236 



The formula (A.l) was given by Herring as applicable only to cases when u is small compared with da/dt 

 but it applies equally well when u is comparable with da/dt provided it is assumed that the bubble is 

 constrained to remain spherical and terms containing powers of 1/h higher than l/h' are neglected. With 

 these assumptions the velocity potential of the flo*/ is 



a^a ^ ua^ cos-*?' ^ a^a / a^ cos d \v. 

 ~ ^2' «h2 \ 2r2 )j 



(A.3) 



where r- and 5' are the co-ordinstes of any point referred to the image centre as origin and the line 

 joining the centre of the sphere with Its image as (?• = 0. The upper sign refers to a rigid plane end 

 the lower to a free surface. 



Near the sphere . • may be written 



a^a ^ Ua^ cos ^ j J a^a ^ a^ar 

 r 2r^ j 2h Uh 



cosg 1 

 2,2 J 



(A.«) 



The kinetic energy T of the whole notion is one-half the kinetic energy of the flow surrounding the spher* 

 and its Image si that U = - JJ T ( '^ \ i^S, the integral being taken over the sphere only. 



p \3 r ; 



Substituting rf and 3 (j!/3 r from equation (A.u) 



T 1.7 / a\ 77 



^^ • 2 TT a-'l' I 1 ± — V — 

 p [ 2hj 



The energy equation is therefore 



i? u' TT a^aU 



3 



(-) 



"np a-' gz + 2 "^p a 



. 1 / a 'v ■" -i ■} TTp a' a- 



a^ / 1 ± — \ ♦ _ p a^ u' ± — r dU = W - G(a) 



\ i^) y 2 h^ 



{*.») 



(A.«) 



where w and G(a) have the same meming as in Section 2 ab»ve. 



The condition that no resultant external force acts on the sphere Implies that the co-efflclent 

 of cos 6 in the expression for the pressure at the surf,ice of the sphere shall vanish. 



d Cf. 

 d t 



fixed point 



- iq' + gz 



gr cos 6 



(A. 7) 



where z is the depth of the centre below the level where the pressure is zero. *lso If B <i>/3 t represent* 

 the rate of variation of <t at a point which moves vertically at the same velocity as the centre of the 

 sphere 



d (p 



d t I fixed point 



Substituting from equation (A.3) in equations (A. 8 and 7) the term containing cos 5 In p/p 

 is found to be 



- uJ cos y slnS — -> 



? t j3 r r B5j 



ion (A.3) in 0( 



J3 ± — 5- I — a^ a < 

 uh'^ I 2 



ting to zer 



(A.8) 



Multiplying this by 2a'' Ani equating to zero it is found that 



(a'u) 



(A.9) 



