## 2 Materials and Methods
### Guided Wave Ultrasonic NDT Techniques  
#### 2.4.1 Analytical and Numerical Investigation  
For bulk wave through-transmission method, it is straightforward that the loss of amplitude will indicate where there is a defect. However, in the case of guided waves, it can be beneficial to understand the physical phenomenon by using numerical simulation investigations to visualize the interaction of ultrasonic guided waves with the different characteristics of the guided medium. It is preferred to build a numerical model, then select experimental setup parameters according to the numerical results and validate. Thus, based on the investigation from the simulation, it would be easy to perform the experiment and to validate the numerical model with regards to the theory. Figure 4: Air coupled through-transmission experimental setup. Figure 3: Experimental set-up: immersion ultrasonic non-destructive testing (NDT) with through-transmission (**a**) and scanning acoustic microscopy (**b**). Therefore, a numerical model of air-coupled excitation and contact reception of guided waves was created, as shown in Figure 5.  
To generate the guided waves in the bonded plate, it is essential to obtain the parameters from the dispersion curves so that the specific modes can be generated. The phase velocity dispersion curves for 1.6 mm thick aluminium 2024 plate were obtained by a semi-analytical finite element (SAFE) method developed in Ultrasound Institute, Kaunas, is shown in Figure 6. To reduce the complexity of the investigation, fundamental modes are preferred. Specifically, A0 mode was selected because it has high out-of-plane displacement when compared to the symmetric S0 mode. Since air is the coupling medium, the incidence angle of 10.84\({}^{\circ}\) to generate A0 mode at 300 kHz was calculated using [PERSON]'s law assuming the velocity of sound in air is 343.21 m/s and the phase velocity of A0 mode at 300 kHz is 1824 m/s. The symmetrical three-dimensional (3D) model as shown in Figure 7 was created in COMSOL Multiphysics. The air is assigned as a pressure-acoustics domain, where aluminium and epoxy are assigned as a solid domain. To reduce the complexity of the numerical model, the actual geometry of the transducers was not considered, but instead, only the air domain for the medium of propagation was considered. The acoustic domain and the solid domain are coupled using the COMSOL Multi-physics node called acoustic-solid interaction, where the necessary equations are loaded automatically. Figure 5: Schema of the simulation model for the air-coupled transducer to contact transducer guided wave. Figure 6: Phase velocity dispersion curves for aluminium 2024 plate with 1.6 mm thickness. For the actuator, a cylindrical air domain with a diameter of 14 mm was considered, and the bottom flat surface was considered for excitation with a 10 period Hanning windowed sinusoidal burst at 300 kHz. For the reception, a point probe was considered to record the guided waves after passing through the bond. The normal displacement of the guided waves is recorded at 65 mm away from the transducer. Two simulations were performed, one case for a perfect bond condition and another case with a defect. Here the defect is considered as a void where a rectangular portion of the epoxy is removed to simulate the artificial debonding within the structure. The size of the void is 12.7 mm \(\times\) 0.063 mm. The change in amplitude of the ultrasonic signal will indicate the presence of debonding, and this effect will be compared with the results of the experiment. To avoid boundary reflection, the Rayleigh damping condition was applied to the edge surface of the plate and the radiating boundary condition was applied for the air domain to simulate an infinite acoustic medium. The mesh size considered in each domain is according to the wavelength of each material at 300 kHz and the criteria of eight elements per wavelength was used. The second-order discretization (quadratic discretization) was used for elements.