203 



Removal of Apf roxiiiat ton. 



Coiisiaer an elenent of the plate situated at Q distant x from the centre of the plate 

 (Pijure 3). T^hh reaction pressure at Q fnay be regarded as the ayyrc-gate of a series of (Elementary 

 pressures genorat'^d in time intervals St". The elementary pressure at time t due to wavelets 

 generated Oetwoen tim-is t - f -i 5t ' ind t - t' + i St" comes from the arc aB of radius 

 r = cf suDtunainj hn in,! 2 9 =-t Q. From the orthodox formul?,, the pressure due to this set 

 of wavelets is:- 



- p ii (l - f) Jr/TT- 



The ne-;n pr^ssurt over the pl^te due to all wavelets generated in the specified interval 

 dt" is thjrifore:- 



" (t - f ) or f ,„fl, ,, 



2 TT Ox 



77 R' 



/3 U (t - f) or F {r/R) 

 ,x = R 



in which F (i) 



77 R' 



Using ft = X + r - 2 xr cos 5, we find F (^) = 1 - i {sin 2 /3 + 2 /S) in which 

 sin-/3 = r/2R so long as r < 2R. wnen r > 2R, F (I) = 0. 



Integratinc, ajain with respect to t' the total mean reaction pressure ovpr the plate is:- 

 ,t 



-pc 



Jo 



(%'' 



) u (t - f) df 



or by integration by parts and taking u(o) = 



P = -pc I u(t) 



t) -is f /i -£^ u(t - f) df 



■7717 



(1) 



Hence if the applied pressure is P^ ."" the equation of motion of the plate (mass m per unit 

 area) ii:- 



du 

 31 



(2) 



Equation (2) an be solved by step by step rrrthods. Also the initial motion can be found 

 approxitiately oy direct integration using:- 



V = -/x 



u-i£ [ u(t) at 



(3) 



which is the approximate form of (l) when t is small. Identifying U(t) dt with displacement S, 

 this gives:- Jo 



2I1 * oc^ - l£sl 



pc 



dt ttR 



P e 

 



M 



the initial conditions being 



r - as . 



For the dinunsions givrn in the example of the main report the solution of (u) is:- 



dS , ■_o 

 dt 



P„ (e-"' * 1.191 e'-*""' - 2.191 e-"-''^"') 



2.508 pc 



Using 



