THEORY OF THE SHOCK WAVE 139 



peak pressure. The peak pressure as a function of distance R can be 

 used equally well for determination of initial conditions from the total 

 differential equation for dPm/dR. This procedure has been used by 

 Kirkwood and Brinkley to extrapolate measured shock wave pressures 

 and compute shock wave energies for TNT and a mixed explosive. 

 These results are discussed in sections 4.7 and 4.8. 



4.7. Comparison of Shock Wave Theories 



Three different theories of shock wave propagation have been out- 

 lined in preceding sections. All of these theories are approximate de- 

 velopments from the basic equations of hydrodynamics, but the methods 

 of formulation and nature of the approximations involved are quite 

 different in the three cases. It is therefore worth-while to compare the 

 results by the different approaches w^ith experimental evidence and with 

 each other, in order to bring out their relative accuracy, flexibility, and 

 fundamental limitations. 



An unambiguous comparison of the various theories with each other 

 and experimentally measured shock wave pressures is unfortunately 

 not possible at the present time. This is so because results of all three 

 theories have been obtained for only one explosive, cast TNT, and even 

 in this case they are not directly comparable because different loading 

 densities and heats of explosion were assumed in the developments. 

 Penney and Dasgupta calculated the shock wave pressures for TNT 

 with initial conditions corresponding to an energy release of 800 cal./gm. ; 

 calculations from the Kirkwood-Bethe theory are based on calculations 

 leading to an energy of 1,060 cal./gm.; and the Kirkwood-Brinkley 

 theory has been used to extrapolate experimental peak pressure meas- 

 urements at low pressures to points closer to the charge. Other dif- 

 ferences are in the range of pressures for which calculations have been 

 made or involve less serious approximations, and are discussed in more 

 detail later in this section. 



The predicted dependences of peak pressure with distance R from a 

 spherical charge of radius ao are shown in Fig. 4.6, the product PmR/cio 

 being plotted against R/ao, using logarithmic scales to keep the graph 

 within bounds. In addition, experimental values from piezoelectric 

 gauge measurements over the pressure range 5,000-16,000 Ib./in.^ are 

 shown, these values being obtained from a number of measurements 

 with a number of sizes of charge at Woods Hole. The close agreement 

 of the curve from the Kirkwood-Brinkley theory for R/ao > 10 is the 

 result of the fact that the parameters of the theory were obtained from 

 the plotted points, and only the part of the curve for higher pressures 

 represents a theoretical prediction. It is seen that this curve lies very 

 nearly parallel to, but lower than, the result obtained from the Kirk- 

 wood-Bethe theory, which lies fifteen-twenty per cent higher than the 



