THE DETONATION PROCESS 101 



3.7. The Approximation of Adiabatic Explosion 

 AT Constant Volume 



An exact solution for the propagation of underwater shock waves 

 should of course be based on a continuation of the solution for the det- 

 onation wave after it reaches the surface of the charge, applying the 

 appropriate boundary conditions. These conditions require a shock 

 wave in the water and a rarefaction wave travelling back toward the 

 center of the charge, and a reasonably good solution taking account 

 of both would be a matter of considerable difficulty. Kirkwood and 

 Bethe (59, I) observe, however, that the exact form of the detonation 

 wave near its front is in most cases of no great importance, for the rea- 

 son that the initial portions of the underwater shock wave, determined 

 by the head of the detonation wave, are rapidly destroyed by the over- 

 taking effect as it progresses outward. In other words, even the front 

 of the shock wave is, in their propagation theory, determined by condi- 

 tions obtaining at times increasingly later than the time of complete 

 detonation. It is therefore assumed in the theory that the actual initial 

 conditions can be adequately approximated by assuming simplified con- 

 ditions in which the explosion takes place adiabatically without change 

 in volume, the explosion products being at uniform pressure throughout 

 the volume. 



Before describing the actual calculation of these simplified condi- 

 tions it is of interest to make a rough estimate of the range in which 

 neglect of the head of the detonation wave involves appreciable error. 

 According to the Kirkw^ood-Bethe propagation theory the condition at 

 the shock front is determined by the pressure on the gas sphere at a 

 time T which increases with the distance R the shock wave has travelled. 

 The value of r is related to R by the equation x = e~^/^i in which, for 

 any specific explosive, the dissipation factor x is a computed function 

 of R/ao (tto is the original charge radius) and di/ao is characteristic of the 

 explosive. Assuming this relation, the value of R/ao corresponding to 

 advance of the head of the detonation wave into the boundary may be 

 computed. The time required is 2/3 ao/D from Taylor's calculation, and 

 the value of x for this time is then x = exp(-2/3 • ao/Si • l/D) . For TNT 

 di/ao = 3.4-10-3 m./sec. D = 6,400 m./sec. and so x = 0.74, which the 

 theory predicts will be realized at R/qo = 2.0. Although the calculation 

 is a very crude one indeed, w^e might infer that the theory will be in- 

 creasingly in error for distances of the order of two charge radii or less. 



Calculations of the initial pressure and density following adiabatic 

 reaction at constant volume have been made by Brinkley, Kirkwood, 

 and coworkers (59, V, VII), using the modified Wilson-Kistiakowsky 

 equation of state. The reaction must take place in such a way that the 

 chemical energy released by the reaction is equal to the increase in in- 



