HYDRODYNAMICAL RELATIONS 37 



in obtaining both solutions is the same, the appropriate numerical data 

 and approximations for anal^^tical purposes are different. 



A. Methods of solution. In order to express the Rankine-Hugoniot 

 conditions in a convenient form for evaluation from experimental data, 

 some manipulation based on purely thermodynamic considerations is 

 necessary. Changes in internal energy E of unit mass of water can be 

 expressed in terms of pressure and temperature variations as 



'«=(i),"+(S)/^ 



E is, however, a point function, its value being independent of the 

 process by which the fluid was changed from its initial condition to the 

 final one. The change of internal energy AE for water can therefore be 

 written as a sum of changes along any suitable paths, and we have 



T 

 AE = 



/:©,/- /:(a- 



the first process being carried out at the initial pressure Po, the second 

 being an isothermal compression at the final temperature T. From the 

 first and second laws, the partial derivatives of E can be expressed in 

 terms of experimentally obtained quantities as 



-),=(a-{a— Ki), 





[dPjr ^[dPjT ^[dP/r ^[dTjp ^ [dPjT 



where Cp is the specific heat at constant pressure, {dV/dT)p is the 

 thermal expansion coefficient, and —{dV/dP)T is the compressibility 

 (y is the specific volume, i.e., the volume of unit mass of water). 

 Substituting, we have for AE 



T P 



AE = f CpodT - PoiV -Vo) - T f (^)/^ 



-{PV -PoVo)+ r V{T)dP 



^ Po 



where one integral has been evaluated and a second integrated by parts. 

 The increase AE across a shock front is, from Eq. (2.28), given by 



