Resonant Response of Harbors (The Harbor Paradox Revisited) 



= I<2m^2:„)(z^ + Z22) - z'^r'lz^z,^! 



= |(Z,^ + Z^) cos ki +j{(bc) + bcZ^Z^} sin ki I | Z„ | , 



(5,5c) 



where (5.5c) follows from (5. 5b) through (5.4). The frequency de- 

 pendence of Gn(/c) Is qualitatively similar to that established in §4, 

 but /c„ - /Cp may not be small. The values of Gp and Qn may be 

 substantially larger than those given by (4.9) and (4, 12); however, 

 (4.17) remains valid for n^ 0, and the results therefore are of 

 limited interest. There also exist modes that correspond to reso- 

 nance of the canal itself, for which x = i is approximately a node 

 and the motion excited in H is small, but these, too, are governed 

 by (4. 17) in the sense that decreasing the channel width does not 

 affect the mean response of the canal to a random input except in 

 the Helnnholtz mode. 



We consider further the special case of Helmholtz resonance, 

 assuming ki « 1 as well as k A « 1. The equivalent circuit then 

 reduces to that of Fig. 4b. Calculating |Vq/Vj ] in this circuit 

 and neglecting terms of O(k^bi) relative to unity, we obtain 



GoU)= {:^(1 +a)^/C^ + [/cA(Ac) - if }''^^ (5.6) 



where 



a -hi /A (5.7) 



Is the ratio of the canal and harbor areas, and 



A(/c) = Ah + (1 + a)AM(ka) + (1 + ia)(i/b). (5.8) 



Resonance is determined by KqA^Kq) = i and yields 



G„= Q. = 2(1 +a)''^:' (5.9) 



and 



Po = i(l +a)''^''^^ (5.10) 



in place of (4.9a), (4.12a), and (4. 17) 



109 



