74 THE DETONATION PROCESS 



In order to carry out the desired proof it is evidently necessary to con- 

 sider the adiabatic P-V relation at points on the R-H curve. The 

 relation between the adiabatic curve and the R-H curve at a given 

 point will of course depend on the position of the point and it is useful 

 to examine the possibility that for some point the two curves have the 

 same slope. For an adiabatic change (dS = 0) we may write 



dE = -PdV 

 and hence 



(i), = - 



But if the slopes of the two curves are the same, the partial derivative 

 at constant entropy must be the same as the corresponding derivative 

 for the R-H curve. From Eq. (2.28), we have on the Rankine-Hugoniot 

 curve 



E - Eo = i(P-{- Po) iVo - V) 



and hence 



Combining Eqs. (3.5) and (3.6) we obtain for the slope of the curves 



P -Po 



(3.7) 



(-1?), 



This is also the slope of the line AB m Fig. 3.1 passing through the 

 initial point PoVo and this line is thus tangent to the R-H curve at B. 

 The detonation velocity for this point is, from Eqs. (3.2) and (3.7), 

 given by 



D = \ 



\ dVjs 'V 



Combining this result with Eq. (3.3) we obtain 



D = c -]- u 

 as was to be proved. 



The proof that the detonation velocity is less than c + w above the point B and 

 greater than c + m below B is somewhat more complicated. We note that the local 

 sound velocity C is related to the slope of the adiabatic F — V curves by 



(-If). = ^- 



