308 MOTION OF THE GAS SPHERE 



under the influence of gravity, and in effect any constraints necessary 

 to preserve this form are imphcitly assumed present in such a way that 

 the total energy is unaltered by them. Herring has examined the ques- 

 tion of the form of the bubble surface when gravity is included, and he 

 finds that to a first approximation the bubble remains spherical ; but the 

 assumption that the gravity effect is small, made in obtaining the result, 

 breaks down in later stages. 



Herring's analysis, although straightforward, will not be given in 

 detail here. Assuming the correction for gravity to be small, the 

 velocity potential, pressure, and radius vector to the bubble surface 

 are expanded in powers of the acceleration of gravity g, use being made 

 of the fact that coefficients in the expansion of (p must be solutions of 

 Laplace's equation and hence can be expressed in spherical harmonics. 

 The boundary condition which must be satisfied at the bubble surface 

 is that the pressure be a function only of the volume V of gas, regardless 

 of the shape and of g. The first order correction to the radius vector 

 turns out to be simply an equal upward displacement of all points of the 

 surface, the velocity U being given by 



U = ^(' aHt 



2^1 

 a' J 



The first order result for U is in agreement with the result of section 

 8.5, which assumes a spherical form. It shows, however, that the up- 

 ward velocity increases with time, particularly when the radius becomes 

 small in the contracting phase, owing to the factor 1/a^. Hence the 

 assumption that the upward displacement due to gravity is small be- 

 comes increasingly poor as the bubble contracts, and the conclusion 

 that the form remains spherical is no longer established. The inference 

 that the bubble may well become unstable is supported by experimental 

 pictures of the bubble (section 8.3), as well as by more detailed analysis 

 of Penney and Price, and of Ward. 



C. Perturbations of the bubble form. More elaborate approximations 

 to the actual form of a bubble in a fluid have been developed by Penney 

 and Price (86), in which departures from spherical symmetry are treated 

 as small perturbations, which on account of the symmetry about a 

 vertical axis can be simply expressed in a series of surface spherical 

 harmonics. Ward (118) has modified this approach somewhat to ob- 

 tain results expressing the departures from spherical form as a series of 

 approximations in increasing powers of the vertical velocity U of the 

 bubble. Ward's development is particularly interesting, as it shows 

 rather well the stages of approximation in which the various theories 



