SECONDARY PRESSURE WAVES 367 



the minimum. Denoting this fraction by e, we have e = E{a)/Y 

 = ka*-^'\ and Eq. (9.20) becomes 



'-^ - m 



and e is therefore a function only of s for a given explosive material. 

 Expressing Eq. (9.18) in terms of e gives 



(9.22) P„ - P„ = i (1^,) e-"(4 - 3.-) 



and the pressure is therefore also determined in terms of s by Eq. 

 (9.21). Differentiation of Eq. (9.22) shows that Pm is maximized for 

 e = 32/33, corresponding to 97 per cent of the total energy returned to 

 the gas. The corresponding value of s is then, by Eq. (9.21), 

 s = 0.151 k*'^. The quantity k* is, however, so small that the maxi- 

 mum is obtained for all practical purposes at s = 0, corresponding to 

 the stabilized position of zero velocity of the gas sphere at the time of 

 minimum size. 



It is sometimes convenient to express the peak pressure Pm in terms 

 of the minimum radius a*, with the result that 



(9.23) Pm- Po = -PoL* ^ 



r a 



T*2 



3A^ 

 4 



The optimum peak pressure Pmax derived more directly in section 9.2 is 

 readily obtained from this equation, for when da*/dt* and db*/dt* are 

 zero, a* = /b*^/^ very nearly, and hence 



P _ P - ^^ d^ _ S^ Y a^ 



■t max -to . ' — . ■. ' 



4 r 47r 4 r 



which is Eq. (9.11) for the case y = 5/4 if a is expressed in cm. 



The underlying reason for the result that optimum peak pressure 

 occurs when the bubble is stationary lies in the fact that only for this 

 condition is all of the originally available energy of the gas products re- 

 turned to them. If the bubble is in motion, however, part of the total 

 energy remains in the surrounding w^ater as kinetic energy of flow and is 

 not returned. The pressure of the gas proves to be the more important 

 factor in determining the peak pressure in the water and the latter is 

 therefore largest when the flow of water is very nearly stopped. 



The variation of the peak pressure factor e^'^ / {4: — ?>e) in Eq. (9.22) 

 with vertical velocity is such that the pressure has a rather flat maxi- 



