. r - 261 



layer of water Is buiU up upon It, It by no means follows that all this energy Is available to 

 cause plastic damage, In fact, this will only occur If the velocities of water and plate are nearly 

 equal at the montent of collision, during the whole motion, which is not probable. 



( «) The second phase of the motion . 



In order to throw some light on this point, we must Investigate the motion of plate and water 

 after cavitation. Attempts were first made to do this by an application of Kennard's(t) theory of 

 the propagation and disappearance of cavitation, but they proved Ineffectual for two reasons. 

 First, Kennard(u) uses Eulerlan co-ordinates, whereas Lagrangian co-ordinates, which enable one to 

 follow the motion of each particle of water, are much more appropriate to this problem. Secondly, 

 It Is not always possible to assume that the volume of "cavity* Is everywhere small compared with 

 the volume of water, and this Invalidates some of Kennard's formulae. 



Let us fix our attention on a particle of water, which, at the Instant of cavitation, was at 

 a distance » from the origin. Cavitation will occur here at a time given by equation (7) and the 

 particle will start towards the plate with a velocity given by eqiation (8). The particle will 

 travel effectively In a vacuum with undiminished velocity until it collides with particles ahead of 

 it. Since the velocity given by equation (8) decreases with Increasing x, the particle cannot 

 overtake thote ahead of it until they have been brought to rest, which can only happen if they collide 

 with the plate or with the layer of "reconstituted" cavity-free water that is being built up on the 

 plate. In other words, when our particle arrives at this layer, the water already in the layer will 

 be just that which, before cavitation, was between our particle and the origin. Thus, neglecting a 

 smell correction, due to the compressibility of the water, the thickness of the reconstituted layer 

 Is just X, Meanwhile, the plate has moved in ^. distance y. so that the time at which our particle 

 arrives at the "reconstituted" layer Is given by the time the cavitation frpnt takes to travel to x, 

 plus the time the "bubbly" water or 'spray' takes to travel back a distance y to reach the layer, 

 which by this time has grown to just the thickness x. The mechanism may be compared with the 

 transfer of a pack of cards, one by one, from hand to hand. The assumption that the same mass of 

 water fills the same volume before and after cavitation is justifiable from the conpresslblllty point 

 of view " Pm ^ "^ Pq '' "•^''^'^ '' ^^^ ordinary condition for the applicability of 'acoust I cT theory. 

 It Is, however, still possible that bubbles may persist for sane time, e.g. If they are due to 

 dissolvaJair or If they are fairly large so t^iat their period of oscillation is appreciable. If we 

 use the "spray" concept, the oscillation of the bubbles would be replaced by the rebound of some of 

 the water from the "reconstituted" layer,. Frcm the mathematical point of view. It is imireterlal 

 whether we use the concepts of "spray" or of "bubbly water', and we cannot yet distinguish very 

 Clearly between them experimentally, 



(5) The equations of motion of the plat e. 



On the basis of the above discussion, and using equations (7) and (8), we obtain one relation 

 between t, x and y. 



-^ - log ^ sinh 



m * '^ '"" [to 



(18) 



where v^. Is given as a function of x by equation (8), The second relation Is obtained Sy momentum 

 considerations. The momentum of the plate and reconstituted water 1$ (p * p x) 2i . The rate 

 at which momentum Is brought up by the "bubbly" water is p ^. v . We thus have:- 



dt 



(P.^PoX'sf 



^0 ^ ''c * '^^ "' >■ = " f»») 



"here •jrjj is the pseudo-frequency of the plate under whatever forces are acting upon It, in our Case 

 presumably plastic forcss. For a plate made of material with yield stress cr ana clamped along a 

 circle of radius R, we navu for ci) the result:- 



Equations (l8) and (l9) are sufficient to determine x and y as functions of t, our initial conditions 

 being that at t = 9^, x » O and y and g^ are given by equation («). A complete discussion of the 

 problem would require numerical Integration of these equations for a set of values of two parameters. 



