Dispersion relations for gravity water flows with two rotational layers

We derive the dispersion relation for periodic traveling water waves propagating at the surface of water possessing a layer of constant non-zero vorticity γ1γ1 adjacent to the free surface above another rotational layer of vorticity γ2γ2 which is adjacent to the flat bed. As a by-product we give necessary and sufficient condition for local bifurcation in the frame-work of piecewise constant vorticity. Moreover, we give estimates on the speed at the free surface of the bifurcating laminar flows. These estimates involve only the vorticity γ1γ1, the mean depth of water dd and the depth at which the jump in vorticity occurs. A stability result for certain laminar flows is also presented.