THEORY OF THE SHOCK WAVE 119 



It might appear that the manipulations and reductions involved in 

 expressing G{a, To) as a function of G{ao) and x have not accomplished 

 much toward a practical solution, as Ko and Ki both involve G{a, To), 

 and the integrals in Ko and Ki depend on To through the value of aa on 

 the gas sphere. Equation (4.11) is, as a result, strictly a transcendental 

 equation for x(to) which would have to be solved by approximation or 

 other means. It turns out, however, that the integrals involving aa 

 are unimportant compared to the other terms. Furthermore, it was 

 shown in section 3.8 that the variation of the function G{a, t) on the gas 

 sphere can be approximated by an equation of the form 



G{a, t) = G{ao)e-^/'^ 



which is accurate for t = 0, but is increasingly in error for larger values 

 of t. With this approximation, Eq. (4.11) may be solved explicitly for 

 a; as a function of xao/R, and hence ao/R, given values of 12 (a^) and ^i. We 

 shall have to expect, however, that the later portions of the shock wave 

 will be somewhat in error from the peak approximation. 



It is also to be remembered that the boundary conditions at the gas 

 sphere, w^hich led to the "peak" approximation, were also somewhat 

 simplified by neglecting reflections at the boundary of the internal rare- 

 faction wave. These reflections must also introduce perturbations of 

 the later portions of the shock wave, and so too detailed an analysis 

 neglecting this effect is not justified. The form of the correction for 

 error in the peak approximation is, however, described in section 3.8. 

 With the approximation and neglecting the integrals over a on the gas 

 sphere, Eq. (4.11) becomes 



(4.14) : ^^' 



1 + (1 + 4.KoKiyi' 



where Ko and Ki are defined by Eqs. (4.13). If the integrals over o- are 

 neglected, Ko and Ki are functions of Z and Za{0) only, which are ex- 

 pressed in terms of xao/R by the relations 



(4.15) Z 1 + 2^(^j l^^^^j + yj ^(^ao)\aox) ^ \_2U{ao)'\aox) j 



^"^^^ = ^ + 2^ + \lKa5 + (21^^) 



B. Evaluation of y. In order to evaluate the time spread parameter 

 7, much the same procedure is used as in determining x. From Eqs. 



