220 
Sec. 5 age 3 Ig. 22-26 
gases is greater than the velocity of the detonation wave relative tc the 
burnt gases, Consequently, if a rarefaction wave, due to any of a number 
of causes, starts behind the detonation wave, it will catch up with the 
detonation front. The rarefaction will then reduce the pressure, causing 
the final P and V to move down the curve toward the point J of Fig. 5-1. This 
explains why points above J on the curve are not stable. The rarefaction 
wave might be started by a deceleration of the piston, by turbulence back of 
the detonation front, or by loss of heat in the burnt gases through conduc- 
tion, etc. At the point J, the velocity of the detonation wave is equal to 
the velocity of sound in the burnt gases plus the mass velocity of those 
Gases, so that rarefaction waves will not then catch up with the detonation. 
The argument which is used to exclude points on the Hugoniot curve be- 
low J is based on the entropy of the products. Consider the two points Y 
and Z on the same straight line from A (Fig. 5-1). Both Y and Z correspond 
to the same detonation velocity D by Eq. (21). It will be shown that the 
products of combustion have a greater entropy at Z than at Y. Consequently 
Z is a more probable state than Y. This is true for all pairs Y and Z% ona 
line from A until Y and Z coincide at J. Therefore points above J are more 
probable than points below J but points above J slide down to J because of 
the effect of the rarefaction wave, so that J represents the stable and 
point. The condition of tangency therefore provides the additional condition 
required to specify the detonation velocity D uniquely. 
The statements made above remain to be proved mathematically and this 
will now be done. First consider the velocity of rarefaction waves, The 
velocity or sound in a medium is given by 
by = Vy (ap /AVp) gy (22) 
the subscript S denoting that the entropy is held constant. It is there- 
fore important to investigate (dP5/AV5 )g for various points on the Hugoniot 
curve. In general. 
mp = ar 23 ) 
Pods, = 40, + Pav, (23 ) 
and by differentiation of the Hugeniot expression, liq. (12), 
H - By = 2 (P, + Po) (Vy - Vo), (2h) 
‘ 
holding eas Vey constant, one gets 
dE, = - 3 (P, + Pav, + 2 (V, - Vp)APp, (25) 
so that 
(aS, | [ ar,’ ) 
a, = 4 (V, - Vo) ¢ = - 1(Py - Po)/(Vq - v2)| (25) 
\av., = l dV | 
Sel tal SEs j 
the subscript H denoting the derivative along the H curve. Now suppose 
that at some point along the H curve the adiabatic expansion curve (P-V curve 
