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Resonant interactions between Kelvin ship waves and ambient waves

We consider the nonlinear interactions between the steady Kelvin waves behind an advancing ship and an (unsteady) ambient wave. It is shown that, for moderately steep ship waves and/or ambient waves, third-order (quartet) resonant interaction among the two wave systems could occur, leading to the generation of a new propagating wave along a specific ray in the Kelvin wake. The wave vector of the generated wave as well as the angle of the resonance ray are determined by the resonance condition and are functions of the ship forward speed and the wave vector of the ambient wave. To understand the resonance mechanism and the characteristics of the generated wave, we perform theoretical analyses of this problem using two related approaches. To obtain a relatively simple model in the form of a nonlinear Schrödinger (NLS) equation for the evolution of the resonant wave, we first consider a multiple-scale approach assuming locally discrete Kelvin wave components, with constant wave vectors but varying amplitudes along the resonance ray. This NLS model captures the key resonance mechanism but does not account for the detuning effect associated with the wave vector variation of Kevin waves in the neighbourhood of the resonance ray. To obtain the full quantitative features and evolution characteristics, we also consider a more complete model based on Zakharov's integral equation applied in the context of a continuous wave vector spectrum. The resulting evolution equation can be reduced to an NLS form with, however, cross-ray variable coefficients, on imposing a narrow-band assumption valid in the neighbourhood of the resonance ray. As expected, the two models compare well when wave vector detuning is small, in the near wake close to the ray. To verify the analyses, direct high-resolution simulations of the nonlinear wave interaction problem are obtained using a high-order spectral method. The simulations capture the salient features of the resonance in the near wake of the ship, with good agreements with theory for the location of the resonance and the growth rate of the generated wave.