Abstract
The waves on a free surface of 2D deep water can be split into two groups: the waves moving to the right, and the waves moving to the left. A specific feature of the four-wave interactions of water waves allows to describe the evolution of the two groups as a system of two equations. The fundamental consequence of this decomposition is the conservation of the “number of waves” in each particular group. The envelope approximation for the waves in each group of counter streaming waves is obtained.
Highlights
A potential flow of an ideal incompressible 2D fluid with a free surface is described by the following classical system of equations:, φxx + φzz
The Zakharov equation for water waves can be derived in two steps: 1
It is known that the five wave interactions do not vanish, see [7]
Summary
A potential flow of an ideal incompressible 2D fluid with a free surface is described by the following classical system of equations:. Where the linear operator kacts in Fourier–space as a multiplication by the modulus of wavenumber k: kf ( x ) ⇒ |k | f k This Hamiltonian is the starting point to derive the so-called Zakharov equation in 1968. In 1994 a remarkable property of the interaction of water waves was discovered, namely, the coefficient of the four-waves interaction in the Hamiltonian vanishes on the resonance manifold, see [2]. This discovery allowed to greatly simplify the Zakharov equation, and the so-called “compact equation” for unidirectional water waves was derived, see [3,4]. In what follows we will present a new system of two equations that describe the evolution of two almost planar counter streaming waves
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