448 - « - 



Co mparison of Fixed '.'xgtrf Disc with Infinite Rigid Plane . 



In Figure 2 the "attraction coefficient" (lO) for a fixea rigid disc has been divtded by -l/ik) , 

 the value for an infinite rigid plane, and plotted against r^ = d/c. It will be seen that the 

 attraction of the disc is always less than that of the plane and falls off quite rapidly with increasing 

 distance. Thus, for an explosion bubble at a distance of one disc radius the attraction is only half 

 that for the infinite plane at the same distance, whfle the fraction is one seventh at a distance of 

 one disc diameter. 



The Attraction of a Moving Rigid Disc . 



The extension of the foregoing to the case of a rigid disc moving with velocity v along the axis 

 of symetry, i.e. towards or away from the explosion centre, will now be made. The co-ordinate system 

 is fixed in space and the surface r = o is made coincident with the disc at the instant considered. 



The velocity potential due to a disc of radius a moving along its axis with velocity v away from 

 the explosion centre at r^ = d/c is 



^ = ^ 9^ (ir) s (12) 



If this is added lo <i> = p. * <p2 ^^^ complete veloc ity potent iai for a moving rigid disc and a unit poi nt 

 source is obtained. Since this added potential is independent of the strength of the source the 

 analysis given tn R.R.L. Note a0m/210/aR8 must be modified slightly. Pressure gradients and velocities 

 along the axis of symmetry will be additive so that the drift velocity or displacement of the bubble 

 due to the motion of the disc may be calculated separati^ly and added to that due to a fixed rigid disc - 

 equation (u). 



Following the physical arguments of the above mentioned paper the effect of this added velocity 

 potential is two fold. First, the mot ion of the disc imparts a dri ft velocity B t^,/3 x towards the 

 disc to the water in the neighbourhood of the explosion centre. Second, a pressure gradient 



3x 



(^) 



Is set up in the water in the neighbourhood of the explosion centre and this gives the bubble a drift 

 velocity towards the disc 



2 f' 3 C^h. 



? o 5! I a t 



Adding these two drift velocities, separating out the geometrical factor, and inserting the value of 

 (^ from (12) gives for the drift velocity u of the- bubble towards the disc due to tne disc's motion . 



-1 



v.ij / a^^ dti (13) 



where r„ = d/c • 



3^ - 1 ^3 



since — -^ = along the line s = 1, i.e. along the axis of symmetry. 



3 X '■^ 3 r 



For convenience of use the geometrical factor in the first bracket in (l3) has been plotted 

 in Figure 3. It will be seen that the influence of the motion of the disc falls off more rapidly with 

 increasing distance than the attraction due to the disc's rigidity. 



If the acceleration dv/dt is a constant independent of time it may be removed from the integral 

 in (13). This integral is now the same as that occurring in the equations of motion of the bubble 

 under the influence of gravity and in the absence of surfaces; in fact the velocity and displacement 

 of the bubble oue to the acceleration of the disc arc a constant fraction k of the velocity and 

 displacement of the Dubble under gravity, where 



