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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Behera, H. Kaligatla, R.B. and Sahoo, T. 2015. Wave trapping by porous barrier in the presence of step type bottom. Wave Motion, Vol. 57, p. 219.

    Seo, Seung-Nam 2014. Transfer matrix of linear water wave scattering over a stepwise bottom. Coastal Engineering, Vol. 88, p. 33.

    Tsai, Chia-Cheng Lin, Yueh-Ting and Hsu, Tai-Wen 2013. On the weak viscous effect of the reflection and transmission over an arbitrary topography. Physics of Fluids, Vol. 25, Issue. 4, p. 043103.

    Martin, P. A. 2011. The horn-feed problem: sound waves in a tube joined to a cone, and related problems. Journal of Engineering Mathematics, Vol. 71, Issue. 3, p. 291.

    Tsai, Chia-Cheng Hsu, Tai-Wen and Lin, Yueh-Ting 2011. On Step Approximation for Roseau's Analytical Solution of Water Waves. Mathematical Problems in Engineering, Vol. 2011, p. 1.

    Wang, Cynthia D. and Meylan, Michael H. 2002. The linear wave response of a floating thin plate on water of variable depth. Applied Ocean Research, Vol. 24, Issue. 3, p. 163.

  • Journal of Fluid Mechanics, Volume 411
  • May 2000, pp. 131-164

Water wave scattering by a step of arbitrary profile

  • R. PORTER (a1) and D. PORTER (a2)
  • DOI:
  • Published online: 01 May 2000

The two-dimensional scattering of water waves over a finite region of arbitrarily varying topography linking two semi-infinite regions of constant depth is considered. Unlike many approaches to this problem, the formulation employed is exact in the context of linear theory, utilizing simple combinations of Green's functions appropriate to water of constant depth and the Cauchy–Riemann equations to derive a system of coupled integral equations for components of the fluid velocity at certain locations. Two cases arise, depending on whether the deepest point of the topography does or does not lie below the lower of the semi-infinite horizontal bed sections. In each, the reflected and transmitted wave amplitudes are related to the incoming wave amplitudes by a scattering matrix which is defined in terms of inner products involving the solution of the corresponding integral equation system.

This solution is approximated by using the variational method in conjunction with a judicious choice of trial function which correctly models the fluid behaviour at the free surface and near the joins of the varying topography with the constant-depth sections, which may not be smooth. The numerical results are remarkably accurate, with just a two-term trial function giving three decimal places of accuracy in the reflection and transmission coefficents in most cases, whilst increasing the number of terms in the trial function results in rapid convergence. The method is applied to a range of examples.

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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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