= Solution
The equation describes small plane-wave <perturbations> of a uniform, stationary <plasma> in <ideal magnetohydrodynamics>, with propagation \b[parallel to the background <magnetic field>]. The nonzero field defines $\widehat{\mathbf b}=\mathbf v_A/v_A$; $c_s$ is the <sound speed> for the assumed thermodynamic closure. Neglect <viscosity>, <electrical resistivity>, <gravity> and microscopic dispersive effects. A uniform background flow would instead introduce its Doppler-shifted <frequency>.
Decompose $\mathbf u=\mathbf u_\perp+u_\parallel\widehat{\mathbf b}$. The displayed matrix equation then becomes
$$
(k^2v_A^2-\omega^2)\mathbf u_\perp=0,\qquad
(k^2c_s^2-\omega^2)u_\parallel=0.
$$
Thus there are two independent transverse <Alfvén wave> <polarizations> with $\omega^2=k^2v_A^2$, restored by <magnetic tension>, and a longitudinal acoustic mode with $\omega^2=k^2c_s^2$, restored by <pressure>. The transverse modes have no first-order <density> compression; a strictly longitudinal displacement along the field does not bend it. These are the <parallel-propagating ideal magnetohydrodynamic wave> branches. In the general oblique problem, fast and slow magnetosonic modes mix compression and magnetic response; their parallel limits coincide with the faster and slower of these speeds. If $c_s=v_A$, all three <velocity> <polarizations> share the same <frequency> magnitude. The <velocity> equation alone does not enumerate nonpropagating <entropy> <perturbations>.
Back to article page
