Response of a Vibrating Plate in a Fluid 



since <dM(X7^^)dM (Xtt^ )) = R(T-T',t-t') dT dt. 



Thus the Zj^ are independent random variables with unit 

 variance and with zero mean if EX^ . = 0. The transform of (26) is, 



X7-* = \ Z-r VR(K,u))e dK dco (27) 



A simple calculation reveals that (27) satisfies (24). 



These results will now be applied to the plate displacement. 

 Thus, the cross-power spectral density (CPSD) of the plate response 

 is given by (asterisk denotes complex conjugate) 



< W(T)W*(T')) . ^ <p^ g) <p, (g) <p, (f ) .,3 (^) < W,„ W^3*> (28) 

 m,n,r,s 



Now if the solution to Eq. (22) is represented as 



^mn ~ /_! ^mnij ^i 



mnij "^ij 



then 



<W^nW*> = ^ V^nij^rskl^^.j^l) 



ijKI 



Further, 



($,^*>=r r dK dK' ^i(^a)4.,(pb)^^.'a)^,(p'b) ^f (^t) f*(K')) 

 ^-00*^-00 T(K)T (K') 



The force F is now identified with X-*.. so that 



<f(K)f*(K')) = [^(K,a))R*(K',oo)]'^^ ( Z^^^ Z^^ ) 



= [r (K ,w)R (K',u)J 6(K - K') 

 In summary then 

 (W„„W*) = I v.„,vL f aK ^ l '-H | (Pb) | R(Kco)|,:(.a),r(Pb) 



497 



