432 



In addition the imaje source produces a presence p at any point in the water. For points 



1—2 . The pressure gradient -^ In the 

 fi t Sx 



neighbourhood of oue to the image source at is thus. 



not too close to the pressure p is simply p -. 



2£i 



9x 



3t 



3 X 



at 



(3) 



Now it is a matter of common experience that a bubble or hollow in a liquid in which there 

 is a pressure gradient due to gravity will move upwards in the direction of the pressure gradient, 

 i.e. from rpjions of high pr-ssure to regions of low. In tht same way the explosion DuBble will 

 tenfl to drift in the direction of the pressure gradient set up by the image source at ; this 

 will be away from the surface AS if e is positive. 



G.I. Taylor has derived by a simple physical argument an equation for the velocity of the 

 bubble U due to the action of a pressure gradient jp. The argument runs as foUoivs. The 

 "floating power" of a spherical hollow of radius e. due to the pressure gradient g/3 is tTT a ojo, 

 and this is therefore equal to the vertical momentum communicated per second to the water flowing 

 round the spherical hollow. The inertia of the nuss of water effectively moving with the bubble 

 is ^ TT a'^p (a well known hydrodyn.imical result), and its momentum is hejice ^t a p U where U Is 

 the velocity of the centre of the bubble. Thus ^ (| tt a^pu) = i rr a"' go, from which follows 

 equation (u). The same equation is obtainea By Conyers Herring by his perturbation method. 

 Using Taylor's equation for U 



ga^dt 



(«) 



It is clear that any pressure gradient, no matter how produced, will cause a similar drift velocity. 



Thus a pressure gradient p — — ■'£- due to the inage source will cause the bubble to acquire 



dt [ d X J at 



a velocity U, towards , where 



3t [ 3 X J at 



(5) 



The total drift velocity of the bubble U towards the rigid surface is thus the sum of 

 U, and U,. 



In the particular case discussed here it can be seen that c^ 

 the bubble towards the rigid plane distance d from the bubble 



and the velocity U of 



1 P a^ A Uh) dt 



3 a 5 ^ 3 



~ ~T — n 



(6) 



on integrating by parts. Equation (6) is that given by Conyers Herring and by Taylor. 

 Extension to t ke General Case., 



The argument used above for the simple case of a rigid plane is quite general, and can 

 easily be extended to the general case. The result Is simply stated here. 



Let there be any set of free or rigid surfaces symmetrical about the axis x of co-ordinates 

 passing through the centre of the explosion bubble 0. Assume that a distribution of inage sources 

 can be found which satisfies the boundary conditions at the given surfaces when a unit source is 

 placed at 0. Let the velocity potential due to these Image sources be (^. Then the drift velocity 

 U o' the bubble in the direction of the x jxis towards the origin of co-ordinates Is 



