  We revisit the exact microscopic equations (in differential, and equivalent
integral form) ruling a relativistic cold plasma after the plane-wave Ansatz,
without customary approximations. We show that in the Eulerian description the
motion of a very diluted plasma initially at rest and excited by an arbitrary
transverse plane electromagnetic travelling-wave has a very simple and explicit
dependence on the transverse electromagnetic potential; for a non-zero density
plasma the above motion is a good approximation of the real one as long as the
back-reaction of the charges on the electromagnetic field can be neglected,
i.e. for a time lapse decreasing with the plasma density, and can be used as
initial step in an iterative resolution scheme.
  As one of many possible applications, we use these results to describe how
the ponderomotive force of a very intense and short plane laser pulse hitting
normally the surface of a plasma boosts the surface electrons into the ion
background. Because of this penetration the electrons are then pulled back by
the electric force exerted by the ions and may leave the plasma with high
energy in the direction opposite to that of propagation of the pulse [G. Fiore,
R. Fedele, U. De Angelis, "The slingshot effect: a possible new laser-driven
high energy acceleration mechanism for electrons", arXiv:1309.1400].
