### Abstract

A fully nonlinear solution for bi-chromatic progressive waves in water of finite depth in the framework of the homotopy analysis method (HAM) is derived. The bi-chromatic wave field is assumed to be obtained by the nonlinear interaction of two monochromatic wave trains that propagate independently in the same direction before encountering. The equations for the mass, momentum, and energy fluxes based on the accurate high-order homotopy series solutions are obtained using a discrete integration and a Fourier series-based fitting. The conservation equations for the mean rates of the mass, momentum, and energy fluxes before and after the interaction of the two nonlinear monochromatic wave trains are proposed to establish the relationship between the steady-state bi-chromatic wave field and the two nonlinear monochromatic wave trains. The parametric analysis on ε_{1} and ε_{2}, representing the nonlinearity of the bi-chromatic wave field, is performed to obtain a sufficiently small standard deviation S_{d}, which is applied to describe the deviation from the conservation state (S_{d} = 0) in terms of the mean rates of the mass, momentum, and energy fluxes before and after the interaction. It is demonstrated that very small standard deviation from the conservation state can be achieved. After the interaction, the amplitude of the primary wave with a lower circular frequency is found to decrease; while the one with a higher circular frequency is found to increase. Moreover, the highest horizontal velocity of the water particles underneath the largest wave crest, which is obtained by the nonlinear interaction between the two monochromatic waves, is found to be significantly higher than the linear superposition value of the corresponding velocity of the two monochromatic waves. The present study is helpful to enrich and deepen the understanding with insight to steady-state wave-wave interactions.

Original language | English |
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Article number | 127104 |

Journal | Physics of Fluids |

Volume | 28 |

Issue number | 12 |

Early online date | 8 Dec 2016 |

DOIs | |

Publication status | E-pub ahead of print - 8 Dec 2016 |

### Keywords

- energy flux
- monochromatic waves
- water particles
- homotopy analysis method
- HAM
- bi-chromatic wave

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## Cite this

*Physics of Fluids*,

*28*(12), [127104]. https://doi.org/10.1063/1.4971252