The propagation of water waves across a laterally sheared current

This paper studies surface gravity waves propagating across a shearing current in water of constant depth according to linear theory. The approach adopted is to represent the current by a series of vortex sheets separating regions of constant velocity. In each such region the method of solution employed is one of expansion in terms of eigenfunctions. These are then matched at the boundaries between regions. This leads to a large set of linear algebraic equations to solve for the coefficients. The results are then compared with those obtained using simpler semi-analytic theories which neglect the evanescent modes and can be described as analogues of the mild-slope equation. The general conclusion is that these simple approaches are accurate when the wavelength of the incident waves is much less than the lateral length scale over which the current varies but become less and less accurate as the wavelength increases.