 Solution

A displacement perpendicular to the plane spanned by $\mathbf{k}$ and $\mathbf{B}$ has $\mathbf{k}\cdot \tilde{\mathbf{\xi }}={\mathbf{v}}_A\cdot \tilde{\mathbf{\xi }}=0$. The algebraic wave equation then gives the Alfvén waveThe remaining displacement lies in the $\mathbf{k}$-$\mathbf{B}$ plane. Setting the determinant of that two-dimensional system to zero givesIts 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.