Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-15T01:52:15.931Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear refraction–diffraction of waves in shallow water

Published online by Cambridge University Press:  20 April 2006

Philip L.-F. Liu,
Sung B. Yoon and
James T. Kirby
Philip L.-F. Liu
Affiliation:
Joseph H. DeFrees Hydraulics Laboratory, School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853
Sung B. Yoon
Affiliation:
Joseph H. DeFrees Hydraulics Laboratory, School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853
James T. Kirby
Affiliation:
Marine Sciences Research Center, State University of New York at Stony Brook, Stony Brook, NY 11794Present address: Department of Coastal and Oceanographic Engineering, University of Florida, Gainesville, FL 32611
Get accessRights & Permissions [Opens in a new window]

Abstract

The parabolic approximation is developed to study the combined refraction/diffraction of weakly nonlinear shallow-water waves. Two methods of approach are used. In the first method Boussinesq equations are used to derive evolution equations for spectral-wave components in a slowly varying two-dimensional domain. The second method modifies the K–P equation (Kadomtsev & Petviashvili 1970) to include varying depth in two dimensions. Comparisons are made between present numerical results, experimental data (Whalin 1971) and previous numerical calculations (Madsen & Warren 1984).

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 153 , April 1985, pp. 185 - 201
Copyright
© 1985 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Booij, N. 1981 Gravity waves on water with non-uniform depth and current. Rep. 81-1, Dept. Civil Engng, Delft University of Technology.
Bryant, P. J. 1982 Two-dimensional periodic permanent waves in shallow water. J. Fluid Mech. 115, 525–532.Google Scholar
Dalrymple, R. A., Kirby, J. T. & Hwang, P. A. 1984 Wave diffraction due to areas of energy dissipation. J. Waterway, Port, Coastal and Ocean Engineering, ASCE 110, 67–79.Google Scholar
Freilich, M. H. & Guza, R. T. 1984 Nonlinear effects on shoaling surface gravity waves. Phil. Trans. R. Soc. Lond. A311, 1–41.Google Scholar
Johnson, R. S. 1972 Some numerical solutions of a variable-coefficient Korteweg-de Vries equation (with applications to solitary wave development on a shelf). J. Fluid Mech. 54, 81–91.Google Scholar
Kadomtsev, B. B. & Petviashvili, V. I. 1970 On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15, 539–541.Google Scholar
Kirby, J. T. 1984 A note on linear surface wave-current interaction over slowly varying topography. J. Geophys. Res. 89, 745–747.Google Scholar
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, 453–466.Google Scholar
Liu, P. L.-F. 1983 Wave-current interactions on a slowly varying topography. J. Geophys. Res. 88, 4421–4426.Google Scholar
Liu, P. L.-F. & Tsay, T.-K. 1983 On weak reflection of water waves. J. Fluid Mech. 131, 59–71.Google Scholar
Liu, P. L.-F. & Tsay, T.-K. 1984a Numerical prediction of wave transformation. J. Waterway, Port, Coastal and Ocean Engrg. Div. ASCE. (to appear).Google Scholar
Liu, P. L.-F. & Tsay, T.-K. 1984b Refraction-diffraction model for weakly nonlinear water waves. J. Fluid Mech. 141, 265–274.Google Scholar
Lozano, C. & Liu, P. L.-F. 1980 Refraction-diffraction model for linear surface water waves. J. Fluid Mech. 101, 705–720.Google Scholar
Madsen, P. A. & Warren, I. R. 1984 Performance of a numerical short-wave model. Coastal Engng 8, 73–79.Google Scholar
Radder, A. C. 1979 On the parabolic equation method for water-wave propagation. J. Fluid Mech. 95, 159–176.Google Scholar
Rogers, S. R. & Mei, C. C. 1978 Nonlinear resonant excitation of a long and narrow bay. J. Fluid Mech. 88, 161–180.Google Scholar
Skovgaard, O., Jonsson, I. G. & Bertelsen, J. A. 1975 Computation of wave heights due to refraction and friction. J. Waterways, Port, Harbors and Coastal Engng Div., ASCE 101, 15–32.Google Scholar
Skovgaard, O. & Petersen, H. M. 1977 Refraction of cnoidal waves. Coastal Engng 1, 43–61.Google Scholar
Tsay, T.-K. & Liu, P. L.-F. 1982 Numerical solution of water-wave refraction and diffraction problems in parabolic approximation. J. Geophys. Res. 87, 7932–7940.Google Scholar
Whalin, R. W. 1971 The limit of applicability of linear wave refraction theory in a convergence zone. Res. Rep. H-71-3, U.S. Army Corps of Engrs, Waterways Expt. Station, Vicksburg, MS.Google Scholar
Yue, D. K. P. & Mei, C. C. 1980 Forward diffraction of Stokes waves by a thin wedge. J. Fluid Mech. 99, 33–52.Google Scholar