On the recovery of traveling water waves with vorticity from the pressure at the bed

TL;DR

This paper develops higher-order formulas to recover water wave surface elevation from bed pressure measurements, accounting for vorticity, and improves upon existing approximation methods.

Contribution

It introduces new approximation formulas that incorporate vorticity effects via the Rayleigh equation, enhancing the accuracy of surface elevation recovery from pressure data.

Findings

Formulas outperform hydrostatic approximation

Good agreement with asymptotic models in zero vorticity case

Effective in recovering surface elevation with vorticity effects

Abstract

We propose higher-order approximation formulae recovering the surface elevation from the pressure at the bed and the background shear flow for small-amplitude Stokes and solitary water waves. They offer improvements over the pressure transfer function and the hydrostatic approximation. The formulae compare reasonably well with asymptotic approximations of the exact relation between the pressure at the bed and the surface wave in the zero vorticity case, but they incorporate the effects of vorticity through solutions of the Rayleigh equation. Several examples are discussed.