- 3 - 165 



The total vertical momentum of the water surroundtnff the bubble. 



During the period when the Bubble is near fts minimum radius at the end of the first oscillation 

 the vertical momentum of the water is approximately constant and equal to 

 7 

 luei L m lbs. feet/second. (6) 



where L is the scale factor in feet and m is non-dimensional. 



TTie minimum radius of the bubble - a^ 



a, (l - ca,~ ) = —J- m (non-dimensional) (7) 



Equation (7) must Be solved graphically. Using equations (5) and (t), a, has been plotted in 

 Figure 3 against the norvdimensional depth z . 



The pressure wave emitted by the collaps\ng and expanding bubble . 



During the period when the bubble is near its minimum radius a pressure pulse is radiated 

 outwards with the velocity of sound. For points not too near the bubble the peak pressure P (lb./ 

 square inch), the total positive impulse I (lb. -second/square inch) and the durat ion of the positive 

 pressure pulse (second) are given by the following formulae:- 



RP 3 J 



^ — , = 1— {l - i ca," ) (R.H.S. non-dimensional) (b) 



0.1J311 L^ « 7T aj 



where R is the distance in feet from the point to the centre of the bubble. 



The right-hand side of equation (8) is plotted against a,, in Figure U for a number of charge 

 weights. Using Figure 3, P^ may be tabulated for various charge weights and depths and is given as 

 a function of the depth in Figure 5. 



2 

 15.1 M^ 

 The impulse I = r — (9) 



R 7 ' 

 



The impulse I is plotted in Figure 6 against charge weight M. 



The duration >; 0. 218 T "^ 



1 

 0.9UU M 



(10) 



or t 



, ■? 



The maximum vertical velocity of the bubble - U feet/ second, 

 i_ HL- 



■I m 



■'"-^ = — 5 (R.H.S. non-dimensional) (ll) 



11.31 [7 aj 



u^ is plotted against the depth in Figure 7. 



The rise of the bubble at the end of the first oscillation - h. 



At the point where the bubble Is at its minimum radius it has risen a distance h (non- 

 dimensional) above the point where the charge was detonated 



1.19 

 h = -^ (non-dimensional) (l2) 







The rise h is plotted against z in Figure 8, 



