In pulsed beam free-electron devices, longitudinal space-charge fields result in collective effects leading to expansion of short electron pulses along their trajectory. This effect restricts application of intense ultra-short electron pulses, and requests a careful theoretical description. In the present work, longitudinal space-charge fields are considered in the framework of a three-dimensional, space-frequency approach. The model is based on the expansion of the total electromagnetic field (radiation and space-charge waves) in terms of transverse eigen-modes of the (cold) cavity, in which the field is excited and propagates. This approach has being successfully applied for analysis of wide-band interactions in free-electron lasers operating in the linear and non-linear regimes.

The basic equations of the model, originally obtained as a solution of the wave equations for the electromagnetic field in an uniform waveguide, are shown to satisfy also Gauss's law. Longitudinal electric field was found in the model analytically for a point-like charge, moving along a waveguide with a constant velocity. This enables consideration and comparison of different components of the resulted longitudinal electric field, such as forward and backward going waves, near and under cut-off frequencies, and so on. Possible simplifications in evaluations of longitudinal space-charge fields in the model are discussed.