= Solution
Let the equilibrium <ideal gas> be at rest, with uniform $\rho_0,p_0$ and $\mathbf B_0=B_0\mathbf e_z$. This local wave calculation neglects gravity, viscosity and magnetic diffusion. Define the <adiabatic sound speed> $c\equiv v_s=(\gamma p_0/\rho_0)^{1/2}$, the <Alfvén speed> $v_a=B_0/(\mu_0\rho_0)^{1/2}$, and a <wavevector> $\mathbf k=k(\sin\theta,0,\cos\theta)$, where $k=|\mathbf k|$ and $\theta$ is its angle to the equilibrium <magnetic field>. Perturbations have dependence $e^{i\mathbf k\cdot\mathbf x-i\omega t}$ and <phase speed> $v=\omega/k$.
The <linearized ideal magnetohydrodynamic equations> for <mass conservation>, <pressure>, induction and momentum give
$$
\delta\rho=\frac{\rho_0}{\omega}\mathbf k\cdot\mathbf u',\qquad
\delta p=c^2\delta\rho,\qquad
\mathbf b=\frac{\mathbf B_0(\mathbf k\cdot\mathbf u')-\mathbf u'(\mathbf k\cdot\mathbf B_0)}\omega,
$$
$$
\omega\rho_0\mathbf u'=\mathbf k\delta p-
\frac1{\mu_0}\left[(\mathbf k\cdot\mathbf B_0)\mathbf b-\mathbf k(\mathbf B_0\cdot\mathbf b)\right].
$$
The relation between <pressure> and <mass density> follows from eliminating $\mathbf k\cdot\mathbf u'$ between the two material evolution equations; it applies to the nonzero-frequency wave sector. The induction result automatically has $\mathbf k\cdot\mathbf b=0$.
The $y$ <MHD wave polarization> is perpendicular to both the field and <wavevector>. It has $\delta p=\delta\rho=0$ and obeys
$$
\omega^2u'_y=v_a^2k^2\cos^2\theta\,u'_y.
$$
Thus the transverse <Alfvén wave> has
$$
\boxed{v_A^2=v_a^2\cos^2\theta}.
$$
In the $x$-$z$ plane the two remaining <velocity> components instead obey the <symmetric matrix> eigenproblem
$$
v^2\binom{u'_x}{u'_z}
=\begin{pmatrix}
v_a^2+c^2\sin^2\theta&c^2\sin\theta\cos\theta\\
c^2\sin\theta\cos\theta&c^2\cos^2\theta
\end{pmatrix}\binom{u'_x}{u'_z}.
$$
Taking its <determinant> gives $v^4-(c^2+v_a^2)v^2+c^2v_a^2\cos^2\theta=0$. Solving this <quadratic equation> in $v^2$ derives both <magnetoacoustic wave> <phase speeds>,
$$
\boxed{v_{f,\mathrm{slow}}^2=\frac{c^2+v_a^2}{2}\pm
\sqrt{\frac{(c^2+v_a^2)^2}{4}-c^2v_a^2\cos^2\theta}}.
$$
The plus sign is the <fast magnetosonic wave>, the minus sign the <slow magnetosonic wave>. The radicand is nonnegative, since it equals $[(c^2-v_a^2)^2+4c^2v_a^2\sin^2\theta]/4$. Both squared speeds are nonnegative. Each nonzero mode has forward and backward propagation; an advected <entropy> disturbance is a separate zero-frequency mode in this static equilibrium.
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