353 



Radius of the Bubble 



The maximum radius of the bubble is foxind by putting 

 a = In equation (3.15) and solving 



(3.21) a^ + ka"^^'^"-'-^ + 3s^/2a^ = 1 



for the root near one. If we expand a in powers of k and 

 solve (3.21) we find that 



3g2 



(3.22) a^ = 1 - I - (X^ + l)k2 - -gi . 



This is the same as formula (1.20). 



To find the minlmxjm radius of the bubble we put 

 a = in equation (3.15) and consider it at the time of 

 minimum size. The equation reduces to 



(3.23) a^ + ka"^^"^"-"-^ + 3iV2a^ = 1 



and the root near zero is the minimum radius. Note that 



(3.23) is the same as (3.21) except for the value of the 

 momentum. The equation simplifies if we put 



(3.24) ^= k^/^'^-^^ 5^ = ^u, e = 3i2/21^ 



term, since a is r 



U-C^-1) + £U-^ = 1 



3 

 and neglect that a term, since a is near zero. We have 



or 



(3.25) u = u^""^ + 6. 



Peak Pressure 



The pressure in the water is given by Bernoulli's 

 Law as 



