Maestrello and Linden 



K- plane 



X / « X 



Fig. 11. Contour for r_- Eq. (31) 



The contour is deformed in the appropriate half-planes and 

 branch cuts are replaced by poles as discussed previously. Thus, 



J_r 



2Tii ■^mn 



= Res 



{^'} ^ n {^} 



-{^>}-Ke.{^} 



+ Res 



f G-(z) \ 



We now invert the matrix S^n ^ Tmn* Let us denote this inverse by 

 Gjj . Then 



where the tilde denotes Hermitian conjugate and sum of p and q is 

 implied. The final step, then, becomes the diagoncdization of the 

 CPSD matrix ( Wm wj) thus giving the PSD for the actual degrees 

 of freedom of the system. 



c) Acoustic Power, Power Radiated 



In computing the power radiated, we nnake use of the unitary 

 matrix Ujj which diagonalized the CPSD matrix. The average 

 power radiated, to the far field, P, is given by 



p-<i-!^*lfi) 



506 



