Dispersion relations for steady periodic water waves with an isolated layer of vorticity at the surface

In this article we obtain the dispersion relation for small-amplitude steady periodic water waves, which propagate over a flat bed with a specified and fixed mean depth, and where the underlying flow has a discontinuous vorticity distribution. This discontinuity takes the form of an isolated layer of constant non-zero vorticity at the surface of the flow, with purely irrotational flow beneath this layer.