3J^0 MOTION OF THE GAS SPHERE 



,T* . , .. „ ^1 



(1 - a*3)i/2^a* = 0.74 



o o 



T* 1 



J (1 - a*3)i/2 



The ratio of the two terms is then 



Surface term Zo* ^ ^^ Zo 



Gravity term = " °-'° 2?? = " °-'V^ "" 



the change to dimensional units introducing the scale factor for length, 

 which is the maximum radius am in the present approximation. The 

 distance —Zo is just Po/poQ, where Po is the hydrostatic pressure at the 

 depth of explosion, and the migration M)i is roughly given by Herring's 

 result : 



(8.67) ^hl = A5 



pogl 



arn~] 



u 



where A6 is the migration under gravity alone. 



This formula shows that the effect of either surface falls off roughly 

 as 1/P, where I is the distance to the surface from the initial center of 

 the bubble, and if the surface is close enough the bubble may even sink 

 during its oscillation, despite its buoyancy. The rise due to buoyancy 

 for charges at a given depth increases with charge weight, the effect 

 being greater near the surface owing to the relatively greater differences 

 in hydrostatic pressure over the larger bubble surface and the longer 

 period of oscillation. For sufficiently large charges, the relatively 

 greater buoyancy cannot be counteracted by the surface repulsion ex- 

 cept for depths so shallow that the bubble breaks surface before con- 

 traction, and a rest position of zero net migration does not occur. The 

 exact charge weight for which this first occurs has not been determined 

 and cannot be calculated with any accuracy from the approximate 

 formulas which have been developed, but is of the order of a few pounds. 



The absence of a rest position is not indicated for a charge fired near 

 the bottom, as the bubble is carried away from the rigid surface by 

 gravity, and in practical cases the bubble is relatively much less buoy- 

 ant than it is near the surface. It is to be noted that this case is more 

 complex because the distance above the bottom is an independent vari- 

 able for a given depth. The calculation of migration given here is of 

 course only a crude first approximation, as only the leading terms in the 

 surface corrections have been used, and internal gas pressure has been 

 neglected. 



