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THE BEHAVIOUR OF AN UNDER':i'ATER EXPLOSION BUBBLE 

 FURTHER APPROXIMATIONS 



A. R. Bryant 



March 1945 



Swnmary , 



A noinoer of approximate formulae relatinj to the Ojhaviour of an unfliirwatcr explosion 

 Dubole are presentiO nero as a supplen-tnt to thv; report "The bohaviojr of an unflerwater 

 explosion oubblc", h'^rjaftor call d Report i., Taylor's non-dimensional units are used 

 throughout. 



The equations and jraphs jiven make possible the calculation of the displacement and 

 momentum of the Duoole t wards a number of rijid surfaces, viz. an infinite plane, an infinite 

 cylinder, a spheri. and a disc. In the case: of the infinite plinc- th^ equations are based on 

 the work of Schiffm-.n, and on O.S.R.D. Report No. 3811, -md ire v,'\l id rijht up to the case where 

 the bubble touches the plane it its maximum r=,dhjs. 



For cofTipleteness two equations, based on jraphs given in Bureau of Ships (u.S. Navy) 

 Report 19U» - 1, are included wnereby the minimum radius i^nd the peak preesure in the pulse 

 emitted by the collapsing Bubbl : can be calculated in the case wher; all surfaces are absent. 



Introductton . 



This note is an extension of Report A in which equations and 'graphs were ^iven enabling 

 some of the principal quantities associated with the underwater explosion bubble to be determined 

 approximately. The equations wer:, cased on G.I. Taylor's thtory of the motion of the bubble 

 tojethor with Conyers H.-rring's theory of tni influence of plane free or rijio surfaces. 



Tne equations of motion of the DuDole near an infinite ri^id plane nave been extended by 

 Schiffman(l), beyond the approximation jivtn by H^rrinj, rijht up to the case whc-re tne bubble 

 touches the plane at its maximjn ralius. The int:i^ration of nis equations to give an approximate 

 formula for the momentum and Jisplaceaent , and a ccmparison with some exact integrations of the 

 equations nave been given in a recent paperCz). In the present note these approximations have 

 been converted into the- more familiar non-dirrfcnsional units given by Taylor, and slightly modified 

 to make them of more general application. 



A number of other approxiirate formulae relating to the motion of the explosion bubble 

 are added nsre, some extracted from the report "A simplified theory of the effect of surfaces 

 en the motion of an explosion bubble", hereafter called Report 3, in order to bring tne 

 collection of approximations in Report A up to date. as in that note, tne equations are 

 collected together in Part I for ease of reference, with thr^ir derivations omitted. Tneir 

 derivations are given in Part II. 



notation . 



The notation employed is tnat given in the previous paper. Taylor non-dimensional units 

 are usee throughout and denoted by small letters, i.e. ill lengths in fe.et are divided by the 

 length scali factor l = lOM , wh.'re M is the weight in lb. of T.N.T. havirig th. same total energy 

 as the bubble under considsriit ion. All times ir. divided by tne time factor (l/g) , whert* g 

 is the acceleration aue to gravity. As before, jll non-dimensional equ-itions will be I'bellcd 

 as such. 



