-9- 9 



then U is real and finite throughout the ranje of integration and may therefore oe eviiluated 

 by any approximate method. Now for thj case /3 = 2t the expression in {} in equation (2k) is 

 nearly constant. In tact it never differs oy more than 3» from - 1.72 x 10"'^ throughout its 

 while ranjc. Hence to this accuracy we may write I3 ■ - "-57 x lO"'' [f? ~ 1) when /3 = 21. 

 Calculating 1^, .j, I, from these formulae and using in (2l), we find that the time required 

 for the bubble to reach its maximum radius is jiven by t = 2l70/j^ R . In the case of 



. 



100 lb. charSe of T.N.T. with P = 9000 atmosphL'res R = 0.7 foot, t = 0.5 second, so that 



' 00 



if the incompressible theory hold rigidly the bubble would return to its original state in one 



second and a series of explosions at intervals of one second would be heard. As a matter of 



fact two reports are oft3n heard but observers generally attribute the second report to the 



breaking of the surface by the bubble. In Figure 3, curve A gives the charge of radius of 



bubble with time according to equation (l) and curve B shows the modified course as calculated 



from the above data. Curve 6 is one half of a periodic curve and shows that during the greater 



proportion of its existence the bubble is of large volume and low pressure thus accounting for 



the scarcity of spout-like upheavals mentioned in the last section. 



