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  • Cited by 7
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    ADAMOU, ALEXANDER T. I. CRASTER, R. V. and SMITH, STEFAN G. LLEWELLYN 2007. Trapped edge waves in stratified rotating fluids: numerical and asymptotic results. Journal of Fluid Mechanics, Vol. 592,

    Johnson, R.S 2007. Edge waves: theories past and present. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 365, Issue. 1858, p. 2359.

    Merzon, A. E. and Zhevandrov, P. N. 1998. High-Frequency Asymptotics of Edge Waves on a Beach of Nonconstant Slope. SIAM Journal on Applied Mathematics, Vol. 59, Issue. 2, p. 529.

    Schäffer, Hemming A. and Jonsson, Ivar G. 1992. Edge waves revisited. Coastal Engineering, Vol. 16, Issue. 4, p. 349.

    Zhevandrov, Peter 1991. Edge waves on a gently sloping beach: uniform asymptotics. Journal of Fluid Mechanics, Vol. 233, Issue. -1, p. 483.

    Miles, John 1990. Parametrically excited standing edge waves. Journal of Fluid Mechanics, Vol. 214, Issue. -1, p. 43.

    Miles, John 1990. Wave motion in a viscous fluid of variable depth. Journal of Fluid Mechanics, Vol. 212, Issue. -1, p. 365.

  • Journal of Fluid Mechanics, Volume 199
  • February 1989, pp. 125-131

Edge waves on a gently sloping beach

  • John Miles (a1)
  • DOI:
  • Published online: 01 April 2006

Edge waves of frequency ω and longshore wavenumber k in water of depth h(y) = h1Hy/h1), 0 [les ] y < ∞, are calculated through an asymptotic expansion in σ/kh1 on the assumptions that σ [Lt ] 1 and kh1 = O(1). Approximations to the free-surface displacement in an inner domain that includes the singular point at h = 0 and the turning point near gh ≈ ω2/K2 and to the eigenvalue λ ≡ ω2gh are obtained for the complete set of modes on the assumption that h(y) is analytic. A uniformly valid approximation for the free-surface displacement and a variational approximation to Λ are obtained for the dominant mode. The results are compared with the shallow-water approximations of Ball (1967) for a slope that decays exponentially from σ to 0 as h increases from 0 to h1 and of Minzoni (1976) for a uniform slope that joins h = 0 to a flat bottom at h = h1 and with the geometrical-optics approximation of Shen, Meyer & Keller (1968).

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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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