## 2 Theory of the Non-Detection Zone
### Non-Detection Zone in Simulation  
To investigate the reasons for the appearance of the non-detection zone, we focus on the simulations of the propagation of an ultrasonic guided wave in a square steel bar with a length of 6000 mm using COMSOL Multiphysics, ver.5.0. As shown in Figure 2, during which a 10-pulse Hanning-window function pulse signal with a frequency of 30 kHz is used as the incident pulse. The Hanning function is used as the window function:  
\[Hanning(t)=(1-\cos(2\pi t/T))/2\]  
The type of excitation signal:  
\[p(t)=\sin(2\pi f_{0}t)Hanning(t)\]  
where \(T=1.6\,[\mathrm{\SIUnitSymbolMicro s}]\). The point (130, 30) is used as the measurement point and is the same as in the actual experiment in Section 3. A transverse flaw with a depth of 5 mm (\(D0\)) and a width of 1 mm is created at the  
Figure 2: The simulation settings. displacement \(L\) with a variable value of \(L\in\left[3500\text{ mm},\,5500\text{ mm}\right]\) along the \(x\) direction from the incident point and the simulation interval is set as \(500\text{ mm}\). (\(D0\) is set as \(5\text{ mm}\) for reducing the superposition of the flaw reflected pulses). A triangular grids (\(\lambda/26\)) are used in this study (\(<\!\!\lambda/20\) recommended minimum mesh size [19]) and the area near the location of the flaw \(L\) is refined using a local mesh. During the simulation, the excitation energy is properly decreased so that \(5\) pulses are reflected. The material parameters of the steel square bar are the following:  
Young's modulus: \(E=200\left[\text{GPa}\right]\)  
Density: \(\rho=7850\left[\text{kg}/\text{m}^{3}\right]\)  
Poisson's ratio: \(\
u=0.3\)  
Taking advantage of the periodicity of the terminal reflection, as shown in Figure 3a, the time domain is divided into the monitoring time Interval I, the monitoring time Interval II, the monitoring time Interval III, and so on. The pulse waves of the multiple reflections are labeled as the incident pulse wave, the first terminal reflection pulse wave, the second terminal reflection pulse wave, and so on. Figure 3: \(Cont.\)  
As shown in Figure 2b, at \(L=3500\) mm, an \(S0\) mode wave is emitted from the incident face and is then reflected by the transverse flaw [20]; then, after the reflection, two different types of waves, \(S0\) and \(A0\), are generated. In Interval I, the peak points of the \(S0\) mode and \(A0\) mode flaw-reflected pulse waves are (0.001598, 0.1206) and (0.00205, \(-0.1729\)), respectively. Based on the simulation results, the \(S0\) mode flaw-reflected pulse wave is calculated to verify whether the detected location of the flaw was accurate. It is known that the first terminal-reflected pulse wave and the initial pulse exhibit the maximum amplitude at ((0.0002036, 1) and (0.002642, 1.195)); the \(S0\) wave group velocity (\(Cg_{S0}\)) was calculated as:  
\[Cg_{S0}=\Delta s/\Delta t=(6-0.13)\times 2/(0.002642-0.0002036)=4814.6325\left[ \mathrm{m}/s\right] \tag{21}\]  
where \(\Delta s\) and \(\Delta t\) represent the propagation distance and the corresponding time respectively. Using the calculated \(Cg_{S0}\), the flaw-induced \(S0\) mode reflected wave is verified. The \(S0\) mode pulse wave  
Figure 3: Waveforms of the reflected pulses in the simulation, (**a**) \(D0=0\) mm; (**b**) \(D0=5\) mm, \(L=3500\) mm; (**c**) \(D0=5\) mm, \(L=4000\) mm; (**d**) \(D0=5\) mm, \(L=4500\) mm; (**e**) \(D0=5\) mm, \(L=5000\) mm; (**f**) \(D0=5\) mm, \(L=5500\) mm. appears at 0.001598 s and \(L_{0}\) denotes the distance between the transverse flaw and the initial face, the following expression can be acquired:  
\[L_{0}=(Cg_{50}\times\Delta t)/2+0.13=Cg_{50}\times(0.001598-0.0002036)/2+0.13=3. 4868\,[\mathrm{m}] \tag{22}\]  
The detected location is quite close to the actual position of the flaw (3.5 m).