electrostatic case. Thus in a free surface the induced charge is of opposite sign, so 

 that there is repulsion; and in a rigid surface the induced charge is of the same sign, 

 and there is attraction. 



The Bjerknes' developed the analogy and brought in the notion of polarization, 

 and this suggests a way of interpolating between free and rigid boundaries . . . within 

 the realm of inviscid, incompressible hydrodynamics. Recently Birkhoff has used the 

 idea of polarization and stressed many other analogies in an article in the von Mises 

 memorial volume. The interpolation can be made in analogy to the behaviour of a 

 homogeneous dielectric. What corresponds to the dielectric constant e is the inverse 

 of the density: p _1 . As an example, consider the model of an explosion bubble as a 

 charge Q in water above a "bottom" taken to be a fluid of density p. The fluid 

 boundary is assumed to remain plane. The square of velocity is neglected, otherwise 

 there is no straightforward electrostatic analogy. The boundary conditions are continuity 

 of velocity, and pressure continuity. These can be met by using images just as in the 

 electrostatic problem of a point charge and a plane dielectric: in the water the potential 

 is (r from bubble; r' from image) 



*i = Q/'r + ( ~ J Q/r' 



and in the bottom fluid 



2Q 1 



P+ 1 r 



In the appropriate limits this agrees with the well known cases. This may be useful to 

 improve calculation of bottom effects on explosion bubbles well away from the bottom. 



One can make a somewhat more realistic model of the bottom, imagining it to 

 be a mixture of two fluids of mean density p, and then proceed in analogy with the 

 microscopic physics of dielectrics. 



The other remark has to do with the behaviour of a rising bubble. Calculations 

 on Herring's theory give consistently larger rates of rise than are observed. Herring's 

 theory constrains the bubble to remain spherical. Work by Kolodner and Keller, in 

 which deformations of the bubble are taken into account, improves the agreement with 

 experiment. The trend is evident from a simple consideration: that the bubble flattens 

 owing to underpressure at its sides, once moving, and as it flattens the virtual mass 

 increases. For a torus, for example, the virtual mass is roughly twice that of a sphere 

 of equal volume. One would then expect the bubble to slow down appreciably com- 

 pared with a sphere. 



G. E. Hudson 



Dr. Snay has commented on the studies of water columns arising from shallow 

 explosions, and he showed us a picture of the Bikini test Baker water column as an 

 illustration. Now this column was vertical, or nearly so, and I should like to point out 

 that this is by no means always the case. In fact Dr. E. Swift and Mr. George Young 

 of the Naval Ordnance Laboratory made a series of experiments in which the ambient 

 atmospheric pressure was varied above a shallow layer of water in which a small charge 

 was detonated at mid-depth. I have spent some time in analyzing the data from these 

 experiments. 



The most striking effect noticeable in high speed motion pictures of the resulting 

 water columns is the variation in slope of the column generators as the pressure changes. 

 The accompanying Figures I A, B, C, D show sequences of frames of the motion pic- 

 tures which illustrate this. The depth d of the water layer was also varied, being 0.39 

 inches in Figure I A, 0.78 inches in Figure I B, 1.17 inches in Figure I C and 1.56 

 inches in Figure I D. 



348 



