\begin{equation}
\left(\frac{\omega}{k}\right)^2 = 
0.5\left[(c_s^2+v_A^2)
\pm\sqrt{(c_s^2+v_A^2)^2-4v_A^2c_s^2{\rm cos}^2\psi}\right]
\end{equation}
where $\omega$ is the frequency, $k$ is the wave vector, $v_A$ is the
Alfv\'en velocity, $c_s$ is the sound speed, cos $\psi$ = ${\bf k}\cdot
{\bf B}_{\circ}/
\|{\bf k}\|\|{\bf B_{\circ}}\|$,
and {\bf B}$_{\circ}$ is the equilibrium magnetic field.