# Numerical Investigation of Excitation of Various Lamb Waves Modes in Thin Plastic Films
## 3 Finite Elements Modeling Results
### Linear Array Model  
Therefore, we decided to check if it was feasible to excite S\({}_{0}\) mode by a properly phased linear array placed close to the film surface. For this purpose, we proposed a single ultrasonic transducer to use linear air-coupled arrays, which can be electronically readjusted to optimally excite and receive the required guided wave modes. This linear air-coupled array can improve the excitation of S\({}_{0}\) mode. Numerical investigation was performed simulating the excitation of the S\({}_{0}\) guided wave mode by a planar linear array. The schematic diagram is presented in Figure 8. The array consists of eight strip-like elements of 1 mm width. All spacings between elements should theoretically be equal to \(\lambda_{S_{0}}/2\) = 5.3 mm. For the excitation of the S\({}_{0}\) modes, the linear air-coupled array elements are excited successively with the time delay necessary for this mode to propagate the distance between the adjacent elements. The delay time between elements is \(\Delta\tau\) = 3.3 \(\upmu\)s and calculated by the following Equation (7):  
\[\Delta\tau\ =\ \frac{\frac{\lambda_{S_{0}}}{2}}{c_{Sph}(f,I)}. \tag{7}\]  
where \(\lambda_{S_{0}}\) is the wavelength of the S\({}_{0}\) mode, \(I\) is the thickness of the thin PVC film, and \(f\) is the frequency. In such case each strip-like piezoelectric element of the linear array excites  
Figure 7: Ultrasonic pulses of the in-plane \(x\) component of the S\({}_{0}\) mode at two different distances from the excitation zone: (**a**) 210 mm; (**b**) 300 mm. the S\({}_{0}\) mode in phase, thus increasing the amplitude of the propagating guided wave. The B scan of the \(x\) component in-plane particle velocity of the array is presented in Figure 9.  
For identification of the excited Lamb wave modes shown in Figure 9, we calculated propagation velocities of the simulated modes similar to the single element case and compared them with velocities determined by the Semi Analytical Finite Element method (Figure 2). From the numerical simulation, it follows that the A\({}_{0}\) mode propagates with a group velocity \(c_{gr}\) = 459 m/s; the A\({}_{0}\) phase velocity is \(c_{ph}\) = 244 m/s. The S\({}_{0}\) mode propagates with the phase velocity of \(c_{ph}\) = 1591 m/s (Figure 9). The ultrasonic pulses at three different distances from the excitation zone (\(x\) = 50 mm, \(y\) = 0 mm) are presented in Figure 10a-c. The first pulse in those figures is the S\({}_{0}\) mode, the second pulse corresponds to the A\({}_{0}\) mode. The ultrasound velocities of the ultrasonic pulses at two different distances from the excitation zone (Figure 10b,c) are calculated by Equations (5) and (6). The obtained S\({}_{0}\) mode phase velocity in this case is \(c_{ph}\) = 1551 m/s, which is very close to the velocity obtained by the Semi Analytical Finite Element method. Figure 8: Simulation of excitation of guided waves by a linear phased array film. Figure 9: The B scan \(x\) component in-plane particle velocity in the thin PVC film. Figure 10: _Cont._  
The particle velocity amplitude at the excitation zone (\(x\) = 50 mm, \(y\) = 0 mm) is 0.05 m/s. The array excitation zone is presented in Figure 8. The planar linear array improves the excitation of S\({}_{0}\) mode eight times in comparison to when the excitation zone is 1 mm. Although the amplitude of the A\({}_{0}\) mode is bigger than the S\({}_{0}\) mode at longer distances, both modes may be exploited for simultaneous measurements because they arrive at different time instants.