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

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 the 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 the solutions of the Rayleigh equation. Several examples are discussed.