8 - 8- 



Hence by (12). (13). (l6), we have when 3n = 1 anJ a = <» 



u^ P R 3 [i _ !el - ^ rr V («5 - ^/) - i^ = 2-P R^ (^) ^ 



r. I D Ot 



J 



Putting R = /5 R 



The bubble ceases to grow wf^en 



Q/3 Pj, = (,5- 1 )/(,/^-/^ ) 

 or since is small compared with P , /? is large and approxiratcly equal to (3 P^/ O) » 



Thus if Q = 2 atmospheres (which will De the case for an ifxplosion at a depth of about 

 30 feet) and P = 9000 atmospheres, 13= 2.4. Thus if the incodiprtssibli; theory held rigidly 

 the bubble would osciUats perpetually citween the radii R and 24 R^. In the actual case energy 

 will be dissipated partly as a pressure wavij and pirtly in the form of heat. But the tendency to 

 oscillate should not Be entirely absent. Tne varied form of the surface effects as the bubble 

 breaks the surface may bo accounted for by the variation in size and internal pressure. The 

 spout-like form of the surface upheaval which occasionally is observed is what we should expect 

 from a high-pressuro, small— volume bubble, Tne occasional spout like form of the surface 

 uphdaval is ref-'rred to by Hillif;r, and he -quotes a cast of two shots of "0 lbs. 40/60 amatol 

 at 3 d^pth of 18 feet in which the upheaval reached a height of nearly 200 feet while in similar 

 shots the upheaval only reached -. height of 60 feet. The comparative rareness of its 

 occurroncG \z in accord with timi cons ider-it Ions ivhicn we now proceed to investigation. 



By equation (20) 



R„ / p I 'GS - i)*'{i -/3 (6'*/3+ i)//3^^* 



where BJ is written for 3 f^/Q since 5 is large. 



The t'-rm involvin.; ?^- will only be of importance when .'^ is large and then /S^ * /^ + 1 = /5^. 



/, p ,1' /?^d e 



'■^ o - p 3/2 1 , rr (21) 



!/?- if C3-' -/5¥ 



The integrand in (2l) is infinite when /3 = 1, /3 = /5 and imaginary whjn /3 > /5 . 

 If we write for the integral Ij + Ij + I, where 



T 3- ; f "^^ = 2 (fl- 1)* (3,f2* 4/3+ e)/i5 05^3- 1)* (22) 



^ - 1) j^ 1/3-1) 





(23) 



(fi-TI^ = ' ^(^0 - 1) - ^o' -/5')* >/3 (^0 - '^■ 



^ '^^■^ (5- l)n^^-^*"(0- 1)' (V- l)*>o- 1'' (^0^-/5'')* 



(24) 

 then 



