New gravity-capillary waves at low speeds. Part 2: Nonlinear geometries

TL;DR

This paper extends exponential asymptotics to nonlinear gravity-capillary wave problems, revealing six classes of solutions and new wave phenomena caused by complex geometries and nonlinear effects.

Contribution

It introduces a nonlinear analysis framework for gravity-capillary waves using exponential asymptotics, uncovering multiple wave classes beyond linear solutions.

Findings

Six classes of gravity-capillary waves identified.

Two classes connect with classical linear solutions.

New wave solutions arise from complex geometries and nonlinear effects.

Abstract

When traditional linearised theory is used to study gravity-capillary waves produced by flow past an obstruction, the geometry of the object is assumed to be small in one or several of its dimensions. In order to preserve the nonlinear nature of the obstruction, asymptotic expansions in the low-Froude or low-Bond number limits can be derived, but here, the solutions are waveless to every order. This is because the waves are in fact, exponentially small, and thus beyond-all-orders of regular asymptotics; their formation is a consequence of the divergence of the asymptotic series and the associated Stokes Phenomenon. In Part 1, we showed how exponential asymptotics could be used to study the problem when the size of the obstruction is first linearised. In this paper, we extend the analysis to the nonlinear problem, thus allowing the full geometry to be considered at leading order. When applied to the classic problem of flow over a step, our analysis reveals the existence of six classes of gravity-capillary waves, from which two share a connection with the usual linearised solutions first discovered by Lord Rayleigh. The new solutions arise due to the availability of multiple singularities in the geometry, coupled with the interplay of gravitational and cohesive effects.