(Solved): Consider linearised MHD Theory in which a small displacement is introduced such that v1= ...

Consider linearised MHD Theory in which a small displacement $?$ is introduced such that ${\overrightarrow{\mathbf{v}}}_1=?\overrightarrow{?}/?t$. If the parallel wave number $k_{?}=\overrightarrow{\mathbf{k}}?\hat{\mathbf{B}}$ is much less than the perpendicular wave number amplitude $k_{?}=?\overrightarrow{\mathbf{k}}\times \hat{\mathbf{B}}?$, where $\hat{\mathbf{B}}=\overrightarrow{\mathbf{B}}/B$, what are the dispersion relations for the Alvén, slow magnetosonic and fast magnetosonic waves?