= Incompressible Hall-MHD wave dispersion
{title2=$(\omega^2-k_\parallel^2v_A^2)^2=\omega^2k_\parallel^2v_A^2k^2d_i^2$}
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 $\omega\mathbf b=-k_\parallel v_A\mathbf u+i k_\parallel v_Ad_i\mathbf k\times\mathbf b$. The <helicity decomposition of a transverse Fourier mode> then yields the displayed <dispersion relation>. Its two positive frequency magnitudes are $|k_\parallel|v_A[\sqrt{1+(kd_i)^2/4}\pm kd_i/2]$, approaching the <Alfvén wave> frequency for $kd_i\ll1$. For $kd_i\gg1$, the fast branch is a <whistler wave> and the slow branch tends to $|k_\parallel|v_A/(kd_i)$.
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