Igor V. Shugan

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

Numerous laboratory and field experiments on nonlinear surface wave trains propagating in deep water (Lake & Yuen; Ramamonjiarisoa & Mollo-Christensen; Mollo-Christensen & Ramamonjiarisoa; Melville) 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 Schrodinger 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

Dynamical cascade generation as a basic mechanism of Benjamin-Feir instability

A novel model of a discretized energy cascade generated by Benjamin-Feir instability is presented. Conditions for appearance of direct and inverse cascades are given explicitly, as well as conditions for stabilization of the wave system due to cascade termination. These results can be used directly for the explanation of the available results of laboratory experiments and as basic forecast scenarios for planned experiments, depending on the frequency of an initially excited mode and the steepness of its amplitude.

Exposure of internal waves on the sea surface

We theoretically analyse the impact of subsurface currents induced by internal waves on nonlinear Stokes surface waves. We present analytical and numerical solutions of the modulation equations under conditions that are close to group velocity resonance. Our results show that smoothing of the downcurrent surface waves is accompanied by a relatively high-frequency modulation, while the profile of the opposing current is reproduced by the surface wave’s envelope. We confirm the possibility of generating an internal wave forerunner that is a modulated surface wave packet. Long surface waves can create such a wave modulation forerunner ahead of the internal wave, while other relatively short surface waves comprise the trace of the internal wave itself. Modulation of surface waves by a periodic internal wavetrain may exhibit a characteristic period that is less than the internal wave period. This period can be non-uniform while the wave crosses the current zone. Our results confirm that surface wave excitation by means of internal waves, as observed at their group resonance frequencies, is efficient only in the context of opposing currents.

Modulational Instability in Directional Wave Fields, and Extreme Wave Events

The paper is based on review of research articles by the authors, with the purpose to demonstrate that the modulationalinstability mechanism is active in typical directional wave fields. If so, possible limits for the wave height due to such mechanism can be outlined. The modulational instability can lead to occurrence of very high waves, which either proceed to the breaking or appear as rogue events, but it was derived for and is usually associated with two-dimensional wave trains. There exists argument, both analytical and experimental, that this kind of instability is impaired or even suppressed in three-dimensional (directional) wave systems. The first part of the paper demonstrates indirect experimental evidences which relate the wave breaking in oceanic conditions to features of two-dimensional breaking waves due to modulational instability. The second section is dedicated to direct measurements of such instability-caused breaking in a directional wave tank with directional spread and mean steepness typical of those in the field. The last section provides conclusions on what is maximal height of an individual wave, depending on the mean wave steepness in a wave train/field, that can be achieved due to such non-linear evolution of wave trains. Copyright

Elastic plate as floating wave breaker in a beach zone

Wave propagation and damping mechanism due to elastic coating of the sea surface is considered. The hydrodynamic performance of an elastic plate is analyzed for various conditions in terms of wave reflection and transmission, plate deflection, and surface strain. Rigidity and geometrical scales of the coating plate essentially affect the wave transmission characteristics. The model of wave propagation and scattering is constructed in the long-wave approximation. The case of elastic plate with fixed edges is considered. It is shown that optimally designed horizontal flexible membrane can be a very effective wave barrier in a beach zone.

Ship wake structure on the finite sea depth in the presence of wind waves

A kinematics model of the ship wake in the presence of surface waves, generated by wind is presented. It is found that the stationary wave structure behind the ship covered a wedge region with the 16.9° half an angle at the top of the wake and only divergent waves are present in a ship wake for co propagating wind waves. Wind waves field directed at some nonzero angle to the ship motion can cause essential asymmetry of the wake and compressing of its windward half. The extension of Whitham-Lighthill kinematics theory of ship wake for the intermediate sea depth is also presented. The ship wake structure essentially depends from the Froude (Fr) number based on the value of the sea depth and ship velocity. For Froude number less than unit both longitudinal and cross waves are presented in the wake region and Kelvin wake angle increased with Fr. For Fr>1 wake angle decreased with Froude number and finally only divergent waves directed almost normally to the ship track are presented in the very narrow ship wake.Copyright

Internal Waves Impact on the Sea Surface

The impact of subsurface currents induced by internal waves on nonlinear Stokes surface waves is theoretically analyzed. An analytical and numerical solution of the modulation equations are found under the conditions close to the group velocity resonance. It is shown that smoothing of the down current surface waves is accompanied by a relatively high-frequency modulation while the profile of the opposing current is reproduced by the surface wave’s envelope. The possibility of generation of an internal wave forerunner, that is a modulated surface wavepacket, is established. Long surface waves can form the wave modulation forerunner ahead of the internal wave, while the relatively short surface waves create the trace of the internal wave. Modulation of surface waves by the periodic internal wave train may have the characteristic period less than the internal wave period and be no uniform while crossing the current zone. Surface wave excitation by internal waves, observable at their group resonance is efficient only on the opposing current.Copyright

Resonant Generation of Internal Waves on Two Muddy Sea Beds by a Progressive Surface Wave

The present work is motivated by recent studies on the interaction between a progressive surface wave and the nearly standing subharmonic internal waves in a two-layer system. It is well known that the loading of progressive surface waves, a silty sediment bed was repeatedly and extensively fluidized. The great interest in understanding this phenomenon was induced by the practical applications in sediment transport, wave attenuation, and the design of marine structures. The nonlinear response of an initially flat sea bed, with two muddy sections, to a monochromatic surface progressive wave was investigated in the present study. Based on an analysis similar to that of Hill & Foda’s (1998), the multiple scale perturbation method was adopted and the boundary value problem was expanded in a power series of the surface-wave steepness. The linear harmonics and the conditions for resonance were obtained by the leading order. While, the temporal evolution equations for the internal-wave amplitudes were investigated by a second-order analysis. It was found that result for equal density of two muddy sections is similar to that of Hill & Foda’s (1998). Two opposite-traveling internal “mud” waves are selectively excited and formed a resonant triad with the progressive surface wave. However for a surface water wave progressing over two different muddy sections, the surface wave will also excite only two opposite-traveling short interfacial waves, forming a nearly standing wave at the interface of the fresh water and the muddy layer. Meanwhile, two opposite-outgoing “mud” waves each with very long wavelength will be simultaneously induced at the interface of two muddy sections. As a result, the amplitudes of the two short internal waves are found to grow exponentially in time. Furthermore, it will be much difficult to excite the internal waves when surface water wave progressing over two muddy sections with the large density gap.Copyright