On the Discovery of the Steady-State Resonant Water Waves
In 1960 Phillips gave the criterion of wave resonance and showed that the amplitude of a resonant wave component, if it is zero initially, grows linearly with time. In 1962 Benney derived evolution equations of wave-mode amplitudes and demonstrated periodic exchange of wave energy for resonant waves. However, in the past half century, the so-called steady-state resonant waves with time-independent spectrum have never been found for order higher than three, because perturbation results contain secular terms when Phillips’ criterion is satisfied so that “the perturbation theory breaks down due to singularities in the transfer functions”, as pointed out by Madsen and Fuhrman in 2012.
Recently, by means of the homotopy analysis method (HAM), an analytic approximation method for highly nonlinear problems, steady-state resonant waves have been obtained not only in deep water but also for constant water depth and even over a bottom with an infinite number of periodic ripples. In addition, steady-state resonant waves were observed experimentally in a basin at the State Key Laboratory of Ocean Engineering, Shanghai, China, showing excellent agreement with theoretical predictions.
In this chapter we briefly describe the history of research of steady-state resonant water waves, from theoretical predictions to their experimental verification. All of these illustrate that the HAM is a novel method which indeed renders something new and different.
KeywordsWave Energy Homotopy Analysis Method Wave Component Resonant Wave Deformation Equation
This work is partly supported by the National Natural Science Foundation of China (Approval No. 11272209 and 11432009) and State Key Laboratory of Ocean Engineering (Approval No. GKZD010063).
- 5.Davies, A.G.: Some interactions between surface water waves and ripples and dunes on the seabed. http://eprints.soton.ac.uk/14524/1/14524-01.pdf (1980)
- 10.Liao, S.J.: On the proposed homotopy analysis techniques for nonlinear problems and its application. Ph.D. dissertation, Shanghai Jiao Tong University (1992)Google Scholar
- 15.Liao, S.J.: An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 15, 2315–2332 (2010)Google Scholar
- 23.Lyapunov, A.M.: General Problem on Stability of Motion (English translation). Taylor & Francis, London (1992) [the original one was published in 1892 in Russian]Google Scholar
- 27.Mei, C.C., Stiassnie, M., Yue, D.: Theory and Applications of Ocean Surface Waves: Nonlinear Aspects, vol. 2. World Scientific, Singapore (2005)Google Scholar