A New Development of Reduced Hamiltonian Equations for Ocean Surface Waves: an Extension From Small to Finite Amplitude

WANG Zhao-ling>; XIAO Heng

doi:10.3879/j.issn.1000-0887.2015.11.002

Volume 36 Issue 11

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2015 > 36(11): 1135-1144

WANG Zhao-ling>, XIAO Heng. A New Development of Reduced Hamiltonian Equations for Ocean Surface Waves: an Extension From Small to Finite Amplitude[J]. Applied Mathematics and Mechanics, 2015, 36(11): 1135-1144. doi: 10.3879/j.issn.1000-0887.2015.11.002

Citation:

PDF( 431 KB)

A New Development of Reduced Hamiltonian Equations for Ocean Surface Waves: an Extension From Small to Finite Amplitude

doi: 10.3879/j.issn.1000-0887.2015.11.002

WANG Zhao-ling>^1,2,
XIAO Heng^2,3

¹School of Mathematics and Information Sciences, Weifang University, Weifang, Shandong 261061, P.R.China;²The State Key Laboratory of Advanced Special Steel(Shanghai University),Shanghai 200444, P.R.China;³Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, P.R.China

Funds: The National Natural Science Foundation of China(11372172)

Received Date: 2015-04-10
Rev Recd Date: 2015-09-08
Publish Date: 2015-11-15

Abstract

Abstract

The reduced 3-wave and 4-wave Hamiltonian equations for ocean surface waves were widely used for the simplified structure with symmetric polynomial kernels and for the conservation of energy, etc. However, according to the related assumption for approximation in derivation, the range of applicability was limited to weakly nonlinear waves of small amplitude. Here the following issue was further studied: for nonlinear waves of finite amplitude within a certain range, was it also possible to obtain reduced Hamiltonian equations with symmetric polynomial kernels in a sense of sufficient accuracy? Because of complicated strongly nonlinear coupling, few development in this significant respect had been made as yet. A new approach was proposed based on the Chebyshev polynomials to best approximate the primitive water wave equations in the exact sense of strongly nonlinear coupling and derive new reduced Hamiltonian equations with symmetric polynomial kernels. The new results exhibit an extension from a weakly nonlinear case in which the product of the wave number and the wave steepness is small to a nonlinear case in which this product goes up to about 1.035. Moreover, within this range, the approximation errors are lower than 5%, and in particular, the new results prove exact whenever the said product lies close to 0.9.
- ocean surface wave,
- finite amplitude,
- new approach for reduction,
- new reduced Hamiltonian equation

FullText(HTML)

References(21)

References

[1]	Bouscasse B, Colagrossi A, Marrone S, Antuono M. Nonlinear water wave interaction with floating bodies in SPH[J].Journal of Fluid and Structures,2013,42: 112-129.
[2]	Belibassakis K A, Athanassoulis G A. A coupled-mode system with application to nonlinear water waves propagating in finite water depth and in variable bathymetry regions[J].Coastal Engineering,2011,58(4): 337-350.
[3]	TAO Long-bin, SONG Hao, Chakrabarti S. Nonlinear progressive waves in water of finite depth—an analytic approximation[J].Coastal Engineering,2007,54(11): 825-834.
[4]	LI Bin. A mathematical model for weakly nonlinear water wave propagation[J].Wave Motion,2010,47(5): 265-278.
[5]	Bateman W J D, Katsardi V, Swan C. Extreme ocean waves—part I: the practical application of fully nonlinear wave modeling[J].Applied Ocean Research,2012,34: 209-224.
[6]	Zakharov V. Stability of periodic waves of finite amplitude on the surface of a deep fluid[J].Journal of Applied Mechanics and Technical Physics,1968,9(2): 190-198.
[7]	Zakharov V E, Kharitonov V G. Instability of monochromatic waves on the surface of a liquid of arbitrary depth[J].Journal of Applied Mechanics and Technical Physics,1970,11(5): 741-751.
[8]	Miles J W. On Hamilton’s principle for surface waves[J].Journal of Fluid Mechanics,1977,83: 153-158.
[9]	Krasitskii V P. Canonical transformations in a theory of weakly nonlinear waves with a nondecay dispersion law[J].Soviet Physics JETP,1990,71(5): 921-927.
[10]	Krasitskii V P. On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves[J].Journal of Fluid Mechanics,1994,272: 1-20.
[11]	Stiassnie M, Shemer L. On modifications of the Zakharov equation for surface gravity waves[J].Journal of Fluid Mechanics,1984,143: 47-67.
[12]	Zakharov V E, Musher S L, Rubenchick A M. Hamiltonian approach to the description of non-linear plasma phenomena[J].Physics Reports,1985,129(5): 285-366.
[13]	Janssen P A E M. On some consequences of the canonical transformation in the Hamiltonian theory of water waves[J].Journal of Fluid Mechanics,2009,637: 1-44.
[14]	Benney D J, Roskes G J. Wave instabilities[J].Studies in Applied Mathematics,1969,48: 377-385.
[15]	Chu V H, Mei C C. On slowly-varying Stokes waves[J].Journal of Fluid Mechanics,1970,41(4): 873-887.
[16]	Davey A, Stewartson K. On three-dimensional packets of surface waves[J].Proceedings of the Royal Society of London(Series A: Mathematical and Physical Sciences),1974,338(1613): 101-110.
[17]	Lavrova O T. On the lateral instability of waves on the surface of a finite-depth fluid[J].Izvestiya Atmospheric and Oceanic Physics,1983,19: 807-810.
[18]	Newell A C, Rumpf B. Wave turbulence[J].Annual Review of Fluid Mechanics,2011,43: 59-78.
[19]	Tanaka M. Verification of Hasselmann’s energy transfer among surface gravity waves by direct numerical simulations of primitive equations[J].Journal of Fluid Mechanics,2001,444: 199-221.
[20]	Annenkov S Y, Shrira V I. Numerical modelling of water-wave evolution based on the Zakharov equation[J].Journal of Fluid Mechanics,2001,449: 341-371.
[21]	黄虎. 近海波-流相互作用的缓坡方程理论体系[J]. 物理学报, 2010,59(2): 740-743.(HUANG Hu. A theoretical hierarchy of the mild-slope equations for wave-current interactions in coastal waters[J].Acta Physica Sinica,2010,59(2): 740-743.(in Chinese))