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New solutions for the propagation of long water waves over variable depth

  • Yinglong Zhang (a1) and Songping Zhu (a1)


Based on the linearized long-wave equation, two new analytical solutions are obtained respectively for the propagation of long surface gravity waves around a conical island and over a paraboloidal shoal. Having been intensively studied during the last two decades, these two problems have practical significance and are physically revealing for wave propagation over variable water depth. The newly derived analytical solutions are compared with several previously obtained numerical solutions and the accuracy of those numerical solutions is discussed. The analytical method has the potential to be used to find solutions for wave propagation over more natural bottom topographies.



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Berkhoff, J. C. W. 1972 Computation of combined refraction-diffraction. In 13th Intl Conf. Coastal Engng, Vol. 1, pp. 471490.
Bettess, P. & Zienkiewicz, O. C. 1977 Diffraction and refraction of surface waves using finite and infinite elements. Intl J. Num. Meth. Engng 11, 12711290.
Christiansen, P. L. 1977 Diffraction of gravity waves by ray methods. In Waves on Water of Variable Depth (ed. D. G. Provis & R. Radok). Lecture Notes in Physics, vol. 64. Springer.
Eckart, C. 1950 The ray particle analogy. J. Mar. Res. 9, 139144.
Eckart, C. 1951 Surface waves on water of variable depth. Scripps Inst. Oceanogr., Lecture Notes, Fall Semester 1950–51, Ref. 51–12, Wave Rep. 100.
Flokstra, C. & Berkhoff, J. C. W. 1977 Propagation of short waves over a circular shoal. Delft Hydraulics Lab. Rep. W. 154-V (in Dutch).
Homma, S. 1950 On the behaviour of seismic sea waves around circular island. Geophys. Mag. XXI, 199208.
Ito, Y. & Tanimoto, K. 1972 A method of numerical analysis of wave propagation - application to wave diffraction and refraction. In 13th Coastal Engng Conf., Vol. 1, pp. 503522.
Kirby, J. T. & Dalrymple, R. A. 1983 A parabolic equation for the combined refraction-diffraction of Stokes waves by mildly varying topography. J. Fluid Mech. 136, 453466.
Lautenbacher, C. C. 1970 Gravity wave refraction by islands. J. Fluid Mech. 41, 655672.
Longuet-Higgins, M. S. 1967 On the trapping of wave energy round islands. J. Fluid Mech. 29, 781821.
Maccamy, R. C. & Fuchs, R. A. 1954 Wave forces on piles: a diffraction theory. US Army Corps of Engineering, Beach Erosion Board, Washington, DC, Tech. Mem. 69.
Mei, C. C. 1989 The Applied Dynamics of Ocean Surface Waves. World Scientific.
Panchang, V. G., Cushman-Roisin, B. & Pearce, B. R. 1988 Combined refraction-diffraction of short-waves in large coastal regions. Coastal Engng 12, 133156.
Panchang, V. G. & Kopriva, D. A. 1989 Solution of two-dimensional water-wave propagation problems by Chebyshev collocation. Math. Comput. Modelling 12, 625640.
Radder, A. C. 1979 On the parabolic equation method for water-wave propagation. J. Fluid Mech. 95, 159176.
Sarpkaya, T. & Isaacson, M. 1981 Mechanics of Wave Force on Offshore structures. Van Nostrand Reinhold.
Shen, M. C. & Meyer, R. E. 1968 Spectra of water waves in channels and around islands. Phys. Fluids 11, 22892304.
Smith, R. 1974 Edge waves on a beach of mild slope. Q. J. Mech. Appl. Maths 27, 102110.
Smith, R. & Sprinks, T. 1975 Scattering of surface waves by a conical island. J. Fluid Mech. 72, 373384.
Spiegel, M. R. 1981 Applied Differential Equations. Prentice-Hall.
Vastano, A. C. & Reid, R. O. 1966 A numerical study of the tsunami response at an island. Texas A & M University Project 471, Ref. 66–26 T.
Zhu, S.-P. 1993 A new DRBEM model for wave refraction and diffraction. Engng Ana. Boundary Elements (in the press).
Zhu, S.-P. & Zhang, Y.-L. 1994 Scattering of long waves around a circular island mounted on a conical shoal. J. Mar. Res. (submitted).
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New solutions for the propagation of long water waves over variable depth

  • Yinglong Zhang (a1) and Songping Zhu (a1)


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