Dispersion relations for fixed mean-depth flows with two discontinuities in vorticity

Abstract We derive here the dispersion relation for small-amplitude steady two-dimensional periodic water waves that propagate over a flat bed with a specified and fixed mean-depth. The water flow has a discontinuous vorticity distribution reflecting an isolated rotational layer of fluid. We then use the dispersion relation to obtain necessary and sufficient conditions for local bifurcation to occur. Moreover, a stability analysis of the laminar flows is presented.