The basic physics behind the plasma frequency The dispersion relation of ordinary wave in plasma is
\begin{equation}
\omega^2=\omega_{p}^{2}+c^{2}k^{2},
\end{equation}
where $\omega_{p}=\sqrt{{4\pi n_{0}e^{2}}/{m}}$ is the plasma frequency.
Mathematically when 
\begin{equation}
\omega=\omega_{p}
\end{equation}
then $k=0$, and will not propagate but what is basic physics behind this?
 A: All $k = 0$ means is that we're talking about the limit of infinite wavelength, i.e. a wave that's constant in space. When you have a dispersion relation that allows $\omega>0$ when $k=0$, that just means that there's a mode that oscillates in time while being constant as a function of position. So all points in the material are oscillating in phase.
In the case of a plasma oscillation, it's pretty easy to visualize this if we think of a large slab of material that's infinite in the $x$ and $y$ directions but finite in the $z$ direction. Suppose the slab consists of a uniform immobile positive charge density (in the usual case where the positively charged entities are much more massive than the negative ones) and an equal density of mobile negative charge. Now imagine grabbing all of the negative charges and moving them en masse a short distance in the $z$ direction. Your slab now has a net unbalanced positive charge density on one surface and an equal and opposite negative charge density on the other surface. This sets up an electric field that tends to counteract the charge imbalance, and the slab of negative charge will accelerate until those surface charge layers are cancelled out. But the slab of negative charge still has momentum, so it overshoots, and a short time later you now have the surface charge layers reversed and it starts accelerating the other way. And it's a very quick derivation to show that this model produces an oscillation frequency equal to the usual formula for the plasma frequency.
I find this picture to be simple and intuitive and a nice complement to the understanding achieved by slogging through the full derivation, with all of the magnetic effects, retardation effects, damping time constants, etc.  Properly speaking, we really should talk about the $k = 0$ limit of the model with all of that physics in place (instead of jumping straight to an interpretation of "infinite wavelength"), but to my mind at least this picture gives me an intuition for why that limit should give us a well-defined oscillation in the first place.
A: That means that the dispersion law for plasma differs from the dispersion of sound waves, for which $\omega(k=0)=0$ holds. It means also that there is a standing wave mode of plasma oscillations. This frequency is the resonant frequency of the interaction between light and electrons. The light can be absorbed at this frequency most efficient without actual acceleration of an electron in a specific direction.
