= U7T a"^ u (p + i pu') at 



«f| (x'*>/'-)x^t 



•f (^X'* 0'X^t) 



The energy stores between R ana i is kinetic and is equal to 

 2 Tip I r^u'or = 2 77PX' ^ = 2"PX' (R-5) 



The work done Dy the DuDble 



PdV 



"TFT 



,,n + 1 



-— 1 



tne law of expansion o-jrnj PV = constant. 

 Equatina (l3) to (U) and (12) 

 2 P 



3np 



R ^ 1 





\* 1 \ X'J^- 0^ 



This equation tojether with 



ot)tain3d from (e) 



and R' + J^ = 



(11) 



(12) 



(W) 



(15) 



(16) 



give three first order equations to determine Xr '/'. "• When is known the required pressure 

 follows from (9). 



If we choose as our units of length and time, for length the initial radius R of the 

 Bubble and for time the time required to traverse 2 R with the velocity of sound, equations (it), 

 (15), (16), nay be written 



^^ * -41 5R-iX ax-^S'A = 



S + SX * -^ 8t - y (17) 



a 



6r + X S'''"^ = ° 

 in *nich A = a P /p c^ (18) 



