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

The central idea of this paper is to make full use of the recently developed theory of random conjugate spaces to establish a basic strict separation theorem that is universally suitable in an arbitrary random locally convex module. A series of interesting corollaries of the basic theorem are also included.