237 



This is Identic*! with Herring's equation (j) except that a term has been added to allow for the effect 

 of gravity. Equations (*.6) and (a. 9) together with the equation 



U = - ii (A.10) 



dt 



may be used as in Report a to find the motion of the bubble using step-by-step methods of numerical 

 integration. For this purpose it is convenient to transform equations (a.6) and (a. 9) into non- 

 dimensional forms by substituting 



lation (a.6) then assumes the form 



^df J u(2- - h'g)2 ^dt-y 2 



2('' - ^•.) 



G(a) 



6 3 



(A. 11) 



•here u' ■ dr'/df, h'^ • Lh^ and h is the value of 2 at the liquid surface, so that the depth of the 

 explosion below the surface is Z- - 'i- = '-(r* - h' )• Equation (a. 9) assumes the form 



( .3 ., 3 - 3a-' d / 3 «a' \ 



- a-'u') « 2a'-' ♦ 5 I a'-* \ 



f «(Z' - h'j2 dt' ^ dfj 



(*.12) 



