THEORY OF THE SHOCK WAVE 113 



where g is an undetermined function. The impulse measured for pro- 

 portional distance and time scales is therefore proportional to the linear 

 dimension ao. 



The practical importance of the principle of similarity lies in the 

 economy of effort it permits in determining the properties of shock 

 waves and in the predictions it makes possible of the effect of a changed 

 scale. For example, the numerical calculations of Penney (83, 85) for a 

 spherical charge of TNT were explicitly made for a weight of 1,800 

 pounds of explosive. By a suitable change of scale, however, they also 

 describe a charge of any other weight. These calculations were ex- 

 tended to a distance of 6 charge radii or 9.8 feet, and they therefore ap- 

 ply equally out to 6 charge radii for any other size charge, which for 

 100 pound weight is 3.8 feet. The more general theory of Kirkwood 

 and Bethe, being based on the equations from which we deduced the 

 validity of the principle, must and does automatically satisfy similarity 

 and therefore predicts shock wave pressures for any size of spherical 

 charge. Experimentally, the principle is economical because the form 

 of the pressure and other functions can be determined by measurements 

 over a range of either distance or charge size only, the effect of the other 

 variable being determined by similarity. Once the form of the function 

 is so determined, its value for other weights or distances is known, and 

 not an independent result. 



More detailed illustrations of the principle of similarity Avill be given 

 later, but its importance and utility can be appreciated from the fore- 

 going discussion. It is therefore important to examine its limitations 

 and the conditions under which it is applicable. The principle will 

 evidently fail in any circumstances for which forces not scaling geo- 

 metrically are involved. One such force neglected in the equation of 

 motion is the effect of viscosity which gives rise to terms of the form 

 ixd-u/dr'^, where /x is the coefficient of viscosity. The derivative is, how- 

 ever, of the second order, and with its inclusion the equation of motion 

 is not satisfied on substituting P(Xr, \t), etc. At a shock front, proc- 

 esses at least similar in effect to macroscopic viscosity and thermal con- 

 duction must be significant and hence it is not to be expected that the 

 principle of similarity applies to the rapid changes in the front itself. 

 These changes do not appear to be of any practical significance and 

 hence the applicability of the principle to this case is academic. 



Another way in which similarity may be expected to fail is in the 

 case that chemical reactions behind the detonation front are important. 

 The rate of reaction does not of course change in proportion to the 

 amount the front has advanced, and a significant effect of delayed reac- 

 tion will make similarity invalid. No attempt is made in theoretical 

 calculations to account for reaction rates, the chemical composition 

 being assumed either in equilibrium or else "frozen" into a fixed relation. 



