Time domain analysis of a charged particle travelling along the axis of a circular waveguide

The aim of this note is to demonstrate that the electromagnetic field, in the time domain, sustained by a point charge moving at a constant velocity along the axis of a circular waveguide can be expanded by means of a Fourier-Bessel series. This expansion is rapidly convergent in the complementary range of the classical integral representations, namely for large values of \z' - vt'\. Quickly convergent expansions are necessary for tracking codes to simulate beam dynamics in accelerators.