Resonant Response of Harbors (The Harbor Paradox Revisited) 



of a scattered wave, ^g) , the non- radiated energy stored in the ex- 

 terior half-space, and the energy stored in the harbor (we also could 

 incorporate an empirical, resistive component in Z^, say R^^, to 

 account for an energy dissipation proportional to R|^|l| ). We infer 

 from the solution of the corresponding acoustical radiation problem 

 [Miles 1948; §3 below] that both R|^ and X|^ are bounded, positive- 

 definite functions of co, by virtue of which we may regard them as 

 single resistive and inductive elements, respectively (although neither 

 R[^ nor X|Y| has the same frequency dependence as its elementary, 

 electrical counterpart). We infer from the analogy with the corre- 

 sponding acoustical resonator [Morse 1948, §23] that Z^ comprises 

 an infinite sequence of parallel combinations of inductance Lp, and 

 capacitance Cn, which bear a one-to-one correspondence to the 

 natural modes of the closed harbor and resonate at the corresponding 

 frequencies, oj^ = (Lp,Cp) , together with a single capacitor Cq, 

 which corresponds to the degenerate mode of uniform displacement, 

 for which co© - 0- The solution within H may be expanded in this 

 infinite set of modes, with the root-mean- square displacement and 

 the kinetic and potential energies In the n'th mode being proportional 

 to the voltage across C^^ and the energies stored In Lp, and C^, 

 respectively. The arguments of the preceding paragraph suggest 

 that the Individual modal Impedances are Important only In the 

 neighborhoods of their respective resonant frequencies, and hence 

 that Z^may be approximated In the neighborhood of co = co^ by a 

 lumped Inductance, say L^^, In series with either Cq or the single, 

 parallel combination of Lp, and Cp,, such that the energy In all modes 

 but the n'th Is proportional to L |lp. The corresponding equivalent 

 circuit Is shown In Fig. 2b (we gfve a quantitative derivation of this 

 equivalent circuit In §§2 and 3). 



Fig. 2. Equivalent circuit for harbor opening directly at coastline: 

 (a) Implied by (3.2); (b) Implied by (3.2) and (4.6). 



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