## 4 Analysis and Discussion
### The Range of the Non-Detection Zone  
The range of the non-detection zone is a major concern in non-destructive inspections. Based on the analysis in Section 2.2, we used a dual-probe echo method for the location of the flaws and a low-frequency of the incident pulse. If the flaw is a certain distance from the terminal, the \(S0\) mode flaw-reflected pulse is superposed by the first terminal-reflected pulse wave, while the \(A0\) mode flaw-reflected pulse is superposed by the multiple reflected pulse wave and the pulse signal trailing in the monitoring interval II; as a result, the non-detection zone occurs. Therefore, the superposition time point of the \(S0\) mode flaw-reflected pulse and the first terminal-reflected pulse wave can be considered as given the occurring conditions. Figure 6 shows the first terminal-reflected pulse wave and the \(S0\) mode flaw-reflected wave. Based on these occurring conditions, the range of the non-detection zone can be described as:  
\[Cg_{S0}\times\left(\Delta h/2\right)/2\geq\Delta L>0 \tag{27}\]  
where \(Cg_{S0}\) is related to the frequency-thickness product (\(f\cdot d\)) of the incident pulse wave, \(\Delta L\) denotes the distance between the flaw and the terminal, \(\Delta h\) denotes the pulse width of the first terminal-reflected pulse wave (Figures 3a and 5), which is connected with the incident pulse wave and the self-characteristic of the customized longitudinal-wave probe, and \(\Delta L\) denotes the distance between the flaw and the terminal reflection boundary. It should be noted that low-frequency ultrasound is normally used as the core frequency of the excitation pulse in long-distance ultrasonic guided wave inspections [24]. This has many advantages such as reducing the excitation mode number and extending the range of the long-distance detection [25]. However, the disadvantages cannot be ignored. One disadvantage is that the ultrasonic pulse width (\(\Delta h\)) is wider at a low frequency  
Figure 6: Schematic of the distance covered by the \(S0\) mode pulse wave in (**a**) the first terminal-reflected pulse wave and (**b**) the flaw-reflected pulse wave in Interval I.  
than at a high frequency. In addition, in a low-frequency ultrasonic guided wave propagated over a long distance [26], the width of the first terminal reflected pulse wave (\(\Delta h\)) increases because of the dispersion effect, thereby expanding the range of the non-detection zone. The first terminal reflection pulse wave is the result of the overlap of the reflected pulse waves from the end surface and the incidence surface. The width of the first terminal reflection pulse wave (\(\Delta h\)) varies with the transmission distance. This is related to the measuring points, the detection distance, the characteristic of the probe, and so on. In this study, the range of the non-detection zone for the inspection of the 30# square steel bar with a length of 6000 mm is used as an example; the width of the first reflected pulse (\(\Delta h\)) is 0.000822 s in the actual experiment (Figure 5) and 0.0005371 s in the simulation (Figure 3a). The governing equation for the Rayleigh-Lamb waves in a plate-like structure [1] for the symmetric mode is:  
\[\tan(qd)/\tan(pd)=-4k^{2}pq/\left(q^{2}-k^{2}\right)^{2} \tag{28}\]  
and for the anti-symmetric mode:  
\[\tan(qd)/\tan(pd)=-\left(q^{2}-k^{2}\right)^{2}/4k^{2}pq \tag{29}\]  
where \(p^{2}=(\omega/c_{L})-k^{2}\), \(q^{2}=(\omega/c_{T})-k^{2}\), the wave number \(k=\omega/c_{P}\), the longitudinal wave velocity \(c_{L}\), the shear velocity \(c_{T}\), the phase velocity \(c_{P}=(\omega/2\pi)\lambda\), the group velocity \(C_{g}\) and the thickness of the plate is \(d\). The formula of the group velocity theory is:  
\[C_{g}=d\omega/dk \tag{30}\]  
By using (Equations (28)-(30)) and selecting the material parameters for the iron plate and the curve of the 50 mode is calculated.