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.

Figure 1

Schematic diagrams of a periodic array of identical, vertical, rigid cylinders standing in water with constant depth $h$ (a), and a water column pierced by a vertical rigid cylinder, which is surrounded by an “effective” water medium (b). The top panels are the side views and the bottom panels are the vertical views.

Figure 2

Reflection and refraction of a plane water wave by an effective water medium.

Figure 3

The water-wave refractive index of periodic cylinder arrays as a function of the filling fraction. $n_e$ is determined from Eq. (7), $n_d$ is determined from dispersion calculations, and $n_r$ is determined by examining the wave fronts of plane wave in Figs. 4a, 4b, 4c, 4d.

Figure 4

Incidence of a plane water wave with wavelength $\lambda =5a$ on cylinder arrays with $r=0.35a$: wave patterns for different incident angles of 20° (a), 40° (b), 60° (c), 80° (d) for a ten-layer cylinder array, and the reflectance as a function of the incident angle $\theta$ for a nine-layer cylinder array (e).

Figure 5

Focusing of a plane water wave with $\lambda =5a$ and with intensity of unit 1 by a biconvex periodic array of 657 cylinders with $r=0.35a$: (a) the spatial pattern of transmitted intensity, (b) the variation of the intensity along the $x$ axis at $y=0$, and (c) the variation of the intensity along the $y$ axis at $x=82.4a$.