Nonlinear Dispersion of Surface Gravity Waves in Shallow Water

T. H. C. Herbers Department of Oceanography, Naval Postgraduate School, Monterey, California

Search for other papers by T. H. C. Herbers in
Current site
Google Scholar
PubMed
Close
,
Steve Elgar Department of Applied Ocean Physics and Engineering, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

Search for other papers by Steve Elgar in
Current site
Google Scholar
PubMed
Close
,
N. A. Sarap Department of Oceanography, Naval Postgraduate School, Monterey, California

Search for other papers by N. A. Sarap in
Current site
Google Scholar
PubMed
Close
, and
R. T. Guza Center for Coastal Studies, Scripps Institution of Oceanography, La Jolla, California

Search for other papers by R. T. Guza in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

The nonlinear dispersion of random, directionally spread surface gravity waves in shallow water is examined with Boussinesq theory and field observations. A theoretical dispersion relationship giving a directionally averaged wavenumber magnitude as a function of frequency, the local water depth, and the local wave spectrum and bispectrum is derived for waves propagating over a gently sloping beach with straight and parallel depth contours. The linear, nondispersive shallow water relation is recovered as the first-order solution, with weak frequency and amplitude dispersion appearing as second-order corrections. Wavenumbers were estimated using four arrays of pressure sensors deployed in 2–6-m depth on a gently sloping sandy beach. When wave energy is low, the observed wavenumbers agree with the linear, finite-depth dispersion relation over a wide frequency range. In high energy conditions, the observed wavenumbers deviate from the linear dispersion relation by as much as 20%–30% in the frequency range from two to three times the frequency of the primary spectral peak, but agree well with the nonlinear Boussinesq dispersion relation, confirming that the deviations from linear theory are finite amplitude effects. In high energy conditions, the predicted frequency and amplitude dispersion tend to cancel, yielding a nearly nondispersive wave field in which waves of all frequencies travel with approximately the linear shallow water wave speed, consistent with the observations. The nonlinear Boussinesq theory wavenumber predictions (based on the assumption of irrotational wave motion) are accurate even within the surf zone, suggesting that wave breaking on gently sloping beaches has little effect on the dispersion relation.

Corresponding author address: Dr. Thomas H. C. Herbers, Department of Oceanography, Code OC/He, Naval Postgraduate School, Monterey, CA 93943-5122. Email: thherber@nps.navy.mil

Abstract

The nonlinear dispersion of random, directionally spread surface gravity waves in shallow water is examined with Boussinesq theory and field observations. A theoretical dispersion relationship giving a directionally averaged wavenumber magnitude as a function of frequency, the local water depth, and the local wave spectrum and bispectrum is derived for waves propagating over a gently sloping beach with straight and parallel depth contours. The linear, nondispersive shallow water relation is recovered as the first-order solution, with weak frequency and amplitude dispersion appearing as second-order corrections. Wavenumbers were estimated using four arrays of pressure sensors deployed in 2–6-m depth on a gently sloping sandy beach. When wave energy is low, the observed wavenumbers agree with the linear, finite-depth dispersion relation over a wide frequency range. In high energy conditions, the observed wavenumbers deviate from the linear dispersion relation by as much as 20%–30% in the frequency range from two to three times the frequency of the primary spectral peak, but agree well with the nonlinear Boussinesq dispersion relation, confirming that the deviations from linear theory are finite amplitude effects. In high energy conditions, the predicted frequency and amplitude dispersion tend to cancel, yielding a nearly nondispersive wave field in which waves of all frequencies travel with approximately the linear shallow water wave speed, consistent with the observations. The nonlinear Boussinesq theory wavenumber predictions (based on the assumption of irrotational wave motion) are accurate even within the surf zone, suggesting that wave breaking on gently sloping beaches has little effect on the dispersion relation.

