Forces on an A.C.V. Executing an Unsteady Motion 



£ ' ( £ , y, z, t) = (- k + (n + 1)L -2<r , y, z, t) for n odd. 



We add <f> to <^ , ^ to ^ , ^ to ^ , and so on. 



This only alters the exponential factor in Eq. (22), which now beco- 

 mes : 



factor = 2 exp|i(w(- a - £' ) + u(y - y 1 )) } ' cos|w( a + £ ) } 2 exp(2inwL) 



V 



Li 



The integral with respect to w of this factor in Eq. (22) can be 

 simplified using the Poisson summation formula to give 



t 



^jf dS, f dT "H dupS r 



*'( £ >y> z >0 = - y^-r// dS ' / dT „?„/dup T (r.y 1 , r ) 



coshjk (z + d)( sinJ7(t--r)f 



• • T71 — J\ ' cos jw (<r + £ )[ 



cosh(k d) 7 l n ' 



n n 



exp{i(w (-a -£') + u(y - y'))| , 



where 



w = 7r n / L 

 n 



2 2 2 



k = w + u 



n n 



and 



^n 2 = gk • tanh(k d) (3 7) 



We now satisfy the condition on the side walls of the tank by 

 including the image ACVs on lines parallel to the tank centerline. The 

 procedure is similar to that just carried out, and if we assume that 

 the pressure distribution is symmetric about the x axis, then 



t 



*(i.Y.^)--~ff^f ^r Z^v, p S r U\y',r)... 



S'. , 



cosh]k (z + d)\ sinK (t - t )\ 



mn ' « mn ' ( , „ . > 



••• ZTi T\ ' cosjw (cr + £H 



cosh(k d) 7 ( n ' 



mn mn 



• exp)i(w J- a -£') + u (y - y*))[ , (38) 



where 



49 



