Towards eliminating the nonlinear Kelvin wake

TL;DR

This paper investigates the nonlinear disturbance caused by a moving force on a liquid surface, demonstrating that the wake can be eliminated through specific forcing strategies and that the resulting wave-free state is stable, with potential experimental observability.

Contribution

It introduces a method to eliminate nonlinear Kelvin wakes by choosing an appropriate forcing distribution and proves the stability of the resulting wave-free solution.

Findings

Kelvin wave pattern characterized as a function of Froude number

Wake can be eliminated with specific forcing distribution

Wave-free solution is stable and observable in experiments

Abstract

The nonlinear disturbance caused by a localised forcing moving at constant speed on the free surface of a liquid of finite depth is investigated using the forced Kadometsev-Petviashvili equation. The presence of a steady v-shaped Kelvin wave pattern downstream of the forcing is established for this model equation, and the wedge angle is characterised as a function of the Froude number. Inspired by this analysis, it is shown that the wake can be eliminated via a careful choice of the forcing distribution and that, significantly, the corresponding nonlinear wave-free solution is stable so that it could potentially be seen in a physical experiment. The stability is demonstrated via the numerical solution of an initial value problem for which the steady wave-free state is attained in the long-time limit.