57 



p(r,t) - Pq - (r^^/r) [p(r^,t-r/c) - p^ ] (9) 



The quantity In brackets can be computed by using non-compresslve 

 theory for the motion Inside r,. The well-known equation which 

 relates pressure to velocity for Irrotational motion of an 

 Incompressible fluid Is 



p-p = elt'L^^ (10) 



<« 3t 2 



where ^ Is the velocity potential and v ■ - V^ Is the 



velocity. This expression should give a fairly reliable value 



for the pressure at all distances from the bubble. Inside r-j^ 



we may substitute the non-compresslve approximations to v and (^ , 



viz. f 



„ _ a2 da , a^ da 



^ - 72 H • i^ ' ? dt 



it is then easily shown { Appendix 6) that (10) reduces to 



The maximum of (11) will occur at the time when a is a minimum, 

 if the gas behaves at all like a perfect gas. It is easy to see 

 that the pressure given by (11) at the minimum of the first 

 contraction, where & * is of the order of twice the initial 

 radius of the explosive, cannot be more than a fraction of the 

 pressure produced ai. the same point when the original shock 

 wave crosses it. By (9) this implies that the peak pressure 

 in the bubble pulse Is at most a fraction of that in the shock 



