65 



the bubble, besides being sucked back and forth periodically by- 

 its image, should be continually repelled from a free surface and 

 attracted to a rigid one. In the case of the bubble of Fig. 2, 

 the repulsion from the free surface was actually great enough to 

 make the bubble sink. As the mathematical details of the theory 

 are rather involved, only a rather loose picture of the mechanisms 

 involved will be given in the present section; moreover, it will 

 be advantageous to discuss first the simpler theory of the migra- 

 tion due to gravity alone. This theory, which is worked out in 

 detail in Appendix 4, can be described by saying that the bubble 

 has associated with it a vertical momentum equal to its velocity 

 of rise times (2n/3)pa-^, and that the time rate of change of this 

 momentum equals the buoyant force (i+n/3)pa-^. This gives 



3 rt ^ ^. (17) 



velocity of rise due to gravity = ^6 I a' at 



aT -'o 



Note that the velocity of rise becomes enormously accelerated 

 during the contraction, when a becomes small while the integral 

 remains large. Although Appendix 4 assumes gravity to be a small 



perturbation, so that its conclusions are rigorously valid only 



21 

 when the velocity of rise is small. Taylor has shown that the 



effect of this rise on the motion can be calculated, to a fair 

 approximation, by assuming that the bubble is constrained to re- 

 main spherical, so that (17) becomes valid at all times. 



21 



See the article by Taylor in this Volume, 



