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TABLE 1 . 

 Values of a and /? for which cavitation is just prevented. 



Thus, It will Dfr seen that the effect of the diffraction at the edges, in "hurrying on" 

 the onset of cavitation, may nade cavitation possible at distinctly larger thicknesses of plate 

 than we should deduce from infinite plate theory. 



Propagation of Cavt tation along the Axis . 



Equation (l C) can also De applied for non-zero values of x. One gets a transcendental 

 equation slightly more complicated than equation (ll), and can then obtain an idea of the extent 

 of the cavitation zone by solving this equation for various values of x. The limit to the 

 cavitation zone is set by the point at which the diffraction wave from the edge (represented by 

 the fourth term in equation (lO)) arrives at the instant when the pressure would otherwise drop 

 to zero. The corresponding value of x should measure the length of the "beard" of bubbles that 

 photographs show near the plates. Unfortunately, it seems that, in practical cases, the 

 length of the "beard" is very sensitive to the other parameters, becoming very long compared with 

 the radius of the plate even when one is not far off the critical conditions for the occurrence 

 of cavitation at the centre of the plate. The reason for this is as follows. The position of 

 the tip of the "beard" is determined by the diffraction wave, travelling a distance (x + R ) 

 withe velocity of sound overtaking the cavitation front, which travels a distance x, slightly 

 faster than sound but is slightly hsndicapped by the fact that cavitation does not begin until a 

 finite tiire after the arrival of the incident pulse, whereas the diffraction wave starts off at 

 this instant. If a is small, cavitation begins almost immediately so the handicap is small, 

 and the cavitation front spreads a great distance into the water before the diffraction wave can 

 overtake it. Since we are concerned, in practice, with snail values of a, we should expect the 

 theoretical length of the "beard" to be sensitive to a and hence to the time-constant of the 

 pulse, and some specimen calculations showed that this was so. Attempts were made to compare 

 the theory with the experimental results on the appearance of cavitation quoted in various U.S. 

 reports (7), but the fact that the experimental knowledge of the time-constants of small charges 

 Is rather uncertain made it impossible to dj more than verify that the theory gives results of 

 the right order of magnitude. 



The "Paraboloid" Approximation . 



An objection to the "piston" approximation is tnat it does not allow for any bending of 

 the plate. Since we should allow for the effect of immobilisation of the edges of a clamped 

 plate. It would be better if we assumed the velocity to be distributed over the plate In such a 

 way that the edge is at rest. The simplest assumption of this kind that one can make is that 

 the velocity is distributed according to a parabolic law „ , ju — -ZJlf f(t) Although the 



° fi^ 

 observed final forms of dished plates are often not unlike this shape, it does not follow that 

 the velocity of a given point is always the same fraction of the central velocity. Indeed, 

 experimental work on the motion of various parts of a circular diaphragm and theoretical work on 

 the mt-chanism cf a "wave of fixity" travelling from edge to centre by which the diaphragm is 

 brought tc rest both make it likely that the approximation is only rough. Its consequences can, 

 however, be investigated without much trouble, and it seems a more natural approximation than the 

 •piston" one. Rayleigh's method for a piston vibrating with frequency — can be modified fairly 



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easily, or we could derive an equation analogous to equation (l). 



The 



