Refraction of water waves by periodic cylinder arrays

We show that in the long wavelength limit, water waves propagate through an array of bottom-mounted vertical cylinders as if the water has an effective depth and effective gravitational constant that depends on the filling ratio of the cylinders, leading to refraction phenomena that can be described by analytic formulas. The results are obtained with rigorous homogenization techniques, as well as the multiple scattering formalism that gives full dispersion relationships. This phenomenon provides a mechanism to control the flow of ocean wave energy, as exemplified by a water-wave focusing lens.