498 - 2 - 



to that region for which experimental knowledge is the most detailed, and the calculations reported upon 

 here were put in hand in order to fill up this gap. 



2. Details o f the Cal cul atiorts . 



The theory used in suDstant ial ly that developed Dy Herring n.O.R.C. Report CU - Sr20 - Oio) and 

 Taylor ("Vertical motion of a spherical Bubble and the pressure surround it"), who both developed formulae 

 to allow for the effect of gravity, and of the proximity of free and rigid surfaces. The effect of 

 dropping the assumption that the bubble remains spherical has been investigated by shiffmann (Applied 

 Mathematics Panel '■(emo. 37-5) =nd, in more detail, Dy Teniperley. The theory of the bubble formed Dy an 

 explosion in contact with a rigid surface was also given by Temperley. The theory of the behaviour cf 

 the bubble from a charge fired between two rigid surfaces is now published for the first time. 



The calculations were all mads for a detonator equivalent to une gram of T.N.T., and a total pressure 

 of 36 feet (_ f water, i.e., the detjnator 3 feet below the surface of the water at atmcspherlcal pressure. 

 In Taylor's njn-dimensional units, which we shall use henceforward, the values of the relevant quantities 

 are as fcllows:- 



Factor to convert non-dimensional lengths to feet (L> = 2.2 feet 



Factcr to convert non-dimensional times to sec -nds (T) = .261 secnd 



Non-dimensional total external pressure (z ) = 16. 3 



NorHdimensional initial depth below free surface (d^^) = 1.3 



5. Comparison of the Calculations with approximate theory . 



These conditions were chosen because they were thought to De fairly representative, but they do not 

 exactly correspond with any of the previous experiments. Wright, Campbell and senior used a No. 6 detonator, 

 which corresponds almost exactly to 1 gram of explosive, but the depth was only 1.6 feet and not 3 feet. 

 Lieutenant Campbell's experiments included work at 3 feet, but he used a No. 8 detonator, which Is stated by 

 him to contain about 0.7 gram of explosive. The best that we can do appears to be to compare the results 

 of this calculation with the aoproximate formulae developed by Herring and Bryant. |f the approximations 



prove satisfactory for the two cases calculated (conditions as above, effect uf free surface first neglected 

 then taken into account) we can then apply them to other cases with confidence. without these calculations 

 by Nautical Almanac office we should have to extrapolate the formulae from the range z = 1.5 to 7.8 (over 

 which they have been checked against the results of direct solution cf the hydrodynamic equations) to the 

 region z = 16, an undesirably long extrapolation. The following table accordingly gives the results of 

 this comparison:- 



