  In this paper we construct periodic capillarity-gravity water waves with a
piecewise constant vorticity distribution. They describe water waves traveling
on superposed linearly sheared currents that have different vorticities. This
is achieved by associating to the height function formulation of the water wave
problem a diffraction problem where we impose suitable transmission conditions
on each line where the vorticity function has a jump. The solutions of the
diffraction problem, found by using local bifurcation theory, are the desired
solutions of the hydrodynamical problem.
