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On the trapping of wave energy round islands

Published online by Cambridge University Press:  28 March 2006

M. S. Longuet-Higgins
M. S. Longuet-Higgins
Affiliation:
National Institute of Oceanography, Wormley, Surrey
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Abstract

It is shown that islands can trap long-wave energy in a way similar to the capture of a particle by an atomic nucleus. The frequencies of the captured waves form a discrete set, being determined by the shape of the island and the contours of the surrounding sea bed. If the depth at great distances tends to a constant value, the trapped modes must leak some energy to infinity, though the consequent rate of decay may be exceedingly small. The initial energy of the trapped modes may be absorbed from incident radiation of the same frequency or from a sharp pulse. The particular example of a rectilinear pulse incident on a circular island is discussed in some detail.

The effect of the rotation of the Earth is to split the frequencies of a pair of waves progressing in opposite directions round the island. The splitting of the frequencies produces slow beats in the waves as seen at any fixed point. Slight asymmetry in the island induces a slow exchange of energy between each pair of progressive modes.

The present investigation was suggested by the occurrence of regular oscillations having a period of 6 min and a beat period of about 3 h in long-wave records taken at Macquarie Island, in the Southern Ocean.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 29 , Issue 4 , 27 September 1967 , pp. 781 - 821
Copyright
© 1967 Cambridge University Press

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References

Albers, V. M. 1960 Underwater Acoustics Handbook, 290 pp. Penn. State Press.
Bartholomeusz, E. F. 1958 The reflection of long waves at a step Proc. Camb. Phil. Soc. 54, 106–118.Google Scholar
Bickley, W. G. 1943 Formulae relating to Bessel functions of moderate or large argument or order. Phil. Mag. 34 (Ser. 7), 37–49.Google Scholar
Bryan, G. H. 1990 On a revolving cylinder or bell Proc. Camb. Phil. Soc. 7, 101–111.Google Scholar
Budden, K. G. 1961 The Wave-Guide Mode Theory of Wave Propagation. Englewood Cliffs, N.J.: Prentice-Hall, 325 pp.
Chambers Li, G., 1965 On long waves on a rotating earth J. Fluid Mech. 22, 209–216.Google Scholar
Debye, P. 1909 Näherungsformeln für die Zylinderfunktionen für grosse Werte des Arguments und unbeschränkt veränderliche Werte des Index Math. Ann. 67, 535–558.Google Scholar
Eckart, C. 1950 The ray-particle analogy J. Mar. Res. 9, 139–144.Google Scholar
Eckart, C. 1951 Surface waves on water of variable depth. U.S. Office of Naval Research. Wave Rept. no. 100. (Notes of lectures given at the Scripps Institution of Oceanography).Google Scholar
Greenspan, H. P. 1956 The generation of edge-waves by moving pressure distributions J. Fluid Mech. 1, 574–592.Google Scholar
Jeffreys, H. & Jeffreys, B. S. 1950 Methods of Mathematical Physics, 708 pp, 2nd ed. Cambridge University Press.
Kajiura, K. 1958 Effect of coriolis force on edge waves (2) Specific examples of free and forces waves J. Mar. Res. 16, 145–157.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th ed. Cambridge University Press.
Lorenz, L. V. 1890 Sur la lumiére réfléchie et réfractée par une sphére transparente Vidensk. Selsk. Skrifter, 6, 1–62. (Original in Danish; French translation in Oevres Scietififiques de L. Lorenz, vol. 1, Copenhagen, Libraire Lehmann 1896; reprinted New York: Johnson 1964.)Google Scholar
Miller, J. C. P. 1964 The Airy Integral Brit. Ass. Math. Tables, Part-vol. B, 56 pp. Cambridge University Press.
Munk, W. H., Snodgrass, F. & Carrier, G. 1956 Edge waves on the continental shelf Science 123, 127–132.Google Scholar
Olver, F. J. W. 1954 The asymptotic expansion of Bessel functions of large order. Phil. Tran A 247, 328–368.Google Scholar
Rayleigh, Lord 1945 The Theory of Sound 2nd ed., 504 pp. New York: Dover.
Reid, R. O. 1958 Effect of coriolis force on edge waves (1) Investigation of the normal modes J. Mar. Res. 16, 109–144.Google Scholar
Schiff, L. I. 1949 Quantum Mechanics, 409 pp. London: McGraw-Hill.
Snodgrass, F. E., Munk, W. H. & Miller, G. R. 1962 Long-period waves over California's continental borderland. Part 1. Background spectra J. Mar. Res. 20, 3–30.Google Scholar
Stokes, G. G. 1846 Report on recent researches in hydrodynamics. Br. Ass. Rept. 1846; see also Coll. Pap. 1, 167.Google Scholar
Ursell, F. 1951 Trapping modes in the theory of surface waves Proc. Camb. Phil. Soc. 47, 347–358.Google Scholar
Ursell, F. 1952a Edge waves on a sloping beach. Proc. Roy. Soc A 214, 79–97.Google Scholar
Ursell, F. 1952b Discrete and continuous spectra in the theory of gravity waves. Proc. Symp. on Gravity Waves, U. S. Nat. Bur. Standards, Circular 521, pp. 1–5.
Watson, G. N. 1922 A treatise on the theory of Bessel functions, 804 pp. Cambridge University Press.