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Dynamics of Surface Waves in Coastal Waters pp 37–51Cite as

Resonant Interactions Between Weakly Nonlinear Stokes Waves and Ambient Currents and Uneven Bottoms

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Abstract

Assuming that an uneven bottom consists of a slowly varying component and a fast varying one, and the nonlinearity parameter is in the same order of magnitude as the modulation one, the nonlinear resonant wave-current-bottom interactions are considered by using the WKBJ perturbation theory, resulting in synchronous resonance, superharmonic one, and subharmonic one which is investigated in detail to obtain a couple of the governing equations for the evolution of the modulated wave groups and for the generation of the second-order long waves. A Stuart-Landau equation obtained in the case of no spatial modulation is dynamically analyzed with respect to a steady current and a weakly oscillatory one.

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References

  1. Agnon Y (1999). Linear and nonlinear refraction and Bragg scattering of water waves. Phys Rev E 59: 1319–1322

    Article  MathSciNet  ADS  Google Scholar 

  2. Heathershaw A D (1982). Seabed-wave resonance and sand bar growth. Nature 296: 343–345

    Article  ADS  Google Scholar 

  3. Huang N E (1999). A review of coastal wave modeling: the physical and mathematical problems. In: Liu P L-F (ed) Advances in coastal and ocean engineering, Vol 4 World Scientific, Singapore

    Google Scholar 

  4. Huang H, Zhou X R (2001). On the resonant generation of weakly nonlinear Stokes waves in regions with fast varying topography and free surface current. Appl Math Mech 22: 730–740

    Article  MATH  MathSciNet  Google Scholar 

  5. Jansson T T, Herbers T H C, Battjes J A (2006). Generalized evolution equations for nonlinear surface gravity waves over two-dimensional topogaphy. J Fluid Mech 552: 393–418

    Article  MathSciNet  ADS  Google Scholar 

  6. Kennedy J F (1963). The mechanics of dunes and antidunes in erodible-bed channels. J Fluid Mech 16: 521–544

    Article  MATH  ADS  Google Scholar 

  7. Kirby J T (1986). A general wave equation for waves over rippled beds. J Fluid Mech 162: 171–186

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. Li Y L, Mei C C (2006). Subharmonic resonance of a trapped wave near a vertical cylinder in a channel. J Fluid Mech 561: 391–416

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Liu P L-F, Mei C C (1976). Water motion on a beach in the presence of a breakwater 1. waves. J Geophys Res-Oceans Atmos 81: 3079–3084

    Article  ADS  Google Scholar 

  10. Liu Y M, Yue D K P (1998). On generalized Bragg scattering of surface waves by bottom ripples. J Fluid Mech 356: 297–326

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Mei C C, Stiassnie M, Yue D K-P (2005). Theory and applications of ocean surface waves. Part 1: Linear aspects; Part 2: Nonlinear aspects. World Scientific, Singapore

    Google Scholar 

  12. Sammarco P, Mei C C, Trulsen K (1994). Nonlinear resonance of free surface waves in a current over a sinusoidal bottom: a numerical study. J Fluid Mech 279: 377–405

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Sammarco P, Tran H, Mei C C (1997). Subharmonic resonance of Venice storm gates in waves. 1. Evolution equation and uniform incident waves. J Fluid Mech 349: 295–325

    Article  MATH  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, P.R. China

    Prof. Hu Huang

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  1. Prof. Hu Huang
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© 2009 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg

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Huang, H. (2009). Resonant Interactions Between Weakly Nonlinear Stokes Waves and Ambient Currents and Uneven Bottoms. In: Dynamics of Surface Waves in Coastal Waters. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88831-4_3

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  • DOI: https://doi.org/10.1007/978-3-540-88831-4_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-88830-7

  • Online ISBN: 978-3-540-88831-4

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