THE DETONATION PROCESS 103 



constant volume. The basic problem is therefore evaluation of the 

 pressure and particle velocity on the boundary as a function of time, in 

 order to determine the kinetic enthalpy G(a, t) defined by 



G{a, t) = a^ait) = a[co(a, t) + iu^{a, t)] 



where o){a, t) is the enthalpy and a the radius of the gas sphere. 



A. The initial conditions. The initial pressure P(0) in the water 

 must equal the initial pressure Pg{0) in the gas sphere. (In what fol- 

 lows, quantities with subscripts g refer to the gas, without, to the water, 

 and the suffix (0) indicates initial values.) This pressure is of course 

 not the same as the adiabatic explosion pressure Pe calculated from 

 thermochemical data, the equalization of pressure being achieved by 

 generation of an outgoing compression wave in the water and an in- 

 going rarefaction wave in the gas products. Both of these waves leave 

 the medium behind them moving with an outward particle velocity, 

 and the second necessary boundary condition is that the particle veloci- 

 ties in the water and products be the same at the boundary. The rare- 

 faction wave may be described as one in which the Riemann function 

 Ng = {(Tg -\- Ug)/2 is initially zero.^ The Riemann variable Gg is, how- 

 ever, a calculable function of density, and hence pressure, of the ex- 

 plosion products. 



A second necessary condition arises from the fact that the pressure 

 and particle velocity in the water are not independent but are related 

 by the Rankine-Hugoniot conditions and the derived tables (see section 

 2.6). Taken together, this condition and the continuity of velocity 

 determine the initial pressure and particle velocity at the boundary, as 

 indicated symbolically by the relations 



(3.23) n = —(jg{Pg), from the Riemann condition 



u = u(P), from the Rankine-Hugoniot condition 



The relation between dg and Pg = P must be determined by cal- 

 culations of the adiabatic expansion for the explosion products, as out- 

 lined in sections 3.5 and 3.7. With these results, the values of P{0) 

 and u{0) consistent with Eq. (3.23) can be determined numerically or 

 graphically. 



B. Pressure and particle velocity at the gas sphere. The equality of 

 pressure and particle velocity at the gas-water boundary must be true 

 at all times after it is first established. The total time derivatives of 



^ It is important to note that Ng = need not be, and in fact is not, true at later 

 stages in a spherical rarefaction wave, nor does the propagation theory used by 

 Kii kwood et al assume this. 



