THE DETONATION PROCESS 73 



(3.2) D = VoV{P - Po)/{Vo - V) 



It is evident from the diagram that this relation may be written 



D = Vo Vtan a 



where a is the angle a line such as AC, joining the initial point A and 

 a point C on the Rankine-Hugoniot (R-H) curve, makes with the 

 (negative) V axis. 



The two possible states for a given value of D, corresponding to 

 points C and C\ will not have the same entropy, the value at C being 

 higher. That this must be true is evident from the fact that, as Becker 

 (7) points out, a transition from C to C describes a shock wave of com- 

 pression with initial state C and final state C, and the velocity D. 

 But, as discussed in section 2.3, the entropy is greater behind a shock 

 front than ahead of it. The greater entropy at the upper point C of the 

 two possible ones for a given D is therefore thermodynamically a more 

 probable state, and we should expect the state of highest pressure and 

 density compatible with other conditions to be realized. The other 

 condition is of course that D must be equal to or greater than c -\- u. 



It is readily shown, however, that D is equal to c + w at the point 

 B corresponding to the minimum possible value of D and is always less 

 than c -\- u ioY any point above B if, as is generally true, the adiabatic 

 P-V relation for the products has positive curvature: 



The more probable states above B are therefore destroyed by rarefaction 

 waves running into the front, in spite of the higher detonation velocity. 

 We therefore expect the state represented by B to be the one realized. 

 The mathematical proof that D is equal to c + ti at 5 is a straight- 

 forward thermodynamical derivation.^ From Eq. (2.29) the detonation 

 velocity is related to the P-V conditions in two states {Po, Vo) and 

 (P, V) and the particle velocity u by 



(3.3) D = 



Vo-V 

 and the velocity of sound behind the front is 



-(fr--(-i): 



2 The proofs given here follow closely the derivations given by Kistiakowsky and 

 Wilson (64). Thermodynamic considerations for detonation waves have also been 

 discussed by Scorah (99) . 



