Stationary waves and particle flow

We found some amazing stationary waves on the beach at St Bees where the stream flows out into the sea. The point in a stationary wave is that there is no energy transfer. But it has been bugging me for a while that particles flow through the system. It is a bit different to a guitar string where the same bit of wire vibrates up and down to form the wave. Here the peak stays fixed, not wobbling up and down. Particles of water flow through it all the time. You can say that if the peak is fixed in place, it doesn't ever break and transfer energy like the progressive waves on the sea. However, surely the particles carry kinetic energy. Maybe it is because the system is a steady state: same mass flow in and out. Still thinking about it...