Page:Threeshockwaves.pdf/9

Having determined $\gamma$ by this relation we can cross the shock (now of a known intensity) OB and arrive in region 4 with a definite direction for the motion of the gas. For an arbitrarily initially assigned value of $\beta$ the direction of motion derived for region 3 after crossing the two shocks OB and OC will not in general agree with that derived for region 4 after crossing the single shock OE. By trial and error we can adjust the angle $\beta$ till the directions of motion in the regions 3 and 4 agree.^[1] In this manner we can determine the direction of the vortex sheet which will be in equilibrium with a shock OB of given intensity and with a specified direction of motion in region 1.

The method described in the preceding paragraph is suitable for the purpose of isolating numerically configurations of three shocks in equilibrium, particularly if tables of solutions of the appropriate Rankine-Hugoniot equations are available. Tables of the Rankine-Hugoniot function for $\gamma =1.4$ are provided in the Appendix. Using these tables the three shock solutions depicted in Figs. 7, 8 and 9 were obtained.

4. Discussion of the Results of Figs. 7, 8 and 9.

Considering first the case of a strong incident shock we find that for $\zeta =7.0225$ (Fig.7) the stationary Mach reflection occurs for an angle of incidence $\alpha =31.25^{\circ }$. This angle is less than the angle at which regular reflection ceases for shocks of intensity, namely $42^{\circ }$. Moreover this stationary Mach reflection occurs when the reflected shock corresponds to a "stable" reflection. For increasing angles of incidence we get the three shock configurations of Fig. 7. It is to be particularly noticed that these solutions terminate for an angle of incidence $\alpha \sim 63^{\circ }$. At this angle the "Mach" shock becomes a continuation of the original shock, while the reflected shock and the vortex sheet disappear. We should, however, note in this connection that for an angle of incidence of $50^{\circ }$ the vortex sheet is inclined to the horizontal by about $20^{\circ }$. Consequently the non-satisfaction of the boundary condition on the reflecting surface in the region behind the Mach front will make the situation actually realized in practice to deviate considerably from those derived here on the basis of the stationary interaction of

1. It is clear that after the adjustments have been made for the directions of motion in regions 3 and 4 to agree we shall still be left in general with a difference in the magnitude of velocity on either side of the common direction of motion; in other words, we have a vortex sheet in this region.