Shock waves in nonlinear transmission lines

Abstract

We consider interaction between the small amplitude travelling waves ("sound") and the shock waves in the transmission line containing both nonlinear capacitors and nonlinear inductors. We calculate for the "sound" wave the coefficient of reflection from (the coefficient of transmission through) the shock wave. These coefficients are expressed in terms of the wave speeds and the wave impedances. When only the capacitors or only the inductors are nonlinear, the coefficients are expressed in terms of the wave speeds only. We explicitly include into consideration of the shocks the dissipation, introducing ohmic resistors shunting the inductors and also in series with the capacitors. This allows us to describe the shocks as physical objects of finite width and study their profiles. In some particular cases the profiles were obtained in terms of elementary functions.