Strongly nonlinear two-speed wave equations for coastal waves and their application

The theory of strongly nonlinear surface waves is presented, which involves a two-speed wave equation for water of a finite depth. The equation takes into account the effects of gravity, nonlinearity, dispersion, vertical motion of the water particles, as well as, surface tension, variable depth and bottom friction. A variety of particular cases of this equation and the well-known Airy, Boussinesq, Green–Naghdi and Camassa–Holm models are considered. The theory describes a coast resonance of ocean waves. Analytical solutions and calculations are presented, which simulate nonlinear transresonant evolution of harmonic and solitary waves near a coast line.