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# MHD Waves

When the time variations are much slower than the speed of light, we can neglect the electromagnetic waves discussed above. However, slower waves are allowed in a magnetised plasma. Unlike the pure sound discussed earlier, the addition of a background magnetic field introduces a preferred direction and the waves are now anisotropic.

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 (6.21) (6.22)

where (6.23)

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 (6.24)

and (6.25)

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 (6.26)

or (6.27)

where is the angle between the equilibrium magnetic field direction and the direction of propagation of the wave and (6.28) 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.   Next: Summary Up: Waves Previous: Electromagnetic Waves
Prof. Alan Hood
2000-11-06