138 THEORY OF THE SHOCK WAVE 



dim (j±rn , -^ Oi^m r 771 7-) 75-1 



The function F is expressed in terms of P„ by Eq. (4.33), which is 



and the coefficients g, shock front velocity U, and density p appearing 

 in (p are known functions of Pm. This pair of ordinary differential 

 equations for Pm and F can therefore be solved by numerical integration 

 subject to a choice of boundary conditions for determination of the con- 

 stants of integration. The time variation of pressure behind the shock 

 front as measured experimentally at constant r is initially given by the 

 derivative 



i-KI-")],, 



corresponding to the peak approximation to the pressure-time curve 

 P{t) = PmC-^/^ (see section 3.8). In terms of the coordinate R used in 

 the present treatment, this becomes 



which can be evaluated with the aid of Eqs. (4.36). 



The two constants of integration can be determined in either of two 

 ways: by the initial conditions at the boundary of the gas spheres fol- 

 lowing detonation, or by the experimental pressure-time curve at a 

 selected distance.^ If the assumption of adiabatic conversion of the 

 explosive to its products is assumed, the first method is the same in 

 principle as that outlined in section 3.7 for application of the Kirkwood- 

 Bethe theory. The two constants may conveniently be taken to be 

 the initial pressure at the boundary, and either the time parameter ju or 

 the total shock wave energy. The explicit formulation of expressions 

 for these quantities has been made by Kirkwood and Brinkley but is 

 omitted here. If the semiempirical approach of using an experimental 

 pressure-time curve to determine the constants is adopted, the natural 

 quantities to choose are the peak pressure Pm and energy flux integral. 

 The energy integral, however, is not accurately calcula])le from experi- 

 mental pressure-time curves because of contributions from noncom- 

 pressive flow energy, and is subject to greater experimental error than 



® Tables and graphs for niimorical aj)i)licati()ii of either type of boundary condi- 

 tion are given in a report by Brinkley and Kirkwood (11). 



