Response of a Vibrating Plate in a Fluid 



Thus only Ig and I^ will contribute. Therefore 



U(z) 



PX;;y<«^m<t>n) = 2^i 2. f.^^ I 



T(z)T*(z - ^ ) 



II 



+ I2 



+ Ziri ) Resi ^4^^ + I, 



j4^ z = ^i/T(z)T*(z - ^)\ 



i^9 



By the same argument which eliminated I| and I3 we may replace 

 I2 and I^ by pretending that z- and Zg are on the physical sheet 

 and thus 



U(z) \ 



Ig = - Ziri Res 



6 ^T(z)T (z - ^) 



and 



1.= - Zui Res I ^^ 1 



^ = '9 ^T(z)T*(z - ^)^ 



To evaluate the residues it is necessary to determine 



^•/ » /I 3 1 iu-Y (3zM k + z M^ - z M - 3kM z - k z - k M) 



1 \2.) — 4z T —I — ! , ■■ . 



Vk^ + (M^ - l)z^ - 2kMz^ 

 The following relationships among the poles are valid 



- ^ + * - ^ + * - J^ + 



Z4 - -^ r Zy. Zj - -^ T Zq Zc - t^ t Z£ 



CO * CO , * 



Z|0 = u^ "•■ Z| Zjl ~ ^ ^2 



■I^LT*(z-i^)J ^ z^-iT*L77:f J^ ^^X'^)^ 



L 



2 _ _ 



♦2, ,2 

 . 4/. Z M \ 



^*., *^ 4^ ^I^V(I--J^) 



T (z) = z - \ + 



Vk^ + (M^ - i)z^ - 2kMz' 



503 



