76 THE DETONATION PROCESS 



shown in Fig. 3.1 this difference in slopes is negative above the point B and positive 

 below it. We must expect that this curve will have this form in general, as the pres- 

 sure must become infinite as the volume approaches zero and must, for the same 

 volume as in the initial state, lie above the point A. It is thus plausible that the 

 R-H curve for any normal fluids has positive curvature much hke an adiabatic curve 

 of compression, and the fact that the sign of {dS/dV)R has the behavior stated can 

 be demonstrated in terms of the curvature of the adiabatics. 



Eqs. (3.9) and (3.10) permit us to write the velocity difference as given by Eq. 

 (3.8) in terms of variables of known sign, with the result 



,.tc-(D-.)^] = (-f)^-(^) 



From what has been said, the right hand side of this equation is positive for points 

 above B and negative for points below B. Hence, as was to be shown, D is less or 

 greater than c + u according as the final state Ues above or below the point B on 

 the R-H curve to which a line from the initial point A is tangent. 



von Neumann (117) has made a more fundamental investigation of 

 the Chapman-Jouget condition in which he considers that the con- 

 servation equations must hold not only for the complete reaction of 

 the detonation front but also for intermediate stages of reaction in 

 the detonation front. The transition from undisturbed material to the 

 final product is therefore described by a series of states, each with its 

 own R-H curve. The detonation velocity for each state is described 

 by a line from the initial point A of Fig. 3.1 before reaction to the ap- 

 propriate point on each curve. In the cases we have to consider the 

 reaction zone is sufficiently narrow that all stages of reaction proceed 

 with the same velocity and a single line intersecting all these curves 

 must represent the detonation velocity. If the line representing the 

 Chapman-Jouget condition does not intersect all the intermediate 

 curves it cannot from this argument describe such states and is there- 

 fore incorrect. In such a case, the correct detonation velocity is, ac- 

 cording to von Neumann, determined by the line which is tangent to the 

 envelope of all possible R-H curves rather than to the final curve. 



Investigations have been made by Lippmann, Brinkley, and Wilson 

 (11) to determine the conditions under which the Chapman-Jouget con- 

 dition is valid for various special cases of solid explosives, which indi- 

 cate that the relation may break down for high densities of the initial 

 material. The significance of the result is, however, impaired by the 

 approximations necessary to permit the calculations. Perhaps the best 

 justifications of the condition in practical cases are the facts that in- 

 creased refinements of calculation lead to wider predicted ranges of 

 validity for it, and the detonation velocities determined with its aid 



