The pressure in a deep-water Stokes wave of greatest height

TL;DR

This paper analyzes the pressure distribution beneath an extreme deep-water Stokes wave, revealing that pressure decreases horizontally between crest and trough and increases with depth, despite free surface irregularities.

Contribution

It provides a rigorous mathematical analysis of pressure behavior beneath extreme Stokes waves, addressing challenges posed by stagnation points and free surface irregularities.

Findings

Pressure decreases horizontally between crest and trough

Pressure increases strictly with depth

Pressure is stationary along crest and trough lines

Abstract

In this paper we investigate the qualitative behaviour of the pressure function beneath an extreme Stokes wave over infinite depth. The presence of a stagnation point at the wave-crest of an extreme Stokes wave introduces a number of mathematical difficulties resulting in the irregularity of the free surface profile. It will be proven that the pressure decreases in the horizontal direction between a crest-line and subsequent trough-line, except along these lines themselves where the pressure is stationary with respect to the horizontal coordinate. In addition we will prove that the pressure strictly increases with depth throughout the fluid body.