Hostname: page-component-6bf8c574d5-r8w4l Total loading time: 0 Render date: 2025-03-09T15:41:37.179Z Has data issue: false hasContentIssue false
  • English
  • Français

Modified Boussinesq equations and associated parabolic models for water wave propagation

Published online by Cambridge University Press:  26 April 2006

Yongze Chen  and
Philip L.-F. Liu
Yongze Chen
Affiliation:
School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853, USA Present address: Center For Coastal Studies, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92093-0209, USA.
Philip L.-F. Liu
Affiliation:
School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 288 , 10 April 1995 , pp. 351 - 381
Copyright
© 1995 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.)

References

Akylas, T. R. 1994 Three-dimensional long water-wave phenomena. Ann. Rev. Fluid Mech. 26, 191–210.Google Scholar
Chen, Y. 1995 Modelling of surface water wave and interfacial wave propagation. Doctoral dissertation, Cornell University, Ithaca, New York.
Chen, Y. & Liu, P. L.-F. 1994 A pseudospectral approach for scattering of water waves. Proc. R. Soc. Lond. A 445, 619–636.Google Scholar
Chen, Y. & Liu, P. L.-F. 1995 The unified Kadomtsev–Petviashvili equation for interfacial waves. J. Fluid Mech. 288, 383–408.Google Scholar
Freilich, M. H. & Guza, R. T. 1984 Nonlinear effects on shoaling surface gravity waves. Phil. Trans. R. Soc. Lond. A 311, 1–41.Google Scholar
Gottlieb, D., Hussaini, M. Y. & Orszag, S. A. 1984 Introduction: Theory and applications of spectral methods. In Spectral Methods for Partial Differential Equations (ed. R. G. Voigt, D. Gottlieb & M. Y. Hussaini), pp. 1–54. SIAM, Philadelphia.
Hammack, J., Scheffner, N. & Segur, H. 1988 Two-dimensional periodic waves in shallow water. J. Fluid Mech. 209, 567–589.Google Scholar
Kirby, J. T. 1990 Modelling shoaling directional wave spectra in shallow water. In Proc. 22nd Int. Conf. on Coastal Engineering, Delft, The Netherlands, pp. 109–122.
Kirby, J. T. 1991 Intercomparison of truncated series solutions for shallow water waves. J. Waterway, Port, Coastal, Ocean Engng, ASCE 117, 143–155.Google Scholar
Liu, P. L.-F. & Tsay, T. K. 1984 Refraction-diffraction model for weakly nonlinear water waves. J. Fluid Mech. 141, 265–274.Google Scholar
Liu, P. L.-F., Yoon, S. B. & Kirby, J. T. 1985 Nonlinear refraction-diffraction of waves in shallow water. J. Fluid Mech. 153, 185–201.Google Scholar
Madsen, P. A., Murray, R. & Sørensen, O. R. 1991 A new form of the Boussinesq equations with improved linear dispersion characteristics. Coastal Engng 15, 371–388.Google Scholar
Madsen, P. A. & Sørensen, O. R. 1992 A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry. Coastal Engng 18, 183–204.Google Scholar
Madsen, P. A. & Sørensen, O. R. 1993 Bound waves and triad interactions in shallow water. Ocean Engng 20, 359–388.Google Scholar
McCowan, A. D. & Blackman, D. R. 1989 The extension of Boussinesq type equations to modelling short waves in deep water. In Proc. 9th Australasian Conference on Coastal and Ocean Engineering, Adelaide Australia, pp. 412–416.
Nwogu, O. 1993a Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterway, Port, Coastal, Ocean Engng, ASCE 119, 618–638.Google Scholar
Nwogu, O. 1993b Nonlinear transformation of multi-directional waves in water of variable depth. (Submitted for publication)
Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27, 815–827.Google Scholar
Rygg, O. 1988 Nonlinear refraction-diffraction of surface waves in intermediate and shallow water. Coastal Engng 12, 191–211.Google Scholar
Radder, A. C. 1979 On the parabolic equation method for water-wave propagation. J. Fluid Mech. 95, 159–176.Google Scholar
Skovgaard, O. & Petersen, M. H. 1977 Refraction of cnoidal waves. Coastal Engng 1, 43–61.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.
Witting, J. M. 1984 A unified model for the evolution of nonlinear water waves. J. Comput. Phys. 56, 203–236.Google Scholar
Wu, T. Y. 1979 On tsunamis propagation – evaluation of existing models. In Tsunamis – Proceedings of the National Science Foundation Workshop (ed. L. S. Hwang & Y. K. Lee), pp. 110–149. Pasadena, CA: Tetra Tech. Inc.
Yoon, S. B. & Liu, P. L.-F. 1989 Stem waves along breakwater. J. Waterway, Port, Coastal, Ocean Engng, ASCE 115, 635–648.Google Scholar