433 



|4 I * 3ah - 4 I aV dt I (7) 



Strictly speaking a dipole source should be added to to allow for the linear motion of the 

 bubble (3), and further mult I poles to allow for its change of shape. The effect of "Induced" image 

 sources due to these will be second order small quantities if the effect of the primary image 

 distribution is a first order small quantity. Similarly, since the drift is a first order small 

 quantity the error due to assuming the distance from bubble to surfaces to remain constant will be 

 a second order small quantity. 



The velocity U, is oscillatory and is directed away from rigid surfaces in the first half of 

 the oscillation, and towards rigid surfaces in the second half, its Integrated effect over a whole 

 cycle being usually negligible. The velocity U, due to the pressure graOlent set up by the image 

 sources is directed away from rigid surfaces as long as the pressure is positiva, i.e. as long as 

 the bubble is greater than the hydrostatic pressure at the same depth. When the Dubble pressure 

 falls below this hydrostatic pressure the pressure gradients are rGvers>;d and the velocity u, 1 = 

 then towards ?. rigid surface. Since the force exerted on a hollow vessel is proportional tc its 

 volume the bubble acquires most of its momentum when it Is large ana as the pressure is below the 

 flomal hydrostatic pressure when the bubble is large the net effect at the end of one osc i 1 1 tton 

 is a drift towards rigid surfaces. These statements may assist an understanding of the phenoiienc 

 found by Temperley in calculating the effect of a deformable target plate on the motion of th" 

 bubble, and in particular the correllation between the sign of the motion and the sign of the 

 pressure in the bubble. 



Approximate Formula for the Pi st'lacement of the Bubbl e. 



The Integral for the momentum of the bubble towards a surface, and that for the vertical 

 msnentum due to gravity both become nearly constant towards the end of the first oscillation. It 

 is in this region that most of the displacement of the bubble occurs. Hence it is a reasonable 

 approximation to assume that the total displacements at the end of the 'Irst period due to the 

 surface and to gravity are in the same ratio as the momenta. Denoting these two displacements 

 by S and H respectively we have 



■5^1 6 J" aV dt 



3 



(R.H.S. Non-dimensional) (8) 



Where the intcyrals are to bt taken over the whole porioa, S Is towards the origin of the 

 origin of the co-ordinatF x. 



In obtaining (8) all lengths and times on the right hand siae hav<^ been converted to Taylor's 

 non-dimensional units, i.e. all lengths are divided by the standard length 1 = (w/gp) i, where W 

 is the energy of the motion, p the density of water. All times are divided by the standard time / i 



r-* has the dimensions L~ . in addition the samll oscillatory term 3a 4 in the expression (7) 

 o X 



has been dropped, since It may be exactly integrated and shown to be very small at the end of a 



period. In Table 1 the value of these integrals are calculated for a number of cases, and compared 



with two suggested approximate formulae. 



Table 1 



