THE DETONATION PROCESS 77 



agree well with experimental values for conditions where calculations 

 suggest that it may not be valid. As we shall see later, the exact form 

 of the detonation wave is not, in most cases, of decisive importance for 

 computation of shock waves and gas sphere motion. Therefore, a more 

 detailed consideration of the possible failure of the Chapman-Jouget 

 condition is not necessary. 



With the assumption of the Chapman-Jouget condition and the 

 Rankine-Hugoniot relations, the pressure, density, and velocity at the 

 head of a detonation wave are determined in terms of the equation of 

 state and thermochemical data for the medium. An explicit calcu- 

 lation can therefore be made and, once this has been done, the condi- 

 tions behind the front can be computed by using these boundary condi- 

 tions for integration of the equations of motion for the "fluid" behind 

 the front. The pressure and velocity distribution behind the front will 

 evidently depend on the shape of the front in space: plane, spherical, 

 or other. It is important to realize, however, that the properties at 

 the head of the wave do not depend on the shape as long as the thick- 

 ness of the front is negligible, as this condition, which results in the 

 Rankine-Hugoniot relations, makes any divergence of the wave im- 

 material. 



If we consider a detonation wave established by the initial motion 

 of a piston or some other cause which subsequently ceases, it is evident 

 that gas at the point of initiation must come to rest, the agency by 

 which this takes place being a wave of rarefaction moving outward. 

 Immediately behind the detonation front the wave has a velocity 

 c -\- u equal to the detonation velocity D by the Chapman-Jouget con- 

 dition, and at points behind the front its velocity falls off to a final 

 value C2 consistent w^ith the adiabatic pressure density relation and the 

 condition that the particle velocity u is zero. For a plane wave, the 

 discussion in section 2.3 of the Riemann equations shows that the pres- 

 sure and density are propagated forward with a velocity c -\- u behind 

 the front. The particle velocity u is given by 



dp 



u — Ui ' 



\'i 



where Ux and pi are the particle velocity and density for some known 

 state, in this case the state at the detonation front. The final state 

 may thus be obtained by setting w = and determining the density 

 P = p2 for which the integral equation is satisfied. 



It will be observed that if the detonation wave has a constant 

 velocity D, the front will travel a distance Dt in time t, the final state 

 will have travelled a distance C2t, and we expect that the whole scale of 



