## 2 Materials and Methods  
A summary of the physical quantities used in this work is listed in Table 1. \begin{table}
\begin{tabular}{c c c} \hline
**Notation** & **Definition** & **Units** \\ \hline \(r\) & Radius of cylinder & m \\ \(L\) & Length of cylinder & m \\ \(h_{0}\) & Mean water level & m \\ \(h\) & Water depth & m \\ \(v\) & Velocity vector & m\(\cdot\)s\({}^{-1}\) \\ \(p\) & Fluid pressure & kg\(\cdot\)m\({}^{-1}\cdot\)s\({}^{-2}\) \\ \(\rho\) & Fluid density & kg\(\cdot\)m\({}^{3}\) \\ \(g\) & Gravitational acceleration & m\(\cdot\)s\({}^{-2}\) \\ \(\zeta\) & Vertical displacement from the mean water level & m \\ \(p_{a}\) & Atmospheric pressure & kg\(\cdot\)m\({}^{-1}\cdot\)s\({}^{-2}\) \\ \(A\) & Cross-sectional area & m\({}^{2}\) \\ \(H\) & Enhancement factor & none \\ \(a\) & Lattice constant & m \\ \(\lambda\) & Wavelength & m \\ \(c\) & Wave velocity & m\(\cdot\)s\({}^{-1}\) \\ \(f_{n}\) & Normalized frequency & none \\ \(N\) & Number of grid & none \\ \hline \end{tabular}
\end{table} 
Table 1: List of notations and units of the physical quantities used in this work. In this section, the working equations for governing the shallow water waves over under-sea structures are derived. Shallow water waves are also called long gravity waves, which appear on the free water surface. Since the wavelength of a shallow water wave is longer than the effective water depth, the propagation of the wave (amplitude as well as phase) is highly sensible to the bottom topology. For example, the giant tsunami wave is known to be induced by sudden alternation of the sea bottom shape. The shallow water wave equation is a simplified fluid dynamics system that can be used to describe the horizontal motion of fluid. [PERSON] and [PERSON] [27] have applied an invariant imbedding method to study the influence of bottom topography on the propagation of linear shallow water waves over a one-dimensional array. The present study extends their formulation to the two-dimensional (2D) case. The surface wave propagating on an interface overlying the 2D triangular periodic lattice of rigid circular cylinders of radius \(r\) standing vertically on the bottom is schematically depicted in Figure 1.  
The water depth from the mean water level (denoted as \(h_{0}\)) to the top of cylinder is denoted by \(h(x,y)\). The motion of an incompressible inviscid fluid is governed by the following Equation (1), [PERSON]'s equation:  
\[\frac{\partial\overrightarrow{v}}{\partial t}+\left(\overrightarrow{v}. \
abla\right)\overrightarrow{v}=-\frac{1}{\rho}\
abla p+\overrightarrow{g} \tag{1}\]  
where \(\overrightarrow{v}\) is the three-dimensional velocity vector, \(p\) is the pressure in the fluid, \(\rho\) is the fluid density, and \(\overrightarrow{g}\) is the gravitational acceleration. The following Equations (2)-(4), show the Equation (1) in its components:  
\[\frac{\partial v_{x}}{\partial t}+v_{x}\frac{\partial v_{x}}{\partial x}+v_{y} \frac{\partial v_{x}}{\partial y}+v_{z}\frac{\partial v_{x}}{\partial z}=- \frac{1}{\rho}\frac{\partial p}{\partial x} \tag{2}\]  
\[\frac{\partial v_{y}}{\partial t}+v_{x}\frac{\partial v_{y}}{\partial x}+v_{y} \frac{\partial v_{y}}{\partial y}+v_{z}\frac{\partial v_{y}}{\partial z}=- \frac{1}{\rho}\frac{\partial p}{\partial y} \tag{3}\]  
\[\frac{\partial v_{z}}{\partial t}+v_{x}\frac{\partial v_{z}}{\partial x}+v_{y} \frac{\partial v_{z}}{\partial y}+v_{z}\frac{\partial v_{z}}{\partial z}=- \frac{1}{\rho}\frac{\partial p}{\partial z}-g \tag{4}\]  
Figure 1: Schematic plot of a triangular lattice of identical, vertical, rigid circular cylinders standing in a sea bottom. The lattice constant is \(a\) and the cylinder radius is \(r\). (**a**) The \(x\)-\(z\) side-view. (**b**) The \(x\)-\(y\) top-view. An example of the Pelamis sea wave energy conversion (SWEC) device is depicted to show the intended deployment.