Recovering the Water-Wave Profile from Pressure Measurements

TL;DR

This paper introduces a novel method to accurately reconstruct water-wave surface profiles from bottom pressure measurements using a nonlocal nonlinear equation derived from the Euler water-wave model, validated through numerical and experimental comparisons.

Contribution

The paper presents a new nonlocal nonlinear equation for surface reconstruction from pressure data, derived directly from the Euler equations without approximation, and analyzes its solvability.

Findings

The nonlocal equation accurately predicts water-wave profiles.

Asymptotic formulas derived from the equation match numerical and experimental data.

The solvability of the equation is rigorously established.

Abstract

A new method is proposed to recover the water-wave surface elevation from pressure data obtained at the bottom of the fluid. The new method requires the numerical solution of a nonlocal nonlinear equation relating the pressure and the surface elevation which is obtained from the Euler formulation of the water-wave problem without approximation. From this new equation, a variety of different asymptotic formulas are derived. The nonlocal equation and the asymptotic formulas are compared with both numerical data and physical experiments. The solvability properties of the nonlocal equation are rigorously analyzed using the Implicit Function Theorem.