Fun book.
The proof that the pulse shape was Gaussian for sinusoidal quench and exponential for square wave is pretty, and so's the proof that although the frequency of the carrier matches that of the input signal, and the waveform is smooth, it's entirely made up of sums of harmonics of the quench frequency.
The other proof, which explains the main reason why superregens aren't commonly used now that gain is cheap and spectrum is expensive, is that their noise gain is fairly high--the "sampling" interval early in the quench cycle, determines what the pulse will look like, but is several times shorter than the quench cycle. Thus the equivalent noise bandwidth is that much larger than in a superhet.
Exponential growth and decay is useful for lots of things, e.g. I've used the ring-down of a quartz crystal oscillator as a calibrator for logarithmic DLVAs. You have to run it at the series resonance, because that's where the mechanical amplitude is greatest. Otherwise you get a abrupt amplitude decrease as soon as you disconnect the oscillator circuitry, before the exponential decay takes over.
Cheers
Phil Hobbs