Magnetic Waves

By neglecting gravity, but now including magnetism, a plasma at rest with constant density, ρ₀, and uniform magnetic field, B₀, undergoes a small perturbation (subscripts 1) resulting in a plasma flow such that
B = B₀ + B₁
ρ = ρ₀ + ρ₁
v = v₁
Differentiating Eqn. 23 with respect to time, and substituting linearised forms of Eqns. 11 (with a large $R_m$), 17 & 21 leads to a second generalised wave equation,

Assuming a plane wave solution, as for Eqn. 39, leads to,