Resonant Response of Harbors (The Harbor Paradox Revisited) 



holds only for the Helmholtz mode and otherwise must be replaced by 

 the weaker paradox that narrowing the harbor mouth has no effect 

 on the mean- square response of the higher modes to a random input 

 in the absence of friction (narrowing the mouth increases friction, 

 thereby decreasing the response, in a real harbor). It follows that 

 the higher modes are not likely to be strongly excited, but that the 

 Helmholtz mode may dominate the response of a harbor to an exterior 

 disturbance that has significant energy in the spectral neighborhood 



of COq. 



Carrier, Shaw and Miyata [ 1970] consider a harbor that 

 communicates with the coast through a narrow canal and find that 

 both Gq arid Qo are significantly increased (as might be inferred 

 from the analogy with the classical Helmholtz resonator; cf. Rayleigh 

 [ 1945] , §307) . We show in §5 that such a canal is analogous to an 

 electrical transmission line and may be replaced by a symmetrical, 

 four-terminal network for the calculation of Vp (see Fig, 3). The 

 analogy with the transmission line rests on the hypothesis that only 

 plane waves are excited in the canal. An examination of the effects of 

 higher modes shows that the elements of the four-terminal network 

 may be appropriately generalized, but that the plane- wave approxi- 

 mation is likely to be adequate if the breadth of the channel is less 

 than a half-wavelength. 



^2 I 



b 



I, > °- ]^1l"^12l 7- |^22"^12] -<; -E 12 



I 



Fig. 3. Canal and equivalent circuit for the plane-wave 

 approximation. The impedances Z|, = Z^p and 

 Z are given by (5.4) 



The precise determination of Z.y^ and Z|^ requires the solu- 

 tion of an integral equation for the normal velocity in M (or, in the 

 case of an intervening canal, a pair of integral equations for the 

 normal velocities across the terminal sections of the canal). The 

 formulation of §§2 and 3 yields variational approximations to Z.y^ 

 and Zu that are invariant under a scale transformation (i.e. a 



99 