Corresponding author address: Dr. Thomas H. C. Herbers, Department of Oceanography, Code OC/He, Naval Postgraduate School, Monterey, CA 93943-5122. Email: thherber@nps.navy.mil

Save
  • Agnon, Y., and A. Sheremet, 1997: Stochastic nonlinear shoaling of directional spectra. J. Fluid Mech., 345 , 7999.

  • Agnon, Y., A. Sheremet, J. Gonsalves, and M. Stiassnie, 1993: Nonlinear evolution of a unidirectional shoaling wave field. Coastal Eng., 20 , 2958.

    • Search Google Scholar
    • Export Citation
  • Agnon, Y., P. A. Madsen, and H. A. Schäffer, 1999: A new approach to high-order Boussinesq models. J. Fluid Mech., 399 , 319333.

  • Chen, Y., R. T. Guza, and S. Elgar, 1997: Modeling spectra of breaking surface waves in shallow water. J. Geophys. Res., 102 , 2503525046.

    • Search Google Scholar
    • Export Citation
  • Dingemans, M. W., 1997: Water Wave Propagation over Uneven Bottoms. Advanced Series on Ocean Engineering, Vol. 13, World Scientific, 967 pp.

    • Search Google Scholar
    • Export Citation
  • Donelan, M. A., J. Hamilton, and W. H. Hui, 1985: Directional spectra of wind-generated waves. Philos. Trans. Roy. Soc. London, A315 , 509562.

    • Search Google Scholar
    • Export Citation
  • Eldeberky, Y., and J. A. Battjes, 1996: Spectral modeling of wave breaking: Application to Boussinesq equations. J. Geophys. Res., 101 , 12531264.

    • Search Google Scholar
    • Export Citation
  • Elgar, S., and R. T. Guza, 1985a: Shoaling gravity waves: Comparisons between field observations, linear theory, and a nonlinear model. J. Fluid Mech., 158 , 4770.

    • Search Google Scholar
    • Export Citation
  • Elgar, S., and R. T. Guza, . 1985b: Observations of bispectra of shoaling surface gravity waves. J. Fluid Mech., 161 , 425448.

  • Elgar, S., T. H. C. Herbers, and R. T. Guza, 1994: Reflection of ocean surface gravity waves from a natural beach. J. Phys. Oceanogr., 24 , 15031511.

    • Search Google Scholar
    • Export Citation
  • Elgar, S., R. T. Guza, B. Raubenheimer, T. H. C. Herbers, and E. L. Gallagher, 1997: Spectral evolution of shoaling and breaking waves on a barred beach. J. Geophys. Res., 102 , 1579715805.

    • Search Google Scholar
    • Export Citation
  • Elgar, S., R. T. Guza, W. C. O'Reilly, B. Raubenheimer, and T. H. C. Herbers, 2001: Wave energy and direction observed near a pier. J. Waterway, Port, Coastal, Ocean Eng., 127 , 26.

    • Search Google Scholar
    • Export Citation
  • Freilich, M. H., and R. T. Guza, 1984: Nonlinear effects on shoaling surface gravity waves. Philos. Trans. Roy. Soc. London, A311 , 141.

    • Search Google Scholar
    • Export Citation
  • Herbers, T. H. C., and R. T. Guza, 1994: Nonlinear wave interactions and high-frequency seafloor pressure. J. Geophys. Res., 99 , 1003510048.

    • Search Google Scholar
    • Export Citation
  • Herbers, T. H. C., and M. C. Burton, 1997: Nonlinear shoaling of directionally spread waves on a beach. J. Geophys. Res., 102 , 2110121114.

    • Search Google Scholar
    • Export Citation
  • Herbers, T. H. C., S. Elgar, and R. T. Guza, 1995: Generation and propagation of infragravity waves. J. Geophys. Res., 100 , 2486324872.

    • Search Google Scholar
    • Export Citation
  • Herbers, T. H. C., N. R. Russnogle, and S. Elgar, 2000: Spectral energy balance of breaking waves within the surf zone. J. Phys. Oceanogr., 30 , 27232737.

