Demonstration for the existence of longitudinal electrostatic oscillations How could I demonstrate that in a linear, homogeneous and isotropic medium without losses but electrically charged, Maxwell's equation admit as solutions longitudinal electromagnetic waves, beyond transversal waves?
 A: Kirk McDonald (Princeton) gives a nice proof and discussion of this possibility in "An Electrostatic Wave" (arXiv). The idea is that a plasma may support an electrostatic field of the form
$$
{\bf E} = E_x {\bf x} e^{i(kx - \omega t)}
$$
provided the electric displacement ${\bf D}$ is zero. Field ${\bf E}$ above satisfies 
$$
{\bf E} = -\nabla V, \;\;V = i \frac{E_x}{k}e^{i(kx - \omega t)}
$$
which implies
$$
\nabla \times {\bf E} = 0  \;\; \rightarrow \;\;-\frac{1}{c}\frac{\partial {\bf B}}{\partial t} = \nabla \times {\bf E} = 0
$$
This means that the magnetic field must be static, possibly null.
From the first Maxwell equation
$$
\nabla \cdot {\bf E} = 4\pi\rho
$$
it also follows that there must exist a charge density. This charge can be consistent with the longitudinal field ${\bf E}$ if it originates in a polarization ${\bf P}$ according to 
$$
\rho = -\nabla \cdot {\bf P}
$$
provided ${\bf P}$ is related to field ${\bf E}$ as 
$$
{\bf P} = -\frac{{\bf E}}{4\pi}
$$
In this case, the electric displacement is simply null, as mentioned above, 
$$
{\bf D} = {\bf E} + 4\pi{\bf P} = 0
$$ 
Polarization ${\bf P}$ also ensures that the polarization current is 
$$
{\bf J} = \frac{\partial {\bf P}}{\partial t} = - \frac{1}{4\pi}\frac{\partial {\bf E}}{\partial t}
$$ 
which means that the 4th Maxwell equation reads
$$
\nabla \times {\bf B} = \frac{4\pi}{c}\left({\bf J} + \frac{1}{4\pi}\frac{\partial {\bf E}}{\partial t} \right) = 0,
$$
and is consistent with a static magnetic field.
Note that the Maxwell equations do not impose a dispersion relation between the wave number $k$ and the frequency $\omega$ of this longitudinal electrostatic wave, hence the phase velocity is left arbitrary at this point. 
See McDonald's paper for further details on such waves in cold and hot plasmas.  
