Parallel-propagating ideal magnetohydrodynamic wave (${\omega }^2=k^2{v}_A^2\text{or}{k}^2{c}_s^2$)

In a uniform ideal-MHD plasma at rest, take the wavevector parallel to a nonzero background field. Projecting the linear velocity equation along and perpendicular to the field gives two transverse Alfvén polarizations at speed $v_A$ and one longitudinal acoustic polarization at speed $c_s$. Compression does not bend field lines for the longitudinal branch. The general oblique fast/slow magnetosonic modes reduce to the corresponding faster/slower parallel branches, with degeneracies when their speeds agree.