Shock waves in nonlinear transmission lines
Eugene Kogan
TL;DR
The paper analyzes shock waves in nonlinear transmission lines composed of nonlinear capacitors and inductors, with a twofold aim: first to quantify how small-amplitude sound waves reflect from or transmit through a shock, and second to incorporate dissipation to justify the jump conditions and to reveal finite-width shock/kink profiles. It derives general reflection/transmission coefficients $R$ and $T$ in terms of wave speeds and impedances, and then specializes to half-nonlinear lines to obtain simplified closed-form results for linear-capacitor and linear-inductor limits. The work further develops the lossy problem by introducing $R_L$ and $R_C$, reducing the traveling-wave dynamics to ODEs and a Newtonian-particle analogy that yields analytic shock/kink profiles for special parameter choices, connecting to Abel-equation and generalized KdV structures. It also provides a framework to recover profiles via quadratures in limiting cases and identifies integrability conditions that produce elementary-function shocks, while clarifying the distinction between shocks and kinks via boundary fixed points and explicit solutions.
Abstract
In the first half of the paper 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 the "sound" wave coefficient of reflection from (coefficient of transmission through) the shock wave. These coefficients are expressed in terms of the speeds of the "sound" waves relative to the shock and the wave impedances. In the second half of the paper we explicitly include into consideration the dissipation in the system, introducing ohmic resistors shunting the inductors and also in series with the capacitors. This allows us to justify the conditions on the shocks, postulated in the first half of the paper. This also allows us to describe the shocks as physical objects of finite width and study their profiles, same as the profiles of the waves closely connected with the shocks - the kinks. The profiles of the latter, and in some particular cases the profiles of the former, were obtained in terms of elementary functions.