## 3 Results and Discussion
### Laboratory Demonstration  
To demonstrate the predicted enhancement of water wave amplitude, we have designed and conducted a reduced-scale laboratory experiment. All the experiments were performed in a rectangular water tank with side and bottom walls made of transparent glass. The experimental set-up and the home-designed under-water periodic structure are shown in Figure 7. The water tank is 5 m in length, 0.8 m in width and 1.2 m in height mounted on the laboratory ground (Figure 7a). The principal \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Lattice** & \(r/a\) & \(h_{0}/h\) & \(f_{n}\) \\ \hline Triangular & 0.2–0.32 & 5–13 & 0.32–0.60 \\ Square & 0.14–0.34 & 4–13 & 0.33–0.64 \\ Hexagonal (\(K\) point) & 0.15–0.24 & 14–16 & 0.36–0.53 \\ Hexagonal (\(M\) point) & 0.16–0.23 & 14–16 & 0.33–0.44 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: List of optimization parameters and their ranges with enhancement factor of global maxima. Figure 6: Enhancement factor of the surface water waves over the triangular, square and hexagonal structured arrays of cylinders as a function of the normalized frequency. facilities include a wave generating system, a back-wave absorber, and an image recording system. Incident plane waves are generated by a self-designed displacement piston wave-maker, which can be used stably and effectively in shallow water wave regions. The driving frequency of the wave-maker can be altered from 0.5 to 1.2 Hz. The back wave reflected from the other end of the water tank is damped by the wave-absorbing sponge beach. Both top-view and side-view images are recorded by a digital camera at running rates of 10-50 frames per second and exposure times 1-10 ms. The wave elevations are estimated in the following way. A stick ruler is attached vertically on the side wall (Figure 7b) and is calibrated with respect to the right perspective of the camera lens. The contrasted videos are recorded before and after placing the structures in the water tank. The film images are analyzed through an image-extraction software and the wave heights are read from the images. The wave amplitude enhancement factor is then calculated from the obtained wave elevations with respect to the structure-free (control) results. The under-water periodic structure consists of solid circular cylinders forming a triangular lattice mounted on the bottom of the water tank (Figure 7c,d). We arrange the cylinders with a lattice constant \(a=4\) cm in \(35\times 35\pi\) cm\({}^{2}\) circular area and set the \(\Gamma K\) direction of the structure parallel to the short side of the structure. The radius of cylinder \(r\) is 1 cm and the length of cylinder \(L\) is 24 cm. The structure is positioned on the central line of the wave-tank at a distance of 120 cm from the wave-maker and 300 cm from the wave absorbing area. In Figure 8 we show one of the main experimental results by suitably adjusting the \(h_{0}/h\) (ca. 9) and \(r/a\) (ca. 0.25) ratios to be closely similar to theoretical values. We can see obvious wave concentration on the central part of the structure (see the Supplementary Videos and the closer-up bird-view in Figure 8a). The amplitude of wave in the middle of the tank is clearly higher than that outside the structure area. By changing the driving frequency from 0.5 Hz with a step of 0.05 Hz, the amplitude enhancement does not appear until the frequency is tuned to 0.7 Hz. The enhancement factor is determined to be  
Figure 7: The under-water periodic structure and the experimental set-up in the water tank. **(a)** The bird-view and **(b)** the side-view of the wave amplitude enhancement. **(c)** The three-dimensional view of the triangular lattice of \(13\times 13\) cylinders with lattice constant \(a=4\) cm in **(d)** the \(35\times 35\pi\) cm\({}^{2}\) circular area. about 1.8 \(\pm\) 0.3 (Figure 8b), which is about one half of the expected value 3.78. This enhancement effect disappears at the frequency of 0.75 Hz. The details can be seen in the Supplementary Videos.