Envelope equation for water waves

Abstract

Water waves have long been a subject of attention of both mathematicians and physicists. The formulation of the problem is simple enough to be considered fundamental, but as of yet many questions still remain unanswered and many phenomena associated with wind-driven turbulence remain puzzling. We consider a “unidirectional” motion of weakly nonlinear gravity waves, i.e., we assume that the spectrum of the free surface contains only nonnegative wavenumbers. We use remarkably simple form of the water wave equation that we named “the super compact equation”. This new equation includes a nonlinear wave term (a la NLSE) together with an advection term that can describe the initial stage of wave breaking. This equation has also very important property. It allows to introduce exact envelope for waves without assumption of narrowness bandwidth.