Figure 9.15 Relationships between the propagation vector k, the underlying field B0, its perturbation SB, and the induced fluid velocity Su in MHD waves. For each mode, the density and field perturbations are also sketched. If k and B0 are parallel, the mode can be (a) transverse or (b) longitudinal. (c) If k and B0 are perpendicular, the mode must be longitudinal. (d) In the most general case, the wave is neither purely transverse nor longitudinal.
Eliminating 5u between (9.67) and (9.68) yields the dispersion relation:
For any plane wave, u/k gives the phase velocity. In our particular case (k parallel to B0, transverse mode), this quantity is VA, the Alfven velocity introduced in Chapter 8:
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