On the Alber equation for shoaling water waves

Abstract

The problem of unidirectional shoaling of a water-wave field with a narrow energy spectrum is treated by using a new Alber equation. The stability of the linear stationary solution to small non-stationary disturbances is analysed; and numerical solutions for its subsequent long-distance evolution are presented. The results quantify the physics which causes the gradual decay in the probability of freak-wave occurrence, when moving from deep to shallow coastal waters.

Freak waves caused by reflection

“…Indeed, while Iusim and Stiassnie (1985) extended the cubic self-interaction Zakharov kernel ( , , , ) to include the effects of slow depth variation, a general form for arbitrary quartets is still unknown. Recently work has been undertaken to extend the narrowbandwidth Alber equation to a sloping bed by Kluczek et al (2021), although this unidirectional model cannot include the effects of reflection.…”