We present in this paper compelling evidence supporting the three-wave traveling-wave theory developed by Pierce fifty years ago. The transition in a Smith-Purcell free-electron laser from low, through moderate amplified spontaneous emission, to strong gain conditions was carefully controlled. Below threshold, the emitted far-infrared power exhibits oscillations with a cubic dependence on the electron beam current. Both characteristics are expected in a three-wave interaction yet, to date, have not been observed.