146 _ J _ 



S = ^ C P„ „ (9) cos {m<^ + 7„ „) (2) 



n ^0 m,n m,n m,n 



but it should be noted that the time factor t,^(t) depends only on the degree n of the harmonic, and 

 not on m. 



The volume of the bubble is the integral of ^ b over a unit sphere, which to the first 

 order is equal to » w a-'/?. In virtue of the orthogonal properties of the harmonics S^. 



We assume that the bubble is filled with an almoet massless perfectly elastic gas; pressure 

 waves in this gas are neglected, so that tne pressure p at any instant is uniform throughout the 

 bubble. Assuming the usual -idiabatic relation between pressure and volume, we have 



p a^y = o ,yy (3) 



where P Is the pressure and a the mean raoius at time t = 0. 



The motion of the surrounding fluid, since it Is generated from rest by pressure, will possess 

 a velocity potential (^, say. Assuming the fluid is inccmpressible, this potential satisfies Laplace's 

 equation and is therefore expressible in the form 



4. = r-' A(t) ♦ f r-'^^ B„(t) S (9. 4,) (4) 



n=l n n 



We shall assume that the surface harmonics S^ In («) are identical with the corresponding harmonics 

 in (l), and that the time factors B (t) in (u) are of the same order of smallness as the b^'s in (l). 

 These assumptions are Justified by the consistency of the subsequent analysis. 



The leading term on the right of {*) is separated from the others because it corresponds 

 to the case when the bubble is accurately spherical, for which the solution is known. 



Spherical harmonics of negative degree only are taken in the expression f:r</), because the 

 velocity, and therefore alsoc^, vanishes at infinity. 



The pressure in the fluid is given by the hyorodynamical equation 



£ = 4* n. 1 (gradcjS^ + F(t) (5) 



P 2 



where n is the potential of the extraneous forces, limited in our case to gravity, so that 

 n= - gz. 



The arbitrary function F(t) is determined in the present case by the conditions at infinity. 

 Taking the pressure at infinity at the original level of*khe bubble to be constant and equal to 5, 

 we have, since tends to zero, 



F(t) = Q/p (6) 



Substituting for c/j from (l) into (5), and retaining terms up to the first order, then gives for the 

 pressure p in the fluid at distance r from the origin, 



£^.0= i-i - .z^-^.l!:!^^ 3„ (., 



Boundary condition ?. 



The conditions to be satisfied by the velocity and the pressure at infinity have already 

 been considered. We now investigate the conditions to be satisfied at the surface of the bubble. 



The condition to be satisfied by the velocity is that its normal component at the surface 

 shall be equal to the normal component of the velocity of the surface as determined by the expression 

 (l) for R. These norral components are, to the first order of snail quantities, equal to the 

 corresponding radial components, and hence, to this order of approximation. 



