HYDRODYNAMICAL RELATIONS S5 



po(U — Uo) {U — Uo) = P — Po 



an expression of conservation of momentum. 



A similar argument for energy requires that the net work done by 

 the pressures P and Po equal the increase in kinetic plus potential energy 

 when the time increment dt becomes infinitesimal. The work done by 

 P is Pudt for unit area of the front, by Po is PoUodt. The kinetic energies 

 per unit mass are iu^ and ^Uo^, and if E and Eo denote the internal ener- 

 gies per unit mass, we have 



Pu - PoUo = Po(U - Uo) [E - Eo-\- hiu" - Uo^)] 



the requirement for energy conservation. 



In further applications of these equations we shall usually be inter- 

 ested in the case of an undisturbed fluid ahead of the front for which 

 Uo = 0. With this simplification and some rearrangement, we obtain 

 more convenient relations 



(2.28) p(U -u) = poU 



P - Po = PoUu 



E -Eo = i{P + Po) 



\Po P/ 



These relations, being obtained for the limit of negligible thickness of 

 the shock front, should be equally valid for plane or spherical shock 

 fronts. 



If the equation of state for the fluid is known, it is possible to deter- 

 mine the increase in internal energy E — Eo SiS Si function of the pres- 

 sures P and Po, and densities p and po. For given values of Po and po, 

 in undisturbed fluid ahead of the shock front, the third of Eqs. (2.28) 

 then defines a relation between P and p immediately behind the front. 

 This pressure-density relationship is usually known as the Rankine- 

 Hugoniot equation, and is a curve of somewhat the same form as adi- 

 abatic or isothermal P-V curves, although it is evidently not the same 

 function. 



The first and second of Eqs. (2.28) can be solved for the shock 

 velocity U and particle velocity u in terms of the pressures and densities 

 behind the front, giving 



(2.29) U 



^\Po/ P — Po 



u = P-^^^.U 



