Effects of Dispersion on Steady State Electromagnetic Shock Profiles

Abstract

In nonlinear electromagnetic media the various portions of a wave can travel with different velocities, which can result in the formation of electromagnetic shock waves. The structure of such a steady state shock is determined by an equilibrium between the velocity differences that tend to sharpen the shock and the sources of dispersion that cause a broadening of the shock. Several nonlinear transmission line models are examined for the nature and existence of a single-valued steady state shock. In all cases a nonlinear shunt capacitance is assumed. If the dispersion arises from the relaxation behavior caused by a resistance in series with the nonlinear capacitance, a steady shock always exists, its width decreasing as the extent of the nonlinearity generated by the shock increases. If the series resistance is itself shunted by another capacitance, the relaxation process is not manifested at very high frequencies. This system yields a critical condition for the existence of a continuous single-valued steady state wave profile. If the line has too little dispersion, the steady state profile is multivalued and therefore physically unrealizable. These dispersion requirements are equivalent to the condition that the velocity of small, high frequency signals ahead of the shock must be greater than the velocity of the shock itself. It is believed that this condition is a broadly applicable criterion for the existence of a stable, single-valued, steady state wave profile. While this hypothesis is not proved analytically, it is supported here by plausibility arguments and by analysis of another system in which the dispersion is included in the linear series inductance rather than in the nonlinear shunt capacitance.