  We investigate the particle trajectories in a constant vorticity shallow
water flow over a flat bed as periodic waves propagate on the water's free
surface. Within the framework of small amplitude waves, we find the solutions
of the nonlinear differential equations system which describes the particle
motion in the considered case, and we describe the possible particle
trajectories. Depending on the relation between the initial data and the
constant vorticity, some particle trajectories are undulating curves to the
right, or to the left, others are loops with forward drift, or with backward
drift, others can follow some peculiar shapes.
