Response of a Vibrating Tlate in a Fluid 



^ y d"^g(K|73)w(T3) 



plate 



where 



(18) 



g(k| r') = -i^ j dr e g( r | r') 



plate 



Substituting these results into Eq. (15) gives 



BT(K)W('k) + BA[W] -Q[W] =f(K) (19) 



where 



X(K ) = k' - -^' - ^P°^'^f - ^'^'^ (1 9a) 



Let \\>n denote the finite Fourier transform of a beam eigenfunction, 

 <Pj^, It follows from the Fourier representation that the ijjn^form an 

 orthogonal set on the infinite interval. Thus, expanding W as 



W(K)=^ W„n4iJora)ij;n(Pb) (20) 



ln,n 



or alternatively, W as 



W(T)= ^ W„„v„{|),„(X) (21) 



m,n 



and introducing these expressions into (19) and subsequently utilizing 

 the orthogonality, gives 



W + / r W = ^ (22) 



where $jnn are the projections of 



.(k") = liM = y $„„^„,(aa)4;n(Pb) 



T(K) i^„ 



491 



