290 



- e 



definea as the function 1 - ^ (sin 27+27). sin 7 ^ 



piston is therefore: 



"l^-^^'^f^'i^) (&)...,"■■• 



The equation of motion of the 



BN' cr„ hy 



2 P^P" 



(9) 



where we have used the same "spring" term as before. N becomes unity for a plate clamped at the 

 edges only. For some purposes it is desiraSle to integrate the middle term once Dy parts, and we 

 obtain: (cf II equation (l)). 



4 



'^ d? ° dt TjR .0 y urt^ J [oi J ^ _ ^. 



8N a hy 

 



1? 



2 P^P' B 



do) 



It is possible to ot.tain the solution of either form of this equation in integral form by use of the 

 Fourier t ransfomv.but the resulting integral is intractable. We can, however, deduce enough about 

 the behaviour of the solution for the purposes of this report by simpler means. 



We wish, first of'all, to establish whether for large R, there is any case where diffraction 

 can be neglc-cte-d. For such a case one coulo. accoraing to the discussion of section 3, deduce at 

 once that cavitation will increase damage. If wc assume that the effect of diffraction is 

 negligible, we neglect the integral trrm in equation (lO) and obtain the case of an infinite plate 

 without cavitation treated by Taylor (3). The approximate solution of this equation is: 



2 p / t t \ 



dt p,,C (l-a) l^'^ '^ ) 



(11) 



which leads tcj the following result for the final deflection: 



ym 



(12) 



It can be shown that for any reasonable size and thickness cf plate the effect of the spring term 

 only introduces a small correction into expression (12). To assess the effect of diffraction we 

 substitute the' solution (ll) into the integral term of equation (lO). Since the expression (ll) i = 

 always positive we can obtain an upper limit for the effect of diffraction Dy replacing the factor 



('-*-)' 



by unity, ano a lower 1 imit by replacing 1 1 oy 1 - — i_- 



For large R these 



two quantities are nearly equal. The value of the integral is thus given in order of magnitude 

 by:- 



ac_e ^ 



TT R (1-a) 



4CJ Pn 



TT R (1-a) 



i- p- 



p- B 



t 

 ■ a + a" 55 



t < 



(13) 



-a-P~ aB 



4, / » 



t > 



2R 



Tables of the basic integral have been prepared. Tnus for equation (ll) to ce a satisfactory 

 solution of equation (lO), tntse two .expressions must be spiall compared with the applied pressure 

 2 p p~ ff . This requires in the Ti^rst place that — 5- should be small, i.e. S must be large 

 compared with the "wave-length" C 5 of the pulse, which is, in general well satisfied for a ship, 

 and fairly well for the bcxwTiodel . This criterion is, however, not sufficient, because, if it 

 is satisfied, it implies that expression (l3) contains an approximately constant term cf order of 



magnitude - _ Pm and it is necessary to verify that the steady deflection such a pressure 



tTW 

 would cause ar^'inst thr spring is Irss than expression (ll), for solution (10) not to oe invalidated. 

 The condition for this is: 



tc e p, 



~R 



K^ cr h 



C' R < tt 77 n' c- h 

 



' P.. ~ 



(!■*) 



Thi s 



