# Assessment of the Length and Depth of Delamination-Type Defects Using Ultrasonic Guided Waves
## 3 Estimation of Reference Dependencies
### Theoretical Formulation of Reference Dataset  
In the case of depth assessment, our goal is to predict the waveform of A\({}_{0}\) mode which propagates within the defect. Equation (3) can be rewritten in a short form as:  
\[ \begin{array}{l}U_{\rm IIA_{0}}(f)\ =\ \Big{(}U_{\rm A_{0}}(f)\cdot H\Big{(}f,c _{p_{0}A_{0}},L-l_{1}\Big{)}\Big{)}\cdot\left(\begin{array}{c}k_{\rm T01A} \cdot H\Big{(}f,c_{p_{1}A_{0}},l_{1}\Big{)}\cdot k_{\rm T10A}+\\ +k_{\rm T02A}\cdot H\Big{(}f,c_{p_{2}A_{0}},l_{1}\Big{)}\cdot k_{\rm T20A}\end{array}\right) \\ =U_{A_{0}}(f)\cdot\Big{(}U_{A\rm 0 ad}(f)+U_{A\rm 0 bd}(f)\Big{)},\end{array}  \tag{8}\]  
where \(U_{\rm A0 ad}(f)\) and \(U_{A\rm 0 bd}(f)\) represent the propagation of A\({}_{0}\) mode above and below the delamination, respectively. Waveforms \(u_{\rm A0 ad}(t)\) and \(u_{\rm A0 bd}(t)\) can be predicted according to the theory of linear acoustics which states that the output waveform of the system is the convolution of the input signal and the system impulse response. Consequently, each of the signals \(u_{\rm A0 ad}(t)\) and \(u_{\rm A0 bd}(t)\) can be expressed by the equations:  
\[u_{A_{0}\rm ad}(t,l_{k},h_{1m},f_{l})\ =\ k_{\rm T01A}\cdot k_{\rm T10A}\cdot \mathrm{Re}\bigg{\{}\mathrm{FT}^{-1}\bigg{\{}\mathrm{FT}(u_{\rm ref}(t,f_{l})) \cdot e^{-\alpha(f)l_{k}}e^{\frac{-\beta l_{f}}{\rho(l_{1m})}}\bigg{\}}\bigg{\}}, \tag{9}\]  
\[u_{A_{0}\rm bd}(t,l_{k},h_{2m},f_{l})\ =\ k_{\rm T02A}\cdot k_{\rm T20A}\cdot \mathrm{Re}\bigg{\{}\mathrm{FT}^{-1}\bigg{\{}\mathrm{FT}(u_{\rm ref}(t,f_{l})) \cdot e^{-\alpha(f)l_{k}}e^{\frac{-\beta l_{f}}{\rho(l_{2m})}}\bigg{\}}\bigg{\}} \tag{10}\]  
where \(u_{\rm ref}(t,f_{l})\) is the theoretical incident signal at a particular excitation frequency (\(l=1\div M\), \(M\) is the total number of frequencies used to drive the emitter); \(\alpha(f)\) is the attenuation function; \(l_{k}\) is the length of the defect (\(k=1\div N\), where \(N\) is the total number of length incrementations); \(c_{\rm p}(f_{1}h_{1m})\) and \(c_{\rm p}(f_{1}h_{2m})\) are the phase velocities for the layer above (\(h_{1m}\)) and below the defect (\(h_{2m}\)) which depend on the level of defect asymmetry; \(h_{1m}\) and \(h_{2m}\) are the thicknesses of the layer above and below the defect; and FT denotes the Fourier transform. The in phase or out of phase interference of these signals defines the peak to peak amplitude of \"direct A\({}_{0}\) transmission\" \(u_{\rm A0 ref}(l_{k},h_{1m}h_{2m}f_{l})\) at excitation frequency \(f_{l}\):  
\[ \begin{array}{l}u_{A\rm{oref}}(l_{k},h_{1m},h_{2m},f_{l})\ =\ \max_{t}\Big{(}u_{A\rm 0 ad}(t,l_{k},h_{1m},f_{l})+u_{A\rm 0 bd }(t,l_{k},h_{2m},f_{l})\Big{)}\\ -\min_{t}\Big{(}u_{A\rm 0 ad}(t,l_{k},h_{1m},f_{l})+u_{A\rm 0 bd}(t,l_{k},h_{2m}, f_{l})\Big{)}.\end{array}  \tag{11}\]  
To illustrate the estimation of reference dependency, let us consider a GFRP plate of thickness \(h=4\) mm with delamination-type defects situated at various depths as follows: \(h_{1m}=mh/8,h_{2m}=h-h_{1m}\), where \(m=1\), \(2\),..., \(3\). Let us assume that the excitation burst central frequency of the incident A\({}_{0}\) mode varies starting from 50 kHz to 200 kHz with increments of 1 kHz. The graphic illustration of the magnitude of A\({}_{0}\) mode \(u_{A\rm{oref}}(l_{k},h_{1m},\,h_{2m}f_{l})\) as the function of excitation frequency \(f_{l}\) at discrete depths \(h_{1m}=mh/8\), \(h_{2m}=-h_{1m}\) is illustrated in Figures 6 and 7. The results are presented at fixed defect lengths of \(l_{1}=50\) mm, \(l_{2}=70\) mm, \(l_{3}=90\) mm and \(l_{4}=110\) mm. Each graph in Figures 6 and 7 represent magnitude variation of forward scattered A\({}_{0}\) mode at different excitation frequencies. It can be observed that the magnitude variation of \"direct A\({}_{0}\) transmission\" depends not only on the depth of the defect (which is represented by solid, dashed and dash-dot lines) but on its length as well. As one might expect, the size of the defect could be estimated from the same magnitude variations.