- 5 



501 



It is clear from these results that the discrepancies are due to the fact that the bubDle never 

 closes up as much as inccmpressible theory indicates, owing presumably tc the Ksses of energy that sesm 

 to occur near the minimum. If the observed values for minimum r.idius are used instead of the calculated 

 ones in the calculation of the maximum velocity, much better agreement is obtained. The values ■ btained 

 thus are shown in brackets, but they are :nly very r.ugh, as they aro inversely proportional to the cube 

 ^f the minimum radius, which i s ni. t known accurately. 



U.3. T he shape of the bubble : This can be dealt with very sh, rtly. Ttt calculations made by Nautical 



Almanac Office applying the the:ry developed by Temporley shows that (contrary tc^ what was at first expected) 

 large departures fr_m the spherical shape might be expected to occur even for the case of a detonat.T 3 feet 

 below the water. This wLuld not occur (even if the incompressible the^'ry were valid right up to the 

 minimum) until a few hundredths of a millisecond before the minimum, so that it wjuld be extremely difficult 

 t. detect experimentally. It may therefore be said that the fact that the photographs show the bubble 

 stayiiy very nearly spherical for several oscillations is not in disagreement with theory. 



li. u. T he behaviour of the bubble from a charge fired in contact with a steel plate : Photographs'of the 



bubble from a detonator fired under these conditions were also obtained by Lieutenant Campbell. The effect 

 of gravity on a bubble in this case was investigated theoretically by Temperley, who found that, for a 

 bubble above a horizontal steel plate the effect of gravity should be to make the bubble become pointed 

 during the early stage of its oscillation, and then to flatten itself against, the plate just before it 

 reaches its minimum. The photographs show that, although the bubble certainly does flatten itself against 

 the plate, this process is fairly gradual and is spread over a considerable proportion of the oscillation, 

 in contradiction with the theory. This discrepancy may be connected with the one already noted, that a 

 charge fired jn contact with a steel plate does not produce so large a bubble as a similar charge in mid- 

 water. This matter seems worth further investigation, as it may have some bearing on the behaviour of a 

 "contact" charge. 



11.5. A charge fired between two steel plates, ("the bubble that splits in hrlf") : Lieutenant Campbell 



investigated the effect of firing 3 detonator exactly mid-way between two vertical steel plates. Although 

 the bubble rushed towards one or other of the plates unless the detonator was fairly accurately centred, 

 it was possible to obtain two other types of behrvi ur. If the plates wpre fc.irly far apart, the bubble 

 behaved very much as if they were not there at all. |f they were fairly close together, the bubble divided 

 just before the minimum into two equal halves, one of which moved rapidly t^^wards each plate, and flattened 

 itself against it. F. r the No. 8 cop, the critical distance between the plates was fvund to lie between 

 12 and is inches. It was decided to investigate this phon^menin mathenatical ly, as it seemed t,. afford a 

 very good Mpp^^rtunity of nvr.king a fairly stringent check ^n the whole theory ..f the dist^rtiin ^f the bubble. 



we consider a bubble with its centre mid-way between two vertical rigid surfaces. we neglect the 

 effect of gravity and of the surface of the water. A rnotion of a few inches parallel to the plates would 

 not affect otion in a perpendicular direction very much. (At the time this theory was developed it was 

 imagined that the effect of gravity in distorting the bubble would not be vzry large, at any rate until the 

 first minimum is reached. as stated above, this is not quite true, but it is sufficiently nearly true for 

 the present investigation to be valid). The method used is a modification of the original image theory 

 due to Herring. we use spherical polar co-ordinates, and take es the 5 axis the line through the centre of 

 the bubble perpendicular to the plates. we take the profile of the buoble to be:- 



R = a+b^ P2(cosi?) ♦ b^ P^ (cos £*) (l) 



The odd harmonics wi 1 1 be absent owing to syrmietry considerations. 



Allowing for possible departures from the spherical shape, we assume the velocity potential expanded 

 as a series of axial harmonics. In order that the boundary conditions at the rigid surfaces should be 

 satisfied, we introduce an infinite series of images exactly equivalent to the compound source that represents 

 the bubble in an infinite sea. If this proceeding were permissible, the velocity potential would take 



A, P, (cos 8) '^ \ 1 



r? ° n=2 (r^ 4. n^d^ + 2nd r Cos &)* (r^ + n^d^ - 2nd r Cos 5) - 



similar series for A , A etc. (2) 



Where 



