440 



The last column is obtainefl Dy putting in the radius of curvature of the cap at its top point instead 



of a. By the us'i».l formula we have 



It is evident that the relation oetween the velocity and the radius of the cap is sensitive to slight 

 departures from the spherical shape of the cap. (it is known that the virtual mass of an ellipsoid 

 of given volume is sensitive to the exact shape of the ellipsoid). It would therefore seem that 

 one would have to go to high powers of 9, and introduce a correspondingly large number of spherical 

 harmonies in the velocity potential in order to determine the exact theoretical shape. It is 

 probaOle that one ould obtain a multiplicity of solutions, and one would then have to determine 

 which are stable by assuming small perturbations. Asa matter of fact, substitution of the apparently 

 better solution (3) for (1) would worsen the agreement with exp>^riment. 



we have introduced a certain degree of arbitrariness by the fact that the origin of 

 co-ordinates is unspecified. The same profile might be specified in quite different ways (as a 

 function of 5) for different choices of origin. It will, in fact, be noticed that the discrepancies 

 between the various values of uA/gp are much smaller than those between the various values of UA'^gil 

 so that it may be that the three solutions we have found are all really first approximations to the 



vestigation, but it has not been 

 vestigation would be lengthy. The 

 Figure 4. It might be thought that 



same actual profile. These points seem well worth further 



thought advisable to hold up the issue of the report ss the 



profiles corresponding to the various solutions are plotted 



it would be an easy rretter to decide exporirr^ntally between them, but even curves 2 and 3 can be 



brought nearly into coincidence over the relevant range of angles (about 60°) by shifting the origin of 



co-ordinates, and choosing the scale so that the radii of curvature of the caps are the same. 



Conclusions . 



Taylor and avieC formula relating upward velocity and radius has been confirmed for radii 

 up to 15 cm,, which is probably near the limit of stability. The available evidence suggests that, 

 after ah underwater explosion bubble has ceased oscillating, it splits up into many small bubbles of 

 about this order of size. 



Reference. 



(1) The rate of rise of large volumes of gas in water. Taylor and Davies. 



