A rigorous solution is obtained for the problem of radiation from an electric line charge that moves, at a constant speed, parallel to an electrically perfectly conducting grating. The relevant vectorial electromagnetic problem is reduced to a two-dimensional scalar one. With the aid of a Green’s-function formulation of the problem, an integral equation of the second kind for the surface current density on a single period of the grating surface is derived. This integral equation is solved numerically by a method of moments. Some numerical results pertaining to the radiation from a moving line charge above a sinusoidal grating are presented.
P. M. van den Berg, "Smith—Purcell radiation from a line charge moving parallel to a reflection grating," J. Opt. Soc. Am. 63, 689-698 (1973)