264 



10 - 



2 2 

 of the total energy falling on the plate, is simply 4S y 



K). 



When the plate has come to rest 



only the water which, before cavit=>tion, lay between the origin and x = x has contributed to damage. 

 Since the velocity v^. is a decreasing function of x, it is clear that, once the plate has come to 

 rest, the pressure c.in never ajain build up to a value large enough to set it in motion. Thus, the 

 kinetic energy of the rsro^ining "bubbly" water is ill wasted. From the expression (3) for the 

 velocity, we ca/i write down the l<inetic energy and integrate from x to infinity, the results also 

 Being entered in Table 2,. Finally, by subtracting all these percentages from 100, we can obtain 

 the energy lost by collision at the rr^lcading front while the plate is actually moving. This can 

 also bo inferred directly from the step-by-step calculations, Dy a numerical Integration of the 



quantity u jj- (g^ - v^j (n) which can easily be shown to represent the rate of loss of energy in this 

 way {the total energy oeing unity). The agreement of these two methods provide a satisfactory 

 over-»ll check on the computations, which appear to be accurate to within 2i at most. Table 3 

 below gives an idea of the dissipation of energy (expressed as a percentage of the total energy) that 

 has occurred up to a given instant. (Time expressed in non-dimensional units). 



TABLE 3. 



Energy lost by col L ision up to a given time 

 (as percentage of total energy). 



The figures for a = 1 ire approximate only. 

 (7) The third phase of the motion. 



It seems advisable here to call attention to yet another effict that is experimentally known to 

 be important for a finite plate, thought it vanishes for an infinite plate. The motion may not be 

 complete, even though all the originally 'bubbly" w;ter has piled up on to the plate. There is still 

 a gap left In the water, ?.s the net result of the processes we have followed so far has been that the 

 plati has been pushed in, and the cavitatcd water has all followed it. In any nctual case, the 

 resulting gap wculd be filled by water thnt has never cavitated, which would fjllow iimiedlately after 

 the last .f the "bubbly" water, and wcyld b; orlven inwards by its hydrostatic pressure. Now we have 

 seen that an infinite plate in general comes to rest long before all the cavitation has disappeared, 

 so that there may well be an Interval between the second and third phases of the motion, as Indeed 

 experimental work suggests that there is. (See, for example, (9) T.M.B. feport R.248. Evidence 

 suggesting the 3.ame thing Is being obtained at Admiralty Undex Works). Whether or when the 'build-up" 

 of pressure caused by the filling up of the gap is enough to force the plate forwards aipin remains a 

 matter for detailed Investigation, but the following rough argument (based on energy considerations) 

 shows that the effect will have to be considered. Consider a plate clamped along a circle of radius R. 

 It receives energy of the order of m'.jnitude ^m ^ r/ R^ 



The mean deflection of the plate is gi 



by equating this to the plastic energy 4 77 i cr y 

 2 "J "^ 



ho that we have:- 



Pc 



8 1 CT 



(24) 



Volume ^cf dish = 77 R y^ and the energy acquired Oy water ...t pressure p^ entering this volume Is 

 P5 77 R y^, and this is comparable with the energy already acquired by the plate If:- 



/2 acr 6' 

 _2 ~ /. o 



p_ / R"^ D C 



(25) 



where 



