Ship waves in the presence of uniform vorticity

Abstract

AbstractLord Kelvin’s result that waves behind a ship lie within a half-angle $\phi _{\mathit{K}}\approx 19^{\circ }28'$ is perhaps the most famous and striking result in the field of surface waves. We solve the linear ship wave problem in the presence of a shear current of constant vorticity $S$, and show that the Kelvin angles (one each side of wake) as well as other aspects of the wake depend closely on the ‘shear Froude number’ $\mathit{Fr}_{\mathit{s}}=VS/g$ (based on length $g/S^2$ and the ship’s speed $V$), and on the angle between current and the ship’s line of motion. In all directions except exactly along the shear flow there exists a critical value of $\mathit{Fr}_{\mathit{s}}$ beyond which no transverse waves are produced, and where the full wake angle reaches $180^\circ $. Such critical behaviour is previously known from waves at finite depth. For side-on shear, one Kelvin angle can exceed $90^\circ $. On the other hand, the angle of maximum wave amplitude scales as $\mathit{Fr}^{-1}$ ($\mathit{Fr}$ based on size of ship) when $\mathit{Fr}\gg 1$, a scaling virtually unaffected by the shear flow.