276 - 8 - 



where y represents the central deflection and S = — • — f- being a rreasure of the energy 



P " 

 required to stretcn the plate plast ically, a^j being the yield stress, (in the "piston" 

 approximation we cannot fix this coefficient unambiguously). 



Comjiarison with the "Piston" Approximation. 



Equation (18) is very similar to equation (8), differing from it only in the numerical 

 coefficients. In a later report it is hoped to examine a case where the corrections to 

 incompressible theory are not negligible, but where cavitation probably does not occur. 

 Equation (l7) differs from equation (2) both in the numerical coefficients and also in the fact 

 that the first order correcting term is proportional to the time integral of y instead of to y 

 itself. The reason for the latter difference is that the edge of the plate is being kept at 

 rest, so that the effect of the diffraction wave is less abrupt, and therefore less important 

 in the very early stages of thf motion, than it is when the velocity jumps discont inuously to 

 zero at the edge. The forrrer difference means that, for a very large plate where both types 

 of correction term are negligible, cavitation will set in according to the 'piston" approximation 

 at precisely the time given by Taylor's (2) simple theory, but according to the "paraboloid" 

 approximation at an earlier time The physical reason for this is that, in the latter case, 

 the velocity of the central point (and therefore also the relief pressure at the centre) is 

 li times larger (relative to the incident pressure) than in the former. As we decrease the 

 radius of the plate the effect of the correction due to diffraction (tending to speed up the 

 motion of the plate and thus to hasten cavitation) will presumably become important In the 

 •piston" approximation before it does in the "paraboloid", so that it tray be that the curves 

 cross. Whether they do or not cannot be settled without a detailed calculation. This brings 

 us to an Important criticism of both models, which renders such a calculation rather superfluous. 



Criticism of the Assumption of "Proportional Motion ", 



In both the models discussed sbove we have made a fundamental assumption, namely, that 

 the law of distribution of velocity over the surface of the plate is the same for alil times. 

 For this reason we get the paradoxical result that an increase in the pressure near the edges 

 tends to increase the velocity of the central point and thus to reduce the pressure there. In 

 the very early stages of the motion when the deformation of the plate is elastic, such a mechanism 

 cannot be ruled gut, as elastic stresses can be propagated from the edge to the centre of the 

 plate faster than can the diffraction wave in the Water. In general, however, any lateral 

 effects propagated elastically at such rates should be very small unless the plate is so thick 

 that shear deflection is important. The situation is even more definite if one assumes that 

 practically the whole of the plate is stretched plastically (as seems likely if it is left with 

 appreciable permanent deformation) since in this case lateral effects will be propagated along 



the plate at the comparatively slow velocity /_o which is only of the order of lOt of the 



^ P 

 velocity of sound in water. It is thus doubtful whether any apprecicable effect on lateral 

 motion can be propagated through the plate from edge to centre as fast as the diffraction wave 

 through the water. If this is a true representation of the situation, then it can only mean 

 that the central portion of the plate will move according to Taylor's (2) original equation 

 and that, apart from small elastic effects in the early stages of the motion, it will only "know" 

 that it is part of a finite plite when the diffraction wave arrives. Although we cannot yet 

 prove it rigorously, as we can in the rigid piston case, it seems fair to deduce that the same 

 criterion for the occurrence of negative pressures will apply, i.e. that cavitation must occur 

 either before the arrival of the diffraction wave at the central point or else not at all. 

 On this assumption, we would deduce that Kirkwood's (6) cavitation criterion (that the Taylor 

 cavitation time must be less than r. j should be rigorously correct and not merely an approximation. 



Experimental Evidenci'. . 



As these "plastic" and "proportional motion" assumptions lead in general to quite 

 different results (see Table l) for the critical thickness of plate at which cavitation occurs, 

 and for the cavitation times, it should be possible to decide betwt^en them experimentally. A 

 little evidence of this type is already available (7), similar charges being fired against 

 various thicknesses of plate and the resulting cavitation photographed.. So far, it appears 



to point 



