Magneto-Acoustic Waves

By taking the scale product of Eqn. 49 with k and B₀ in turn leads to a pair of equations which combine to give,

and (for k.v₁ = 0) the Alfvén wave of Eqn. 53. As Eqn. 54 is quadratic in (ω²/k²) the solutions are simply,

yielding two outwardly propagating waves termed the fast-mode and the slow-mode (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 slow-mode 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 fast-mode becomes a sound wave or Alfvén wave respectively and the slow-mode vanishes.

Table 2.1: Summary of magneto-acoustic 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, vA
Fast	Isotropic	Magnetic Pressure	Gas Pressure
		~vA	~cS
Slow	approximately	Gas Pressure	Magnetic Tension
	along B	~cS	~vA