Near-linear Dynamics for Shallow Water Waves

TL;DR

This paper demonstrates that shallow water surface waves with high-frequency energy behave nearly linearly over long timescales, due to dispersive averaging of nonlinearity, confirmed by analysis and simulations.

Contribution

It introduces an averaging method showing near-linear behavior of high-frequency shallow water waves over extended periods.

Findings

High-frequency energy leads to near-linear wave dynamics.

Nonlinearity is effectively averaged out by dispersion.

Numerical simulations confirm theoretical predictions.

Abstract

It is shown that spatially periodic one-dimensional surface waves in shallow water behave almost linearly, provided large part of the energy is contained in sufficiently high frequencies. The amplitude is not required to be small (apart from the shallow water approximation assumption) and the near-linear behavior occurs on a much longer time scale than might be anticipated based on the amplitude size. Heuristically speaking, this effect is due to the nonlinearity getting averaged by the dispersive action. This result is obtained by an averaging procedure, which is briefly outlined, and is also confirmed by numerical simulations.