Type of dike using C-shaped vertical cylinders

The present study investigates a way to design dikes which can filter the wavelengths of ocean surface waves. This offers the possibility to achieve a structure that can attenuate waves associated with storm swell, without affecting coastline in other conditions. Our approach is based on low-frequency resonances in metamaterials combined with Bragg frequencies for which waves cannot propagate in periodic lattices.

Figure 1

(a) Top view of a two-dimensional rectangular lattice with lattice vectors ${\mathit{a}}_{\mathit{1}}$ and ${\mathit{a}}_{\mathit{2}}$, composed of C-shaped cylinders. (b) Periodic cell of the lattice spanned by ${\mathit{a}}_{\mathit{1}}$ and ${\mathit{a}}_{\mathit{2}}$, with $d_o$ the outer diameter, $d_i$ the inner diameter, and $l_{sr}$ the width of the slit of cylinders. The points $\mathrm{\Gamma }XM{X}^{\prime }$ denote the extrema of the irreducible Brillouin zone and the dashed lines denote directions that bound this zone.

Figure 2

Band diagrams corresponding to the configurations mentionned in Table 1. The dashed lines stand for bands associated with a square lattice $L=0.325\text{m}$ of plain cylinders with diameter $d_o=0.150\text{m}$.

Figure 3

Schematic top view of the experimentally tested arrays. Array with smaller size C-shape cylinder (gray) corresponding to configuration 1 in Table 1. Array with larger size C-shape cylinder (red with hatch) corresponding to configurations 2 and 3.

Figure 4

Comparison of experimental measurements of the transmision coefficient with numerical results for incident water waves along $\mathrm{\Gamma }X$ with wave steepness $H/\lambda =2\%$.

Figure 5

Schematic side view of the wave tank located at engineering school Centrale Marseille.

Figure 6

Experimental measurements of the transmission coefficient (T), reflection coefficient (R), normalized amplitude inside a cylinder of the first column (Ac), and dissipation coefficient (Ed) [see Eq. (8)] for configuration 3. The amplitude inside the cylinders for other columns is similar but attenuated.

Figure 7

Photos of a cylinder at different time steps and for different wave steepnesses. Photos are side views where the waves propagate from the left (arrows), the dashed yellow lines are the plain side of the cylinder, and the dashed green lines indicate the side with the split. Cylinders are transparent such that we see water inside and behind. (a) Water comes out of the cylinder for $H/\lambda =2\%$, (b) water comes in the cylinder for $H/\lambda =2\%$, and (c) wave breaking in the cylinder for $H/\lambda =5\%$. The vortices are identified inside the red circles.

Figure 8

Dissipation coefficients for all the studied configurations. When the amplitude of waves is important in the cylinder ($kL\lesssim 2.7$), we note that, the thinner the slit (configurations 1 and 2), the higher the dissipation.

Figure 9

Comparison of experimental reflection and transmission coefficients for different wave steepnesses (shown in parentheses). Results for higher wave steepnesses are represented with markers (circles for configuration 1 and squares for configuration 3).