S4 



HYDRODYNAMICAL RELATIONS 



Rankine and by Hugoniot,^ and are easily obtained by considering 

 regions immediately ahead of and behind the discontinuity. If an ob- 

 server moves with the velocity U of such a front (see Fig. 2.2) into a 

 region of particle velocity Uo and density po, the apparent velocity of the 

 fluid toward him is U — Uo and in a time dt, a mass of fluid po{U — Uo)dt 

 will enter the unit area of the front. The apparent velocity of fluid 

 leaving the front is — (U — u), where u is the particle velocity relative 



(U-u) 



P,p 



(U-Uo) 



SHOCK 

 FRONT 



Fig. 2.2 Conditions at a shoe front moving with velocity U. 



to fixed coordinates, and a mass of fluid p{U — u)dt will leave the front 

 in time dt, p being the density. If the front is discontinuous, we can 

 shrink the time dt to infinitesimal values for which the mass flow into 

 the front must approach that away from it. In the limit the two be- 

 come equal so that 



po{U — Uo) = p{U — u) 



an expression of conservation of mass. 



The mass flow into the front in time dt has momentum 

 po{U — Uo)uddt, and the mass flow out has momentum po{U — Uo)udt. 

 The difference, which is the change in momentum, must equal the im- 

 pulse of the net force per unit area in the limit c// — > 0. If the pressures 

 ahead of and behind the front are Po and P, we obtain 



^ The argument which follows was first developed by Rankine (89) in determining 

 the conditions necessary at a discontinuous front. Rankine considered the equations 

 of mass and momentum. Hugoniot in later work gave the equation for increase in 

 internal energy, assuming the existence of a discontinuity. See Lamb (65), p. 484ff. 



