= Solution
A displacement perpendicular to the plane spanned by $\mathbf k$ and $\mathbf B$ has $\mathbf k\mathbin\cdot\widetilde{\boldsymbol\xi}=\mathbf v_A\mathbin\cdot\widetilde{\boldsymbol\xi}=0$. The algebraic wave equation then gives the <Alfvén wave>
$$
\boxed{\omega^2=(\mathbf k\mathbin\cdot\mathbf v_A)^2}.
$$
The remaining displacement lies in the $\mathbf k$-$\mathbf B$ plane. Setting the determinant of that two-dimensional system to zero gives
$$
\boxed{\omega^4-(v_s^2+v_A^2)k^2\omega^2
+v_s^2k^2(\mathbf k\mathbin\cdot\mathbf v_A)^2=0}.
$$
Its larger root is the <fast magnetosonic wave>, in which gas and magnetic pressure act together. Its smaller root is the <slow magnetosonic wave>, whose motion is guided more strongly along the field. Both are compressive, whereas the Alfvén mode is transverse and incompressible.
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