Resonant response of harbours: an equivalent-circuit analysis

The surface-wave response of a harbour to a prescribed, incident wave is calculated on the hypotheses of shallow-water theory, an ideal fluid, and a narrow mouth,M.An equivalent electrical circuit is constructed, in which the incidentwave displacement inMappears as the input voltage and the flow throughMappears as the input current. This circuit contains aradiation impedance, ZM, which comprises resistive and inductive terms, and aharbour impedance, ZH, which comprises an infinite sequence of parallel combinations of inductance and capacitance that bear a one-to-one correspondence with the natural modes of the closed harbour, together with a single capacitor, which corresponds to the degenerate mode of uniform displacement and dominates the response of the harbour as a Helmholtz resonator. Variational approximations toZHandZMare developed. The equivalent circuit exhibits parallel resonance at the resonant frequencies of the closed harbour, ωn, and series resonance at a second set of frequencies,$\tilde{\omega}_n $, where$\tilde{\omega}_n\downarrow \omega_n > 0 $and$\tilde{\omega}_0\downarrow 0 $asM→ 0;$\tilde{\omega}_0 $corresponds to the Helmholtz mode. A narrow canal between the coastline and the harbour is represented by a four-terminal network betweenZMandZH. It is shown that narrowing the harbour mouth and/or increasing the length of the canal does not affect the mean response of the harbour to a broad-band, random input except in the Helmholtz mode, but that it does increase significantly the response in that mode, which may dominate tsunami response. The general results are applied to circular and rectangular harbours. The numerical calculation ofZHfor an arbitrarily shaped harbour is discussed.