we obtain 



THE DETONATION PROCESS 



-KS)'7ivrT['^a,0-^)' 



© 



This equation determines values of z near r = i^ in terms of ^, yp, and 

 with these initial values, successive increments of ^, \p are computed by 

 stepwise integration of Eqs. (3.21) for d^/dz, d\j//dz. The final results 

 obtained by Taylor are shown in Fig. 3.4 and, although they are quite 

 similar to the curves for a plane wave, the pressure and particle velocity 

 fall off more rapidly behind the detonation front and the final pressure 

 for ^^ = is smaller. The excess over the final value is negligible for 

 r < 3/5 R. This point is the crux of a basic assumption in some theories 

 of shock-wave propagation and a numerical example may indicate more 

 clearly the order of magnitude involved. If the detonation velocity D 

 is 6,400 m./sec. the time required for the tail of the wave at r = S/5 Rto 

 reach the point R is 2/3 R/D. For a change of radius 25 cm. (approx. 

 300 pounds TNT) this time is then roughly 25 microseconds, a com- 

 paratively small quantity relative to the time scale of the pressure wave 

 developed in the water. 



The calculations of Taylor for the spherical detonation wave in TNT 

 suffer in accuracy from the fact that an approximate adiabatic relation 

 for the explosion products was used. Appreciable errors in the cal- 

 culated detonation wave result from small errors in this adiabatic be- 

 cause the function / = (p/c'^) dc^/dp is sensitive to small differences in 

 values for P{p). Dasgupta and Penney* have therefore carried out 

 revised calculations, in which tabular values from Jones' calculations 

 were used, and the initial conditions at the head of the wave redeter- 

 mined to give the observed detonation velocity, taken as 6,780 m./sec. 

 In order to satisfy this and the Chapman-Jouget condition, a higher 

 pressure of 196 kilobars was necessary, as obtained by extrapolation of 

 Jones' adiabatics, the corresponding sound velocity and particle velocity 

 being c = 4,870 m./sec, Uo = 1,920 m./sec. 



The detonation wave was then computed by stepwise numerical 

 integration of Eqs. (3.21), which are solved for d\l//d^ and dz/d^ to avoid 

 the singularities at the detonation front encountered using z as an inde- 

 pendent variable. The pressure and particle velocity values finally 

 obtained are plotted as dashed curves in Fig. 3.4. It is seen that the 

 pressure, while higher than Taylor's, decays more gradually and a con- 

 stant value is reached for distances less than half the charge radius from 

 the center. 



Dasgupta and Penney have also computed, by the same method, the 



* H. K. Dasgupta and W. G. Penney, British Report RC 373. 



