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Set $ϵ=v_a^2/c^2\ll 1$. Expanding the square root in part (a) givesHence, to first absolute order in $ϵ$,The high-pressure limit of magnetosonic waves can also be expressed as nonnegative speeds. Retaining the next squared-speed term gives the first relative correction for the slow mode:The exact Alfvén speed along the propagation direction remains $v_a|\cos \theta |$.

The fast magnetosonic wave is predominantly an acoustic wave, with velocity almost parallel to $\mathbf{k}$ and appreciable mass density and gas pressure variations. Its magnetic pressure correction is weak. The Alfvén wave is exactly incompressible, polarized perpendicular to the $\mathbf{k}$-${\mathbf{B}}_0$ plane and restored by magnetic tension.

The slow magnetosonic wave is polarized inside that plane and almost perpendicular to $\mathbf{k}$. Indeed, the leading sound speed part of the matrix in (a) annihilates a vector perpendicular to $\mathbf{k}$, so only the smaller magnetic restoring force acts on that MHD wave polarization. Its compression is smaller by order $ϵ$, with gas pressure changes almost cancelling magnetic pressure changes. It is often called the pseudo-Alfvén wave MHD wave polarization in this high pressure limit. Thus two distinct almost-incompressible magnetic MHD wave polarizations have the same leading speed, while the third is predominantly sound. For exactly perpendicular propagation the Alfvén wave and slow magnetosonic wave frequencies vanish; for parallel propagation the two transverse magnetic MHD wave polarizations are degenerate.