THE DETONATION PROCESS 95 



of some interest to examine the consequences of the simphfied case of a 

 plane wave for wliich the products are assumed to be ideal gases. 



A. Plane wave, ideal gases. If we assume ideal gases in which the 

 ratio of specific heats is independent of temperature, the adiabatic rela- 

 tion at any point behind the front is P = kp^ , where k and 7 are taken 

 to be constant. The detonation velocity, for such a mixture, was shown 

 in section 3.3 to be 



D = (l + y) J^ 



where Pi, pi are the pressure and density at the head of the wave. The 

 velocity of sound c in these gases is given by 



[dp ). 



= kyp^ ^ = 7 — 

 P 



and the velocity of sound Ci at the detonation front is C\ = 'VyPi/pi. 

 From this relation and the Chapman-Jouget condition, the particle 

 velocity Ui at the front is Mi = D — Ci = Ci/y. 



If the detonation is a plane wave, Riemann's analysis shows that 

 behind the front constant values of particle velocity, density, and sound 

 velocity travel with a speed c -\- u and u is given by 



(3.16) ui-u= I c^ 



/:■ 



p 



-y-i 

 From the adiabatic law, c = ci (p/pi) ^ and substituting for c in the 

 integral gives 



2 



(3.17) ui - u = 7 (ci - c) 



7-1 



If the detonation is initiated by any cause of duration short enough that 

 the gases can be considered at rest for all times immediately after initi- 

 ation, a given value of ii will be propagated a distance r in time t such 

 that r = (c -\- u)t. The detonation front will have travelled a distance 

 R = (ci -\- ui)t and we have 



— - — = (ci - c) - {ui - u) 



