202 -'- 



Equation (3) and {*) jive strai jnt forwarC solutions which change form at intervals T. 

 By measuring ti.nt in each case from tho o^jinning of ^JCh of these intervals we havu:- 







P e""^ - pc U 



P^ e-"^ e-"* - pc {Uj - Uj) 



P e-^"^ e'"' - pc (U, - U ) 



t < T 



T < t < 2T 

 2T < t < 3T j 



(5) 



Integritinj an^ adJing, we get 



<^ '-tc 



■' 



1 - e 

 or the final velocity is:- 



pcT '^ 



P P 



^ ' n (m + /9cT) ° n (n + p**) 



(6) 



This result is inaeoondent of c and (to the approximation made above) should agree with the 

 solution given in tne report "Danage to ship's plates By underwater explosions". According to 

 equation (22) of that pap.:r the value of U^ is P^/n (m + j^f pS) . 



The factor t% arises through taking into account tne differences in reaction pressures 

 over the plate area. 



The way in whicn the final velocity is reached is illustrated in Figure 1 which holds for 

 the case 



r. = 2(^.27 ^rins/sg.cra. (1.023 incnes iron plate). 



n - 626.5 Seconds 



c = 50CC f.p.s. 



R = 3.19 feet. 



The value of n is approximately that occurring on the Detonation of 750 16. torpex. The 

 remaining figures are chosen so as to jive the round values pc/.r.n = 12 and nT = O.uo thus simplifying 

 the calculations. 



In Figure 1 the velocity curv.^ up to nt = O.U is that of Taylor's theory. If water will not 

 stand tension then th" plate acquires the rnaxinum velocity shown in this stage oT the curvu, and the 

 rest of the curve is irriilcvant, but if water will stand tension then the effect of reduction of 

 relief pressure becomes ap?ar-nt after t = T nnO thj vrlocity shays a further incrc.ss, tending oy 

 a s-^ries of os-illations to the v?lje given By (6). 



Figure 2 shows the nett pressure accelerating the plate. For the theory to hold, the 

 tensions shown on that curve must not exceed those which water can stand. 



The theory qiw^n is only approximate and depends on whether the centre point can be taken 

 as typical in representing the whole reaction pressure over the plate. It does, hcwever, show in 

 what way the infinite plate theory is linked with tnat cf a finite plate in an incompressible fluid. 



Appendix 



