THEORY OF THE SHOCK WAVE 115 



be simply related to its value 12[a(T), r] on the gas sphere boundary at 

 an earlier retarded time r given by 



(4.3) r = t- ( -*- 



where a^r) is the radius of the gas sphere, and cir, r) is the velocity with 

 which the function r^ is propagated. This relation was shown in sec- 

 tion 2.4 to be 



fi(r, = — n{a, t) 

 r 



As an illustration of how the pressure-time curve is related to these 

 quantities, consider the asymptotic case of pressures at a point R suffi- 

 ciently far from the charge that u'^ « P/p. In such cases P = po ^{R, t), 

 where po is the density of undisturbed water, and from Eq. (4.2) is given 

 by 



(4.4) PiR,t)=('^]P{r) 



■"=© 



where a^ is the initial radius of the gas sphere, and 



P{r) =p„^Q(a,T) 



If the particle velocity term is not negligible, the pressure must be 

 obtained from the kinetic enthalpy using the equation of state for water. 

 In either case, it is evidently necessary to evaluate the enthalpy 0(a, t) 

 at the gas sphere, determined in the manner described in section 3.8, at 

 the retarded time r. In order to do this it is necessary to make a num- 

 ber of approximations, the first of which is to represent the propagation 

 velocity c of 12 by the velocity c -\- a and represent 12 by (co + a^/2) . It 

 might seem more logical to use the particle velocity u rather than the 

 Riemann function a, but as shown in the Appendix the latter represents 

 a better approximation for the difference a — u in the water behind the 

 shock front. Furthermore, as shown in section 2.6, a is also a con- 

 venient variable for expression of the properties behind the front. The 

 question of importance with regard to this approximation is, of course, 

 what error results from its use in evaluating 12(a, r), and the fact that 

 the error proves on investigation to be small is the justification for the 

 approximation. The detailed analysis, described in the Appendix, is 

 based on consideration of the error in terms of {d/dr (rl2)),., which is not 

 zero for the approximate r, and can be written as a function of known or 



