= Strong-collision ion-neutral Alfvén modes
{title2=$\omega_\pm=\pm a/\sqrt2-ia^2/(8\mu)$}
For $|a|\ll\mu$, the equal-density ion-neutral cubic has a rapidly damped mode $\omega_d=-2i\mu+ia^2/(4\mu)+O(a^4/\mu^3)$ and two propagating modes $\omega_\pm=\pm a/\sqrt2-ia^2/(8\mu)+O(a^3/\mu^2)$. The reduced <phase speed> uses the total <density> $\rho_i+\rho_n=2\rho_i$. The exact <displacement> ratio is $\xi_n/\xi_i=\mu/(\mu-i\omega)$. It is nearly $-1$ for the rapidly damped counterflow and nearly $1\pm ia/(\sqrt2\mu)$ for the propagating modes. Thus the latter have small ion-neutral slippage of relative size $|a|/\mu$, responsible for <ambipolar damping>.
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