Response of a Vibrating Plate in a Fluid 



T(-H- + z*)T-'(z ) T(if + 4)T"'(Z2) 



l-(zJ 



In a similar, though simpler fashion, we nmay evaluate the integral 

 in Eq. (3 2). 



^[^m©] = Am(K) +BjK)e- 



iKa 



where 



AJK) = -llip(Xm)' [v - cos xm - eh Xm iKal 

 f" a \ a / L^m sm Xm ^ sh Xm J 



and 



B (K) = ^^^ /Xm\^ r Xm (cos Xm sh Xm "^ sin Xm ch Xm ) - ^Ka sin Xm s^i Xml 

 "" a V a / L sinxm +shXm ^ 



Thus ijjn •^[ "^ml niay be decomposed into factors analytic in upper 

 and lower half- planes , respectively. Denoting these terms by G 

 and G" we have , 



F - C G-^(K) +G"(K) ,^ 

 ^'"" ~ J^ ^TK] ^^ 



where 



G^K) = Am(K)(Sn(K) + T^We'"^) 

 and 



G"{K) = Bm(K)(Tn*(K) + S*(K)e) 

 The contour of integration is indicated below. 



505 



