The fast and slow modes are distinguished by magnetic and gas pressure oscillations that are either in-phase or anti-phase, respectively.
The fast and slow magnetosonic waves are defined by a bi-quadratic dispersion relation that can be derived from the linearized MHD equations.
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.
Setting the determinant to zero gives the dispersion relation where is the magnetosonic speed.
From the eigenvalue equation, the y-component of the velocity perturbation decouples from the other two components giving the dispersion relation ω2A = v2Ak2∥ for the Alfvén wave.
The remaining bi-quadratic equation is the dispersion relation for the fast and slow magnetosonic modes.
The phase velocities of the fast and slow magnetosonic waves depend on the angle θ between the wavevector k and the equilibrium magnetic field B0 as well as the equilibrium density, pressure, and magnetic field strength.
This is due to the differences in the signs of the thermal and magnetic pressure perturbations associated with each mode.
Conversely, for the slow mode v2− < c2s cos2 θ, so magnetic and thermal pressure perturbations have opposite signs.
[2][4] In an incompressible fluid, the density and pressure perturbations vanish, ρ1 = 0 and p1 = 0, resulting in the sound speed tending to infinity, cs → ∞.
Under the assumption that the background temperature is zero, it follows from the ideal gas law that the thermal pressure is also zero, p0 = 0, and, as a result, that the sound speed vanishes, cs = 0.
In this limit, the fast mode is sometimes referred to as a compressional Alfvén wave.
When the wavevector and the equilibrium magnetic field are perpendicular, θ → π/2, the fast mode propagates as a longitudinal wave with phase velocity equal to the magnetosonic speed, and the slow mode propagates as a transverse wave with phase velocity approaching zero.