Speaker: Tom Bridges, University of Surrey
Date: January 27, 2022
Title: Reappraisal of Whitham's 1967 theory for wave-meanflow interaction in shallow water
Abstract: The first part of the talk will review Whitham's 1967 paper on the modulation of Stokes waves in shallow water coupled to wave-generated meanflow, based on Whitham modulation theory, extracting insights, recasting the equations, and re-evaluating the outcomes. For example, the modulation equations can be recast as coupled shallow water equations, and the amplitude of the meanflow and wave are, in general, independent. In the second part of the talk, the coalescing characteristics in the modulation equations discovered by Whitham, whose unfolding is associated with the Benjamin-Feir instability transition at kh=1.363, are studied in more detail. The theory is then taken a step further showing that the unfolding of the coalescing characteristics leads to a nonlinear modulation equation of the form of a generalized two-way Boussinesq equation. Analysis of this equations leads to new insights, such as the multifarious solutions that emerge near kh=1.363, and how a natural phase shift emerges. This is joint work with Dan Ratliff and Olga Trichtchenko.