134 THEORY OF THE SHOCK WAVE 



functions only of the pressure P. If a fourth relation among the partial 

 derivatives and the shock front pressures can be obtained from other 

 information about the shock wave, the set of equations can be formu- 

 lated as a pair of ordinary or total differential equations adapted to 

 numerical solution. 



Before considering how this fourth relation is obtained, the trans- 

 formation of the first three equations into more convenient form will be 

 outhned. Instead of using r, the position in space of a point at which 

 the derivatives are evaluated, Kirkwood and Brinkley choose as a vari- 

 able R, the position in the undisturbed fluid (density po) of a volume 

 element which has at time t the position r. To effect the transformation 

 of Eqs. (4.26), it will be noted that by definition {dr/dt)R = u, and a 

 spherical shell of fluid with thickness dR and mass ^TpoR'^dR has at time 

 t a thickness dr and mass ^TrpvHr, hence {dr/dR)t = poR'^/pr'^. Using 

 these relations, Eqs. (4.26) become 



pr^ /du\ , 2i/ ^ _ J^/^\ 

 PoR^XdR/t r pc'-ydt/R 



valid at any point behind the shock front. At the front itself r = R 

 and the equations are further simplified. The conservation of mass at 

 the shock front requires that Pm = poUum where U is the shock front 

 velocity and Pm, Um are the values of P, u at the shock front. This can 

 be written as a relation between partial derivatives by noting that the 

 derivative of Pm or Um for a displacement dR of the shock front from a 

 point Ro is given by 



KdRjR-Ro \dR)t^\dt)R\ 



\dR)t U\dtjR 



dRJi 



where U is the shock front velocity, and is determined as a function of 

 peak pressure Pm for any fluid by the other two Rankine-Hugoniot con- 

 ditions and the equation of state. Applying this operator to the equa- 

 tion Pm = poUum gives 



(A OQ,) ^ _1_ TJ ^ ^ ^^ _ l£^ = 



^ ^ ^ dt dR Poll dl podR 



where a = I — poU — ? ^rid is determined by the value of Pm- 

 dP 



