Confinement of hydrodynamic modes on a free surface and their quantum analogs

A subtle procedure to confine hydrodynamic modes on the free surface of a fluid is presented here. The experiment consists of a square vessel with an immersed square central well vibrating vertically so that the surface waves generated by the meniscus at the vessel boundary interfere with the bound states of the well. This is a classical analogy of a quantum well where some fundamental phenomena, such as bonding of states and interference between free waves and bound states, can be visualized and controlled. The above mentioned interference leads to a novel hydrodynamic transition from quasiperiodic to periodic patterns. Tight binding numerical calculations are performed here to show that our results could be transferred to design quantum confinements exhibiting electronic quasiperiodic surface states and their rational approximants for the first time.