ABSTRACT
In this chapter, we present a derivation of the equations of a weak shock ray theory from those of WNLRT, a physically realistic conservation form of NTSD and finally, some results of numerical solutions of these equations for converging shock fronts starting from various kinds of initial geometry. Distributions of the shock strength µ and the normal derivative µ1, defined in (9.5.1), have been varied in order to bring out some interesting results. The effects of changing the initial strength of the shock or that of the normal derivative, and also the effect of initial curvature on the formation, propagation and separation of kinks have been studied. Then we have studied the ultimate shape and decay of shocks with initially periodic shapes and plane shocks with a dent and bulge, and interpreted these results as corrugational stability of a shock front. Finally, we have presented a comparison of these results with those obtained from other theories.
A system of equations of shock ray theory consists of the ray equations derived from a shock manifold partial differential equation and an infinite system of compatibility conditions along a shock ray. We emphasize again that unlike the well known geometrical optics theory for the propagation of a one parameter family of wavefronts across which wave amplitude is continuous, the shock ray theory with
infinite system of compatibility conditions is exact. This is because geometrical optics requires high frequency approximation which is satisfied exactly for a shock front. Even under the formulation of the NTSD with just two compatibility conditions, the equations for a curved shock propagation are very complex and so far, only one attempt (Singh and Singh (1999)) has been made to use them to compute shock geometry and amplitude distribution on it at a later time for a shock of arbitrary strength. However, much more work, especially in the formulation of conservation laws from the two compatibility needs to be done. This problem did not arise in the case of one-dimensional piston problem dealt in chapter 8 where the NTSD was found to be very successful even in dealing with a strong shock produced by an accelerating piston. We have seen at in the end of the last chapter that, under suitable assumptions, these equations for a weak shock reduce to a rather simpler set of equations (9.5.12 - 15) which we shall use in this chapter.
