Water wave problem with inclined walls

TL;DR

This paper develops models for small amplitude nonlinear water waves in a 2-D triangular channel with inclined walls, extending linear theory to analyze mode interactions and spectral properties.

Contribution

It introduces a new modeling approach for nonlinear water waves in inclined geometries, including explicit normal mode construction and spectral analysis methods.

Findings

Explicit normal mode construction for inclined channels

Simplified system for low frequency mode interactions

Analysis of spectral truncation structures

Abstract

We study free surface water waves in a 2-D symmetric triangular channel with sides that have a 45o slope. We develop models for small amplitude nonlinear waves, extending earlier studies that have considered the linearized problem. We see that a combination of heuristic small amplitude expansions lead to a relatively simple system that we then use to study interactions of low frequency modes. The formalism relies on an explicit construction of the normal modes of the linear problem and a new way to represent the free surface. We argue that the construction can be applied to more general geometries. We also examine the structure and some dynamical features of spectral truncations for the lowest even modes.