702 



Where H(a) = —-, ^ 



.f.i^ 



ana Xjj 15 

 in (3) is 



e initial slevaticn o' the element 05, and is ijiven Dy equation (i). The integral 

 De taken over the whole of the rejion occuoieo Oy the initial cavity. 



In oenney's methca of evaluating (i) two aooroxi mat ions are mafle. Firstly, the integration 

 ii r=-trict?J to that :crti;n of tre cavity 5nc»n in Fiijrr i where the «at?r surface Is 

 initially t)tlow the unnisluroea level, i.e. r V5ries from zero to/^h. Seconoly an aooroximate 

 exoression foro) is usoo, which is illustrates in Figure 2. Here a clan of the water surface is 

 shown, Being the vertical axis through the Charge, ana = being the ooint distant R from C, at 

 which the w;V5 height is tc oe calculatso. The circle of radius h/T reoresents the area within 

 which the water surface is initially deoressed. Then in (3) O) is aooroximated Dy the quantity 

 R - X which tends too) when S is large compared with h. With these two aooroximaticns (3) may 

 be reduced tc 



>^ 



(U) 



and where 



Sumerx cal values . 



It was assumed that the energy in the bubble motion o' "olar Ammcn gelignite was 

 ma calories cer gram, i.e. the same as that used in most calculations of bubble behaviour for 

 T.N.T. This gives for a 32-10. charge, h = 11.3 feet. The maximum bubble radius h is calculated 

 from the formula given In <-. crevlcus oaoor'^'. The wave height y was calculated at a distance 

 R = 5h = 56,5 feet. uo to t = 3.75 seconds equation (j) was used, while for greater times 

 equ'ition (4) was jsfd. In tnis latter equation the Integral was oerformed numeric^lly by dividing 

 the range of z into twenty equal stecs. The integration is estimated tc be accurate to at least 

 51; for most of the range the errors are orobaCly not more than 24. 



The calculated wave amclitude is shown in Figure 3 by a full curv= uo to about 9.25 seconds, 

 when the third trough has just reached the coint under consideration. The slight discontinuity at 

 t = 3.75 seconds In this curve is due to the change from equation (2) to equation (4). Equation 

 (2) takes account of the whole of the cavity while (4) ignores the effects of that oortion of the 

 initial disturbance outside a radius of /2h, where the water is initially above the undisturbed 

 water level. 



Coniparj son with experimental observations. 



"enney's theory acclies strictly only when tne total deoth of the water is very large 

 comcared to the bubble radius. In the exceriments described' ' the total decth o' water was about 

 15 feet. The wavelengths observed Old not exceed 25 feet so that as far as the orocagation of 

 waves is concerned the difference between the exoeriment and the case in which the decth was assumed 

 infinite will be =1 ight.- The chief effect of the bottom will be to oistcrt the shaoe of the cavity 

 though It should not greatly alter its volume. 



The case whore the charge was fired at 8 feel deoth is most nearly 1' .ivisaged by 

 the theory, since at this decth the to= of the bubble will just reach the done the bubble attains 

 its maximum volume. Accordingly the wave amolitudes observed at a distance o' -j feet from a 

 3?-lD. charge 8 fert aceo have been taKen from the excerimental oaoer' ' and blotted in Figure 3 by 

 a broken line. As the zero of time for this curve was not known it has been arbitrarily chosen to 

 give the best fit. 



It will be seen that the agreement between theory and exoeriment is remarkably good, at 

 least uo to the second crest. From the calculations it seemed likely that the lack cf agreement 

 for times later than about 9 seconds might be due tc the actual cavity being stallower than the 

 assumed cavity owing to the crescnce of the bottom. To test this idea a cavity, having the same 

 volume as, but shallower than, that assumed by 'enney was artitrarily chosen. This is shown by 

 a broken line in Figure 1. A few coints, corresconoing to two cres:s and two troughs were 

 recomouted using this modified cavity shaoe, ana are clotted in Figure } by circles. These are 

 in better agreement witn the observed wav^s, excect for the crest at 8 seconds,. 



It may 



