36 HYDRODYNAMICAL RELATIONS 



The first of these equations is frequently useful, as together with a 

 knowledge of the equation of state it permits calculation of shock front 

 velocity U in terms of the pressure P behind the shock front and the 

 initial conditions Po, po- Similarly, the second equation permits cal- 

 culations of the particle velocity u. Considered more generally, Eqs. 

 (2.28) provide three relations among the four variables, P, p, U , u, by 

 means of which any three can be expressed in terms of the fourth, given 

 the initial values Po, po. 



In the preliminary discussion, it was shown that shock waves of 

 rarefaction, in which the pressure is less than that in the fluid ahead of 

 the disturbance, cannot be expected to maintain themselves in any 

 normal liquid or gas. This conclusion also follows from the Rankine- 

 Hugoniot relations. It is evident that a rapid compression of any fluid 

 must, as a result of dissipative processes, leave it at a higher tempera- 

 ture than would be the case for adiabatic compression to the same final 

 volume. For any normal fluid, this irreversible process also means a 

 higher final pressure than the adiabatic value, and it is easily shown that 

 such a state is consistent with the energy requirements for a shock front 

 of compression. This excess of temperature over the adiabatic value is, 

 of course, the result of degradation of energy to heat which cannot be 

 returned to the fluid as available energy. A reversal of the process as 

 an expansion, which would be required to satisfy the conditions for a 

 shock front of rarefaction, cannot therefore take place and such fronts 

 are impossible in normal fluids. 



2.6. Properties of Water at a Shock Front 



We have seen from section 2.5 that the requirements for conservation 

 of mass, momentum, and energy at a shock front lead to three equations 

 for the pressure, density, and particle velocity behind the front in terms 

 of the corresponding quantities ahead of it and the velocity of the front. 

 In the case of shock waves travelling into undisturbed water, these con- 

 ditions therefore provide three relations among the four variables de- 

 scribing the front at any point. The necessary fourth relation to solve 

 for the properties of the water as a function of any one variable, is evi- 

 dently provided by a suitable equation of state and specific heat data. 

 One objective of such a solution is to provide relations suitable for ex- 

 plicit calculation of shock wave pressures from explosions on the basis 

 of the conditions at the boundary of the gas sphere and the propagation 

 theory of Kirkwood and Bethe. For this purpose, solutions of reason- 

 able accuracy over a broad range of pressures, from to 500,000 Ib./in.^ 

 or higher, are needed. A second desired type of solution is one of 

 greater accuracy over a more limited, lower pressure range, which will 

 be suitable for example, in comparison of experimental shock wave 

 velocity and pressure measurements. Although the general procedure 



