U HYDRODYNAMICAL RELATIONS 



-f [(;.)■-'] 



\dpjs ' \Po) 



-=/:f=.-^[(r-'] 



Po 



If (7 is used as an independent variable, we find by substitution that 



CO = Cocr 1 H <j 



L 4c« J 



From Table 2.1 for properties of water at the shock front it is evident 

 that (7 is approximately equal to u. Substituting u = cr/ we obtain for 

 the kinetic enthalpy 12 (from its definition) and the shock front velocity 

 (from the Rankine-Hugoniot conditions) the relations 



The propagation velocity c + o- as a function of o- also follows im- 

 mediately, and with the notation jS = (n + l)/4co we obtain the rela- 

 tions employed by Kirkwood and Bethe in their propagation theory 



(2.32) c + a = Co [1 + 2/3(7] 



t/ = Co [1 + /3(7], where ^ = '-^^4^, c^^ = ('^\ 



4c. \dp ho 



12 = CoC7[l + /5a] 



The values of c + o-, t/, and 12 obtained by these relations with 

 n = 7.0 and values of a from Table 2.1 are compared for several pres- 



^ It might appear that replacing w by o- is an additional approximation, but the 

 errors in the propagation theory of Kirkwood and Bethe are actually reduced, as 

 discussed in the Appendix. 



