Group and phase velocities in the free-surface visco-potential flow: new kind of boundary layer induced instability

TL;DR

This paper analyzes the dispersion relation of visco-potential water waves without simplifying assumptions, revealing a new boundary layer induced instability affecting group velocities.

Contribution

It introduces a comprehensive analysis of complex integro-differential equations for wave velocities, validating previous approximations and discovering a novel instability mechanism.

Findings

Boundary layer creates disintegrating modes in group velocity

Imaginary part of phase velocity remains negative over time

Validates previous qualitative results

Abstract

Water wave propagation can be attenuated by various physical mechanisms. One of the main sources of wave energy dissipation lies in boundary layers. The present work is entirely devoted to thorough analysis of the dispersion relation of the novel visco-potential formulation. Namely, in this study we relax all assumptions of the weak dependence of the wave frequency on time. As a result, we have to deal with complex integro-differential equations that describe transient behaviour of the phase and group velocities. Using numerical computations, we show several snapshots of these important quantities at different times as functions of the wave number. Good qualitative agreement with previous study [Dutykh2009] is obtained. Thus, we validate in some sense approximations made anteriorly. There is an unexpected conclusion of this study. According to our computations, the bottom boundary layer creates disintegrating modes in the group velocity. In the same time, the imaginary part of the phase velocity remains negative for all times. This result can be interpreted as a new kind of instability which is induced by the bottom boundary layer effect.