Characteristic wave fronts in magnetohydrodynamics

Abstract

The influence of magnetic field on the process of steepening or flattening of the characteristic wave fronts in a plane and cylindrically symmetric motion of an ideal plasma is investigated. This aspect of the problem has not been considered until now. Remarkable differences between plane, cylindrical diverging, and cylindrical converging waves are discovered. For instance, when the adiabatic index γ is 2, the magnetic field does not affect the behaviour of plane waves, but does affect cylindrical waves. As the field strength increases, the time tc taken for the shock formation varies monotonically for plane waves, while for cylindrical waves, in some situations tc exhibits a unique minimum for diverging waves and a unique maximum for converging waves. For cylindrical converging waves, a shock formation takes place if and only if, γ and the field strength are restricted to certain finite intervals. Moreover, tc is bounded in all cases except for cylindrical diverging waves. The discontinuity in the velocity gradient at the wave front is shown to satisfy a Bernoulli-type equation. The discussion of the solutions of such equations reported in the literature is shown to be incomplete, and three general theorems are established.