wave has velocity perpendicular to the plane of the background magnetic field and wavevector; the magnetosonic waves have velocity in that plane. A real fixed polarization vector gives a linear oscillation. Two orthogonal components of equal amplitude in quadrature give a circular trajectory. For propagation parallel to a uniform field, combining the two degenerate transverse linear polarizations produces circular polarization; at finite amplitude its constant transverse magnetic-field magnitude removes the longitudinal magnetic pressure gradient.

A uniform conducting fluid supports three propagating magnetohydrodynamic wave branches. Ideal magnetohydrodynamics couples fluid velocity to the magnetic field through the induction equation; the Lorentz force then supplies magnetic tension and magnetic pressure restoring forces. Let $\theta$ be the angle between wave vector $\mathbf{k}$ and background field ${\mathbf{B}}_0$, and define Alfvén speed $v_A=B_0/\sqrt{{\mu }_0{\rho }_0}$ and adiabatic sound speed $c_s^2=\gamma p_0/{\rho }_0$.

The shear Alfvén wave hasIts velocity perturbation is perpendicular to both $\mathbf{k}$ and ${\mathbf{B}}_0$. It bends field lines without first-order compression, so magnetic tension is the restoring force. Its group velocity follows the background field; the displayed phase speed along $\mathbf{k}$ is not the full group speed.

The fast magnetosonic wave and slow magnetosonic wave involve coupled compression and magnetic perturbations, with displacement in the plane of $\mathbf{k},{\mathbf{B}}_0$:The plus sign is the fast branch. Parallel propagation gives characteristic speeds $c_s,v_A$; perpendicular propagation gives fast speed $\sqrt{c_s^2+v_A^2}$ while the other two phase frequencies vanish. Degenerate directions require care in assigning polarizations, rather than claiming three distinct nonzero frequencies everywhere.

These waves transport energy and momentum through magnetized stellar atmospheres and astrophysical plasmas. Their restoring mechanisms distinguish them from ordinary acoustic waves. Resistivity, viscosity, ion-neutral friction and kinetic effects introduce damping or change the model; an ideal undamped dispersion relation alone does not calculate coronal heating.

Write $\mathbf{u}={\partial }_t\mathbf{\xi }$. In the uniform equilibrium, the linearized ideal magnetohydrodynamic equations reduce toIntegrating from the undisplaced reference state, and using the uniform background magnetic field, givesThese are the perturbations induced by the fluid displacement; independent time-independent changes to the reference state are excluded. The linear Lorentz force density is $(\nabla \times \delta \mathbf{B})\times \mathbf{B}/{\mu }_0$. Its magnetic pressure and magnetic tension parts giveFor a Fourier mode, put $s=\mathbf{k}\cdot \mathbf{\xi }$, $q=\mathbf{B}\cdot \mathbf{\xi }$ and $K=\mathbf{k}\cdot \mathbf{B}$. Then $\delta p=-i\gamma ps$ and $\delta \mathbf{B}=i(K\mathbf{\xi }-\mathbf{B}s)$. Substitution givesIf $s=q=0$, the fluid displacement is perpendicular to both the wave vector and the magnetic field. Only magnetic tension restores it, andThis is the Alfvén wave. The Alfvén velocity is the field-directed propagation vector: $\omega =\pm \mathbf{k}\cdot {\mathbf{v}}_a$. The signed phase velocity normal to the wavefront is $\pm v_a\cos \theta \hat{\mathbf{k}}$; the group velocity is $\pm {\mathbf{v}}_a$. The distinction matters for oblique propagation. For generic directions the Alfvén wave has one transverse polarization, with $\delta \rho =\delta p=0$.

To obtain the other magnetohydrodynamic waves, take the scalar products with $\mathbf{k}$ and $\mathbf{B}$:The determinant condition for a nonzero pair $(s,q)$ isThus the fast magnetosonic wave and slow magnetosonic wave haveTheir fluid displacements lie in the plane of $\mathbf{k}$ and $\mathbf{B}$ and are generally compressive. In the fast magnetosonic wave, gas and magnetic pressure provide the stronger restoring combination; in the slow magnetosonic wave, their perturbations oppose one another. The phase speeds depend on direction. They satisfy $v_{\mathrm{sl}}\le \min (c_s,v_a)$ and $v_{\mathrm{f}}\ge \max (c_s,v_a)$. For parallel propagation the two speeds are $c_s$ and $v_a$, with a degeneracy between a transverse branch and the Alfvén wave. For perpendicular propagation $v_{\mathrm{f}}^2=c_s^2+v_a^2$ and $v_{\mathrm{sl}}=0$; the latter is a nonpropagating limiting disturbance. These special directions require interpreting the polarizations by continuity rather than assuming three distinct nonzero frequencies.

A physicist.^[ref]