Shock waves in nonlinear transmission lines

TL;DR

This paper analyzes shock wave interactions in nonlinear transmission lines, deriving reflection coefficients, incorporating dissipation effects, and describing shock and kink profiles as physical objects with finite width.

Contribution

It introduces a detailed analysis of shock wave behavior in nonlinear transmission lines, including reflection coefficients and the effects of dissipation on shock profiles.

Findings

Reflection coefficients expressed via wave speeds and impedances

Dissipation effects justify shock conditions and profile shapes

Explicit shock and kink profiles obtained in elementary functions

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.