HYDRODYNAMICAL RELATIONS 43 



front. Penney also used an adiabatic form of the Tait equation which 

 be fitted to Bridgman's data for pure water with the result which may 

 be written 



_j/a7\ 1 



V\dP)s 7.47(P + 2.94) 



where P is in kilobars. The slightly different constants, as compared 

 to Kirkwood's values, lead to results which differ by one or two per 

 cent only for pressures up to eighty kilobars. 



It is interesting to note that any adiabatic equation of the form 



(2.30) leads to a simple numerical relation between the Riemann func- 

 tion 0- and the sound velocity c behind the front, the relation being 



(2.31) a = ^- (c - Co) 



n — I 



where Co is the sound velocity in the undisturbed medium ahead of the 

 shock front. A relation of this kind considerably simplifies numerical 

 integration of the Riemann equations or analytic formulations, and 

 both Penney and Kirkwood make use of such relations. 



C. Approxi7nate relations for adiabatic changes. In application of 

 the Kirkwood-Bethe theory for propagation of spherical shock waves 

 it is necessary to utilize relations for the various variables behind the 

 shock front. The adiabatic modification of the Tait equation provides 

 a relation convenient for this purpose, and it is also found useful to 

 employ the Riemann function a as an independent variable. Although 

 the final state behind a shock front should properly be calculated from 

 the Rankine-Hugoniot relations, the results so obtained differ little 

 from those obtained assuming an adiabatic change. This is plausible 

 from the small values of entropy change at the shock front (see Table 

 2.1), and will be shown explicitly from the adiabatic results which we 

 proceed to derive. 



If the pressure Po in the undisturbed fluid is taken to be zero (a 

 pressure of at most a few atmospheres being negligible for the present 

 interest), the adiabatic Tait equation gives a relation for the parameter 

 B in terms of the velocity of sound at zero pressure Co, 



B = -^-^j where 



n \ap /So 





Assuming that n and B are constants, the adiabatic equation gives 

 the following relations 



