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Edge waves on a gently sloping beach

  • John Miles (a1)

Abstract

Edge waves of frequency ω and longshore wavenumber k in water of depth h(y) = h1 Hy/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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Ball, F. K.: 1967 Edge waves in an ocean of finite depth. Deep-Sea Res. 14, 7988.
Buchholz, H.: 1953 Die Konfluente Hypergeometrische Funktion, p. 143. Springer.
Keller, J. B.: 1958 Surface waves on water of non-uniform depth. J. Fluid Mech. 4, 607614.
Keller, J. B.: 1961 Tsunamis – Water waves produced by earthquakes. Tsunami Hydrodynamics Conference (IUGG Monograph no. 24).
Miles, J. W.: 1985 Surface waves in basins of variable depth. J. Fluid Mech. 152, 379389.
Minzoni, A. A.: 1976 Nonlinear edge waves and shallow-water theory. J. Fluid Mech. 74, 369374.
Shen, M. C., Meyer, R. E. & Keller, J. B., 1968 Spectra of water waves in channels and around islands. Phys. Fluids 11, 22892304.
Shen, M. C. & Keller, J. B., 1975 Uniform ray theory of surface, internal and acoustic wave propagation in a rotating ocean or atmosphere. SIAM J. Appl. Maths 28, 857875.
Stokes, G. G.: 1846 Report on recent research in hydrodynamics. Brit. Ass. Rep. Part 1; Mathematical and Physical Papers (1880), vol. 1, pp. 167187. Cambridge University Press.
Ursell, F.: 1952 Edge waves on a sloping beach. Proc. R. Soc. Lond. A 214, 7997.
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Edge waves on a gently sloping beach

  • John Miles (a1)

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