Nonlinear Interaction in One-Dimensional Wave Field

Abstract

Full nonlinear equations for one-dimensional potential surface waves were used for investigation of evolution of the initially homogeneous train of exact Stokes waves with steepness \({AK} = 0.01 - 0.42\). The numerical algorithm for integration of non-stationary equations and calculation of exact Stokes waves is described. Since the instability of the exact Stokes waves develops very slowly, a random small-amplitude noise was introduced in the initial conditions. Development of instability occurs in two stages: In the first stage, the growth rate of disturbances was close to that established for small steepness by Benjamin and Feir in (1967) and for medium steepness—by McLean (1982). For any steepness, Stokes waves disintegrate and create a random superposition of waves. For \({AK} 0.13\), one of the waves increases its height and finally comes to a breaking point. For the large steepness \({AK} > 0.35\), the rate of growth is slower than for medium steepness, but it does not turn to zero, as it was predicted by McLean (J Fluid Mech 114:315–330, 1982) on the basis of linearized equations for disturbances. The data for spectral composition of disturbances and their frequencies are given. The model is used for investigation of evolution of the wave field initially assigned as a train of harmonic waves. It is shown that a harmonic wave of any amplitude quickly generates new modes which undergo complicated evolution. These modes cannot be referred to neither as bound waves nor as free waves. The results of numerical simulation of adiabatic evolution of the waves assigned in the initial condition with empirical spectrum are presented. It is shown that wave spectrum is subject to strong fluctuations. Most of such fluctuations are reversible; however, a residual effect of the fluctuations causes downshifting of the spectrum. The rate of downshifting depends on the nonlinearity.