172 



APPENDIX 



Effect of the Proximity of a Free or Rigid Surface . 



Conyers Herring has shown that if the charge Is exploded at distance d (non-dimensional units) 

 from an Infinite free or rigid surface the Bubble acquires a velocity v towards that surface, given 

 by 



V ^ f 1. [ 3 . a** (2|)^ dt (non-d i, liens ional) (23) 



a^ w' °' 



where the upper sign is taken for a free surface, the lower for a rigid surface. Since 

 usually the two most important surfaces are horizontal (e.g. the sea surface and the sea-bed) this 

 velocity may be added to the term- g| in equation (13) for the motion, i e. the g^avity term. Its 

 effect on the minimum radius, the peak vertical velocity and the peak pressure in the bubble collapse 

 pressure wave may be allowed for by re-defining the momentum constant m which determines them (see 

 e.g. equations [t) , (8) and (ll)). Hence if m" is the vertical momentum constant for a charge 

 detonated 6 non-dimensional units below the sea surface or above the sea-bed, then 



/ a^ [l--^ a (^)^ ] dt (non-dimensional) 



4d'^ °^ 



Using the method which led to the approximate value of m (equations (iB) and (5)) -^ives 



m' = m (l- 0.52 -2! 2_ ) (non-dimensional) (5a) 



d^ 



•here the constant 0.52 is determined from one full step by step solution of the equations 

 of motion of the bubble (13). 



As regards the effect of the surface on the rise of the bubble Conyers Herring gives a simple 

 approximation which m.ay be ^rxpressed in the following way:- 



Rise of the bubble at end of first oscillation, in proximity to the sea's 

 surface or the sea bed 



h' = h(l - i -■,■■) (non-dimensional) (l2a) 



5 a' 



where h .s the rise in the absence of the surface effect (see equation (l2)). 



Finally Herring has shown that the presence of a surface alters the period of the motion, 

 so that if T' is the period in proximity to a surface, and T is the value given by equation (3), 



T' = T (iT|) (25) 



where a is the average value of the bubble radius (non-dimensional) over a complete 

 oscillation the upper sign refers to a free surface, the lower to a rigid one. Using the assumptions 

 involved in computing the vertical momentum constant m (page (>) , a is a constant fraction of the 

 maximum radius a , so that (28) may be written 



f = T (1 T 0.21 -i? ) (3a) 



