THEORY OF THE SHOCK WAVE 111 



fi r2 ix' n 



|-,v(r',0 = -^-^grad'P(r',n 

 at p {T , t ) 



where grad' indicates differentiation in the primed coordinates. We 

 note that all the indicated derivatives are of the same order and we can 

 therefore write 



■^ v(Xr, \t) = grad P(Xr, \t) 



at p 



the scale factor X cancelling out. Hence the same differential equation 

 is satisfied by v(Xr, X^ etc. as by v(r, t). A similar result is obtained for 

 the second of Eqs. (4.1) and we conclude that the equations are satisfied 

 by values of P, p, v measured in scaled coordinates. At the shock front, 

 Eqs. (4.1) are not valid, but the Rankine-Hugoniot equations (Eqs. 

 (2.28) of Chapter 2) are easily seen to be satisfied by the same change of 

 scale. We can therefore conclude that if the principle of similarity is 

 true at any value (rj, ^i), e.g., P(ri, ^i) = P(Xri, X^i), it is true for all 

 values of r and t. The validity of the principle therefore depends on 

 whether similarity is found to hold true in the initial stages of the ex- 

 plosion. 



As was shown in Chapter 3, the process of detonation, once estab- 

 lished, leads to a shock front advancing outward with constant speed 

 and intensity. If the hydrodynamical equations (4.1) describe the 

 situation behind the front then, by the argument just presented for the 

 shock wave, the profile of the detonation wave is spread out in propor- 

 tion to the amount the wave has advanced, but has the same form ex- 

 cept for this change in scale. If the time required to establish the 

 steady condition is negligible, the profile of the wave is the same for all 

 geometrically similar charges, provided the scales of length and time 

 used to specify it are proportional to the linear dimensions of the charge 

 and the origins of time and distance are at the point of initiation. This 

 is just the necessary condition for similarity to be established in the 

 water shock wave and the remaining question is as to whether the 

 boundary conditions at the interface of the explosion products and 

 the water are compatible with similarity. In the absence of viscosity 

 effects (shear), these conditions require continuity of pressure and 

 normal components of particle velocity and hence of their total time 

 derivatives, as described in section 3.8. We can easily convince our- 

 selves that these conditions are satisfied if pressures and particle veloci- 

 ties are scaled geometrically, and the approximate relations so far de- 

 veloped to account for the shock wave are thus all consistent with the 

 principle. 



Deferring for the moment an examination of circumstances in which 



