134 



is snell the first term in (l») is much greater than the others so that (m) may be Integrated 

 approximately jivinj 



f = ^ /T^a'^'^ = 1.0025 6-'''^ (15) 



Substituting for a' (6) Becomes 



_ 2£: = _|_ I f"^df = ii f. (16) 



df f'" 



which may be integrated to 



2.0 



(17) 



Using (is) and (17) up to t' = 05 and subsequently solving (l4) and (6), step-by-step, 

 the results for z' - 2.0 are shown in Figure 1. The radius of the bubble, the height of Its 

 centre and the level of its highest point are shown. 



It will be seen that the bubble at first rises slowly but that its centre jumps rapidly 

 through « height about equal to twice its maximum radius during the short period while its radius 

 Is only half the maximum radius. 



The minimum radius is given by a' = 0.211. This seems strangp, because if the vertical 

 motion is neglected the approximation in which G(a)/w is also neglected implies that the Bubble 

 collapses completely into a point; thsre is, in fact, no gas pressure to prevent this collapse. 

 When the vertical motion is taken into account the collapse is (theoretically) preventad by the 

 fact that at a certain minimum radius the whole energy is concentrated in the flow due to the 

 rapid vertical motion of the spherical hollow. 



Similarity on varying scale s. 



Since Figure 1 is non-dimensional it is possiole to assign to it any des'red linear scale. 

 The scale is, in fact, determined only by W and for any given explosive this appears to be 

 proportional to M. Thus for T.N.T. the unit length is, from (?) and (10), 



li r ,n "li 



1.85 



L .j^JlJ . 1^ ^-"^ " !" J m' = 65.91 M» (18) 



when M is expressed in gm. of explosive and L is in cm. 



The depth below a point 33 feet ibove the sea is in this case 2L. Thus Figure 1 represents 

 a single series of explosions in which J depth of wa^er * 33 feet ^ j^ f;^ga. 



(charge diameter) 



* set of scales of depth and the corresponding positions of the sea surface fn relation 

 to the explosive are shown at the side of Figure 1. Scales are given for M = 2200 lb., 300 lb., 

 10 lb., and 1 oz. It will be seen that the level of the sea surface is well clear of the bubble 

 through the greater part of the motion for M = 2200 lb. at 103 feet and 300 lb, at 50 feet depth 

 and that these two explosions might therefore be expected to give similar pressure distributions. 

 The 30 Id. charge at 14 feet would give a similar Bubble during the greater part of its expansion, 

 but n^ar its greatest expansion it would be -iBove the surf-.ce ^ind similarity would break down. 

 It is evidently impossible to place a cnarge of less than 30 lb. in water so as to give a regime 

 which is similar to that produced by the two larger charges. On the other hand, if it were 

 possible to reduce the atmospheric pressure to less than 33 feet of water, small-scale models could 

 Be made; moreover by using the correct pressure a model experiment could be carried out which would 

 represent either of the large explosions. Thus if a 1 oz. charge were exploded at a depth of 

 10.0 - 2.U = 7.6 feet below the surface of Aater and the air pressure above the water surface were 

 reduced to that of 2.4 feet of water, the resulting OuOble would (in the present approximation) Be 



similar ,.... 



