= Solution
A uniform conducting fluid supports three propagating <magnetohydrodynamic wave> branches. <Ideal magnetohydrodynamics> couples fluid <velocity> to the <magnetic field> through the induction equation; the <Lorentz force> then supplies magnetic tension and magnetic pressure restoring forces. Let $\theta$ be the angle between <wave vector> $\mathbf k$ and background field $\mathbf B_0$, and define <Alfvén speed> $v_A=B_0/\sqrt{\mu_0\rho_0}$ and <adiabatic sound speed> $c_s^2=\gamma p_0/\rho_0$.
The shear <Alfvén wave> has
$$
\boxed{\omega^2=k^2v_A^2\cos^2\theta.}
$$
Its <velocity> perturbation is perpendicular to both $\mathbf k$ and $\mathbf B_0$. It bends field lines without first-order compression, so magnetic tension is the restoring force. Its <group velocity> follows the background field; the displayed phase speed along $\mathbf k$ is not the full group speed.
The <fast magnetosonic wave> and <slow magnetosonic wave> involve coupled compression and magnetic perturbations, with displacement in the plane of $\mathbf k,\mathbf B_0$:
$$
\boxed{\frac{\omega_{f,s}^2}{k^2}=\frac12\left[c_s^2+v_A^2
\pm\sqrt{(c_s^2+v_A^2)^2-4c_s^2v_A^2\cos^2\theta}\right].}
$$
The plus sign is the fast branch. Parallel propagation gives characteristic speeds $c_s,v_A$; perpendicular propagation gives fast speed $\sqrt{c_s^2+v_A^2}$ while the other two phase frequencies vanish. Degenerate directions require care in assigning polarizations, rather than claiming three distinct nonzero frequencies everywhere.
\Image[/past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2015/paper-1-mhd-phase-speeds.png]
{title=Angular dependence of the three ideal magnetohydrodynamic phase speeds}
{description=Original homogeneous ideal-fluid calculation for sound speed 0.7 times the Alfvén speed. The curves are phase speeds and do not depict group-velocity rays.}
These <waves> transport <energy> and <momentum> through magnetized stellar atmospheres and astrophysical plasmas. Their restoring mechanisms distinguish them from ordinary acoustic waves. Resistivity, viscosity, ion-neutral friction and kinetic effects introduce damping or change the model; an ideal undamped <dispersion relation> alone does not calculate coronal heating.
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