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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 314 / 3 / b / Solution

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The fast magnetosonic mode uses gas and magnetic pressure as reinforcing restoring forces and is the faster root of the magnetosonic dispersion relation.


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 Incoming links: Fast magnetosonic wave