## 2 Materials and Methods
### Mode Selection of Lamb Waves for Defect Detection in Underwater Plates  
Figure 1 shows the dispersion curves of Lamb waves propagating in a free aluminum plate, which is calculated based on the Lamb wave dispersion equations [30]. In the region _fd_ < 2 MHz\(\cdot\)mm, only A\({}_{0}\) and S\({}_{0}\) modes can be generated, where the A\({}_{0}\) mode has strong dispersion. The weak dispersion region of the S\({}_{0}\) mode exists in the low-_fd_ range around 1 MHz\(\cdot\)mm, as shown in the dotted frame in Figure 1. When the plate is fully immersed in water, the dispersion curves of S\({}_{0}\) and A\({}_{0}\) mode are almost invariant [27; 31]. The energy of A\({}_{0}\) mode partly converts into a Scholte wave, which decreases exponentially away from the surface into water. Utilizing the dispersion equations, the displacement components of both S\({}_{0}\) and A\({}_{0}\) modes in the plate are also calculated, and the normalized amplitudes at 1 MHz\(\cdot\)mm are shown in Figure 2.  
Figure 1: Dispersion curves of Lamb waves in a free aluminum plate.  
In Figure 2, \(u_{x}\) and \(u_{z}\) represent the normalized in-plane and out-of-plane displacement in a plate with thickness \(d\), respectively. Clearly, the out-of-plane component of the S\({}_{0}\) mode is much smaller than its in-plane component. However, the A\({}_{0}\) mode is the opposite, having a higher out-of-plane component than its in-plane component. The high out-of-plane displacement of the A\({}_{0}\) mode at the surface of plate leads to a significant attenuation in underwater plate, and thus will significantly limit its propagation distance.  
The attenuation curves are calculated using the finite element method (FEM) software COMSOL(r) in a 2D model for mode selection [32; 33; 34]. Considering an aluminum plate with a thickness of 0.2 mm and a length of 150 mm, all boundaries are in contact with water whose thickness is 0.5 mm. The outside of the water layer is set to be outwardly radiative, so that it can be considered as infinite water. The interface between the water and the aluminum plate is simulated using the acoustic structure-coupling boundary condition. The material properties of aluminum and water used are shown in Table 1. The aluminum plate and water layer are meshed by quadrangle elements. To ensure the accuracy of simulation, the maximum of the quadrangle elements is less than \(L_{e}\) = 80 \(\upmu\)m, guaranteeing that at least 8 elements are allocated per wavelength [35]. In the water layer, acoustic pressure is only applied in the \(y\)-axis, as shown in Figure 3, because fluid is unable to withstand shear loads in the \(x\)-axis. The interface between the water layer and the aluminum plate is simulated using an acoustic-structure boundary provided by COMSOL(r). This coupling constraint ensures that the motions of particles at the interface are continuous. The simulation is carried out in the frequency domain for \(fd\) varying from 0.5 to 1.5 MHz\(\cdot\)mm. For precise excitation at the corresponding \(fd\), the generation of both S\({}_{0}\) and A\({}_{0}\) mode are simulated by loading the displacements at the left side of aluminum plate, including in-plane and out-of-plane components, at each specific \(fd\).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Density (kg/m\({}^{3}\))** & **C\({}_{\text{T}}\) (km/s)** & **C\({}_{\text{L}}\) (km/s)** & **Young’s Modulus GPa** \\ \hline Aluminum & 2700 & 3103 & 6176 & 70 \\ Water & 1000 & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties of aluminum and water used in the simulation and experimental measurements.  
Figure 3: Schematic of 2D model: aluminum plate underwater.  
Figure 2: Normalized displacement components of Lamb waves in the plate of thickness \(d\): (**a**) the S\({}_{0}\) mode; (**b**) the A\({}_{0}\) mode.