270 -24- 



the time of the niinimuzi size of the bubble, the above 

 integral becomes 



4p C 

 (2.51) I = -^ . ^ . [a^a], 



where [a a] is to be evaluated at the time when (a a) = 0, 



2» 4 



or a a is a maximum. 



Equations (2.8), (2.9) and (2.16) yield 



„4»2 , 1/4 33^ 4 

 aa =a-ka^ - — n " ^ 

 2a 



and 



,4-2. T k ^ 3s^ .3 

 (a a ) = 1 -7-37^ + -3- - 4a . 



4a ' a 



Setting the last expression equal to zero in order to find 



2* 2 



the maximum of (a a) , we obtain an equation for a. If 



both k and s were equal to zero, the solution of this 

 equation would be a = .eS. In a more realistic case when 



k = .2, 3 = .06, one obtains a = .61. This illustrates 



2* 

 that the value of a when a a is a maximum is not very 



sensitive to changes in the momentum s, since s is generally 



quite small in actual casesi Using a = .61 and k = .2, 



the value of [a a] is .55. 



By (2.7), equation (2.31) becomes 



I = ,73 _!^ — = *^ . . — — — atmosphere-secondso 

 R ■p. -'-/ci R 



o 



Taking a typical case of a depth of 100 ft., so that 0^=133, 

 we find that 



(2.52) I = .37 -~ — atmosphere-seconds. 



R 



i-c See Part Hi. 



