## Abstract

The extended Korteweg-de Vries equation which includes nonlinear and dispersive terms cubic in the wave amplitude is derived from the water-wave equations and the Lagrangian for the water-wave equations. For the special case in which only the higher-order nonlinear term is retained, the extended Korteweg-de Vries equation is transformed into the Korteweg-de Vries equation. Modulation equations for this equation are then derived from the modulation equations for the Korteweg-de Vries equation and the undular bore solution of the extended Korteweg-de Vries equation is found as a simple wave solution of these modulation equations. The modulation equations are also used to extend the solution for the resonant flow of a fluid over topography. This resonant flow occurs when, in the weakly nonlinear, long-wave limit, the basic flow speed is close to a linear long-wave phase speed for one of the long-wave modes. In addition to the effect of higher-order terms, the effect of boundary-layer viscosity is also considered. These solutions (with and without viscosity) are compared with recent experimental and numerical results.

Original language | English |
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Pages (from-to) | 263-288 |

Number of pages | 26 |

Journal | Journal of Fluid Mechanics |

Volume | 221 |

Publication status | Published - Dec 1990 |

## Keywords

- INTERNAL SOLITARY WAVES
- MOVING DISTURBANCES
- PERTURBATION-THEORY
- STRATIFIED FLUID
- 2-LAYER FLOW
- SOLITONS