66 



The migration due to a neighboring free or rigid surface 

 can be explained with similar concepts. The image of the bubble 

 in the surface affects conditions in the neighborhood of the real 

 bubble in two ways; it gives to the water there a velocity normal 

 to the surface, and it creates a pressure gradient in this direction. 

 As is shown at the close of Appendix 5, the migration can be cal- 

 culated, correctly to the second order in l/h, by a simple momentum 

 argument. To begin with, the pressure gradient due to the image 

 will impel the bubble in the direction of lower pressure, just as 

 the buoyancy of the bubble does when the pressure gradient is due 

 to gravity. The normal momentum due to this effect can therefore 

 be obtained by integrating the product of the image pressure 

 gradient by the volume of the bubble, and the corresponding normal 

 velocity can be computed. The sum of this velocity and the velocity 

 field of the image, evaluated at the position of the center of the 



bubble, turns out to give the correct value of the velocity of 



2 

 migration normal to the surface, to order l/h , as determined by 



the more rigorous calculation given earlier in Appendix 5. This 



velocity is 



dk . . 2a! ^ - -K f a'^ (^)^ dt (18) 



for a bubble at a distance h from a free surface; if the free sur- 

 face is replaced by a rigid one, the sign of dh/dt is reversed. 



