# Assessment of the Length and Depth of Delamination-Type Defects Using Ultrasonic Guided Waves
## 2 Method for Assessment of Delamination Parameters
### Method to Estimate the Depth of Defect  
As it was shown by Equation (3), the \"direct A\({}_{0}\) transmission\" is a superposition of the waves traveling at the sub-laminates below and above the damage which does not convert to other modes, neither at the leading, nor at the trailing edge of the defect. If the delamination is asymmetrical across the thickness of the laminate (\(h_{1}\
eq h_{2}\) in Figure 1), the phase velocities at sub-laminates are different (\(c_{\text{p1}}\
eq c_{\text{p2}}\)), and they depend on the excitation frequency and thickness of each sub-laminate. Beyond the defect, the modes from the upper and lower sub-laminates interfere with each other, which results in the single forward scattered A\({}_{0}\) mode \(u_{\text{IIA0}}(t)\). The magnitude of this mode is proportional to the result of interference which can be either in phase, out of phase or intermediate. If the transducer is driven by a set of different frequencies, a collection of magnitude variation patterns can be obtained and used for damage depth extraction. The use of multiple excitation frequencies enables one to exploit the frequency dependent phase velocity of A\({}_{0}\) mode, especially in the case of the presence of dispersion. Thus, the idea of the proposed method to estimate the depth of damage relies on the magnitude measurements of \"direct A\({}_{0}\) transmission\" \(u_{\text{IIA0}}(t)\) at burst signals with different central frequencies (\(f_{1},f_{2},\,\ldots\,f_{n}\)). We should note that this technique is only valid if the delamination is asymmetrical \(h_{1}\
eq h/2\
eq h_{2}\) and \(c_{\text{p1}}\
eq c_{\text{p2}}\). For ideally symmetrical delamination, there will be no out-of-phase interference of A\({}_{0}\) mode as the velocities at the layer above and below the defect will be very close. To extract the absolute values of the defect depth, a database of reference dependencies is required which would represent the collection of magnitude variation patterns at different excitation frequencies, the depth and the length of delaminations. Then, the estimated reference dependency can be compared to the experimental measurement in order to extract the depth of delamination. It is presumed that the  
Figure 4: Illustration of A\({}_{0}\) mode interaction with delamination: B-scan images of vertical \(u_{x}\) (**a**) and longitudinal \(u_{z}\) (**b**) component of particle velocity along the top surface of 4 mm thickness GFRP sample. reference dependency which best matches the experiments is the one that gives the closest definition of the damage depth in the structure. The step-by-step procedure of the damage depth estimation can be outlined as follows:  
1. The source of guided waves \(E\) is driven by burst with Gaussian envelope of central frequency \(f_{1}\) to introduce A\({}_{0}\) mode in the investigated structure. 2. Time trace \(u_{\rm A0}(t)\) is received with sensor \(R_{\rm ref}\), which represents the structure without the damage (the reference signal). Meanwhile, the other receiver \(R_{1}\) captures time history \(u_{\rm IIA0}(t)\) which represents the response of the structure at the monitoring point beyond the defect. 3. Ratio \(u_{\rm A0 expf1}(t)\) of peak-to-peak amplitudes of waveforms \(u_{\rm A0}(t)\) and \(u_{\rm IIA0}(t)\) is estimated at the burst central frequency \(f_{1}\). 4. Steps 1-3 are repeated over for other excitation frequencies \(f_{\rm N}=\langle f_{1},f_{2},\ldots f_{\rm n}\rangle\). As a consequence, the experimental dataset U\({}_{\rm A0 exp}(f)=\langle f_{\rm N}\), \(u_{\rm A0 expf1}(t)\), \(u_{\rm A0 expf2}(t)\), \(\ldots\), \(u_{\rm A0 expfn}(t)\rangle\) is collected, where \(f_{\rm N}\) denotes the burst central frequencies used for the excitation of A\({}_{0}\) mode. 5. The experimental dataset U\({}_{\rm A0 exp}(f)\) is compared to the prescribed database of reference dependencies U\({}_{\rm A0 ref}(f,h,l)\) (where \(h\), \(l\) are the depth and the length of damage, respectively).