## 4 Analysis and Discussion
### Approximation Based on Envelope Comparison  
Although the reflected pulses show complex superposition patterns in Interval II, the propagation speed of the fundamental mode remains relatively stable and the superposition state varies correspondingly as the flaw location \(L\) changes along the \(x\)-direction. If the energy of the excitation pulse signal, the flaw location \(L\), and \(D0\) were fixed, the propagation energy would attenuate. Compared with the condition without any flaws, the amplitude of the pulse overall decreases but the amplitude of the pulse reflected by the flaw increases. The difference in the envelope between these two different conditions (with and without the flaw) of the pulse waves can be used to identify the reflected pulse. Because the ultrasonic reflected pulse wave from the flaws located in monitoring Interval I is invalid, we decided to use the envelope wave of the pulse that is extended to monitoring Interval II to locate the flaws. To resolve the interference problem caused by the trailing of the actual experimental data and the unexpectedly generated waves, firstly, we use the envelope of the waveforms of two areas (for example, \(D0\) = 0 mm and \(D0\) = 10 mm) in the simulation and in the experiment and calculate the difference in the envelope amplitude of the two areas; these are used as basic data for comparison. Firstly, by observing these envelopes shown in Figure 9a-f, a coarse division of the flaw areas is proposed based on the analysis in Section 4.2. * Situation 1: in the actual experimental envelope during monitoring Interval II, the reflected pulse with only one envelope amplitude that is larger than the amplitude of the terminal reflected pulse wave is categorized into a zone ranging from \(L\) = 5800 mm to \(L\) = 5900 mm;
* Situation 2: the reflected pulses with two similar highest envelope amplitudes are categorized into a zone ranging from \(L\) = 5600 mm to \(L\) = 5700 mm;
* Situation 3: the remaining flaw detection pulses with different amplitudes are categorized into a zone from \(L\) = 5400 mm to \(L\) = 5500 mm. Secondly, we selected the effective time domain intervals for comparison (from 7.649 \(\times\) 10\({}^{-4}\) s to 4.653 \(\times\) 10\({}^{-3}\) s intervals within the monitoring Intervals I and II) to avoid the interference caused by the trailing of the incident pulse and the second terminal-reflected pulse wave, as shown in Figure 9f. Thirdly, the envelope theory (ET) is known as an auxiliary field method [28; 29]; therefore, we could use this convenient method for the wave comparison between the simulation and the measurements. The envelope correlation algorithm is applied to compute the maximum correlation coefficient in order to choose the most similar envelope waveforms in a quantitative approach and locate the flaws. The discretization cross-correlation function can be represented as follows:  
\[R_{se}(k)=\sum_{n=0}^{N-1}s(n)e(n-k)\;k=0,1,2,\cdots,M-1 \tag{31}\]  
where \(R_{se}(k)\) denotes the correlation function for the simulation data \(s(n)\) and the experimental data \(e(n)\). \(N\) is the cumulative average number and \(k\) is the time-lapse sequence number, \(M\) is a positive integer. The formula for the envelope correlation is expressed as follows:  
\[\rho=R_{se}(0)/\sqrt{\sum_{n=0}^{N-1}s^{2}(n)\sum_{n=0}^{N-1}e^{2}(n)} \tag{32}\]  
The scan comparisons are not required (\(k=0\)) in Equation (31). The envelope correlation coefficient (\(\rho\)) of the experimental data and the simulation data are obtained using Equation (32), and the envelope whose simulation data is closest to the experimental data envelope is selected as the indicator for locating the corresponding flaw. Figure 9: _Cont._  
Figure 9: Envelope calculation results of the pulse wave at different locations of flaw: (**a**) \(L\) = 5400 mm \(D0\) = 10 mm (**b**) \(L\) = 5500 mm \(D0\) = 10 mm (**c**) \(L\) = 5600 mm \(D0\) = 10 mm (**d**) \(L\) = 5700 mm \(D0\) = 10 mm (**e**) \(L\) = 5800 mm \(D0\) = 10 mm (**f**) \(L\) = 5900 mm \(D0\) = 10 mm.