THEORY OF THE SHOCK WAVE 121 



With the peak approximation G{a, t) = G{ao)e~^/^', this becomes 



T — TO 



Q{R,t-to) =^%^e-^ 

 K 



Remembering the definitions 



t - to 



T — To = 



y 

 G{a, To) = xG{ao) = xao^M 

 we obtain 



(4.17) ^ir,t- to) = xf^y-'-^ 



At the surface of the charge (R/ao = I), the pressure and particle 

 velocity are known directly from the boundary conditions; this analysis 

 in the approximation of Kirkwood and Bethe is described in section 3.8. 

 At later times the pressure and other variables are determined from the 

 kinetic enthalpy 12 by the equation of state and shock front conditions. 

 The necessary relations, obtained numerically by Kirkwood and others, 

 are discussed in section 2.6 and reproduced in part in Tables 2.1 and 2.2. 

 These relations become increasingly uncertain near the surface of the 

 charge, owing to lack of knowledge of the equation of state, but the 

 initial approximations of the boundary value theory are such as to make 

 the theory inaccurate in this region. 



At sufficiently large distances from the charge, the particle velocity 

 term u'^/2 in the expression 12 = P/p + u'^/2 becomes unimportant, and 

 the density p can be approximated by po, the initial value in undisturbed 

 water. With this approximation, sufficiently accurate at distances 

 greater than twenty-five charge radii for ordinary high explosives, the 

 pressure is given by 



(4.18) P(r, t-to)= P{0)x (^\ e " ^" 

 where P(0) = p(0)12(ao) and 6 = yd^ 



The theory therefore predicts that the pressure-time curve at a point R 

 decays exponentially after the initial peak. The parameters x and 6 

 both depend on (R/ao), however, as does the velocity of propagation U. 

 The quantities 12(ao) and di/ao depend only on the explosive ma- 

 terials, and when their values are assigned, the values of x and y are 

 determined only by the ratio R/ao and not by either R or a^ separately. 



