Ship waves in the presence of uniform vorticity

Lord Kelvin’s result that waves behind a ship lie within a half-angle φ K≈ 19∘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’ Frₛ=VS/g (based on length g/S² 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 Frₛ beyond which no transverse waves are produced, and where the full wake angle reaches 180^∘ . Such critical behaviour is previously known from waves at finite depth. For side-on shear, one Kelvin angle can exceed 90^∘ . On the other hand, the angle of maximum wave amplitude scales as Fr⁻¹ ( Fr based on size of ship) when Fr 1 , a scaling virtually unaffected by the shear flow.