In an idealized model of a traveling wave amplifier operating in the linear regime, it is assumed that all the transients have decayed. This implies that the electromagnetic wave has an amplitude which is constant in time but it may vary in space according to the interaction process. In an idealized model of an oscillator, the situation is reversed. The amplitude of the electromagnetic wave is constant in space and i t may vary in time. We present a generalized formulation of the interaction in a traveling wave tube which includes reflections and spatial and temporal transients. Within this framework i t is shown that, in an amplifier, the reflections cause time variations of the amplitude that are ultimately revealed as broadening of the spectrum. The "transition" from an amplifier to an oscillator is also investigated. In the case of an oscillator, it is shown that in addition to the well known temporal transient (lethargy) there is a spatial transient. This is due to the fact that it takes some distance for the "fresh" electrons entering the oscillator to get bunched. Beyond this transient the amplitude is constant in space, as anticipated by idealized theory.
John A. Nation,
"Microwave amplifiers and oscillators: a unified study", Proc. SPIE 1629, Intense Microwave and Particle Beams III, (1 April 1992); doi: 10.1117/12.137172; https://doi.org/10.1117/12.137172