63 



an asymptotic one which should be a good approximation when h 

 is two or three times ^^a^^ ' ^ clue as to how T varies for 

 smaller values of h is provided by considering the case h = 0. 

 The motion of the water for this case must be the same as the 

 motion to one side of an imaginary plane drawn through the center 

 of the bubble produced by the explosion in free water of double 

 the charge used. Since T is proportional to the cube root 

 of the charge, 



3_ 



T(0) = T{oo ) v2 (16) 



VJhen free and rigid surfaces are simultaneously present 

 in the neighborhood of the bubble, the effect on the period, 

 though calculable, is not simply the sum of the effects due to 

 the various surfaces separately. For example, it is shown in 

 Appendix 5 that when the explosion takes place halfway between 

 the surface and a horizontal rigid bottom, and at distance h 

 from either, the effects of surface and bottom do not cancel 

 each other; instead we have 



a log 2 

 T-T (oo) (1 --^;h- ) 



Similar calculations can be made when a free and a rigid surface 

 intersect at an angle. In general, when both kinds of surface 

 are present the effect of the free surface tends to predominate. 



