= Equal-density ion-neutral Alfvén dispersion relation
{title2=$\omega^3+2i\mu\omega^2-a^2\omega-i\mu a^2=0$}
For equal <ion> and neutral mass densities, let $\mu=\mu_{in}=\mu_{ni}$, $a=k_\parallel B_0/\sqrt{4\pi\rho_i}$ and use Fourier time dependence $e^{-i\omega t}$. <Solenoidal> <displacements> obey $\ddot\xi_i=-a^2\xi_i-\mu(\dot\xi_i-\dot\xi_n)$ and $\ddot\xi_n=-\mu(\dot\xi_n-\dot\xi_i)$. For nonzero <frequency>, the neutral equation gives $\xi_i=(1-i\omega/\mu)\xi_n$. Substitution in the <ion> equation gives
$$
\omega^3+2i\mu\omega^2-a^2\omega-i\mu a^2=0.
$$
Equivalently, the <growth rate> $s=-i\omega$ obeys $s^3+2\mu s^2+a^2s+\mu a^2=0$. A time-independent neutral <displacement> with zero <velocity> is a relabeling of a uniform fluid, not an additional <wave> root of the <velocity>/magnetic-field system.
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