3S HYDRODYNAMICAL RELATIONS 



AE = UP + Po) {V - Vo)o 



Therefore, equating the two values of AE results in an implicit relation 

 between the initial and final conditions of pressure and temperature in 

 terms of known properties of water. 



Kirkwood and co-workers^ have devised effective methods of suc- 

 cessive approximation to determine the temperature increase satisfying 

 the equation for an assumed pressure increase. Knowing the tempera- 

 ture of the final state, the density can be computed from the equation 

 of state. These data then determine the enthalpy co and Riemann 

 function o- used in Kirkwood's formulation of shock wave propagation, 

 which are defined by the relations 



CO = 



— J 0" = I "dp where c^ = ( -7- ) 

 P P J P \dp/^ 



L O Po 



Although, strictly speaking, these quantities should be evaluated isen- 

 tropically for a transition to the final state, the error incurred by using 

 the Rankine-Hugoniot P-V relation is small. This can be shown by 

 calculation of the change in entropy involved at the shock front, the 

 results being given in the next section. 



Other variables of interest, such as the particle velocity and shock 

 front velocity, can also be computed from the Rankine-Hugoniot rela- 

 tions, and thermodynamic functions such as the changes in entropy or 

 internal energy can be calculated in a straightforward manner by ap- 

 propriate integrations in terms of known functions. 



B. Solutions for high pressures. Kirkwood et al. (59, II and III) 

 have made calculations for pressures up to 50 kilobars in pure water and 

 pressures up to 90 kilobars in sea water (taken to be a 0.7 molal solution 

 of NaCl) at 0, 20°, 40° C. The differences between the two conditions 

 are not large and only the results for salt water will be described here. 

 Extensive work has been done on the P-V-T relation for water at pres- 

 sures up to 350,000 lb. /in. 2 by Bridgman and Gibson. Gibson has made 

 measurements to determine appropriate constants for a Tait form of 

 equation of state and the effect of salinity. The Tait equation may be 

 written 



V(T, P) = V{T, 0)h - ilog (1 + 1)1 



-m = I 



yapjr n[P + B{T)] 

 See Reference (59), Reports II and III. 



or 



