MOTION OF THE GAS SPHERE 287 



atmospheric), and the position of the surface must in general change in 

 time to satisfy this condition. 



A consequence of potential theory which is useful for later develop- 

 ments is a modified form of Green's theorem relating the variations of 

 any potential function (p satisfying AV = throughout a volume of 

 fluid to its values on surfaces enclosing the fluid. As is shown in stand- 

 ard references/ this integral theorem may be written 



///[©■ -O' -(!)>=-// 4:- 



where the volume integral is extended throughout the volume of fluid 

 and the surface integral is evaluated over all boundary surfaces, d/dn 

 indicating differentiation of (p in the direction of a normal to the surface 

 (taken as positive away from the fluid). The volume integrand, how- 

 ever, is double the kinetic energy of unit mass of fluid and the integral is 

 thus 2/po times the total kinetic energy T of the fluid, which gives 



(8.19) ^=-|Jj4?^ 



The kinetic energy can therefore be evaluated from the velocity poten- 

 tial and its variation at the boundaries. 



8.5. Motion of a Gas Sphere under Gravity 



A rigorous solution from noncompressive theory for the motion of 

 the gaseous products and surrounding water after an explosion should 

 start from the initial form of the boundary surface and distribution of 

 pressure and velocity, from which these quantities would be determined 

 at later times by solution of the dynamical equations. Initially the 

 gas boundary is spherical or nearly so, but there is no guarantee in the 

 appropriate boundary conditions that a spherical surface is the form in 

 equilibrium with the gas pressure of the interior. Detailed investi- 

 gations show, in fact, that this is not the case if gravity is considered, 

 and the motion may even become dynamically unstable under some 

 conditions. Attempts to develop a solution determining the actual 

 state of motion without artificial restrictions on the geometrical form 

 of the gas water interface rapidly become extremely complicated and 

 have so far only yielded qualitative information about limited phases of 

 the motion. Some of the conclusions which have been drawn from such 

 analysis are given in section 8.7. 



From a more empirical point of view, the experimental evidence that 

 a spherical boundary is in fact a good approximation to the actual out- 



* See, for example, Lamb, p. 46 (65). 



