THEORY OF THE SHOCK WAVE 131 



(4.23) N = \{a + u) 



Jp 

 - dp 

 p/ 



The usefulness of these functions hes in the fact that the increments dN 

 and dQ in a short interval dt and displacements dr can be found from the 

 values of N and Q throughout the gas products and surrounding water 

 at a time t. For the displacements dr indicated these increments are, 

 from Eqs. (2.18), 



(4.24) dN = -- dt, if dr = {C + u) dt 



r 



dQ = dt, if dr = —{c — u) dt 



r 



Calculation of dN and dQ for a series of values of u and r at time t gives 

 values at a time dt later. These values are of course only approximate, 

 their accuracy increasing with the smallness of the interval chosen. 

 Successive applications of the method then permit, with sufficient labor, 

 building up the solution to any desired time. At each stage, the neces- 

 sary values of u for the next stage are recovered from the relation 

 u = N — Q, and the pressure is determined from a = N -\- Q, which 

 must be a known function of density and hence pressure in the water 

 and gaseous products. 



The starting point of the step-by-step calculations is determined by 

 the conditions in the gas products after detonation is just complete. 

 The position of the shock wave front is determined by the Rankine- 

 Hogoniot conditions, as R{t -\- dt) = R{t) + Udt and the shock front 

 velocity [/ is a known function of pressure (see section 2.5). Although 

 Eqs. (4.24) determine N and Q through most of the fluid, the displace- 

 ments dr Sit which they are known after time dt leaves small shells in 

 which they are not. The function Q, for instance, is an inward moving 

 wave and its values at and near the shock front are progressively lost as 

 t increases. Values at the shock front can, however, be recovered from 

 the value of N and ii, both of which are determined because the Rankine- 

 Hugoniot conditions must l)e satisfied. The gap left in values of P and 

 u on either side of the initial gas sphere boundary can be filled in by 

 interpolation, and the solution built up. 



In Penney's original calculation, this step-by-step process was 

 started from an assumed initial pressure and density of the products 

 corresponding to adiabatic conversion of the explosive at constant vol- 

 ume to its final state, the adiabatics used being those computed by 

 H. Jones (see section 3.5), for which the initial pressure was 90 kilobars. 



