Resistive-wall wake and impedance for non-ultrarelativistic beams

The usual formulae for the resistive-wall wake field are derived considering ultra-relativistic beams, travelling at the speed of light. This simplifies the calculation, and it leads to a cancellation between electric and magnetic fields. However, for proton beams below 10 GeV and for many heavy ion beams, the velocities may significantly differ from the speed of light. In this paper, we compute the longitudinal and transverse wake fields for velocities smaller than c, and examine under which conditions non-relativistic effects become important. We illustrate our results by a few examples.