HYDRODYNAMICAL RELATIONS 





t-v ; .^\ 



the integration being along a path of constant r. If we set the variable 

 T equal to t' , we have 





(2.26) r = / , , ^ 



■) 



The quantity r is thus assigned the dimension of time and, as consider- 

 ation of the function Ga{r) shows, plays the role of a retarded time. 

 For the values r = t, r = a(t), Eq. (2.25) becomes 



Gait) = G[ait),t] 



and the function Ga{t) is thus simply the value of G on the gas sphere at 

 time t. The solution G{r, t) is then expressed in terms of Gair) evalu- 

 ated at the retarded time r. The kinetic enthalpy 12 (r, t) is, from the 

 definition of G, similarly expressed by the relation 



r r r 



where Q^air) = ^[a{r)j r] is evaluated on the gas sphere boundary at 

 time r. 



As Kirkwood and Bethe develop it, the solution of the propagation 

 problem is therefore reduced to a determination of the boundary con- 

 ditions at the gas sphere and evaluation of the retarded time r. The 

 explicit calculation of ^aij) is described in Chapter 3, but it is instruc- 

 tive to consider qualitatively the value G{R, Q at the front of the shock 

 wave, which has been propagated to the point R at time to. The time 

 to is given by 



.R 



to= { 



dr 



U{r, T(r)) 



where U is the velocity of the shock front and ao is the initial radius of 

 the gas sphere. The time To at which Gairo) determines the value 

 G{R, to) is, from Eq. (2.26), given by 



^^'^^^ '" )u[r,r{r)] J 



dr 



airf^'' ^J 