    • Search Google Scholar
    • Export Citation
  • Kaihatu, J. M., and J. T. Kirby, 1995: Nonlinear transformation of waves in finite water depth. Phys. Fluids, 7 , 19031914.

  • Laing, A. K., 1986: Nonlinear properties of random gravity waves in water of finite depth. J. Phys. Oceanogr., 16 , 20132030.

  • Liu, P. L-F., S. B. Yoon, and J. T. Kirby, 1985: Nonlinear refraction-diffraction of waves in shallow water. J. Fluid Mech., 153 , 185201.

    • Search Google Scholar
    • Export Citation
  • Longuet-Higgins, M. S., and O. M. Phillips, 1962: Phase velocity effects in tertiary wave interactions. J. Fluid Mech., 12 , 333336.

  • Madsen, P. A., and H. A. Schäffer, 1998: Higher-order Boussinesq-type equations for surface gravity waves: Derivation and analysis. Philos. Trans. Roy. Soc. London, A356 , 31233184.

    • Search Google Scholar
    • Export Citation
  • Madsen, P. A., R. Murray, and O. R. Sørensen, 1991: A new form of the Boussinesq equations with improved linear dispersion characteristics. Coastal Eng., 15 , 371388.

    • Search Google Scholar
    • Export Citation
  • Mase, H., and J. T. Kirby, 1992: Hybrid frequency-domain KdV equation for random wave transformation. Proc. 23rd Int. Conf. on Coastal Engineering, Venice, Italy, American Society of Civil Engineers, 474–487.

    • Search Google Scholar
    • Export Citation
  • Masuda, A., Y-Y. Kuo, and H. Mitsuyasu, 1979: On the dispersion relation of random gravity waves. Part 1. Theoretical framework. J. Fluid Mech., 92 , 717730.

    • Search Google Scholar
    • Export Citation
  • Mitsuyasu, H., Y-Y. Kuo, and A. Masuda, 1979: On the dispersion relation of random gravity waves. Part 2. An experiment. J. Fluid Mech., 92 , 731749.

    • Search Google Scholar
    • Export Citation
  • Norheim, C. A., T. H. C. Herbers, and S. Elgar, 1998: Nonlinear evolution of surface wave spectra on a beach. J. Phys. Oceanogr., 28 , 15341551.

    • Search Google Scholar
    • Export Citation
  • Peregrine, D. H., 1967: Long waves on a beach. J. Fluid Mech., 27 , 815827.

  • Phillips, O. M., 1960: On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech., 9 , 193217.

    • Search Google Scholar
    • Export Citation
  • Schäffer, H. A., P. A. Madsen, and R. Deigaard, 1993: A Boussinesq model for waves breaking in shallow water. Coastal Eng., 20 , 185202.

    • Search Google Scholar
    • Export Citation
  • Stive, M. J. F., 1984: Energy dissipation in waves breaking on gentle slopes. Coastal Eng., 8 , 99127.

  • Thornton, E. B., and R. T. Guza, 1982: Energy saturation and phase speeds measured on a natural beach. J. Geophys. Res., 87 , 94999508.

    • Search Google Scholar
    • Export Citation
  • Thornton, E. B., J. S. Galvin, F. L. Bub, and D. P. Richardson, 1976: Kinematics of breaking waves. Proc. 15th Int. Conf. on Coastal Engineering, Honolulu, HI, American Society of Civil Engineers, 461–476.

    • Search Google Scholar
    • Export Citation
  • Wei, G., J. T. Kirby, S. T. Grilli, and R. Subramanya, 1995: A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. J. Fluid Mech., 294 , 7192.

    • Search Google Scholar
    • Export Citation
  • Whitham, G. B., 1974: Linear and Nonlinear Waves. Wiley Interscience, 636 pp.

  • Willebrand, J., 1975: Energy transport in a nonlinear and inhomogeneous random gravity wave field. J. Fluid Mech., 70 , 113126.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 585 193 41
PDF Downloads 386 126 21