THEORY OF THE SHOCK WAVE 117 



lation is somewhat involved and some of the details will be omitted in^ 

 this presentation of the argument. As was shown in section 2.6, the 

 various quantities associated with the propagation are conveniently 

 wTitten in terms of the Riemann function cr, defined as 



(7 = I —dp, where c = \ — - 



Po 



In the approach to be outlined, we are interested in regions at or 

 near the shock front. In this case, a is not greatly different from the 

 particle velocity u, and we may use the relations obtained in section 2.6 

 from the equation of state and the Rankine-Hugoniot conditions, which 

 are 



c = c + o- = Co[l + 2/30-] 



(4.7) u = Co[l + (3ct], where ^ = -^^-^ 



4co 



n = CoO- [1 + /3(7] 



It will be noted that the propagation velocity c is approximated by 

 c -]- a rather than c -\- u and it is therefore not assumed that a = u. 

 In terms of a, the basic propagation equation for G(r, t) = rQ(r, t) be- 

 comes 



(4.8) G{r, t) = CoMl + ^u) = a(T)12(a, r) = G{a, r) 



A. Evaluation of 12 [a, rj. From Eq. (4.5), the retarded time To at 

 the shock front is expressed as the sum of two integrals, the first inte- 

 gration being performed at the shock front, propagated from the initial 

 radius a{o) of the gas sphere to the point R, and the second being car- 

 ried out at constant Tq. These integrals can be expressed in terms of o- 

 and G[a, r] by differentiating Eq. (4.8) w^hich gives 



dr = ^(^. r) 1 + 2^a ^^ _^ 1 dG{a, r) 



a' [1 + (Serf c. cr (1 + I3a) 



Substituting this value of dr and noting that dG[a, r] = for the 

 second integration, we find after some rearrangement 



(4.9) To = ^,[f'^ ^^'^''7. , ^:::""'' da + 



2G{a, To) - G{a, T{a)) ^,_ , G(a, Tq) - G{ao) 



{ao) 



<j{ao) 



G{a, To) 



+ 



o-(a) 



r^^" da 1 



