A method to recover water-wave profiles from bottom pressure measurements

A fully nonlinear, closed-form mapping from time series of pressure measurements at an arbitrary depth in the fluid column to time series of surface displacement is derived from the full Stokes boundary value problem for water waves. The formula is implicit and nonlocal and must be solved numerically, albeit to machine precision. We further use it to derive several explicit, asymptotic formulae and compare the fully-nonlinear formula, the asymptotic formulae, and the two main formulae presently used by engineers to numerically generated exact solutions of Euler’s equations and to our laboratory experiments on solitons, wave groups, periodic (cnoidal) waves and reflected waves.