## 2 Ultrasonic Double-Probe Penetration Detection Method  
The double-probe penetration detection method uses two probes (one transmitting and one receiving) to judge the propagation of the crack in the inspected components according to the strength of the penetrating acoustic signal. As shown in Figure 1, Pr-wave refers to the height of the transmitted wave, and Pe-wave to the height of the penetrating wave. The ultrasonic wave emitted by the transmit probe will diffuse into the material under test. When the wave encounters a defect, part of the wave will be reflected by the defect, and the remainder will propagate to the receiving probe. The receiving signal will be reduced compared with that without a defect and the degree of reduction depends on the amount of wave that is reflected. According to the strength of the received signal, the internal crack propagation can be judged. According to the conditions of the measured object, ultrasonic testing could be carried out by P-wave, S-wave, or surface wave. In this testing, the use of S-waves is more suitable.  
\(K\) value is one of the parameters of the angle probe, which is defined as the tangent value of angle \(\beta\) between the incident wave and an interface normal line, as shown in Figure 1. When the \(K\) value of the angled probe and the thickness of the steel plate t were both determined, the distance between the transmitting point and the receiving point could be calculated using Equation (1) when the wave height of the receiving signal is maximum (Figure 1).  
\[L_{0}=2 tK \tag{1}\]  
Figure 1: Diagram of the double-probe penetration method.  
where \(L_{0}\) is the distance between the incident point of wave and the receiving point at the maximum transmission signal wave in a crack-free state; \(t\) is the thickness of steel plate; and \(K\) is the \(K\) value of transmitting (receiving) probe.  
In actual detection work, there is a certain proportional relationship between the received wave height \(R_{a}\) and the maximum wave height \(R_{\max}\). When the axis of the ultrasonic beam just passes through the crack tip, the ratio of the two is \(\omega_{K}\) (Equation (2)). According to the maximum wave height received by two sets of probes with different \(K\) values (\(K_{1}\) and \(K_{2}\)) and the corresponding probe position, the position of the crack tip is determined. The depth \(d_{c}\) and deflection angle \(\alpha_{c}\) of the crack can be deduced according to their geometric relationship (Equations (3)-(6)). A schematic representation thereof is shown in Figure 2.  
\[\omega_{K}=\frac{R_{a}}{R_{\max}} \tag{2}\]  
\[\left\{\begin{array}{l}\beta_{1}=\arctan K_{1}\\ \beta_{2}=\arctan K_{2}\\ \Delta=a_{K2}-a_{K1}\end{array}\right\} \tag{3}\]  
\[\left\{\begin{array}{l}x_{0}=a_{K2},y_{0}=0\\ x_{A}=\frac{\cos\beta_{1}\sin\beta_{2}}{\sin(\beta_{2}-\beta_{1})}\cdot\Delta, y_{A}=\frac{\cos\beta_{1}\cos\beta_{2}}{\sin(\beta_{2}-\beta_{1})}\cdot \Delta\end{array}\right\} \tag{4}\]  
\[d_{OA}=\sqrt{\left(x_{A}-x_{O}\right)^{2}+\left(y_{A}-y_{O}\right)^{2}} \tag{5}\]  
\[\alpha_{A}=\arctan\left|\frac{x_{A}-x_{0}}{y_{A}-y_{0}}\right| \tag{6}\]  
where \(a_{K1}\) and \(a_{K2}\) are the distance between the wave incident point and the crack surface point. When the acoustic beam axis passes through the crack tip, \(\beta_{1}\) and \(\beta_{2}\) are the included angle between the incident wave and the perpendicular line. \(x_{0},y_{0},x_{A}\), and \(y_{A}\) are the coordinates of the crack endpoint and \(\Delta\) is the distance between the two incident points.