Properties of shock adiabats in the neighborhood of jouguet points

Abstract

Two well-known properties of shock adiabats in a gas [1] are proved for shock adiabats corresponding to discontinuous solutions of hyperbolic systems of equations expressing “conservation laws”. If the state on one side of a discontinuity is fixed, then at the point of extremum of the discontinuity velocity on the shock adiabat the velocity of the discontinuity is equal to one of the velocities of the characteristics on the other side of the discontinuity and vice versa. If for the systems there is defined an entropy flux or mass density of entropy, then at the points of extremum of the velocity there is an extremum of the entropy production at the discontinuity and the entropy mass density. If the system is a symmetric hyperbolic system [2, 3], then the extrema of the entropy production at the discontinuity correspond to extrema of the velocity. These properties may be helpful in the study of discontinuities in complex media, since the sections of a shock adiabat whose points can correspond to actually existing discontinuities are frequently bounded by points corresponding to discontinuities whose velocity is equal to the velocity of a characteristic on one of the sides of the discontinuity (see, for example, [1, 4, 5]).