Abstract
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.
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Notes
- 1.
The same problem in finite-depth water can be solved in a rather similar way. For simplicity, let us first consider the problem in deep water.
- 2.
As pointed out by Liu and Liao [20], such an assumption is not absolutely necessary: only the nonlinear resonance criterion given by Eq. (3.69) must be satisfied, but the linear ones given by Eq. (3.1) are unnecessary. In other words, most of the conclusions reported in this chapter about the steady-state resonant waves qualitatively stand up as long as the nonlinear resonance criterion given by Eq. (3.69) is satisfied, even if the linear resonance criterion given by Eq. (3.1) is violated.
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Acknowledgements
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).
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Liao, S., Xu, D., Liu, Z. (2016). On the Discovery of the Steady-State Resonant Water Waves. In: Tobisch, E. (eds) New Approaches to Nonlinear Waves. Lecture Notes in Physics, vol 908. Springer, Cham. https://doi.org/10.1007/978-3-319-20690-5_3
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