56 



HYDRODYNAMICAL RELATIONS 



conditions of the fluid. From section 2.5, a suitable representation of 

 the pressure-density relation for adiabatic compression is 



P + B = kp-" 



where B is approximately 2.94 kilobars (= 42,670 Ib./in.^) and 7 = 7.25 

 from Penney's calculations. The ratio of reflected to incident pressures 

 is then 



(2.38) 



P' - Po 

 P - Po 



(0 



1 



fcT ' 



The pair of equations, (2.37) and (2.38), permit simultaneous solution 

 for the final pressure and density in terms of the values behind the 

 incident wave. It can readily be seen that the ratio {P' — Po)/{P — Po) 

 is always greater than two, and approaches this value for weak shock 

 waves as it should. Penney and Dasgupta (85) have computed the 



Table 2.4. Pressure increase by normal reflection at a rigid boundary 

 (from calculations of Penney and Dasgupta) . 



ratio P' jPo for various values of Po and some of their results are given 

 in Table 2.4. It is evident that even for relatively enormous pressures 

 the departures from acoustic "pressure doubling" are not very large. 

 It should also be pointed out that the particle or flow velocities behind 

 shock fronts in water are always subsonic, i.e., less than sound velocity, 

 as the values given in section 2.5 show. These results are of course for 

 water; in more compressible media, such as air, supersonic flow and 

 non-acoustic differences occur for practical cases of importance. 



The case considered here is clearly very special and, if the wave has 

 a finite duration, applies only to the pressures at the boundary im- 

 mediately after impact. The pressure at later times and the impulse, 

 or time integral of pressure, for waves of finite duration have been com- 

 puted by Finkelstein^° for linear and exponential decay. He finds that 

 the pressure at a rigid boundary decays more rapidly than in the inci- 

 dent wave and that the impulse is less than predicted by acoustic 



^° R. Finkelstein, U. S. Navy Bureau of Ordnance Explosives Research Report 



