5 - 



(l) The effect of cavitation on diffraction. 



287 



Owing to th-s simultaneous Qc-curr^nce of cavitation and diffraction, it has, up to the 

 present, been impossiPle to extend the work of I to a finite plate, but one can bring forward 

 qualitative arguments which suggest that, for a sufficiently large plate, cavitation probably 

 interferes seriously with the contribution to danege due to diffraction. Although the complete 

 mapping out of the cavitation region would be a difficult matter, one can deduce enough about 

 it for present purposes. In II we concluded that the cavitation time is probably independent 

 of the radius of the plate, so that criterion for cavitation is simply that this time must be 

 less than the time ■;. for the diffraction wave to arrive at th& centre of the plate. By an 

 extension of the arguments that led to this conclusion we deduce that cavitation occurs, at the 

 Taylor cavitation time, over just that portion of the plate which the diffraction wave has not 

 reached Oy then. In II wo also deduced that the position of the tip of the cavitation zone is 

 sensitive to the thickness of the plate, so that even if the plate was a little below the 

 critical thickness, the length of the "beard" would be several times the radius of the plate. 

 We may therefore regard the cavitation zone as being, roughly, a cylinder. Tne diffraction 

 waves ircident on the surface of this zone will be reflected as at a fret surface. Sinc^; we 

 are cissuming that water cannot stand any tension the effect will be to cause further cavitation 

 and throw drops of water inwards at rijht angles to the surface. Since this surface is nearly 

 a cylinder, the drops will only teve a small velocity component towards the plate, and will 

 contribute little or nothing to daiiiage. We have also to consider the effect of diffraction on 

 the outside portion of the plate, where cavitation does not occur. The breadth of this portion 

 is equal to the velocity of sound multiplieo by the cavitation time, and its area therefore 

 becomes a steadily smaller fraction of the total .area of the plate, as Its radius Increases. 

 It remains to consider wheth';r diffraction would have any effect after cavitation has disappeared. 

 In 1 we found that the plate came to rest (with any reasonable assumption about the strength of 

 the spring) only at a i;inne several times greater than the characteristic time (9 of the explosion 

 pulse, and that even th-^n the layer of water built up on the plate is only about half the "wave- 

 length" C ^ of the pulse. Reference to the theory of diffraction at a rigid piston shows that 

 the main contribution of diffraction occurs within a time of the order of -»- so that if 

 diffraction is to niake any appreciable contribution after cavitation has disappeared R must be 

 several times as big as C i9. We shall show later that, if this is so, the effects of cavitation 

 and "follow-up" (without any assistance from diffraction) are by themselves sufficient to make 

 damage greater in the case when cavitatiotv occurs. 



Since we are trying to assess the effect of cavitation on damage for large plates it 

 seems fair to compare the case treated in I (large rigid piston with cavitation occurring over the 

 whole area) with the case part ially treated in II (rigid piston moving in an aperture in a rigid 

 wall, water bt'ing able to stand tension) as both these cases can Be treated rigorously. A 

 comparison of thi' behaviour of these two models as the radius becomes large should give a fair 

 idea of the critical conditions for the increase or decrease of damage due to tht: occurrence of 

 cavitation. The complete neglect of diffraction in the cavitational case is, for the reasons 

 just given, probably justified as a first approximation. 



(2) The "spring" approxinat ion . 



When we .re considering Urge plates the introduction and choice of a spring to replace 

 the plastic forces in the plate needs some care. we shall consider two types of case. In the 

 first we shall suppose the plate clamped at the edges only and surrounded by a rigid baffle. 

 This will be referred to as the "box-model" set of assumptions. In the second we shall suppose 

 the platt to be supported at intervals by stiffeners, clamped at the edges, and surrounded by 

 water on all sides. This latter case corresponds closely to the case of an actual ship's 

 bottom, so will be referred to as the "ship" set of assumptions. It is customary to take 

 account of the difference between "baffle" and "no baffle" simply by dropping the factor 2 

 representing the doubling in pressure in cas's whsre diffraction is important but actually the 

 point seems to merit a more careful investigation. 



A vecond point that requires examination, if we are using the 'box model" assumptions, 

 is that the plate is actually brought to rest by plastic waves travelling in from the edges. 

 For it to be possible to replace these by a spring one must first be certain not only that the 

 ratio between deflection and energy for the spring is the same as that for the plate, but also 



that 



