Write

u = \partial_{t} ξ

. In the uniform equilibrium, the linearized ideal magnetohydrodynamic equations reduce to

\partial_{t} δ ρ = - ρ \nabla \cdot \partial_{t} ξ, \partial_{t} δ p = - γ p \nabla \cdot \partial_{t} ξ, \partial_{t} δ B = \nabla \times (\partial_{t} ξ \times B) .

(1)

Integrating from the undisplaced reference state, and using the uniform background magnetic field, gives

δ ρ = - ρ \nabla \cdot ξ, δ p = - γ p \nabla \cdot ξ, δ B = (B \cdot \nabla) ξ - B \nabla \cdot ξ .

(2)

These are the perturbations induced by the fluid displacement; independent time-independent changes to the reference state are excluded. The linear Lorentz force density is

(\nabla \times δ B) \times B / μ_{0}

. Its magnetic pressure and magnetic tension parts give

ρ \partial_{t}^{2} ξ = - \nabla (δ p + \frac{B \cdot δ B}{μ _{0}}) + \frac{( B \cdot \nabla ) δ B}{μ _{0}} .

(3)

For a Fourier mode, put

s = k \cdot ξ

,

q = B \cdot ξ

and

K = k \cdot B

. Then

δ p = - iγ p s

and

δ B = i (K ξ - B s)

. Substitution gives

ρ ω^{2} ξ = k [(γ p + \frac{B ^{2}}{μ _{0}}) s - \frac{K q}{μ _{0}}] + \frac{K}{μ _{0}} (K ξ - B s) .

(4)

If

s = q = 0

, the fluid displacement is perpendicular to both the wave vector and the magnetic field. Only magnetic tension restores it, and

ω^{2} = (k \cdot v_{a})^{2}, v_{a} = \frac{B}{μ _{0} ρ} .

(5)

This is the Alfvén wave. The Alfvén velocity is the field-directed propagation vector:

ω = \pm k \cdot v_{a}

. The signed phase velocity normal to the wavefront is

\pm v_{a} cos θ k

; the group velocity is

\pm v_{a}

. The distinction matters for oblique propagation. For generic directions the Alfvén wave has one transverse polarization, with

δ ρ = δ p = 0

.

To obtain the other magnetohydrodynamic waves, take the scalar products with

k

and

B

:

ρ ω^{2} s = k^{2} [(γ p + \frac{B ^{2}}{μ _{0}}) s - \frac{K q}{μ _{0}}], ρ ω^{2} q = γ p Ks .

(6)

The determinant condition for a nonzero pair

(s, q)

is

ω^{4} - k^{2} (c_{s}^{2} + v_{a}^{2}) ω^{2} + k^{4} c_{s}^{2} v_{a}^{2} cos^{2} θ = 0, c_{s}^{2} = \frac{γ p}{ρ} .

(7)

Thus the fast magnetosonic wave and slow magnetosonic wave have

v_{f, sl}^{2} = \frac{c _{s}^{2} + v _{a}^{2}}{2} \pm \frac{( c _{s}^{2} + v _{a}^{2} ) ^{2}}{4} - c_{s}^{2} v_{a}^{2} cos^{2} θ .

(8)

Their fluid displacements lie in the plane of

k

and

B

and are generally compressive. In the fast magnetosonic wave, gas and magnetic pressure provide the stronger restoring combination; in the slow magnetosonic wave, their perturbations oppose one another. The phase speeds depend on direction. They satisfy

v_{sl} \leq min (c_{s}, v_{a})

and

v_{f} \geq max (c_{s}, v_{a})

. For parallel propagation the two speeds are

c_{s}

and

v_{a}

, with a degeneracy between a transverse branch and the Alfvén wave. For perpendicular propagation

v_{f}^{2} = c_{s}^{2} + v_{a}^{2}

and

v_{sl} = 0

; the latter is a nonpropagating limiting disturbance. These special directions require interpreting the polarizations by continuity rather than assuming three distinct nonzero frequencies.

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 Incoming links: Phase speed