274 -28- 



tne b-oquation (2.13), we obtain 



(3.3) ^[2a^(^+f2)b - 2a^f3_a] 



a 

 2? 



df. 



d06 



<ifn 



df 



- 2 — = ab + 



d06 dOL 



2 *2 

 b 



*3^ 



where 

 (3.4) 



06 = 



^ 



and the quantities f , f., f^ are fvuictlons of 06 only. 



The differential equations (3.2), (3.3) are to be 

 integrated, subject to the following Initial condition; 



at t = 0, 



a = 



b = b^, b = 0, 



where a is the smallest root (near k ^ ) of 



a + ^iv = 1. Of course, these initial conditions are not 



exactly realistic, since at the very "beginning a shock wave 

 Is formed by the explosion. But the time interval required 

 for incompressible flow to set in is relatively minute and 

 may be ignored. ' 



The quantity in the brackets on the left-hand side 

 of (3.3) is the linear momentum s of the system. The 

 first tenii on the right-hand side of (3.3) is due to the 

 presence of the rigid wall, and the second term is due to 

 gravity. 



By integrating equation (3.3) we obtain the momentum 

 s at the end of the period of pulsation of the bubble: 



•«• See [4]. 



