MOTION OF THE GAS SPHERE 327 



where z is the distance of the center below the surface of zero hydrostatic 

 pressure and U is the upward velocity, from the bottom and toward 

 the surface. 



Comparing this result with Eq. (8.23) for motion under gravity alone, 

 it is seen that the effect of the rigid boundary is represented by the 

 quantities /«, fi, fi, which are functions of the ratio a/26 (6 = distance 

 from bubble center to the wall) and vanish for a/26 = 0, corresponding 

 to negligible influence of the wall on the flow. 



Energy considerations thus provide one of the two equations neces- 

 sary to determine the radius and displacement of the bubble with time. 

 In the absence of boundary effects, a second equation was obtained by 

 momentum considerations (see Eq. (8.25) and the discussion following 

 it). While the same approach could be used in the present more gen- 

 eral case, it is simpler to use the more powerful method of the LaGrange 

 formulation of the equations of motion in terms of the kinetic and po- 

 tential energies of the system, employing the radius a and displacement 

 6 as generalized coordinates. The validity of this general method of 

 mechanics for the motion of a sphere in a fluid is discussed in Lamb's 

 Hydrodynamics (65), and is assumed here without proof. 



If gravity is the only force of origin external to the sphere and V is 

 the potential energy of hydrostatic buoyancy plus internal energy of 

 the gas, the LaGrange function S = 7" — F satisfies the differential 

 equations 



\da) 



dt\dal da 



dt\du) 



02 

 db 



where d = da/dt, U = dh/dt. Either of these equations may be used 

 to give the second relation in addition to the energy integral, Eq. (8.49). 

 The second of the two proves more convenient and gives directly the 

 equation corresponding to the translational momentum under influence 

 of gravity alone (Eq. (8.25)). Substitution and carrying out the dif- 

 ferentiations gives, letting a = a/26, 



(8.50) 



I [5 ..„ + 3/.,. - M, (f )] = - I [I (f )■ + If V 



