MOTION OF THE GAS SPHERE B99 



r'' 



of the integral I a'W during the contracted phase, beginning when 



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the bubble has contracted to about half its maximum size and contin- 

 uing beyond the next minimum. This integral, however, determines 

 the vertical momentum acquired by the water, which thus remains 

 nearly constant during the contraction. This constant value permits 

 a simple expression of the velocity of the bubble center, for if we let 



m = I a'Mt', the momentum equation (Eq. (8.26)) becomes 



a result appHcable in the contracted phase if m can be evaluated. 



The value of 7n in Eq. (8.29) can, of course, be found exactly for the 

 calculated cases. An approximate evaluation for other values of the 

 parameters can be obtained if it is assumed that the shape of the radius- 

 time curve is essentially the same, no matter what the migration, when 

 the radius is large and the constant value of m acquired. With this 

 assumption, a'^ = a'm^ F{t' /T'), where F is a function independent of 

 charge weight and depth, and T' is the period. We therefore have 



3r 

 m = I an," p\ tir. \ai = Ka-m'^T' = ^ 



= J* a'JF(Jr^dl' = k'a', 



5/6 



where /c is a constant. The value of k required to fit numerical cal- 

 culations is estimated in the RRL report to be 0.70, but a better value 

 from calculations made by the Nautical Almanac Office would be from 

 0.57 to 0.67. Taking a value 0.6 gives 



(8.30) '» = 0.6^, 



With this value of ??z (and hence dz'/dt') known, the vertical velocity 

 and minimum radius a' of the bubble can be computed from Eq. (8.26). 

 Inserting the vertical velocity dz'/dt' from Eq. (8.29) and setting 



da' jdi' = gives 



(8.31) a'^l - ca'-'") = ~ m' 



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