98 THE DETONATION PROCESS 



to establish the detonation is small. Hence all the quantities behind 

 the front can be functions only of the ratio r/t. The time t is given by 

 R/{ci + Ui) = R/D, where, by assumption, D is constant. An equiv- 

 alent statement is therefore that the properties behind the front are 

 functions only of the ratio r/R, which may be considered a statement of 

 the principle of similarity for a spherical detonation wave. This propo- 

 sition, that the properties of the phenomenon are unchanged if both 

 length and time scales are changed by the same factor, is of course not 

 restricted to the spherical case, being equally true for any other sym- 

 metry, provided the time for the steady state condition to be established 

 is negligible. 



In order to integrate the equations for the spherical case, Taylor 

 introduces the variables 



^ = ~, \J/ = —, z = log X, where x = - 

 X c t 



in terms of which Eqs. (3.20) become 



(3.21) ci^ ^^se-ii-^yr- 



dz r<i - 0' - ^ 



#^ ,,2^jii:!(i_:i_i)/ 

 dz yp\i - ^y - e 



where/ = {p/c^)dc^/dp and is a definite function of p from the adiabatic 

 relation behind the front. These equations Taylor solves for TNT by 

 step-by-step numerical integration back from the front, thus determin- 

 ing ^ and xp in terms oi z. A difficulty presents itself at the starting point 

 in that both d^/dz and dx/z/dz become infinite for r = R, where 

 X = ci -\- Ui. The values of ^, xp and dxp/d^ are all finite, however, and 

 the integration can be begun from a slightly smaller value z' obtained 

 by a series expansion in the neighborhood of R: 



-'-(l).,<— Kl) 



Taking z = log r/R, as is permissible since z appears in the equations 

 only as a differential and noting that (dz/d^^i = 0, this becomes 



2 WJ,, 



(^ - ^i) 



This expression can be evaluated from Eqs. (3.21). Noting that 



= 1 , i//i = — = 1 



Cl + Ui po Ci Pc 



