Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-28T09:48:34.285Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear dispersion for ocean surface waves

Published online by Cambridge University Press:  16 November 2018

Raphael Stuhlmeier Open the ORCID record for Raphael Stuhlmeier [Opens in a new window]  and
Michael Stiassnie
Raphael Stuhlmeier*
Affiliation:
Centre for Mathematical Sciences, University of Plymouth, Drake Circus, Plymouth PL4 8AA, UK
Michael Stiassnie
Affiliation:
Faculty of Civil and Environmental Engineering, Technion, Haifa 32000, Israel
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Two expressions for the nonlinear dispersion relation for gravity waves on water of constant depth are derived, one for wave fields with discrete amplitude spectra, the other for wave fields with continuous wavenumber energy spectra. Numerical examples for wave quartets and for two-dimensional Pierson–Moskowitz spectra are given, and an important possible application is discussed.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 859 , 25 January 2019 , pp. 49 - 58
Copyright
© 2018 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

Dyachenko, A. I. & Zakharov, V. E. 1994 Is free-surface hydrodynamics an integrable system? Phys. Lett. A 190, 144–148.Google Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 1980 Tables of Integrals, Series, and Products, 4th edn. Academic Press.Google Scholar
Hogan, S. J., Gruman, I. & Stiassnie, M. 1988 On the changes in phase speed of one train of water waves in the presence of another. J. Fluid Mech. 192, 97–114.Google Scholar
Holthuijsen, L. H. 2007 Waves in Oceanic and Coastal Waters. Cambridge University Press.Google Scholar
Kinsman, B. 1984 Wind Waves. Dover.Google Scholar
Leblanc, S. 2009 Stability of bichromatic gravity waves on deep water. Eur. J. Mech. (B/Fluids) 28 (5), 605–612.Google Scholar
Longuet-Higgins, M. S. 1962 Resonant interactions between two trains of gravity waves. J. Fluid Mech. 12, 321–332.Google Scholar
Longuet-Higgins, M. S. & Phillips, O. M. 1962 Phase velocity effects in tertiary wave interactions. J. Fluid Mech. 12 (3), 333–336.Google Scholar
L’vov, V. S. & Nazarenko, S. 2010 Discrete and mesoscopic regimes of finite-size wave turbulence. Phys. Rev. E 82, 056322.Google Scholar
Madsen, P. A. & Fuhrman, D. R. 2006 Third-order theory for bichromatic bi-directional water waves. J. Fluid Mech. 557, 369–397.Google Scholar
Madsen, P. A. & Fuhrman, D. R. 2012 Third-order theory for multi-directional irregular waves. J. Fluid Mech. 698, 304–334.Google Scholar
Mei, C. C., Stiassnie, M. A. & Yue, D. K.-P. 2018 Theory and Applications of Ocean Surface Waves, 3rd edn. World Scientific.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9 (2), 193–217.Google Scholar
Qi, Y., Wu, G., Liu, Y., Kim, M.-H. & Yue, D. K.-P. 2018 Nonlinear phase-resolved reconstruction of irregular water waves. J. Fluid Mech. 838, 544–572.Google Scholar
Stiassnie, M. 1991 The fractal dimension of the ocean surface. In Nonlinear Topics in Ocean Physics: Proceedings of the International School of Physics Enrico Fermi, Course CIX 1988 (ed. Osborne, A. R.), pp. 633–647. North-Holland.Google Scholar
Stiassnie, M. & Gramstad, O. 2009 On Zakharov’s kernel and the interaction of non-collinear wavetrains in finite water depth. J. Fluid Mech. 639, 433–442.Google Scholar
Stiassnie, M. & Shemer, L. 2005 On the interaction of four water waves. Wave Motion 41, 307–328.Google Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Tick, L. J. 1959 A nonlinear random model of gravity waves. Part I. J. Math. Mech. 8 (5), 643–651.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9 (2), 190–194.Google Scholar
Zakharov, V. E. 1992 Inverse and direct cascade in a wind-driven surface wave turbulence and wave-breaking. In IUTAM Symposium (ed. Banner, M. L. & Grimshaw, R. H. J.), pp. 69–91. Springer.Google Scholar
Zakharov, V. E. 1999 Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid. Eur. J. Mech. (B/Fluids) 18 (3), 327–344.Google Scholar
Zakharov, V. E., L’vov, V. S. & Falkovich, G. 1992 Kolmogorov Spectra of Turbulence I. Springer.Google Scholar
Zhang, J. & Chen, L. 1999 General third order solutions for irregular waves in deep water. J. Engng Mech. ASCE 125 (7), 768–779.Google Scholar