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By taking the scale product of Eqn. 49 with k and B_{0} in turn leads to a pair of equations which combine to give,

(54) 
and (for k.v_{1} = 0) the Alfvén wave of Eqn. 53. As Eqn. 54 is quadratic in (ω^{2}/k^{2}) the solutions are simply,

(55) 
yielding two outwardly propagating waves termed the fastmode and the slowmode (defined by their phase velocity) in addition to the Alfvén wave. The characteristics of each type of wave in a strong and weak magnetic field are summarised in Table 2.1. Essentially, the fast and slowmode waves may be regarded as a sound wave modified by the magnetic field, and an Alfvén wave modified by the pressure. In the limits of B = 0 or ρ = 0, the fastmode becomes a sound wave or Alfvén wave respectively and the slowmode vanishes.
Table 2.1:
Summary of magnetoacoustic modes. Direction of propagation, and dominant restoring force and velocity of waves with strong and weak magnetic fields are given. (after Stein & Leibacher 1974)
Mode of Wave 
Propagation Direction 
Magnetic Field 
Strong 
Weak 
Alfvén 
along B 
Magnetic Tension, v_{A} 




Fast 
Isotropic 
Magnetic Pressure 
Gas Pressure 


~v_{A} 
~c_{S} 
Slow 
approximately 
Gas Pressure 
Magnetic Tension 

along B 
~c_{S} 
~v_{A} 
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James McAteer
20040114