Alfvén Waves

Next we assume that there are no pressure or density variations so
that (4.6) implies that
and
the motion is * incompressible*. Again taking Fourier components,
the incompressible assumption reduces to

From (4.30) it is clear that

so that the motion is transverse to the direction of the equilibrium magnetic field. Using vector identities the right hand side of (4.30) can be built up. Thus,

since . Next,

Finally,

since . Hence, (4.30) reduces to

We define the Alfvén speed, , as

so that (4.32) can be written as

(4.34) describes Alfvén waves that are anisotropic (due to the term. Note that

- is perpendicular to both , the
equilibrium magnetic field, and , the direction of
propagation. Thus, Alfvén waves are transverse waves.
- There are no disturbances in the pressure and density and so that the motion is incompressible.

From Figure 4.1 we have
so that the phase speed is

The group velocity is
so that

Hence, the period, , of oscillation is

Thus, oscillations in a coronal loop with these properties should have a period of approximately 50 seconds.