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  • Journal of Fluid Mechanics, Volume 288
  • April 1995, pp. 351-381

Modified Boussinesq equations and associated parabolic models for water wave propagation

  • Yongze Chen (a1) (a2) and Philip L.-F. Liu (a1)
  • DOI:
  • Published online: 01 April 2006

The modified Boussinesq equations given by Nwogu (1993a) are rederived in terms of a velocity potential on an arbitrary elevation and the free surface displacement. The optimal elevation where the velocity potential should be evaluated is determined by comparing the dispersion and shoaling properties of the linearized modified Boussinesq equations with those given by the linear Stokes theory over a range of depths from zero to one half of the equivalent deep-water wavelength. For regular waves consisting of a finite number of harmonics and propagating over a slowly varying topography, the governing equations for velocity potentials of each harmonic are a set of weakly nonlinear coupled fourth-order elliptic equations with variable coefficients. The parabolic approximation is applied to these coupled fourth-order elliptic equations for the first time. A small-angle parabolic model is developed for waves propagating primarily in a dominant direction. The pseudospectral Fourier method is employed to derive an angular-spectrum parabolic model for multi-directional wave propagation. The small-angle model is examined by comparing numerical results with Whalin's (1971) experimental data. The angular-spectrum model is tested by comparing numerical results with the refraction theory of cnoidal waves (Skovgaard & Petersen 1977) and is used to study the effect of the directed wave angle on the oblique interaction of two identical cnoidal wavetrains in shallow water.

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Akylas, T. R.1994Three-dimensional long water-wave phenomena. Ann. Rev. Fluid Mech.26, 191210.

Chen, Y. & Liu, P. L.-F.1994A pseudospectral approach for scattering of water waves. Proc. R. Soc. Lond. A 445, 619636.

Chen, Y. & Liu, P. L.-F.1995The unified Kadomtsev–Petviashvili equation for interfacial waves. J. Fluid Mech.288, 383408.

Freilich, M. H. & Guza, R. T.1984Nonlinear effects on shoaling surface gravity waves. Phil. Trans. R. Soc. Lond. A 311, 141.

Kirby, J. T.1991Intercomparison of truncated series solutions for shallow water waves. J. Waterway, Port, Coastal, Ocean Engng, ASCE117, 143155.

Madsen, P. A., Murray, R. & Sørensen, O. R.1991A new form of the Boussinesq equations with improved linear dispersion characteristics. Coastal Engng15, 371388.

Madsen, P. A. & Sørensen, O. R.1992A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry. Coastal Engng18, 183204.

Madsen, P. A. & Sørensen, O. R.1993Bound waves and triad interactions in shallow water. Ocean Engng20, 359388.

Nwogu, O.1993aAlternative form of Boussinesq equations for nearshore wave propagation. J. Waterway, Port, Coastal, Ocean Engng, ASCE119, 618638.

Rygg, O.1988Nonlinear refraction-diffraction of surface waves in intermediate and shallow water. Coastal Engng12, 191211.

Skovgaard, O. & Petersen, M. H.1977Refraction of cnoidal waves. Coastal Engng1, 4361.

Witting, J. M.1984A unified model for the evolution of nonlinear water waves. J. Comput. Phys.56, 203236.

Yoon, S. B. & Liu, P. L.-F.1989Stem waves along breakwater. J. Waterway, Port, Coastal, Ocean Engng, ASCE115, 635648.

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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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