62 



out quantitatively by the method of images, and it is shown that 

 when the center of the bubble is a distance h below the surface, 

 the period T(h) is related to the period T{a3 } in the absence 

 of a surface (but for the same p ) by 



T(h) - T(oo) (1 - ]^) + O(^) (14) 



where a is the time average of a over a complete period. 

 Near the maximum, the radius-time curve is contracted in the 



time direction by the factor 

 with the curve for h ■ oo. 



1 . !mx + o(^) 

 ^ 4h ^^^2' 



as compared 



The theory of Appendix 5 also predicts that when the 

 explosion takes place at distance h from a plane rigid surface 

 the period will be longer than in the absence of such a surface: 

 qiiantitatively 



T(h) - T(oo) (1 + ^) + O(^) (15) 



The qualitative explanation follows the pattern outlined above 



for a free surface. The stream lines avoid the rigid surface, 



da 

 so that for given s. dt » ^^® kinetic energy is greater than in 



the absence of the surface. Thus the effective inertia of the 



water is increased and the period lengthened. The formula (15) is 



