What happens when a slow wave reaches lower hybrid resonance? Lower hybrid resonance occurs when $n_{\perp}^2$ goes to infinity, and it occurs only for the slow wave solution, not the fast wave.  Since $n_{\perp}$ is proportional to $k_{\perp}$, and $k = \frac{2 \pi}{\lambda}$, it means that the wavelength of the wave goes to zero.  But what physically happens when the slow wave reaches the lower hybrid resonance?
I should mention that I'm talking about in the cold plasma model, where the fast and slow wave modes have meaning.  
 A: Well, I am not sure if your statements are entirely accurate because the fast mode can approach the lower hybrid resonance.  In fact, in this regime, it becomes effectively indistinguishable from an electrostatic whistler mode.  At low frequency and oblique angles, the fast (or magnetosonic) modes are right-hand polarized (with respect to $\mathbf{B}_{o}$) electromagnetic waves, which happen to be on the same branch of the dispersion relation as whistler mode waves.
In any case, in the limit as $\mathbf{k}_{\parallel}$ $\rightarrow$ 0 the wave will become electrostatic and, for all intents and purposes, be a form of ion-acoustic wave (not to be confused with the much higher frequency electrostatic version that has $\mathbf{k}_{\perp}$ ~ 0) or lower-hybrid wave.  At such oblique angles, the only important things are the wave frequency and that it is electrostatic.  The name given to the mode is really just semantics.
Edits/Additions
I realized after the fact that I had forgotten to say that $\mathbf{k}$, in addition to knowledge of $\omega$ and polarization (e.g., electrostatic), are the only important things for these modes.  The reason is that these properties let you know how and with what particles these modes interact.
At the lower hybrid resonance, an electrostatic wave can simultaneously couple and exchange energy/momentum with both electrons and ions.  This is why the waves are so popular in current dissipation theories, since $\mathbf{j}$ = $\sum_{s} n_{s} \ q_{s} \ \mathbf{V}_{s}$ .  If the waves can transfer energy/moment from(to) electrons to(from) ions, then have the capacity to limit $\mathbf{j}$.  If the interaction is stochastic in nature, then the result can be an irreversible form of energy transformation (i.e., energy dissipation).
