Incompressible Hall-MHD wave dispersion ($({\omega }^2-k_{\parallel }^2{v}_A^2{)}^2={\omega }^2{k}_{\parallel }^2{v}_A^2{k}^2{d}_i^2$)

For a uniform guide field, write the transverse perturbation in velocity units as $\mathbf{b}$. Linear momentum gives $\omega \mathbf{u}=-k_{\parallel }{v}_A\mathbf{b}$. Linear induction gives $\omega \mathbf{b}=-k_{\parallel }{v}_A\mathbf{u}+i{k}_{\parallel }{v}_A{d}_i\mathbf{k}\times \mathbf{b}$. The helicity decomposition of a transverse Fourier mode then yields the displayed dispersion relation. Its two positive frequency magnitudes are $|k_{\parallel }|v_A[\sqrt{1+(k{d}_i{)}^2/4}\pm k{d}_i/2]$, approaching the Alfvén wave frequency for $k{d}_i\ll 1$. For $k{d}_i\gg 1$, the fast branch is a whistler wave and the slow branch tends to $|k_{\parallel }|v_A/(k{d}_i)$.