77 



APPENDIX Z. PULSATIONS OF A BUBBLE IN AM 



INCOMPRESSIBLE FLUID 



V/e have to Integrate the energy equation (5) of 



Section 3. This equation is 



2rpa^(^^)% i:^ p^^G(a) s W (1) 



Consider first the case ivhere the potential energy G of 

 the gas is neglected. Solving (1) for dt and integrating 

 v/e have for the expanding phase 



t4n- t = /--'-^ ^L^ (2) 



v; 



W 2 Poo -<3 



ST? " 3 e 



a 



where t^ is the tine of maxiraiaa size. (It is convenient 

 to choose trie naximum as origin because the approximations 

 of this simplified theory are most nearly fulfilled near 

 the maximtan. ) Putting 



y E _§L_. s a -f/ 4r P~ <3) 



%iax y 3 ~W 



we have from (2) A / 



Jj V 1 - y^ 



The most convenient way to evaluate the integral for values 

 of the lov;er limit different from zero is to transform It 

 into an incomplete beta function. If we set 



z 



VT- y^ ■ (5) 



we have y 2 4 z (1-z) 



dz s |. _S-~^-. eo that 

 " Vl » y'" 



