Kelvin wake pattern at large Froude numbers

Abstract

Gravity waves generated by an object moving at constant speed at the water surface form a specific pattern commonly known as the Kelvin wake. It was proved by Lord Kelvin that such a wake is delimited by a constant angle $\simeq 19.47^\circ$. However a recent study by Rabaud and Moisy based on the observation of airborne images showed that the wake angle seems to decrease as the Froude number $Fr$ increases, scaling as $Fr^{-1}$ for large Froude numbers. To explain such observations the authors make the strong hypothesis that an object of size $b$ cannot generate wavelengths larger than $b$. With no need of such an assumption and modelling the moving object by an axisymmetric pressure field, we analytically show that the angle corresponding to the maximum amplitude of the waves scales as $Fr^{-1}$ for large Froude numbers, whereas the angle delimiting the wake region outside which the surface is essentially flat remains constant and equal to the Kelvin angle for all $Fr$.