Journal of Fluid Mechanics


On the steady-state fully resonant progressive waves in water of finite depth

Dali Xua2, Zhiliang Lina2, Shijun Liaoa1a2 c1 and Michael Stiassniea3

a1 State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

a2 School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

a3 Faculty of Civil and Environmental Engineering, Technion IIT, Haifa 32000, Israel


The steady-state fully resonant wave system, consisting of two progressive primary waves in finite water depth and all components due to nonlinear interaction, is investigated in detail by means of analytically solving the fully nonlinear wave equations as a nonlinear boundary-value problem. It is found that multiple steady-state fully resonant waves exist in some cases which have no exchange of wave energy at all, so that the energy spectrum is time-independent. Further, the steady-state resonant wave component may contain only a small proportion of the wave energy. However, even in these cases, there usually exist time-dependent periodic exchanges of wave energy around the time-independent energy spectrum corresponding to such a steady-state fully resonant wave, since it is hard to be exactly in such a balanced state in practice. This view serves to deepen and enrich our understanding of the resonance of gravity waves.

(Received January 14 2012)

(Reviewed June 30 2012)

(Accepted July 17 2012)

(Online publication September 07 2012)

Key Words:

  • surface gravity waves


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