# Regular Wave Effects on the Hydrodynamic Performance of Fine-Mesh Nettings in Sampling Nets
## 2 Material and Methods
### Hydrodynamic Experiment of Fine-Mesh Nettings in a Wave Tank
#### 2.1.3 Determination of the Hydrodynamic Coefficients  
A total of 12 regular waves were employed to evaluate the hydrodynamic performance of the fine-mesh nettings. The wave characteristics were approximated using linear wave theory [22; 40; 41] and described in terms of the velocity potential (\(\varphi\)) and surface elevation (\(\eta\)):  
\[\varphi=\frac{\pi H}{Tk}\frac{\cosh k(z+h)}{\cosh(kh)}\text{sin}(kx-2\pi ft) \tag{1}\]  
Figure 1: Schematic illustration of the hydrodynamic experimental setup for fine-mesh netting in a wave tank: (**a**) experimental setup and (**b**) the netting model.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Wave Case** & \multirow{2}{*}{**Wave Period \(T\) (s)**} & **Wave Height \(H\)** & **Wave Length \(L\)** & **Wave Steepness** \\
**No.** & & **(m)** & **(m)** & **(H/L)** \\ \hline
1 & 1.3 & 0.10 & 2.49 & 0.040 \\
2 & 1.3 & 0.13 & 2.49 & 0.052 \\
3 & 1.3 & 0.16 & 2.49 & 0.064 \\
4 & 1.3 & 0.19 & 2.49 & 0.076 \\
5 & 1.8 & 0.10 & 4.03 & 0.025 \\
6 & 1.8 & 0.13 & 4.03 & 0.032 \\
7 & 1.8 & 0.16 & 4.03 & 0.040 \\
8 & 1.8 & 0.19 & 4.03 & 0.047 \\
9 & 2.3 & 0.10 & 5.49 & 0.018 \\
10 & 2.3 & 0.13 & 5.49 & 0.024 \\
11 & 2.3 & 0.16 & 5.49 & 0.029 \\
12 & 2.3 & 0.19 & 5.49 & 0.035 \\ \hline \end{tabular}
\end{table}
Table 2: Wave parameters of the wave cases for the hydrodynamic experiment.  
\[\eta=a\cos\left(kx-2\pi ft\right)=\frac{H}{2}\text{cos}(kx-2\pi ft) \tag{2}\]  
where \(a\) is the wave amplitude (\(a=H/2\), \(H\) is wave height), \(f\) is frequency (the reciprocal of \(T\), where \(T\) is wave period), \(k\) is the wave number (\(k=2\pi/L\), where \(L\) is wave length), \(h\) is water depth, \(z\) is the vertical position in the water column, and \(x\) is the horizontal position.  
Therefore, the fluid element velocity at position \(x\) (\(u_{x}=\frac{\partial\varphi}{\partial x}\)) and its corresponding acceleration (\(\frac{\partial u_{x}}{\partial t}\)) were obtained:  
\[u_{x}=(\frac{\pi H}{T})\frac{\cosh k(z+h)}{\text{sinh}kh}\text{cos}(kx-2\pi ft) \tag{3}\]  
\[\frac{\partial u_{x}}{\partial t}=(\frac{2\pi^{2}H}{T^{2}})\frac{\cosh k(z+h) }{\text{sinh}kh}\text{sin}(kx-2\pi ft) \tag{4}\]  
In this study, the horizontal wave force acting on the netting and the corresponding wave surface elevation were measured simultaneously. The Morison equation was applied to calculate the drag coefficient (\(C_{D}\)) and inertia coefficient (\(C_{M}\)) of the nettings under wave conditions. The hydrodynamic force (\(F\)) exerted on a submerged cylindrical element, whose diameter is small relative to the wavelength, is expressed as:  
\[dF=dF_{D}+dF_{M}=d\left[\frac{1}{2}C_{D}\rho A\left(\overset{\rightarrow}{u _{x}}\right)\left|\overset{\rightarrow}{u_{x}}\right|+C_{M}\rho V\frac{ \partial\overset{\rightarrow}{u_{x}}}{\partial t}\right] \tag{5}\]  
where \(C_{D}\) and \(C_{M}\) are drag and inertial coefficients, respectively, \(A\) is area projecting the vertical section, \(V\) is volume of the submerged cylinder, and \(\rho\) is fluid density.  
The Reynolds number (\(Re\)) and Keulegan-Carpenter number (\(KC\)) are defined as:  
\[Re=u_{m}d/v \tag{6}\]  
\[KC=u_{m}T/d \tag{7}\]  
where \(u_{m}\) is the maximum horizontal velocity of the water element and \(v\) is the kinematic viscosity.