112 THEORY OF THE SHOCK WAVE 



the principle of similarity fails, we consider what can be inferred with 

 its aid about the form of the shock wave. The fact that the pressure 

 and other properties are unchanged if the linear dimensions of the source 

 and scales of length and time are all changed in the same ratio does not 

 of course specify what the values are without other information. It is 

 possible, however, to learn something about their functional dependence 

 on charge size and distance. If the linear dimensions of the charge are 

 specified in terms of a length, a^, the principle can be satisfied only if the 

 pressure depends on distance and time only as a function of the ratios 

 r/tto, t/ao. The truth of this statement is evident from the fact that 

 fixed values of these ratios correspond to the scaling which gives identi- 

 cal values of the pressure. The pressure Pm at the head of the shock 

 wave (peak pressure) may therefore be expressed 



Pm = 



-'if) 



the form of the function / being undetermined. If the quantity 6 is 

 used to represent any measure of time duration of the wave, e.g., the 

 time constant of an exponential decay, it is evident that S/ao can be a 

 function only of the ratio ao/r. Another important property of such a 

 wave is the impulse associated with it, which measures the momentum 

 imparted to the water by its passage. For unit area of the wave front 

 the impulse / is given by 



/(r,0 = fP(r,t)dt 



where the origin of titne is taken to be arrival of the shock front at r. 

 The time t' to which the integration is carried should, for consistency, 

 be taken proportional to the scale factor and we write f = Kao, where 

 X is a function only of ao/r, and the pressure P depends on r and t only 

 by the ratios ao/r, t/ao. 



We may therefore write 



J ^ V «-/ \ao/ 



and, the integral being a function only of ao/r, we obtain 



