Response of a Vibrating Plate in a Fluid 



^X) -f aK ^(f) f dK' ^ (r,K)F*(K')> 



To make the discussion concrete let 



<F(K)F*(K')) = P(co)5(K' - (K - ^) ) 



which corresponds to a spatially uncorrelated pressure field with 

 convection velocity Ug and power spectrum P(to). 



Thus 



.00 4;*(aK)4;s(aK - ^) 



< 4>,<^s) = PM \ dK ^ ^^ (34) 



^-00 T(K)T (K ~) 



c 



The major contribution to the integral for Fmn. Eq. (31), comes about 

 when the peak of 4^n is close to the peaks of l/T(K); since ijjn is a 

 highly oscillatory function (period = Zir/a) with a peak at Xn A ^.nd de- 

 caying with the distance from this point and l/T(K) is a non-oscilla- 

 tory function with peaks whenever K equals the real part of the poles 

 which are roughly located at y times the four roots of unit and the 

 trapped wave poles near the branchpoints (k/(l +M ) )(k/(M- 1)). But 

 for the frequencies we are considering (up to 3000 Hz) only the pole 

 near k/(l+M) lies on the physical sheet. For this frequency range 

 the trapped wave pole is bounded bv and 0.15 and the pole near y 

 by and 0.6, Now, the peak of ^n is given by Xm A which is numer- 

 ically (see Appendix B) 0.155, 0.2b, 0.36, 0.465, 0. 57, 0.67, ... 

 and the period is 0.206. The height of the peaks decays roughly as 

 (1/Ka)3 is 0.05 of the first. The function l/T(K) also has peaks that 

 decay in the same manner. Thus , the infinite matrix F^n l^S-s appreci- 

 ably non-zero entries only in the upper left-hand corner. Consequently, 

 we need only compute Fmn for the first 4 or 5 modes, say, and then 

 invert this matrix + I to obtain the upper left square matrix of order 

 4 or 5 of Gn,n and thus for higher modes 



So that 



^mn - Smn 



<W„W*> = <(^X> 



for other than the first few modes. 



The contour for the integral in (34), Fig. 8, is similar to one 



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