118 , THEORY OF THE SHOCK WAVE 



In order to obtain G{a, To) (which equals a{To) fi(a, To)) from this 

 equation, it is convenient to define a new parameter x(to) by the relation 



G{ao) 



where G(ao) is the initial value at time ^ = on the gas sphere. 



We note from Eq. (4.8) that the quantity ^a can be expressed in 

 terms of (xao/R) by the relation 



(4.10) /3C7 = I 



V'-^«^(?-)-'] 



and hence solution for x in terms of (xao/R) determines a; as a function 

 of ao/R. Substitution of x in Eq. (4.9) and a fairly tedious reduction 

 leads to the equation^ 



(4.11) Kox' ^x - Ki = 



The functions Ko and Ki are expressed for compactness in terms of an 

 auxiUary variable Z defined by 



13(7 



and this function with a subscript a refers to values on the gas sphere, 

 so that 



(4.12) z.(M = i±M^), z„(o) = L±M^ 



It is obvious from Eq. (4.10) that Z and Za are functions of xao/R and 

 xao/a{ro). With this notation, the functions Ko and Ki can be shown 

 to be 



(4 13) K =-' ^^""'^ ~ ^^^^ ^"^ i ^"^^^ ~ ^ + ^^ f"^"""^ da 



■^ c. L ^«(0) U ^aW/ Az' Z.HO))]j 



^ ^ 1 , «o G(ao) - G(a, To) i ZaiO) - 1 _Co_ f'^^^W) , ) 

 Co J>,,,),, t 2Z.(0) 2.(a.) J ^^^^ a.^ ^"/ 



2 For the derivation of this relation and the defining equations (4.13), the original 

 report of Kirkwood and Bet he (59, I) should be consulted. 



