Shock waves in nonlinear transmission lines (2408.01463v6)

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.