HP) 



Response of a Vibrating Plate in a Fluid 



Ua-)[BA(.,P)-«(.,P)] / 2ipoc(k-.M) \ 



\V k^+(M2- 1 )a^- ZkMa- p^ / 



where or = A' - it. So that 



^mnrs = J ^^^ Z residues dP + J 4^n(Pb)I(P) dP 



The branch-cut integral is exponentially damped rather oscillatory, 

 so it may be readily performed numerically using Laguerre-Gauss 

 quadrature. The second integral is more difficult, it oscillates with 

 a period 2ir/b. 



In solving Eq. (22) maximum values of the indices are polstu- 

 lated. This is justified, since the index is inversely proportional to 

 some length on the panel. Now there certainly exists , from the 

 experimental point of view, a smallest length to which a disturbance 

 may be localized. After having solved Eq, (22) the plate displace- 

 ment is simiply the Fourier transform of (21), thus, 



W(T) = ^ W„n<Pm© <Prs (^) (23) 



m,n 



where <p^ is a beam eigenfunction. 



To find the sound pressure level in the cavity the expression 

 for W from Eq. (23) is inserted into Eq. (5b) to give 



2 v^ Y^ cos ^cos , *^ cos k^n(z + d) 



m,n r,s 



where I^^ and J^g are given in the appendix B. 



Similarly, the radiation may be computed from Eq. (12). 



The force is not a deterministic function as has been im- 

 plicitly assumed from the outset, but a stochastic variable whose 

 correlation properties are known, either via a model or directly 

 from experimental data. Thus, it would only be meaningful to com- 

 pute statistical averages of the response based on statistical averages 

 of the force (i.e. cross-correlation). The procedure to be used is 

 an amendment of a procedure due to Rosenblatt [ 1962] , the notation 

 is that of Rosenblatt. 



495 



