Response of a Vibrating Plate in a Fluid 



field be denoted by ^» By applying the Fourier transform on the 

 (x,y) coordinates, one gets the ordinary differential equation 



^^^^>^>^) +CS(«,P,2)=0 



(6) 



dz' 



where 



2 2,2^2 Z 



t, = k + (M - i)a - 2kMa - |3 



il;(x,y,z) = \ \ d« dp e*^''**^^^ i(«.P,z) 



k = a>/c, M is the flow Mach number and c the speed of sound in 

 the region above the plate. Only the positive exponential solution to 

 Eq. (6) is chosen, since it is the solution representing outgoing 

 waves. Thus, 



ilj(cv,p,z) = A(a,p)e 



iCz 



(7a) 



The boundary condition, arising from the continuity of normal dis- 

 placement is 



d4;(a,p,z) 

 dz 



= - icLW 



z=0 



z=0 



where the differential operator 



L = k + iM -^ 

 ox 



Thus, 



r\ 



ijj(a,p,z) = - c -y— e * 



(7b) 



Now, since 



LW = C r dx' dy' e"*^"'*^^'^ LW(x',y') 



then 



487 



