Steady-state waves at class II Bragg resonance

This paper examines class II Bragg resonance in the interaction between surface gravity waves and bottom ripples and derives the corresponding steady-state wave system based on the homotopy analysis method. Specifically, we study the resonance between a nonlinear oblique incident wave and a doubly periodic cosinusoidal bottom. The bottom extends infinitely in the horizontal direction, and its two bottom-wave components are collinear. We analyze the influence of the relative water depth, slope of the incident wave, slope of the bottom topography, and amplitude ratio of the two bottom-wave components on the energy distribution of the wave system. The energy distribution can be classified into two types. In type 1, the incident and resonant wave components have the same energy. In type 2, the energy of each component is different. There are two situations in both type 1 and type 2. The incident and resonant wave components account for most of the energy of the wave system, whereas the energy of other wave components cannot be ignored in the other system. These two situations were not identified in our previous study [Xu et al., "Equilibrium states with finite amplitudes at exactly and nearly class-I Bragg resonances," J. Math. 2021, 9986114]. We then investigate the effects of different physical parameters (including the incident wave slope and bottom undulations) on the energy distribution, wave slope, and angular frequencies of the wave system. The results of this study deepen our understanding of class II Bragg resonance and suggest several directions for future research.