## 2 Theory of the Non-Detection Zone
### Non-Detection Zone in Simulation
The relative deviation \(\sigma_{0}\) can be calculated by:  
\[\sigma_{0}=(3.5-3.4868)/3.5=0.3771\% \tag{23}\]  
As shown in Figure 2b, the \(A0\) mode flaw-reflected pulse wave first appears at 0.00205 s. An S0 mode ultrasonic pulse wave was excited to the transverse flaw at 3.5 m and the mode changed from S0 to \(A0\) when passing the asymmetric flaw. Therefore, the propagation time of the \(A0\) mode ultrasonic pulse wave after mode transmission is equal to:  
\[Cg_{A0}=\Delta s/\Delta t=(3.5-0.13)/(0.00205-3.5/Cg_{50})=2547.2411\,[ \mathrm{m/s}] \tag{24}\]  
The flaw at 4000 mm (Figure 3c) can also be calculated in the same way so that its location can be determined using the reflected \(A0\) mode and \(S0\) mode waves. As shown in Figure 3d,e, the flaw at 4500 mm and 5000 mm can be only located by the \(S0\) mode reflected pulse in the conventional inspection because the \(A0\) mode pulse wave, which is in the superposition, is not in Interval I. As shown in Figure 3f, the times of the \(S0\) mode and \(A0\) mode flaw-reflected pulse waves in Interval I when \(D0\) = 5 mm at 5500 mm are then calculated. The appearance time of the \(S0\) mode pulse wave \(T_{S0}\) can be calculated using:  
\[T_{S0}=0.0002026+(5.5-0.13)\times 2/Cg_{50}=0.0024333\,[\mathrm{s}] \tag{25}\]  
The appearance time of the \(A0\) mode pulse wave \(T_{A0}\) can be calculated using:  
\[p(t)=\sin(2\pi f_{0}t)Hanning(t)\]0  
This analysis shows that, when the value of \(L\) increases along the \(x\)-direction and approaches the length of the terminal, the appearance time of the \(S0\) mode flaw-reflected pulse wave and the \(A0\) mode flaw-reflected pulse wave gradually approach the time of the first terminal reflection pulse wave, even then the appearance time of the \(A0\) mode flaw-reflected pulse appears in the monitoring time Interval II at a given value of \(L\). For the above simulation, at \(L\) = 5500 mm, the \(S0\) mode flaw-reflected pulse wave is superimposed by the first terminal reflection pulse wave and the \(A0\) mode flaw-reflected pulse wave is superimposed by these reflection pulse waves of the monitoring time Interval II. In particular, the calculation (see Equation (26)) indirectly suggests that, in the monitoring Interval II, the highest reflected pulse is not the delay of the \(A0\) mode flaw-reflected pulse in the conventional inspections, as shown in Figure 3f. Under ideal conditions, even without the interference of the signal trailing, the actual \(S0\) or \(A0\) mode flaw reflection pulses cannot be easily identified and the conventional detection method based on a single mode flaw-reflected pulse wave is invalid. For actual measurements, the trailing phenomenon of the pulse wave has to be considered. This is the state of the non-detection zone referred to in this study. It is known that the information of the flaw is superimposed by the first terminal reflection pulse wave and various types of reflection pulse waves in the monitoring time Interval II described in this section. In the following section, we observe the actual waveform of the ultrasonic pulse wave under the condition of the non-detection zone in an actual experiment.