On phase kinks, negative frequencies, and other third-order peculiarities of modulated surface waves

Abstract

Numerous laboratory and field experiments on nonlinear surface wave trains propagating in deep water (Lake & Yuen 1978; Ramamonjiarisoa & Mollo-Christensen 1979; Mollo-Christensen & Ramamonjiarisoa 1982; Melville 1983) have showed a specific wave modulation that so far has not been explained by nonlinear theories. Typical effects were the so-called wave phase reversals, negative frequencies, and crest pairing, experimentally observed in some portions of the modulated wave train. In the present paper, in order to explain these modulation manifestations, the equations for wavenumber, frequency, and velocity potential amplitude are derived consistently in the third-order approximation related to the wave steepness. The resulting model generalizes, for instance, the well-known nonlinear Schrödinger equation theory, to which it transforms at certain values of the governing parameters.The stationary solutions to the derived set of equations are found in quadrature and then analysed. Within well-defined ranges of the model parameters, these solutions explicitly manifest the above-mentioned wave modulation effects. In particular, they show the wave phase kinks to arise on areas of relatively small free-surface displacement in complete accordance with the experiments.The model with deeply modulated wavenumber and frequency permits one also to analyse the appropriately short surface wavepackets and modulation periods. In this case, a variety of new interesting wave solutions arises revealing complicated alteration of smooth and rough portions of the free surface. Of special importance are solitary waves, naturally generalizing envelope solitons of the nonlinear Schrödinger equation, but having a varying frequency (as a principle of the proposed theory) and a non-zero wave ‘pedestal’ at infinity. These new types of modulated surface waves should be also observable in laboratory tanks and under field conditions, because the relevant free parameters of theory are not extreme.

An analytical model of the evolution of a Stokes wave and its two Benjamin–Feir sidebands on nonuniform unidirectional current

“…To satisfy the Laplace equation to second order, Yuen and Lake (1982), Voliak (1998), andHwung et al (2009) suggested an additional phase-shifted term with a linear and quadratic z correction in the representation of the potential function ϕ:…”

Stokes waves modulation by internal waves

“…The solution to the problem, uniformly valid to O ( ɛ 3 ), is found by a two‐scale expansion with the differentiation: Substitution of the velocity potential in its linear form, ϕ = ϕ 0 e kz sin θ , satisfies the Laplace in the first order in ɛ due to and gives the following additional terms in the second order O ( ɛ 2 ): To satisfy the Laplace equation in the second order, Yuen and Lake [1982] and Shugan and Voliak [1998] suggested additional phase shifted term with a linear and quadratic z correction in the representation of the potential function ϕ : Exponential decaying of wave's amplitude with z is accompanied by the second order subsurface jet due to slow horizontal variations of the wave number and amplitude of the wave packet.…”

“…yields the known wave action conservation law. Modulation – are closed by the equation of wave phase conservation that follows from as a compatibility condition [ Phillips , 1977]: The derived set of – in the absence of a current coincide with those of Shugan and Voliak [1998].…”

“…To satisfy the Laplace equation in the second order, Yuen and Lake [1982] and Shugan and Voliak [1998] suggested additional phase shifted term with a linear and quadratic z correction in the representation of the potential function f:…”

Nonlinear resonance interaction between three capillary-gravity waves on the plane charged fluid surface

“…In a half a century a more efficient method of the investigation, namely, different-time scale method was developed [3,4]. Thereafter, investigations of nonlinear waves on the charged fluid surface directed to finding soliton solutions were also begun [5][6][7][8]. The effects of viscosity [9], electric charge [10] and its relaxation [11], surface active agent relaxation [12], fluid stratification [13], and tangential jump in the velocity field on the stratification boundary [14] on implementation of the nonlinear wave motion and location of internal nonlinear resonances were studied.…”