MOTION OF THE GAS SPHERE 339 



In this equation, the bracket on the left side represents the trans- 

 lational momentum, the bracket on the right represents the surface 

 "forces," and the last term is the hydrostatic buoyancy. The variable 

 Z* may represent the distance to the free surface or the bottom, and the 

 values of fo, fi, /2 as obtained by the method of images depend on which 

 case is considered. The analysis shows that dfi/dl*, /2, 6/2/81* are of 

 order 1/Z*^ or higher, and will be neglected in the present approxima- 

 tion. The values of dfo/dl* and /i for a free surface above the charge 

 and for a rigid bottom below it are seen by inspection of the formulas of 

 section 8.9 to be 



the upper sign being for a free surface and the lower for a rigid bottom. 

 Inserting these values in Eq. (8.66) and integrating to obtain the up- 

 ward velocity dh'^/dt* gives 



(^ ^ 3 0^2^ _ J_ "^* 



/S"(i^>+.-i7/-* 



If the distance /* to boundaries can be assumed fixed, the first term 

 can be immediately integrated a second time to give a displacement of 



±^J«*'W-««(0)] 



This is a periodic term, which can be thought of as representing the 

 attraction or repulsion of the bubble by its image. Its value becomes 

 greatest at the bubble maximum, and vanishes at the minimum 

 {a*^{T) = a*^(0)). Its magnitude is in any case small and will there- 

 fore be neglected. 



The integrals in the last two terms increase chiefly while the bubble 

 radius a* is large. It is therefore reasonable to calculate them ap- 

 proximately for the first period T by neglecting the internal energy, in 

 order to determine the nature of the surface correction introduced by 

 the second term. The integrals can be evaluated by using the relation 

 da*/dt* = (1 — a*^y^^ a*"^^^^ obtained from the energy equation by 

 neglecting the internal energy of the gas, and taking twice the integral 

 over the first half cycle from a* = to a* = 1 (a* is measured in units 

 of the maximum radius am obtained neglecting the internal energy). 

 Both integrals can be expressed in terms of /3-functions, giving 



