26 



i.e., that the average diffusion into the cavity is one-sided. The signifi- 

 cant conclusion that may be reached from the experimental and theoretical work 

 cited above is evidently that air or gas diffused into a transient cavity 

 plays an increasingly important role as the number of cycles increases, and 

 is of especial importance in determining the maximum pressures that can be 

 developed at the time the minimum radius is reached. For sea water, with its 

 high air content (both dissolved and entrained — especially near the surface), 

 the presence of air in large quantities in even the initial stages of the mo- 

 tion is a distinct possibility. Furthermore, the change in the properties of 

 the water through action of surface-active agents may assist the diffusion of 

 gases into the bubbles. 



ANALYTICAL DESCRIPTION OF THE MOTION 



The hydrodynamics of the motions of individual cavities have been 

 analyzed for parts of the cycle by various writers. Lord Rayleigh 50 consid- 

 ered the problem of the collapse of a spherical cavity in a homogeneous, in- 

 compressible fluid, the cavity supposed empty and the pressure at an infinite 

 distance remaining constant. Plesset 51 extended these computations to include 

 the case of a single empty cavity moving in a field in which the external pres- 

 sure is a function of the time (corresponding to the cavities of Knapp and 

 Hollander's experiments). Since Plesset 's computation includes the Rayleigh 

 case, only the former will be discussed here. The radius of the cavity at any 

 time t is taken as R, and the distance to any point in the liquid as r, the 

 origin of coordinates being chosen at the center of the sphere. The velocity 

 potential 4> for the motion of a liquid with spherical symmetry is given by 

 Lamb 52 as: 



r 

 where R = -rr . The equation of motion for irrotational, incompressible flow 



-£-*MT^-*£ 



where p(r) is the pressure at r, 



P(t) is the pressure at infinity, and 

 A 4> is the gradient of <j>. 



Using these equations, the following are obtained 



