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.