THE DETONATION PROCESS 71 



3.2. Detonation Waves 



We have already mentioned that even a plane shock wave Avill ulti- 

 mately die out as a result of dissipation processes at the shock front 

 unless energy is continually supplied at some point behind the shock 

 front. If, however, the passage of the wave involves a release of 

 chemical energy in the medium it becomes possible to realize a self- 

 sustaining wave, which after its initiation builds up to a stable limiting 

 rate of propagation characteristic of the medium. Such a wave can 

 develop in an explosive, which is thermodynamically an unstable sub- 

 stance capable of reacting to form a more stable product with release 

 of energy, and is called a detonation wave. This self-sustaining wave 

 differs in two important respects from a shock wave. In the first place, 

 although the Rankine-Hugoniot conditions still relate conditions im- 

 mediately behind the shock front, the chemical energy appropriate to 

 the pressure and density of the transformed matter behind the front 

 must be included in calculations of these conditions. The second dif- 

 ference is that the propagation of the wave is not controlled by the 

 conditions behind the front, such as the motion of a boundary surface 

 without which a shock wave could not exist, but is rather determined 

 by the internal conditions in the material just behind the front. The 

 three equations expressing conservation of mass, momentum, and energy 

 at the shock front do not suffice to determine the four unknown quanti- 

 ties behind the detonation front : pressure P, density p, particle velocity 

 u, and velocity of propagation D. A fourth condition on these vari- 

 ables is therefore necessary to define which of the otherwise possible 

 waves will actually be established. Chapman (19) arbitrarily assumed 

 that the detonation velocity in any particular case was the minimum 

 velocity consistent with the Rankine-Hugoniot conditions for w^hich a 

 self-sustained wave could exist, and later Jouget (53) postulated the 

 condition 



(3.1) D = c + u 



where u is the particle velocity and c the velocity of small amplitude 

 Avaves in the reaction products behind the shock front. These assump- 

 tions are, as we shall show, equivalent and Eq. (3.1) is the mathematical 

 expression of w^hat is known as the Chapman- Jouget condition. 



There is little doubt that the Chapman-Jouget condition correctly 

 predicts detonation velocities for the cases of interest to us and it is 

 generally accepted, but as yet there is apparently no valid theoretical 

 proof of its necessity in all practical circumstances, von Neumann 

 (117) has made a detailed analysis of the conditions at a detonation 



