1^26 APPENDIX 



The basic approximation of the Kirkwood-Bethe theory can be considered to be 

 the assumption that the function r(co + a'^/2) is propagated with velocity c -\- <t 

 behind the shock front. This is readily shown to be equivalent to the assumption 

 that the difference Q = \{(t — u) \^ given by 



(A.4) ^('■'') = "2^-?/ 



1 C' 9 '■(.= - u^) ^, 



where the integration over t is performed at constant r and to is the time of arrival of 

 the shock wave at r. 



The propagation of r(co + 0-^/2) with velocity c + o- if Q is given by Eq. (A.4) 

 can be shown as follows. Substitution from Eq. (A. 2) in Q = ^(o- — w) gives 



O - - - 1-4- i- — 



and equating this relation to Eq. (A.4) gives 



•' d 



^+l? + ^/.^^'^^^-"^^'^ 



Differentiating with respect to i, we have 



(A.5) ,_ + _--_ + I _,(,._„>) =0 



In order to express this equation in terms of r(co + 0-^/2) we note that 

 du/ dt = cda/dt, and hence 







dr dt ' dr 2 



the last step following from Eq. (A. 2). Substituting in Eq. (A.5) gives 



[i^Thi^^^-^O-^ 



which is the equation for propagation of r{co + (t^/2) with velocity c + a. 



The question of importance with regard to this result is, of course, what error is 

 incurred by the approximation that the difference ^(a — u) is given by Eq. (A.4). 

 In the acoustic approximation, for which we have shown that (o- — u)/2 is given by 

 — ^/2r2, the error in Q is measured by the integral in Eq. (A.4) , which is readily shown to 

 be at least of order l/r^. The error in Q is therefore small for increasingly large r. 

 Kirkwood has estimated the value of Q from Eq. (A.4) on the basis of the Kirkwood- 

 Bethe theory and obtains values of the same order as those computed by Penney by 

 direct integration of the Riemann equations (2.17). This is hardly conclusive evi- 

 dence in favor of the approximation, as both calculations are approximate, but does 

 furnish some support for it. More exact theoretical estimates of the error in the 

 enthalpy have not been made and would, as mentioned at the outset, imply an exact 

 solution of the propagation equations. 



