# Delamination Localization in Multilayered CFRP Panel Based on Reconstruction of Guided Wave Modes
## 3 Results and Discussion
### Localization of Delamination Using an Analytical Model  
The interaction of guided waves with the delamination located at 250 mm was analyzed. To determine the position of the delamination (\(X_{d}\)), two different receiver positions were used. The GW signals were received (\(X_{r}\)) at 20 mm and 10 mm from the leading edge of the delamination. In this specific scenario, the A\({}_{0}\) mode, which is reflected from the leading edge of the delamination, was analyzed for the purpose of localizing the delamination. During the analysis of the reflected A\({}_{0}\) mode from the delamination, the A\({}_{0}\) mode that is initially excited and propagates directly to the receiver is referred to as the direct A\({}_{0}\) mode (\(t_{1}\)). Similarly, the reflection from the leading edge of the delamination is denoted as \(t_{2}\). Taking into account the time of flight (ToF) between \(t_{1}\) and \(t_{2}\), it is possible to determine the location of the delamination. During guided wave inspection, the reflection coefficients of the excited A\({}_{0e}\) mode change based on the delamination position within the specimen thickness. As a result, the reflection obtained from the leading edge of the delamination tends to have a lower amplitude. Delamination estimation using these lower amplitude modes is complex. For this reason, an analytical model was developed to match the lower amplitude A\({}_{0r}\) mode, and then the ToF of the reflected mode (\(t_{2}\)) is estimated. Figure 12 shows the Hilbert envelope of the asymmetrical mode of guided wave signals received at various receiver positions. The envelope of the guided wave signal, represented by \(H(t)\), is obtained using the Hilbert transform and can be expressed using the following equation [45]:  
\[H(t)=|\text{Hilbert}[x(t)]| \tag{6}\]  
Figure 11: B-scan images of guided wave propagation: defect-free (a) and delamination (b). By calculating the time of flight (ToF) between \(t_{1}\) and \(t_{2}\), the delamination location can be estimated based on the time difference between direct and reflected A\({}_{0}\) mode. \[Xe=\frac{\Delta t\cdot c_{g}}{2}\, \tag{7}\]  
where \(c_{g}\) is the group velocity and \(\Delta t=t_{2}-t_{1}\) is the time difference between direct and reflected A\({}_{0}\) mode. The distance from the receiver to delamination estimated using an analytical model is presented in Table 3. In this case, the location of the delamination was estimated by taking into account the reflection from the leading edge of the delamination. The absolute error to estimate the delamination location was calculated by considering the difference between the actual and measured value by the following equation:  
\[\varepsilon_{a}=\mid D_{a}-D_{m}\mid \tag{8}\]  
where \(\varepsilon_{a}\) is the absolute error, and \(D_{a}\) and \(D_{m}\) are the actual and measured values, respectively. In the case where the delamination is between the excitation source and the receiver, GW signals were received at distances of 280 mm, 290 mm, and 300 mm from the leading edge of the CFRP plate. The distance between the delamination and the receiver (\(X_{dr}\)) was measured as 10 mm, 20 mm, and 30 mm. In this case, when the A\({}_{0}\) mode propagates above and below the delamination, part of it is transmitted A\({}_{0}\) and part of it is converted into S\({}_{0}\) mode. The converted S\({}_{0}\) mode is received at the receiver as the first wave packet (\(t_{1}\)), while  
Figure 12: GW signal received in case the delamination is located further than the receiver. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**Receiver**} & \multicolumn{2}{c}{**Distance from**} & \multirow{2}{*}{**Propagation**} & \multirow{2}{*}{**ToF (dt) (us)**} & \multicolumn{2}{c}{**Estimated**} \\  & & **Delamination to** & & **Time \(t_{2}-t_{1}\) (μs)** & & **Distance (mm)** & **Absolute Error** \\ \hline
1-1 & R1 & 20 & 87.2 \(-\) 62.8 & 24.4 & 19.8 & 0.2 \\
1-2 & R2 & 10 & 81.1 \(-\) 69.2 & 11.9 & 9.6 & 0.4 \\ \hline \hline \end{tabular}
\end{table}