The equations governing the voltage distribution along a parametric travelling-wave amplifier can, under certain conditions, be solved in terms of elliptic functions.The solution shows that there is a spatially periodic transfer of energy between the signal, idling and pumping waves. The maximum saturated power gain that can be achieved is1 + ω1/ω3pump input power/signal input powerwhere ω1and ω3are the angular frequencies of the signal and of the pump, respectively. The positions of these gain maxima can be calculated explicitly without invoking small-signal conditions.