Investigation On Nonlinear Interactions Of Bidirectional Waves

Bidirectional waves,known as crossing waves,means the interactions of two wave trains with different propagation directions.Such waves can happen when the swell interacts with locally wind-generated waves,or when two swells interacting with different propagation directions or in the case of a rapidly turning wind.The sea states of bidirectional waves not only could bring hazards to sailing ships and marine structures,they are also one of the main mechanisms for the generation of freak waves.Thus,the study of bidirectional waves has significant practical and theoretical values.Although both off-resonance and resonance theories for bidirectional regular waves have been derived,these theories have their inherent limitations and are approximated at only third-order nonlinearity.Besides,for resonant interactions the variatins of wave patterns and velocity profiles have not yet been investigated.Previous studies on crossing random waves mainly focus on statistical properties,and the inherent evolution mechanisms and velocity variations are still not clear.To seek these answers,more detailed and more basic research work is of necessity.The main work of this paper is to investigate the nonlinear evolutions of bidirectional regular waves and bidirectional focusing waves with a fully nonlinear non-hydrostatic model and physical experiments.Firstly,the quasi-resonant interactions between bidirectional regular waves are studied numerically,and the intensity of quasi-resonance is analyzed for different crossing angles.It is shown that if the crossing angle is larger than the resonant angle,the intensity of quasiresonance decreases rapidly with the growth of the crossing angle;if the crossing angle is smaller than the resonant angle,the intensity of quasi-resonance weakens mildly with the decrease of the crossing angle.Thus,the quasi-resonant interactions between bidirectional regular waves are stronger when the crossing angle is approaching to or smaller than the resonant angle,and are weaker when the crossing angle is much larger than the resonant angle.Secondly,the exactly resonant interactions between bidirectional regular waves are studied numerically,and the evolution of the resonant wave,wave patterns and velocity profiles are compared and analyzed for small(?=0.03,0.05)and moderate(?=0.08,0.1)initial steepness.It is found that if the primary waves' steepness is moderate,the applicable distance of the third-order resonance theory decreases with the increase of primary waves' steepness,or this theory would overestimate the resonant wave's growth;the wave patterns near the wavemaker boundaries are uniform and regular,while with the increase of the propagation distance,the wave crest gradually bends and splits,and the extent of the curvature strengthens with the growth of the primary waves' steepness and the resonant wave's propagation distance,and the trough is similar;the horizontal velocity components at the maximum wave crest grow significantly with the propagation distance,especially the component towards x-axis,which is related to the propagation direction of the resonance wave.Thirdly,the resonant interactions between bidirectional regular waves are studied through physical experiments,in which the resonant wave' relative propagation distance is up to kd=396.9(product of wavenumber and propagation distance).It is observed that although the linear resonance conditions are satisfied exactly,with the increase of the primary waves' steepness,the resonant wave's amplitude no longer keeps linear growth along its propagation direction,but grows first and then decreases,or even changes periodically.Its period decreases with the increase of the primary waves' steepness,but the maximum amplitude does not always grow with the increase of the primary waves' steepness.In the meantime,periodic energy exchange occurs between the primary waves and the new resonant waves.By reducing the crossing angle,non-exactly resonant cases are studied as well.With the primary waves' steepness being equal,the larger the absolute value of the wavenumber mismatch,i.e.,more deviation from exact resonance conditions,the maximum growth of the resonant wave is smaller,and its variation period is shorter.Lastly,the nonlinear evolution of bidirectional focusing waves is simulated numerically,and the nonlinear evolution charateristics of bidirectional focusing waves with wide-banded spectrum,narrow-banded spectrum,different crossing angles,and different spectra are investigated successively.It is shown that for the downstream shifting of focusing position and time,the deviation of bidirectional and narrow-banded focusing waves is much larger than bidirectional and wide-banded focusing waves,but both of them are smaller than unidirectional focusing waves.The crossing angle not only affects the intensity of energy transfer in focusing process,but also affects the rules of energy transfer in defocusing process.For bidirectional focusing waves with strong nonlinearity,the third-order resonant interactions would bring about significant changes in wave patterns,such as newly emerged side-waves and the bending and splitting of wave crests.