4. Theory of the Shock Wave 



4.1. The Principle of Similarity 



Before discussing detailed solutions of the hydrodynamical equations 

 for shock wave propagation, it is worth-while to consider what can be 

 inferred more generally about the solution from the form of the equa- 

 tions and the boundary conditions. As shown in Chapter 2, the basic 

 equations describing the motion of a fluid are 



(4.1) -^ = --gradp, 



where c^ = 1^1 



■(: 



dp ,. \dp /s 



— = -pdiv V, 

 dt 



if viscosity and heat conduction can be neglected. The pressure P is 

 a function only of the density p, the relation between the two being 

 determined by the equation of state and heat capacity of the fluid. 

 These equations have solutions of the form P(r, t), u{x, t), where r is the 

 vector distance from an arbitrary origin. 



Suppose that measurements of pressure have been made at a dis- 

 tance r from a charge of specified dimensions at a time t after it is ini- 

 tiated and that a new experiment is arranged in which all the linear 

 dimensions of the charge are changed by a factor X. The principle of 

 similarity^ asserts that the pressure and other properties of a shock wave 

 will be unchanged if the scales of length and time by which it is meas- 

 ured are changed by the same factor X as the dimensions of the charge. 

 For example, the pressure and duration of the shock wave measured 

 ten feet from a cubical charge one foot on an edge will be the same as 

 the pressure and duration measured twenty feet from a charge two feet 

 on an edge in units of time twice as large. The duration in absolute 

 units is therefore doubled at the doubled distance for the charge of twice 

 the linear dimensions (eight times larger weight). The principle does 

 not state, nor is it true, that the pressiu'e at any other distance and time 

 than the scaled ones obeys any such scaling law regardless of other fac- 

 tors. 



In order to examine the validity of this proposition, we observe first 

 that the differential equations (4.1) are satisfied if the scales of measure- 

 ment of both length and time are changed by a factor X. For example, 

 the first of Eqs. (4.1) becomes, on writing r' = Xr, t' = \t, 



^ The earliest general statement of the principle of similarity for shock waves 

 known o the writer is given in the classic paper of Hilliar (47) . 



110 



