Viscous damping of gravity-capillary waves: Dispersion relations and nonlinear corrections

TL;DR

This paper investigates how viscosity affects the nonlinear propagation of surface waves at a fluid-air interface, deriving a modified nonlinear Schrödinger equation that accounts for damping effects in gravity and capillary waves.

Contribution

It introduces a modified boundary condition approach and derives a nonlinear Schrödinger equation incorporating viscous damping for both short and long waves.

Findings

Linear dissipation dominates in damping gravity and capillary waves.

Nonlinear damping effects are generally small corrections.

The approach justifies conventional damping models in wave propagation literature.

Abstract

We discuss the impact of viscosity on nonlinear propagation of surface waves at the interface of air and a fluid of large depth. After a survey of the available approximations of the dispersion relation, we propose to modify the hydrodynamic boundary conditions to model both short and long waves. From them, we derive a nonlinear Schr\"odinger equation where both linear and nonlinear parts are modified by dissipation and show that the former plays the main role in both gravity and capillary-gravity waves while, in most situations, the latter represents only small corrections. This provides a justification of the conventional approaches to damped propagation found in the literature.