On bi-chromatic steady-state gravity waves with an arbitrary included angle

Abstract

Being encouraged by the recent study by Yang et al. [“On collinear steady-state gravity waves with an infinite number of exact resonances,” Phys. Fluids 31, 122109 (2019)] on two primary waves traveling in the same/opposite direction, we investigate two primary waves traveling with an arbitrary included angle by solving the water-wave equations as a nonlinear boundary-value problem. When the included angle is small, there exists an infinite number of nearly resonant wave components corresponding to an infinite number of small denominators in the framework of the classical analytic approximation methods, like perturbation methods. At a certain included angle, it will cause the classical third-order resonance, corresponding to a singularity. Fortunately, based on the homotopy analysis method (HAM), this type of problem can be solved uniformly by choosing a proper auxiliary linear operator with which the small denominators and singularity can be avoided once and for all. Approximate homotopy-series solutions can be obtained for the two primary waves traveling with an arbitrary included angle. The solutions bifurcate at the angle of classical third-order resonance. Regardless of the resonance at the third-order, it is found that when the acute angle between the two primary wave components becomes smaller, the wave energy slowly shifts from the primary wave components to the high-order wave components, and the increase in wave amplitude strengthens this energy transfer. Moreover, when the included angle is close to or smaller than the third-order resonant angle, the third-order quasi-resonant interaction has a greater influence on the energy distribution, especially if the overall amplitude is relatively large. All of this illustrates the validity and potential of the HAM to be applied to rather complicated nonlinear problems that may have an infinite number of singularities or small denominators.