347 



Section III 

 Derivation or pormuias 



Consider the motion of a gas bubble In water or fi- 

 nite depth, taking into account the effects of the bottom, 

 the free surface and gravity. As before, we let A be the 

 radixis of the bubble at the time T, B the distance of its 

 center from the bottom, and H the distance from the sur- 

 face. V/e put Z = H + Z." where Z""' is the height of the 



0" 



B 



_i 



water column whose pressiire equals atmospheric pressvire, 

 so that the pressure at the center of the bubble is pgZ • 

 Just as in Report 57. IR we find that if we use coor- 

 dinates R and to describe the motion of the water, the 

 velocity potential describing the flow is 



(5.1) 



2 ' 



5 = A A 

 * R 



A^b' cos 9 X 



+ 7^ 7T- + «n 



where §-, is the "image" potential necessary to satisfy 

 the boundary conditions on the bottom and on the surface, 

 The primes indicate time derivatives. 



By classical hydrodynamics we can show (see Report 

 37. IR) that the kinetic energy of the water is given by 



