-29- 275 



(3.5) 



=-/; 



4 

 _a 



2b^ 





dt 



/ 



1 / 3 



where the integration is extended over the full period 

 ? of the pulsation, from the explosion to the time of 

 minimum slze» 



2« The approximate evaluation of the period and the momentum * 



The results of a numerical integration of equations 

 (3.2), (3.3) are tabulated for special cases in appendix II. 

 They serve as a check on the approximations which will now 

 be made to evaluate the period t and the momentum s. 



The bubble expands to a maximum size before contracting 

 again. Indicate the value of a quantity at the time of maxi- 

 mum size by a subscript 1. Thus t, is the time of maximum 

 size, s, is the linear momentum, etc. We shall introduce 

 the following approximations which are especially accurate 

 when the bubble is in its balanced position: 



1. The time C and linear momentum s at the minimum 

 size of the bubble is twice the corresponding quantity at 



the maximum size; i. e., t = 2t^, s = 2s-. This assumption 

 agrees very closely v/ith the numerical integration of the 

 equations. It means that the motion is approximately 

 symmetric about the time t, of maximum size. This would 

 be exactly correct if the b-coordlnate did not vary. 



2. During the first half of the period of pulsation, 

 until time t,, b is small and can be taken as zero, so that 

 b remains equal to b . This agrees satisfactorily with 

 the numerical integration, as well as with experimental 

 evidence. 



