# Numerical Investigation of Excitation of Various Lamb Waves Modes in Thin Plastic Films
## 3 Finite Elements Modeling Results  
### Single Transducer Model  
The main idea of the performed numerical investigation was to analyze possibilities to excite not one but a few guided wave modes possessing very different propagation velocities and distributions of displacements with the same linear phased array. For such purpose symmetrical S\({}_{0}\) and antisymmetrical A\({}_{0}\) modes fit very well. The biggest challenge is the excitation of the S\({}_{0}\) mode, as in this mode, the in-plane displacement component is much bigger than the off-plane component, e.g., normal to the surface of the film. For excitation of those guided waves we shall use a linear air-coupled array consisting of eight strip-like elements with radiating apertures of 5 \(\times\) 1 mm. The model of the thin PVC film was meshed using shell elements and the propagation of guided waves was first simulated by the 3D method. The thickness of the PVC film was \(l\) = 135 \(\upmu\)m. The dimension of the finite elements was 62.5 \(\upmu\)m, the numbers of elements in the model were 6 816 600 and only a quarter of the film with symmetry boundary conditions was modelled. Such a dimension is close to 1/20 th of the S\({}_{0}\) wavelength \(\lambda\) = 10.6 mm at the analyzed 150 kHz frequency. In order to evaluate the influence of ultrasonic field spreading due to a diffraction in the plane of the film, calculations by the 3D method were performed for a single array element with a rectangular aperture with dimensions 5 \(\times\) 1 mm. The air-coupled ultrasonic transducer usually is placed very close to the film surface, for example 1 mm. In such case the simulation of the propagation of ultrasonic wave through the air gap can be replaced by the force acting on the surface of the film. This force is uniformly distributed in the rectangular area corresponding to the radiating aperture 5 \(\times\) 1 mm of the array single element. The simulated spatial distributions of the particle velocity \(y\) component at the time instant \(t\) = 200 \(\upmu\)s is shown in Figure 5a. The B scan of this in-plane particle velocity component along \(y\) axis is presented in Figure 5b. The waves are more strongly radiated along the \(y\) axis direction due to directivity properties of the excitation zone. In the B scan (Figure 5b) we have the signals measured at many positions along the same direction, the phase velocity can also be estimated by taking two signals measured  
Figure 4: Spectrum of the excitation signal. at neighboring positions and estimating the propagation time of a particular phase point between those positions. Figure 5: Simulation of guided waves in the thin plastic film excited by a single element by the 3D method: (**a**) spatial distributions of the particle velocity of magnitude (\(X\), \(Y\) and \(Z\) components) at the time instant \(t=200\) μs; (**b**) the B scan of the \(y\) component in-plane particle velocity. Such an estimation gives the group velocity 424 m/s, what allows concluding that a single array element excites mainly a strong A\({}_{0}\) mode in the thin film. There are very weak traces of the S\({}_{0}\) mode propagating with the velocity 1540 m/s, however they look so weak that are not suitable for any measurements. This 3D simulation was performed using shell elements; therefore, we decided to check if the simulation by 2D FEM exploiting conventional finite elements allows for the revealing of a better S\({}_{0}\) mode. On the other hand, the 2D FEM requires a smaller number of elements and much shorter simulation time (Table 1). The schematic diagram with one excitation zone is shown in Figure 1b. The length of the simulated PVC film is 300 mm. The excitation zone was placed at the 50 mm from the origin of the coordinates. The B scan of the in-plane particle velocity component in the PVC film obtained by 2D simulation is presented in Figure 6. The ultrasonic pulses of the in-plane particle velocity at two different distances from the excitation zone (x = 50 mm, y = 0 mm) are shown in Figure 7a,b.