120 THEORY OF THE SHOCK WAVE 



(4.7) and (4.8), the particle velocity u and velocity (c + a) can be ex- 

 pressed in terms of a and G{a, t), and substitution in Eq. (4.6) gives 



aa 2/3 dG(a 



If) C — 



J -a 



Coil + 2/3(7a) C.2 dro J (7(1 + iScr) (1 + 2)3(7)2 



Making the substitution Z = (1 + ^a)/^(T and carrying out the inte- 

 gration yields 



(4.16) 



^ ^ _ 1 _ 2^ 6/G(a, To) L Z 4(Z - Z.) 





l3Co{Za + 1) C.2 (/To { ^Za (Z + 1) (Z. + 1) 



With the peak approximation G{a, t) = G{ao)e~^/^' this becomes 



1 , 2ao/312(ao) (, Z 4(Z - Za) \ 



T = 1 - ^ ,^ , ,x + -TT ^log 



0^1 Co \ 



^Co{Za + 1) C.^1 Co { "^Za (Z + 1) (Z^ + l)j 



As before, Z is a function of xao/R given by Eq. (4.15). The quan- 

 tity Za is strictly a function of xao/a{T^, but it usually suffices to neglect 

 the change in radius a(T^ for increasing Tq and evaluate Za by the re- 

 lation 



i_ , /_!_ , r_i_T 



a„)x \Q(a„)x L2Q(a<.)2^J 



obtained by letting R = ao'm. Eq. (4.15). With this approximation, 7 

 is determined as a function of xao/R and x for given values of l^(ao) and 



C. Calculation of the pressure-time curve. From the results of the 

 preceding paragraphs, it is readily possible to compute the shock wave 

 pressure-time curve. Given the values 12(ao) and di, an assumed value 

 of xao/R may be used to calculate the corresponding value of x. This 

 value of X determines the point ao/R which has the value and, knowing 

 x and (xao/R), the corresponding value of 7 may be computed. From 

 the basic equation of the propagation theory, the kinetic enthalpy at a 

 distance R as o, function of time t — to after arrival of the shock wave 

 at time to is given by 



a(R, /-/„) = ^(''' '- ^-^ 



R 



