- 9 - 291 



This result indicates a difference between the "ship" case and the "Dox-model " case. If we put 

 N = 1. it is iiTipossible to satisfy the criterion (lu) except if tlie thickness of the plate is 

 several times fts radius. The only practical case of this is a piston-type gauye, (such as the 

 Hilliar copper crusher). In such a case -p- is not smaH, and the investigation of II then shows 

 that cavftation cannot occur. It follows therefore that diffraction must always be considered in 

 the box-model case, criterion (lu) showing that solution (ll) cannot be the correct limiting form 

 for large R. The physical reasons for the two criteria are fairly obvious. Equation (ll) implies 

 that the plate is brought to rest by tension in the water rathi.-r than by the plastic forces. For 

 tension to occur at all, we must havt- R at least comparable with C 6, as was shown in II. Criterion 

 (l») implies that when the plate is brought to rest by tension, the plastic forces corresponding 

 to this deflection are sufficient to prevent any further movement of the plate due to diffraction. 

 We have seen that this impl ies that there must be considerable sub-division .of the panel of plating. 

 If we fix the size of the sub-panel, thi s impl tes fixing t, and cri terion (ll) is then sati sf ied 

 in the limiting case of larc;e R. If rtc take -^ as one foot, corresponding roughly to a destroyer, 

 and h = i inch, N comes out at about 20 - JO, corresponding to d^rragt- extending over an area of 

 the order of 1500 square feet of the bottom. This is a not unreasonable value, but the criterion 

 becomes increasingly difficult to satisfy for a ship with larger frame-spacing. 



It might appear that we need another criterion in addition to (it). If we replace the 



spring by plastic waves, it is necessary for (ll) to be a satisfactory approximation, that the 



plastic waves should have been able to reach every part of the plating by the time the diffraction 



wave has reached the centre. For ^ simple plate (n = l) this is obviously impossible, because 



the plastic waves only travel at a spe.-'d/^ , which is about lOS of t the velocity of sound in 



/ p- 



wate"-. Thus again, we deduce that N must be of the order of 20 ond thf- criterion is practially 



the same for plastic waves 3S it is for the equivolent spring. 



4 failure of criterion (l«) implies that the time for the plate to come to rest under 

 plastic forces is greater than ■^. Then if is small compared with _, we may deduce that 



2-^ wil 1 not change much in o time of the order of ^ and we can take -^ outsi de the i nteqral in 



equation (?) and neglect the (Jifference between t and t - t". The integral can now be evaluated 

 and the equation of motion is just that given by »lncompressiblg theory, equation (5). The 

 solution of this equation is given by equation (6), which for large R reduces to one term. To 

 estimate the error caused by assuming incompressible theory we substitute this solution in 

 equation (9) or (lO) the integril \.CTm involving Bessel and Struve functions of order unity of 

 the variable — j- . Reference to tables of these functions shows that for —2^ - 1 the error 

 is of the order of 104. For incompressible theory to fie a reasonable approximation in the absence 

 of cavitation for a large plate {a > Z 8) we therefore have the approxinete criterion ^^ < 1, or 



p„ C R > 12 7r N^ a h 



This criterion is not the exact reverse of (l4) which means that there is an intermediate region 

 where neither incompressible theory nor equation (ll) are satisfactory approximations. Actually 

 the two theories give very mucn the same results in this region (they agree when p C^ R = ^ n^ct h), 

 which indicates that diffraction is probably not very important unless criterion (l5) is satisfied. 



It must be pointed out that the criterion (l5) has nothing whatevtr to do with the criterion 

 for the validity of incompressible theory established in II for small platvs, as the latter applies 

 »»ien 5 is large compared with j., but criterion (l5) is not applicable, unless, in addition, is 

 small compared with ^ so that it is possible to regard the velocity as commjnicated impulsively. 

 These are both particular cases of Kennaros "principle of reduction" (5). 



(3.1) Circumstances in which cavitation reouces damage . 



8y comparison of the various models it is now a simple matter to decide the effect of 

 cavitation on damage if we assume that diffraction does not contribute significantly to damage 

 when cavitation occurs. W; may list the various possibilities as follows:- 



(a) 



