320 MOTION OF THE GAS SPHERE 



C Approxwiate solutioris for a free surface and rigid bottom. The 

 boundary condition to be satisfied at a free surface is that the pressure 

 be the same at all points on the surface. From section 8.4, the pressure 

 at any point in a fluid is given by Bernoulli's equation 



- + ^2 = -77 - Kgrad cpY 



Po 01 



B 



e 



Fig. 8.15 Distribution of dipole sources for a sphere of finite radius and 



infinite rigid wall. 



where z is the depth in feet of water which is equivalent to the hydro- 

 static pressure. At a free surface, P = PoQZ everywhere and the ap- 

 propriate boundary condition is therefore 



(8.42) 



'dt 



i(grad (fY = 



The velocity potential in the fluid is, however, of order 1/r, where r is 

 the distance from the bubble center and, for sufficiently large distance 

 d to the true surface, (grad <py is of order l/d"^, and hence of order 1/d^ 

 compared to d(p/dt. Hence, unless the bubble is very near the surface, 

 the boundary condition on (p at the surface can be taken to he (p = 0. 

 This condition is equivalent to requiring that the streamlines be per- 

 pendicular to the surface, and the appropriate image of a source below 

 the surface is a like source of ecjual strength at the same distance above 

 the surface. 



