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Resonant response of harbours: an equivalent-circuit analysis

Published online by Cambridge University Press:  29 March 2006

John W. Miles
John W. Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, La Jolla
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Abstract

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 in M appears as the input voltage and the flow through M appears as the input current. This circuit contains a radiation impedance, ZM, which comprises resistive and inductive terms, and a harbour 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 to ZH and ZM are 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 $ as M → 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 between ZM and ZH. 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 of ZH for an arbitrarily shaped harbour is discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 46 , Issue 2 , 29 March 1971 , pp. 241 - 265
Copyright
© 1971 Cambridge University Press

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References

Bartholomeusz, E. F. 1958 The reflexion of long waves at a step Proc. Camb. Phil. Soc. 54, 106–18.Google Scholar
Carrier, G. F., Shaw, R. P. & Miyata, M. 1970 The response of narrow mouthed harbors in a straight coastline to periodic incident waves. J. Appl. Mech. (in the press).Google Scholar
Garrett, C. J. R. 1970 Bottomless harbours J. Fluid Mech. 43, 433–49.Google Scholar
Hwang, L. S. & Tuck, E. O. 1970 On the oscillations of harbours of arbitrary shape J. Fluid Mech. 42, 447–64.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Lee, J. J. 1971 Wave induced oscillations in harbours of arbitrary geometry J. Fluid Mech. 45, 375–93.Google Scholar
Miles, J. W. 1946 The analysis of plane discontinuities in cylindrical tubes. Parts I and II. J. Acoust. Soc. Am. 17, 259–71, 272–84.Google Scholar
Miles, J. W. 1948 The coupling of a cylindrical tube to a half-infinite space J. Acoust. Soc. Am. 20, 652–64.Google Scholar
Miles, J. W. 1967 Surface-wave scattering matrix for a shelf J. Fluid Mech. 28, 755–67.Google Scholar
Miles, J. W. & Munk, W. H. 1961 Harbor paradox J. Waterways Harb. Div., Am. Soc. Civ. Engrs 87, 111–30.Google Scholar
Morse, P. M. 1948 Vibration and Sound. McGraw-Hill.
Rayleigh, Lord 1945 Theory of Sound. Dover.
Sommerfeld, A. 1949 Partial Differential Equations in Physics. Academic.