Resonant Response of Harbors (The Harbor Paradox Revisited) 



is a dimenslonless measure of (the square of) the frequency (similarly, 



' ^ ' ' i , we obtai: 

 of 2Ci> by 



Kn - kn-A). Invoking (3.3b) on the hypothesis a /A « 1 , we obtain 

 Oj = iJVj I for the (temporal) meam-square elevation of 



virtue of which (4. 2) reduces to 



= o-f^G,^/c). (4.5) 



2 



cr 



The hypotheses (1.3a,b) imply |Zn| « | Z|^ + Z^\ for each 

 of the modal impedances in the summation of (2.13) except in the 

 neighborhood of /f = K^t where the sum may be approximated by 



2, 



Zh ^ <J" A > [ ^ + H^n(/fn - k) ]' (4. 6a) 



where 



K7,fm' (4. 6b) 



^H = I 



and 



where 



m = being excluded from the summation. Invoking (2»14), (3,6), 

 and (4,6) in (4,3), we obtain 



Go(/c) = ji/ + [/cAo(/c) - 1]^!''^^ (4.7a) 



Gn(/C) = \if \{{K- k/ + [ (/C - /C„)A„ - H•^l^(''^^ (4.7b) 



Ao(/c) = AH + AJka), (4,8a) 



A„ = A„ + A^(k^a) (n gt 0). (4, 8b) 



The peak values of Gn are given by 



~-l -1/2 



Go = 2/Co and fi^ = 2^/ A„ (n ^t 0) , (4,9a^b) 



where K = K^ is the series- resonant point determined by 



7coA(/Co) = 1 and /c^ = «„ + fx^A' (n^tl). (4.10a,b) 



The amplification factor drops off sharply on both sides of K = ACp 



105 



