Derivation from linearized MHD equations[1][2][3]
In an ideal electrically conducting fluid with a homogeneous magnetic field B, the closed set of MHD equations consisting of the equation of motion, continuity equation, equation of state, and ideal induction equation (see Magnetohydrodynamics § Equations) linearized about a stationary equilibrium where the pressure p and density ρ are uniform and constant are:
where equilibrium quantities have subscripts 0, perturbations have subscripts 1, γ is the adiabatic index, and μ0 is the vacuum permeability.
Looking for a solution in the form of a superposition of plane waves which vary like exp[i(k ⋅ x − ωt)]
with wavevector k and angular frequency ω, the linearized equation of motion can be re-expressed as
And assuming that ω ≠ 0, the remaining equations can be solved for perturbed quantities in terms of v1:
Without loss of generality, we can assume that the z-axis is oriented along B0 and that the wavevector k lies in the xz-plane with components k∥ and k⊥ parallel and perpendicular to B0, respectively. The equation of motion after substituting for the perturbed quantities reduces to the eigenvalue equation
where cs = √γp0/ρ0 is the sound speed and vA = B0/√μ0ρ0 is the Alfvén speed. Setting the determinant to zero gives the dispersion relation
where
is the magnetosonic speed.
This dispersion relation has three independent roots: one corresponding to the Alfvén wave and the other two corresponding to the magnetosonic modes. From the eigenvalue equation, the y-component of the velocity perturbation decouples from the other two components giving the dispersion relation ω2
A = v2
Ak2
∥ for the Alfvén wave. The remaining bi-quadratic equation
is the dispersion relation for the fast and slow magnetosonic modes. It has roots
where the upper sign gives the fast magnetosonic mode and the lower sign gives the slow magnetosonic mode.