Nonlinear hyperbolic equations and run-up of huge sea waves

Abstract

Application of the theory of nonlinear hyperbolic systems to a description of climbing of large amplitude sea waves on a beach is discussed. Taking into account the large scales of huge sea waves such as tsunami, storm waves, the nonlinear shallow water theory is used as a basic model. Exact analytical solutions of the corresponding hyperbolic system are obtainrtl and their relation to classical solutions by Carrier, Greenspan ant1 Jeffrey is discussed. Two limiting cases of a vertical wall and a beach of small slope are considered. Nonlinear dynamics of a moving tongue of flood due to long wave run-up is described in closed form. Criteria of sea wave breaking at moving edge are found. Quasi - simple wave approximation for description of wacc transformation in the case when the effect of reflection can be neglected is used to find parameters of wave breaking far from the shoreline.

Keywords:

• nonlinear hyperbolic systems
• characteristics
• tsunami