Page:Threeshockwaves.pdf/6

 More exact and extensive calculations of the type indicated have since been carried out by Dr. R. Seeger, whose results are summarized in Figure 5. An examination of this curve reveals that for $$\zeta_{1,2}>2.26$$ the stationary Mach effect occurs for an angle of incidence $$\alpha_M$$ which is less than the corresponding $$\alpha_{\mathrm{extreme}}$$. On the other hand for $$\zeta_{1,2}<2.26$$ the stationary Mach reflection occurs as a branch along the sequence of the so-called 'unstable' regular reflections. Consequently, for strong shocks with $$\zeta_{1,2}>2.26$$ we may expect the Mach effect to start at an angle of incidence smaller than that at which regular reflection becomes kinematically impossible. The basis for this suggestion is that we may reasonably expect the sequence of Mach-reflections to join the sequence of the regular reflections at a point corresponding to the stationary Mach reflection. However, when we turn to weak shocks the situation becomes somewhat peculiar: The stationary limit of the sequence of Mach reflections and this does join the sequence of the  regular reflections. The question thus arises whether for the reflection of shock waves with $$\zeta_{1,2}<2.26$$ the 'unstable' reflections with angles of incidence $$\alpha$$ greater than $$\alpha_M$$ cannot after all be realized. We shall return to these questions after we have considered in $$\S 3$$ the general conditions for the existence of three shocks together with  a vortex sheet to be in equilibrium.

3.



Consider a configuration of 3 shock waves (OB, OC, OE) and a vortex sheet (OD) as shown in Figure 6. We shall consider the phenomenon in a frame of reference in which the line of intersection of the shock waves and the vortex sheet is at rest. Finally, let OA define the direction of motion of the gas in region 1 in the frame of reference chosen.