MOTION OF THE GAS SPHERE S09 



already discussed are obtained, and is outlined in the following, together 

 with some of the results of Penney and Price. 



We begin by noting that for symmetry about a vertical axis the 

 velocity potential <^(r, t) for the fluid can be expressed as an expression 

 in powers of 1/r and Legendre polynomials Pn (cos 6) :^ 



(8.37) 



A Pi(cos d) P2(cos 6) 

 (p{r, t)= h ^1 h i52 h • 



where d is the angle with the vertical axis, and the coefficients A, Bi . . ., 

 are functions only of time. We further assume that the radius vector 

 R from the center of the bubble to its surface can be written 



(8.38) R{t) = a + 52P2(cos 6) + h^P^{Gos 6) -{- • - - 



where ho, hs . . . , are functions of time, and bi = because of the 

 choice of origin. As is shown in the development, the coefficients 

 62, 63 are of order U^, U^ . . . . 



The radial velocity Ur of the surface is given by { — d(p/dr)r = R, and 

 since the distance to any point R of the surface differs from the value a 

 by a quantity of order U^, the error in identifying Ur with {dR/dt + U 

 cos 6), to make the normal velocity in the fluid equal that of the bound- 

 ary, will be of order U"^. We therefore have 



dR I d(p\ ^j _ 



dt 



- i-fX 



i.+ff. +§■'■.+<'»■') 



the term U cos d being necessary to account for the velocity U of the 

 bubble center chosen as origin for R. The derivative dR/dt can, how- 

 ever, be expressed in terms of the Pn by Eq. (8.38), and if this is done 

 coefficients of like P's must be equal, as the expression must be true for 

 any 6. Substituting for R and dR/dt then gives the relations 



(8.39) Po:A=a^'^, Pi : ^Yl - ? ^A = WU 



dt \ 5 a/ 



^ db2 , 2A , 3B2 j3 dbz , 2A , 4P3 18 Bi 



dt a^ a^ dt a^ a^ 5 a^ 



where terms of order U^ have been dropped. 



^ The double use of P, in connection with these polynomials and for pressure, is 

 to be noted, but should cause no confusion. 



