Consider a perturbation to a uniform state at rest, with magnetic
field and density denoted respectively by

both uniform, that generates a small flow and a perturbed magnetic field so that

We linearise the MHD equations

and, assuming no pressure gradient term and incompressible perturbations , we obtain

where

Since the coefficients are again constant we look for solutions of the form

for a constant amplitude (each variable has its own constant amplitude), so that

and

Now we make use of the equations

Now we take the scalar product of (6.24) with . Thus, we obtain

and so we have

These two homogeneous equations only have a solution if and are related through the dispersion relation

or

where is the angle between the equilibrium magnetic field direction and the direction of propagation of the wave and

is the field strength squared and is the Alfvén speed. These magnetic waves propagate at the Alfvén speed that depends on the magnetic field strength and the density.

Thus, a magnetised plasma allows the propagation of Alfvén waves which propagate at a speed equal to the Alfvén speed when directly along fieldlines but at a slower speed when propagating at an angle to the field.