## 4 Analysis and Discussion
### Experimental Validation  
In order to localize the flaw in the non-detection zone, the different envelopes of the simulated envelope waveforms (Envelopment 2-Envelopment 1) are used as the evaluation standard to compare with those of the experimental data. This section describes the validation of the experiment. In order to depict the validation results, instead of strictly following the steps mentioned in Section 4.3, the preliminary classification in Step 2 was omitted. The variable value of \(L\) is selected from an appropriate range (\(L_{1}\) = 5400 mm, \(L_{2}\) = 5900 mm) in the simulation. \(D0\) = 10 mm is used as an example of the experimental data to determine the location of the flaw (\(L\)). Figure 11a shows the experimental pulse waveform of the square steel bar without the flaw (\(D0\) = 0 mm) and the experimental and simulated pulse waveforms of the square steel bar with the flaw (\(D0\) = 10 mm). Using the envelope calculation, it can be observed that the envelope waveform in monitoring Interval II reduces the interference of the signal trailing in the experiments. Compared with the simulation data, it is evident that the peak points of the difference envelopes in the experiment are consistent, as shown in the red dotted line. The time relative deviation \(\sigma_{1}\) can be calculated by:  
\[\sigma_{1}=(0.003187-0.003107)/0.003187=2.5101\% \tag{33}\]  
The flaw's reflected information that is hidden and the superposition envelope of the flaw-reflected pulse in the trailing are extracted (Envelopment 2-Envelopment 1). Figure 11: Location of the flaw through the envelope approximation method: (**a**) The envelope of experimental data processing (**b**) The comparison of multiple envelopes in the effective comparison interval. Subsequently, the results of the different envelopes (Envelopment 2-Envelopment 1) in the simulation and the experiment are extracted in the comparison interval, as shown in Figure 11b. The correlation coefficients (\(\rho\)) of the experimental data and the different simulation data are determined. Because the simulated envelope waveform is used as the evaluation standard, the noise introduced by the real application cannot be eliminated by the correlation algorithm; therefore, the correlation coefficient is relatively low. However, we can still determine that the correlation coefficient, which is negative, should be discarded in the range from \(L\) = 5800 mm to \(L\) = 5900 mm; in the range from \(L\) = 5600 mm to \(L\) = 5700 mm, the envelopes of the waveform are not well matched. However, in the range from \(L\) = 5400 mm to \(L\) = 5500 mm, the approximated simulation waveforms in the flaw area (\(L\) = 5500 mm) have a significant correlation coefficient of 0.3430; therefore, \(L\) = 5500 mm is selected as the estimated value of the location of the flaw. If these calculating steps are strictly carried out according to the solution in the previous section, based on the preliminary rough classification, it is found that the actual measurement data range from \(L\) = 5400 mm to \(L\) = 5500 mm and we can obtain similar results rapidly to approximate the real value of \(L\) in further simulations. Therefore, the flaw in the non-detection zone can be identified using the inverse correspondence. Using an equivalent mode and approximation algorithms, this methodology provides estimations of the locations of flaws relatively accurately. Based on the analysis in Section 2.1.1, when the equivalent plate thickness (d) is constant, as long as the frequency is below the cutoff frequency of \(A1\) and \(S1\), the excitation frequency in this range can be used for low-frequency ultrasound mode selection. For example, the proposed analysis is applicable to the ultrasonic probe core frequency range of 20-60 kHz low-frequency ultrasonic guided wave detection. The concept of detecting flaws using simulations also applies to different types of flaws. In this study, we extended the analysis to a triangular flaw. This approach also resulted in a similar positioning effect.