## 2 Materials and Methods
When the horizontal scale of motion is much faster than the vertical motion, the hydrostatic balance becomes a statement that gravity is balanced by a pressure gradient force, implying that vertical accelerations are negligible. Introducing the above hypotheses, the Equation (4) becomes the following Equation (5):  
\[\frac{\partial p}{\partial z}=-\rho g \tag{5}\]  
We define \(\zeta(x,y,t)\) as the vertical displacement from the mean water level which describes the shape of the surface wave. By integrating the Equation (5), one can get the following Equation (6):  
\[p=p_{a}+\rho g[\zeta(x,y,t)-z] \tag{6}\]  
where \(p_{a}\) is the constant atmospheric pressure. We assume approximately uniform flow across any vertical section through the water. This condition yields that the horizontal velocities components remain in phase with the surface elevation. By substituting Equation (6) into Equations (2) and (3), we get the following Equations (7) and (8):  
\[\frac{\partial v_{x}}{\partial t}+v_{x}\frac{\partial v_{x}}{\partial x}+v_{y }\frac{\partial v_{x}}{\partial y}=-g\frac{\partial\zeta}{\partial x} \tag{7}\]  
Figure 1.0  
When the horizontal velocities are depth-independent, using a control volume consisting of water, the mass continuity equation is given by the following Equation (9):  
Figure 1.1  
where \(\delta A\) is the cross-sectional area element. By expanding Equation (9) we obtain the continuity Equation (10):  
Figure 1.2  
Equations (7), (8) and (10) are the nonlinear shallow water wave equations. If the attention is restricted on small wave amplitudes (\(v_{x}\), \(v_{y}\) and \(\zeta\) are very small), Equations (7) and (8) can be linearized in the following Equations (11) and (12):  
Figure 1.3  
Figure 1.4  
and the mass continuity equation, can therefore be expressed by the following Equation (13):  
Figure 1.5  
By taking time derivative on Equation (13) and eliminating the velocities, the wave equation for \(\zeta\) becomes:  
Figure 1.6  
Let us try the solution of \(\zeta\) in the form, \(\zeta(x,y,t)=H(x,y)e^{-\omega t}\), the equation satisfied by \(H\) is the following Equation (15):  
Figure 1.7Therefore, by designing \(h(x,y)\) through deployment of under-sea structures, we can obtain the enhancement factor (\(H\) with respect to the free amplitude) and search for the optimal parameters for constructive interference to occur. A simplifying factor in our model is neglecting the stress effect of the structure on the fluid mechanics. Therefore, there is no solid-fluid coupling considered herein. If the structure is being made of strengthened concrete, the hydraulic pressure induced by the wave motion is too small to affect the geometry of the solid structure. Moreover, the concerned activity occurs within the sea surface level which is far from the surface of the structure, as compared to the expected strain size. Therefore, we expect minor stress effect on the wave amplitude enhancement. On the other hand, the lattice geometry is expected to significantly alter the wave amplitude, similar to the case in photonic crystals. Wave energy is in general extracted by a SWEC device from the wave-induced motion that is converted into mechanical energy, and then transformed to electricity. This can be achieved by using hydraulic systems such as the Pelamis shown in Figure 1.8 Pelamis is a surface attenuator designed to operate within the sea surface circumstance. It consists of semi-submerged cylindrical sections, aligned perpendicular to the wavefront of the incoming wave and utilizes the relative motion among the sections [28]. The sections move relative to one another as waves pass the machine. This wave-induced motion causes the cylinders to pump oil through hydraulic motors, which in turn drive generators. Electricity is then transmitted along a connecting cable to a junction on the seabed. Therefore, it is the wave height that determines the generated power of such a device. The Pelamis is thus interacting with the under-sea structure through the wave height enhancement.