116 THEORY OF THE SHOCK WAVE 



small quantities behind the shock front. Numerical estimates by Kirk- 

 wood of the error in rl2 lead to the conclusion that the approximate 

 value will give too large values of pressure, the error not exceeding 

 twenty per cent and in most cases probably much less. 



A further approximation in the evaluation of 12(a, r) lies in the as- 

 sumption that its value for points not far behind the shock front may be 

 obtained sufficiently accurately from a Taylor's series expansion about 

 the value To corresponding to the shock front. If the shock front has 

 reached a point R at time to, we have 



--©. 



^, ) (t -Q = To+~ (t - to) 



7 



dr 



c -\- a 



where I/7 = (dT/dt)R, and from Eq. (4.3) 



JR J ^R 



a.^ ('•' ^) J airy 

 Differentiating Eq. (4.3), we have for 7 



(4.6) , = fi-^^ = 1 _ -^^« - f"" _L_ri (c + Ajr 

 \dT/R c + adr J ^^^^ [c + ay Idr 



The parameter 7, which is a measure of the time scale behind the shock 

 front relative to that on the gas sphere, increases rapidl}^ as the shock 

 front travels outward. This is because [d/dr (c + a)]r is negative at 

 all times following passage of the shock wave if 12(a, r) decreases with 

 time during emission of the shock wave from the gas sphere boundary. 

 This increase of 7 with distance from the source means that the profile 

 of the shock wave broadens as it is propagated outward. The increase 

 ing breadth made plausible by the Riemann formulation of the propa- 

 gation equations thus appears naturally in this approach also. The 

 limiting pressure-time relation, Eq. (4.4), may be written 



P{R, t - t 

 where 



•'=©'■ 



r. + - (/ - /.: 



P{t) - Po^^n{a,t) 

 a{o) 



With these approximations for c and the retarded time r, it becomes 

 practical to compute fl(a, r) and 7 from the equation of state for Avater 

 and numerical values of the enthalpy on the gas sphere. The actual 

 reduction of the expressions for To and 7 to a convenient form for calcu- 



