310 MOTION OF THE GAS SPHERE 



The necessary condition on the boundary of the surface is that the 

 pressure P{R) be uniform and equal to the gas pressure Pg. The pres- 

 sure is determined by BernoulH's equation (Eq. (8.18)) 



-° - gR cos e + (^) - i(grad *,)«^ = ^ 



p« 



[aOn * 



where Po is the hydrostatic pressure at the bubble center. Substitution 

 for R and (p then gives a series of terms in the P n (cos 6) . (In carrying 

 out this substitution, it must be remembered that the origin of R is in 

 moving coordinates and d(p/dt is to be evaluated in fixed coordinates.) 

 Use is then made of the orthogonality relations for Legendre poly- 

 nomials, namely that 



.+1 



2 .„ 



n m = n 



+1 2n + 1 



(cos d)Pm{cos d)dicos 6) = 



ii m 9^ n 



Four equations therefore result from multiplying the Bernoulli equa- 

 tion by Po, Pi, P2, -P3 in turn and integrating over cos 6, these being 



2*(«^^)-i''Jf* + ^^^^^^=^« 



Id , „,. 6 CB^,^ , ^,,„, .,.3 



a^ dt a^ dt 4: ... i . 



a- df^ J a 



}_dBs _iBsda _ldM C^ 

 a^ dt a^ dt o? dt^ J a 



dt + OilJ^ = 



The various stages of approximations of the theories already dis- 

 cussed are nicely shown from these equations. In the zeroth approx- 

 imation neglecting gravity, t/ = and the first of Eqs. (8.40) becomes 





Po 



which is equivalent to Eq. (8.4), obtained for motion of a sphere neg- 

 lecting gravity. The last two of Eqs. (8.40) are equivalent to the rela- 

 tions 



