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From relations (8b") and (8c') it follows that F,,(P) 
and Yel) are monotonically increasing functions of p, while 
their slopes are monotonically decreasing functions of p. 
Before proceeding we would like to call to attention that 
Pr (By) = - Alp,) and W,(P,) = -f{P,), each of which simply 
states that the differerce of material velocity (V, - V,) = -(V,-V)), 
for a shock-wave or for a rarefaction-wave, respectively. 
Let us consider the case of two forward shocks following 
each other. These will divide the x-axis into three regions, 
left, mj.ddle, and right which we will designate 1, m, and r. 
From (A) we have the inequalities Pj > Fees >Pp and V] > Vn > Vn 
These will indicate the relative positions of 1, m, and r on the 
V , p diagram. After the interaction 
Fig. 28 3) 
Fife 29 ; 3 We, 
- 
We 
has taken place the regions 1 and r remain unchanged, while 
a new region m! appears at the center to take the place of mo 
The region m' is connected to region 1 by a backward wave, qu, 
