97 



APPENDIX 4 EFFECT OF GMVITY 



Whenever the radius of the pulsating bubble is 

 small enough compared with the height of the coluran of 

 water necessary to produce pressuro pco, we may expect 

 the effect of c^avity on the pulsations to be snail. Ac- 

 cordingly let us expand the. velocity potential ^ and the 

 flhape of the bubble in poivors of c Choosinc the Initial 

 position of the center of the bubble as cricin of coordi- 

 nates, v;e may denote the radial distance to a point on 

 the boundary cf the bubble by R( ), where 6^^ is the 

 angle with the vertical. *Ve thus set 



P (r,t) =P„-/-GP,-^ ..... (2) 



R(^,t) z a(t) -f- cRjC-^ ,t) -^ ....„ (5) 



Takinr;, Poo to be the pressure at infinity in the plejie z -0, 

 the equation of motion is 



and the first order p&rt of this is 



i-gz (4) 



» ■n 



^T^ -Q T a .r q ^ "^ ^^! 



To integrate this v;e must use three additional relations: 

 (i) Since y <^_ - , Che expansion of S ^ in 

 spherical hariiionics must have ths fo7.'r?i 



