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As discussed in Section 2.2.3 the Lorentz force can be interpreted as the sum of a magnetic tension (B^{2}/μ_{0}) and a magnetic pressure (B^{2}/2μ_{0}). By analogy with a string under tension, this will permit transverse waves with an Alfvén velocity, v_{A}, given by,

(50) 
When the magnetic field is the only restoring force, Eqn. 49 reduces to,

(51) 
which can be reduced by vector identity to,

(52) 
which has two distinct solutions. For k.v_{1} = 0, Eqn. 51 (after multiplying out the vector products) becomes

(53) 
where θ_{B} is the angle between k and B_{0}. This is the shear Alfvén wave, with a phase velocity which is just the Alfvén velocity along the magnetic field and zero perpendicular to it, and a group velocity which is always directed along the magnetic field. The k.v_{1} = 0 property means that the velocity is perpendicular to the direction of propagation, hence the waves are transverse. The second solution, ω =kv_{A}, is the compressional Alfvén wave, which propagates isotropically.
Next: MagnetoAcoustic Waves
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James McAteer
20040114