338 MOTION OF THE GAS SPHERE 



evaluation for k* = 0.20 by Shiffman and Friedman, similar to the 

 method used for the period, gives 



(8.63) 



/0.113 0.019\ , 1 /^^^, , 0.148\ 



As this result is written, it includes only the effect of a rigid bottom. 

 The effect of a free surface is readily included by using the more general 

 expressions for fo and dfo/da. This will not be done here, a more ap- 

 proximate but simpler discussion being given in part D. It is seen from 

 Eq. (8.63) that s can, by suitable choices of 0^* and 6^*, be made zero. 

 This corresponds to a bubble which has no vertical motion at the time 

 of the first minimum, the "stabilized" position in which the effect of 

 gravity is compensated by the rigid bottom. The depth d* for which 

 s = is given by 



(8.64) d* ^ 6.25/2 _|_ 3 35^* _|_ 0.4 



The migration A5* of the bubble due to gravity and the bottom is 

 found by Shiffman and Friedman to be given by 



(8.65) A6* = 19s (1 - 62s2) 



where s is determined by Eq. (8.63). These formulas for the migration 

 and stabilized position as written here are expressed in units proportional 

 to the maximum bubble radius. The actual values obtained for specific 

 cases and their relation to experimental results are considered in section 

 8.11. 



D. Ayproximate formulas for migration near surfaces. Although the 

 analysis employed by Shiffman and Friedman for the influence of the 

 bottom on bubble migration can be extended without difficulty to in- 

 clude a free surface above the bubble, the formulas are rather cumber- 

 some. Instead of carrying out this derivation, a derivation due to 

 Herring (46) will be outlined here which shows more explicitly the na- 

 ture of the effect. The vertical momentum equation including surface 

 terms is the starting point in this analysis, which is, from Eq. (8.53), 

 given by 



(8.66) -^ r^' (1 + 6/.) ^* - 2a*3/, ^-^*1 = a*-^ \^1^(^^\ 



/da^\ /r/5*\ / Of, \ /db*\n _ 1 



2^^!-:- " "^ l + l -T II .- 1 I - - a' 



