# Assessment of the Length and Depth of Delamination-Type Defects Using Ultrasonic Guided Waves
## 2 Method for Assessment of Delamination Parameters
### Method to Assess the Length of Defect  
If the depth of the defect is determined, its length can be estimated from the delay between \"direct A\({}_{0}\) transmission\" \(u_{\rm IIA0}(t)\) and \"first A\({}_{0}\) arrival\" \(u_{\rm IA0}(t)\) which can be denoted as \(\Delta t_{\rm A0}\). Delay \(\Delta t_{\rm A0}\) is proportional to the excitation frequency and the propagation path of S\({}_{0}\) mode, which itself is actually determined by the length of the defect (Figure 5). The time delay between two neighboring wave packets can be estimated according to:  
\[\Delta t_{A_{0}} = \underset{t}{\rm argmax}\Big{[}\mathrm{HT}\Big{|}u_{\rm IIA_{0}} (t)\Big{|}\Big{]}-\underset{t}{\rm argmax}\Big{[}\mathrm{HT}\Big{|}u_{\rm IA 0}(t)\Big{|}\Big{]}, \tag{6}\]  
where \(u_{\rm IA0}(t)\) and \(u_{\rm IIA0}(t)\) are the wave packets of \"first A\({}_{0}\) arrival\" and \"direct A\({}_{0}\) transmission\", and HT denotes the Hilbert transform. Then, the length of the defect at a particular depth \(h\) can be calculated as:  
\[L_{\rm def} = \underset{f}{\rm mean}\Bigg{\{}\Delta t_{A_{0}}(h,f).\frac{c_{ \rm gA_{0}}(f)c_{\rm gS_{0}}(f)}{c_{\rm gS_{0}}(f)-c_{\rm gA_{0}}(f)}\Bigg{\}}. \tag{7}\]The Equation (7) calculates mean defect length from a set of defect lengths obtained at different excitation frequencies. If multiple burst central frequencies are used to excite the guided waves, the group velocity values of A\({}_{0}\) and S\({}_{0}\) modes should be taken at different central frequencies, which allow one to consider the dispersion.  
As the \"direct A\({}_{0}\) transmission\" \(u_{\text{IIA}0}(t)\) is the result of the interference between the wave packets propagating above and below the defect, the maximum of the wave after the exit of delamination (\(u_{\text{IIA}0}(t)\)) may be shifted and no longer correspond to the wave energy velocity. Hence the delay \(\Delta t_{\text{A}0}\) in Equation (6) will be estimated with some error. The best way here would be to calculate the delay according to the mean energy (mass center) of the signal [32]. However, in such case the time window covering entire signal should be selected. Such a task is not trivial due to the dispersion of A\({}_{0}\) mode signals, presence of other modes and reflections from object boundaries. Additionally, the different frequency components of the dispersed A\({}_{0}\) mode are redistributed in the envelope of the signal. The lower frequency components are concentrated at the tail of the signal. So, the energy of wave packet \(u_{\text{IIA}0}(t)\) should be related to a particular frequency and velocity which are used for estimation of the time of flight. By using a maximum or Hilbert envelope of the signal it can be assumed that it corresponds to the dominant frequency of the signal and gives sufficient ToF accuracy.  
It is noteworthy that, for the reliable operation of the method, direct \(u_{\text{IIA}0}(t)\) and converted modes \(u_{\text{IA}0}(t)\) must be separated in the time domain to an adequate extent in order to be able to measure the time delay between the two wave packets. Thus, there exists the minimum size of a defect that can be evaluated by using this approach. Such a minimum detectable defect is frequency dependent, so higher frequency signals can be used to improve sizing accuracy. It is important that the depth of the defect strongly influences the accuracy of the delamination length estimation, especially in the presence of significant dispersion. If the depth of the defect is unknown, such an approach becomes valid in non-dispersive regions only. However, by using the proposed delamination depth estimation technique, the thicknesses and velocities in the upper and lower sub-laminate can be determined, which leads to the improved accuracy of the delamination length evaluation.