Mites 



and is 0(l/An) for \k - Kn\ » i/A^. The point K = K^ corresponds 

 to parallel resonance (Zn = co) , for which the total flow through M 

 vanishes (I = 0) whilst (r remains of the same order as a-f. We 

 define the Q of the resonant response near /C = ?Cn as the ratio of 

 the resonant frequency to the half-power bandwidth, such that [the 

 frequencies at the half-power points are proportional to /c}, (1 * "I Qn )] 



G[«„(l ±q;')] = 2-'^'5„. (4.11) 



Substituting (4.7) into (4.11) and invoking (4.10), we obtain the first 

 app roximations 



Oq= 2.K0 =Go (4.1 2a) 



and 



<^n=^^'X^ = -^'^^n^l• (4. 12b) 



Now suppose that the incident wave is random with the power 

 spectral density Sj (f) , such that 



o-f = \ Si(f) df (co = 2Trf), (4.13) 



where f is the frequency. Generalizing (4.5), we obtain 



.^,Y 



r^^ 2 



0- = > \ Si(f)|G,(Ae=)| df (4.14) 



for the power spectral density in the harbor. Substituting (4.7) into 

 (4. 14), invoking co = cac/VA, and calculating the contribution of the 

 resonant peaks at oj =00^ on the hypothesis that their bandwidths are 

 small compared with those of Sj(f), we obtain 



o-^=(gh/A)'^ ^P„Si(?J, (4.15) 



n 



-I -1/2 2 f** 2 2-1 



Pp = (4Tr) ^n Gn \ [ ^ + (QnAn) (^ - 1^^) ] d/C (4. 16a) 



1 "^ I /2 I '^ 2 

 ~Z«n QnGn iOn/i^rT 00) (4. I6b) 



106 



where 



