-m 



da 



dt (2) 



49 



where a(t) Is the radius of the bubble. Under our assumptions 

 the total energy, which is constant, equal a the kinetic energy 

 of the water, plus the potential energy of the compressed gas, 

 plus the potential energy due to expansion against the pressure 

 p of the water far from the bubble. The kinetic energy is 



I if=v^,4 7rr2dr = 2?r('a3j||j (3) 



/a 



The work done against p is, to within an additive constant 



^^ P (4) 



Denoting the potential energy of the gas for the present merely 

 by G, the energy equation is 



where W is constant in time. This equation can be solved for 



~ and integrated to get the motion. The term G(a) complicates 

 dt 



the integration considerably'; fortunately however it can be 

 shown that for the first pulse «» two the calculated period la 

 very little altered *by setting G ,r 0« The reaaon for this is 

 that G ia appreciable only in stages of the motion when the 

 bubble is small > and these stages occupy only a small fraction of 

 a period. A rough estimate of the error introduced by ignoring 

 G can be obtained from Appendix lo For the later pulsations 

 the amplitude of the motion is much loss, and eventually the 



