Interaction of edge waves with swell on a beach

Excitation of edge waves on a beach by incoming swell is considered on the basis of shallow-water model. Subharmonic resonance mechanism of interaction is analyzed by multi-scaled expansion asymptotic techniques. The generation of edge waves between wave breakers is found to have a dynamic threshold. It is defined by intensity and frequency of incoming swell, geometry of a shore zone. Nonlinear no stationary wave solutions for the envelope of interacting edge waves are described by generalized Sine-Gordon model. An infinite set of exact solutions are received by the Lamb method for the phase synchronism regime of wave’s interaction

Experimental study of breathers and rogue waves generated by random waves over non-uniform bathymetry

We present experimental evidence of formation and persistence of localized waves, breathers, and solitons, occurring in a random sea state and uniformly traveling over non-uniform bathymetry. Recent studies suggest connections between breather dynamics and irregular sea states and between extreme wave formation and breathers, random sea states, or non-uniform bathymetry individually. In this paper, we investigate the joint connection between these phenomena, and we found that breathers and deep-water solitons can persist in more complex environments. Three different sets of significant heights have been generated within a Joint North Sea Wave Observation Project wave spectrum, and the wave heights were recorded with gauges in a wave tank. Statistical analysis was applied to the experimental data, including the space and time distribution of kurtosis, skewness, Benjamin–Feir index, moving Fourier spectra, and probability distribution of wave heights. Stable wave packages formed out of the random wave field and traveling over shoals, valleys, and slopes were compared with exact solutions of the nonlinear Schrödinger equation with a good match, demonstrating that these localized waves have the same structure as deep-water breathers. We identify the formation of rogue waves at moments and over regions where the kurtosis and skewness have local maxima. These results provide insights for understanding of the robustness of Peregrine and higher-order Akhmediev breathers, Kuznetsov–Ma solitons, and rogue waves, and their occurrence in realistic oceanic conditions, and may motivate analogous studies in other fields of physics to identify limitations of exact weakly nonlinear models in non-homogeneous media

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

An analytical weakly nonlinear model of the Benjamin–Feir instability of a Stokes wave on nonuniform unidirectional current is presented. The model describes evolution of a Stokes wave and its two main sidebands propagating on a slowly varying steady current. In contrast to the models based on versions of the cubic Schrödinger equation, the current variations could be strong, which allows us to examine the blockage and consider substantial variations of the wave numbers and frequencies of interacting waves. The spatial scale of the current variation is assumed to have the same order as the spatial scale of the Benjamin–Feir (BF) instability. The model includes wave action conservation law and nonlinear dispersion relation for each of the wave's triad. The effect of nonuniform current, apart from linear transformation, is in the detuning of the resonant interactions, which strongly affects the nonlinear evolution of the system. <br><br> The modulation instability of Stokes waves in nonuniform moving media has special properties. Interaction with countercurrent accelerates the growth of sideband modes on a short spatial scale. An increase in initial wave steepness intensifies the wave energy exchange accompanied by wave breaking dissipation, resulting in asymmetry of sideband modes and a frequency downshift with an energy transfer jump to the lower sideband mode, and depresses the higher sideband and carrier wave. Nonlinear waves may even overpass the blocking barrier produced by strong adverse current. The frequency downshift of the energy peak is permanent and the system does not revert to its initial state. We find reasonable correspondence between the results of model simulations and available experimental results for wave interaction with blocking opposing current. Large transient or freak waves with amplitude and steepness several times those of normal waves may form during temporal nonlinear focusing of the waves accompanied by energy income from sufficiently strong opposing current. We employ the model for the estimation of the maximum amplification of wave amplitudes as a function of opposing current value and compare the result obtained with recently published experimental results and modeling results obtained with the nonlinear Schrödinger equation

Research Note—Channel Structure with Knowledge Spillovers

We study two main questions in this paper: (1) How do spillovers of knowledge created by manufacturers' investments in process innovation affect channel structure and effort investment incentives? (2) What are the interactions between organizational incentives to form joint ventures and strategic alliances with competitors, and coordinate decisions vertically with downstream channel members? We focus on situations where spillovers are involuntary, firms' innovative activities are nonoverlapping, and firms benefit directly from the results of competitors' innovations. Under these conditions, we find that spillovers in process knowledge increase the likelihood of observing decentralized channel structures. Surprisingly, decentralized manufacturers invest more in process innovation than perfectly coordinated manufacturers do when spillovers are large. Moreover, in industries where large spillovers exist, horizontal cooperation among manufacturers induces higher levels of process innovation investments than channel coordination does. From a public policy perspective, however, the desirability of such cooperative arrangements among competitors depends on channel structure: joint ventures among decentralized manufacturers are more likely to meet the regulators' criteria of raising effort investments than cooperation among integrated manufacturers would be. Investment incentives are best provided when firms share their process knowledge and are buffered from subsequent price competition by independent retailers.channel coordination, process innovation, spillovers, research joint ventures