  Electron accelerators with higher and higher longitudinal field gradients are
desirable, as they allow for the production of high energy beams by means of
compact and cheap setups. The new laser-plasma acceleration technique appears
to constitute the more promising breakthrough in this direction, delivering
unprecedent field gradients up to TV/m. In this article we give a quantitative
description of the impact of longitudinal wake fields on the electron beam. Our
paper is based on the solution of Maxwell's equations for the longitudinal
field. Our conclusions are valid when the acceleration distance is much smaller
than the the overtaking length, that is the length that electrons travel as a
light signal from the tail of the bunch overtakes the head of the bunch. This
condition is well verified for laser-plasma devices. We calculate a closed
expression for the impedance and the wake function that may be evaluated
numerically. It is shown that the rate of energy loss in the bunch due to
radiative interaction is equal to the energy emitted through coherent radiation
in the far-zone. Furthermore, an expression is found for the asymptotic limit
of a large distance of the electron beam from the accelerator compared with the
overtaking length. Such expression allows us to calculate analytical solutions
for a Gaussian transverse and longitudinal bunch shape. Finally, we study the
feasibility of Table-Top Free-Electron Lasers in the Vacuum Ultra-Violet
(TT-VUV FEL) and X-ray range (TT-XFEL), respectively based on 100 MeV and 1 GeV
laser-plasma accelerator drivers. Numerical estimations presented in this paper
indicate that the effects of the time-dependent energy change induced by the
longitudinal wake pose a serious threat to the operation of these devices.
