I am looking for an intuitive explanation of the response of cold plasmas with refractive index $n=\sqrt{1-\frac{\omega_p^2}{\omega^2}}$ to electromagnetic waves.
All of the explanations I can find say that for $\omega<\omega_p$ the electrons react 'quickly enough' to cancel the incident wave so it is fully reflected, and for $\omega>\omega_p$ the electrons cannot respond quickly enough to fully cancel it so it is able to propagate.
However, these explanations don't match up with the mathematical result for the transmitted electric field (found by setting $n=\sqrt{1-\frac{\omega_p^2}{\omega^2}}$ in the Fresnel equation for a normally incident wave), $$E_t=\frac{2}{1+\sqrt{1-\frac{\omega_p^2}{\omega^2}}}E_i.$$ This shows that the electrons actually reinforce the field, so surely the explanations which talk about them cancelling cannot be correct.
