Reappraisal of Whitham's 1967 theory for wave-mean flow interaction in shallow water

TL;DR

This paper revisits Whitham's 1967 theory on wave-mean flow interaction in shallow water, revealing new insights into the coupled equations, independence of wave and mean flow amplitudes, and implications for Benjamin-Feir instability.

Contribution

It demonstrates that Whitham's modulation equations can be reformulated as classical shallow water equations with a modified gravity term and explores the independence of wave and mean flow amplitudes.

Findings

Recasting of Whitham's equations into classical shallow water form

Independence of wave amplitude and mean flow amplitude

Analysis of coalescing characteristics related to Benjamin-Feir instability

Abstract

The modulation equations for Stokes waves in shallow water coupled to wave generated mean flow, derived in Whitham (1967), based on an averaged Lagrangian are revisited. Firstly, it is shown that they can be recast into two coupled classical shallow water equations, with modified gravity having the sign of the Whitham index: sign($\omega_0'' \omega_2$). Secondly, it is shown that the amplitude of the mean flow and amplitude of the wave are, in general, independent. Thirdly, the implications of the coalescing characteristics, whose unfolding is associated with the Benjamin-Feir instability, are studied.