Response of a Vibrating Flate in a Fluid 



-i T^(K) = N^ 



(Xm + i^m^ 



lK) / cos Xm ch Xm \ 



MaK)^ - vi (aK)- + vl/ 



Kx. -iaK)( -V% ^ ^> O l 



with Ofp and Xm defined in Appendix B. 

 Now 



4;*(K)4;,(K - -if) = L(K) + U(K) 



where 



■'(K-S)' 



L(K) = S*(K)(s„(K - -g-) + T„(K - ^)e "' ) 



and 



U(K) = e \{K) (S„(K --g-) + T„(K - ^}e ^ ) 



Thus , we have 



00 



<^m^n> =P(u.) \ dK -iii^ 



U(K) 



'-00 T(K)T (K - -g-) 



The above expression is evaluated as a sum of residue contributions 

 plus branch- cut integrals. 



5 



^l-^ <*m<) = 2iriY Res \ ^^ [ 



'^'^^ H /t(z,)T*(z. - 4-)S 



(zj)T (Zj - u^) 

 L(z ) 



12 



+ Ziri ) Res 



j^ ^ = ^j fT(z)T(z - ^) 



j=9 

 where the z. are identified in the above figure. 



f I, +I2 



^ ^3 -^ ^4 



501 



