On the steady-state nearly resonant waves

Abstract

The steady-state nearly resonant water waves with time-independent spectrum in deep water are obtained from the full wave equations for inviscid, incompressible gravity waves in the absence of surface tension by means of a analytic approximation approach based on the homotopy analysis method (HAM). Our strategy is to mathematically transfer the steady-state nearly resonant wave problem into the steady-state exactly resonant ones. By means of choosing a generalized auxiliary linear operator that is a little different from the linear part of the original wave equations, the small divisor, which is unavoidable for nearly resonant waves in the frame of perturbation methods, is avoided, or moved far away from low wave frequency to rather high wave frequency with physically negligible wave energy. It is found that the steady-state nearly resonant waves have nothing fundamentally different from the steady-state exactly resonant ones, from physical and numerical viewpoints. In addition, the validity of this HAM-based analytic approximation approach for the full wave equations in deep water is numerically verified by means of the Zakharov’s equation. A thought experiment is discussed, which suggests that the essence of the so-called ‘wave resonance’ should be reconsidered carefully from both of physical and mathematical viewpoints.