# Assessment of the Length and Depth of Delamination-Type Defects Using Ultrasonic Guided Waves
## 2 Method for Assessment of Delamination Parameters
### Interaction of A\({}_{0}\) Mode with Delamination-Type Defect
\begin{array}{ll}U_{\text{IA}_{0}}(f)&=U_{\text{SA}_{0}}(f)\cdot H\Big{(}f,c _{p_{0}A_{0}},l_{0}\Big{)}\cdot k_{\text{T01A}}\cdot H\Big{(}f,c_{p_{1}S_{0}},l _{1}\Big{)}\cdot k_{\text{T10S}}\cdot H\Big{(}f,c_{p_{0}A_{0}},l_{2}\Big{)}\\ &+U_{\text{SA}_{0}}(f)\cdot H\Big{(}f,c_{p_{0}A_{0}},l_{0}\Big{)}\cdot k_{\text {T02A}}\cdot H\Big{(}f,c_{p_{2}S_{0}},l_{1}\Big{)}\cdot k_{\text{T20S}}\cdot H \Big{(}f,c_{p_{0}A_{0}},l_{2}\Big{)}\\ &=\Big{(}U_{\text{SA}_{0}}(f)\cdot H\Big{(}f,c_{p_{0}A_{0}},L-l_{1}\Big{)} \Big{)}\cdot\left(\begin{array}{c}k_{\text{T01A}}\cdot H\Big{(}f,c_{p_{1}S_{0 }},l_{1}\Big{)}\cdot k_{\text{T10S}}+\\ +k_{\text{T02A}}\cdot H\Big{(}f,c_{p_{2}S_{0}},l_{1}\Big{)}\cdot k_{\text{T20S} }\end{array}\right),\end{array}  \tag{1}\]  
\[H\Big{(}f,c_{\text{P}},l\Big{)}\ =\ e^{-\alpha(f)x}e^{\frac{-\beta f}{c_{\text{P}} (f)}}, \tag{2}\]  
where \(U_{\text{SA0}}(f)\) is the frequency spectrum of the excitation signal of A\({}_{0}\) mode; \(H(f,c_{\text{P}},l)\) is the transfer function of the waves propagating in the corresponding layer; \(c_{\text{PA0}}(f)\), \(c_{\text{P50}}(f)\) are the phase velocities of A\({}_{0}\) and S\({}_{0}\) modes, respectively; \(l\) is the propagation distance (\(l_{0}\), \(l_{2}\)-propagation distances in the defect-free area before and after the defect, respectively, \(l_{1}\)-length of the defect); \(L\) is the separation between the source and the monitoring points (\(L=l_{0}+l_{1}+l_{2}\)); and \(\alpha(f)\) is the frequency dependent attenuation function; \(k_{\text{T01A}}\), \(k_{\text{T02A}}\), \(k_{\text{T10S}}\) and \(k_{\text{T20S}}\) are the transmission coefficients for A\({}_{0}\) and S\({}_{0}\) modes at the leading and trailing edges of the delamination, respectively, for the layer above and below the defect. The estimation of such reflection and transmission coefficients was out of the scope of this study. They essentially depend on the guided wave mode, the delamination depth, the material properties and the total thickness of the plate, and they should be determined for each particular case separately. The wave packet described by Equation (1) is A\({}_{0}\) mode, which is a superposition of the signals traveling at upper and lower sub-laminates. This wave-packet converts from A\({}_{0}\) mode to S\({}_{0}\) mode at the leading edge and back to A\({}_{0}\) at the trailing edge of the defect. Due to this reason, this wave packet \(U_{\text{IA0}}(f)\) arrives earlier compared to the direct transmission of A\({}_{0}\) mode which does not convert within the defected area. This happens due to the reason that S\({}_{0}\) mode possesses a greater group velocity at low frequencies compared to A\({}_{0}\) mode. In this research, such a wave-packet shall be referred to as \"first A\({}_{0}\) arrival\". Consequently, the frequency spectrum of the direct transmission of A\({}_{0}\) mode can be written  
The \"direct A\({}_{0}\) transmission\" described by Equation (3) is the wave-packet which does not convert to other modes at the leading and trailing edges of delamination. The scattering of the incident A\({}_{0}\) mode at the delamination-type defect is further illustrated with B-scans in Figure 4 which were simulated by using the Finite Element plane-strain model on a 2000 mm \(\times\) 4 mm 2D GFRP plate. These B-scans represent a centered delamination-type defect located at a depth of 1.5 mm with respect to the top surface of the sample. A\({}_{0}\) mode was excited with a Gaussian envelope tone burst of three cycles and a central frequency of 80 kHz. The signals which are analyzed in this paper are denoted as \(u_{\text{IIA0}}(t)\) (\"direct A\({}_{0}\) transmission\") and \(u_{\text{IA0}}(t)\) (\"first A\({}_{0}\) arrival\") in Figure 4.