Shock Waves in Nonlinear Transmission Lines and Their Effect on Parametric Amplification

The propagation of a periodic signal on a transmission line with a nonlinearity in the distributed capacitance is examined. The signal is deformed during its propagation and electromagnetic shock waves are generated. It is pointed out that the shock wave will form in a distance which is short for any parametric amplification purposes. The subsequent growth of the shock and its decay, due to the inevitable dissipation associated with a shock, are analyzed assuming that the capacitance variations are small compared to the total capacitance. The propagation of a small deviation from a signal which is perfectly periodic in time is also examined, and it is shown that the small deviation may spread out in time but cannot be changed in its sign. This result was invoked in an earlier paper demonstrating the impossibility of parametric amplification on dispersionless nonlinear lines.