## 2 Method
### Synthetic Focusing Imaging for Circumferential Scanning  
The whole procedure of the inspection with the transducer at one location can be described as follows:  
1. The transducer generates guided waves towards the defect;
2. The defect is motivated and generates scattering waves;
3. Parts of the scattering waves are received by the transducer. When the defect is motivated by the incident pure SH0 waves, other modes may exist in the scatter waves because of the mode conversions. These modes are not dominant and they can hardly be detected by the MST in the experiments. Therefore, the mode conversion is not considered in this study. Because of the reciprocity of the sound field and the constant group velocity \(c_{g}\) of the SH0 mode, Steps 1 and 3 can be described in the same way. The defect can be regarded as a sound source that generates guided waves at time \(t=0\); then, the guided waves propagate to the transducer. Since the sonic path in this assumption is half of the actual path, the group velocity \(\dot{c}_{g}\) is taken as half of the actual value \(c_{g}\) to maintain the flight time of the defect, i.e., \(\dot{c}_{g}=c_{g}/2\). The above equivalent model is also called the exploding reflector model (ERM) [41]. As in Figure 1, the guided waves generated by the defect at the radius \(R_{2}\) expand before being received by the transducer moving at the radius \(R_{1}\). If \(R_{1}\) is equal to \(R_{2}\), there will be no expansion of the guided waves, which are simply generated from the defect at the time \(t=0\). An optimal lateral resolution can be obtained from the guided wave fields at the radius \(R_{2}\) with the time term \(\left.\mathrm{e}^{-i\omega t}\right|_{t=0}=1\). The relationships of the guided wave fields at different radii can be deduced as follows. In Figure 1, the polar coordinate \(O_{2}\) is constructed without considering the direction along the thickness, in which the displacement of the SH0 mode is uniform. The wave equation in a polar coordinate can be written as [42]:  
\[\left[\frac{\partial^{2}}{\partial R^{2}}+\frac{1}{R}\frac{\partial}{\partial R }+\frac{1}{R^{2}}\frac{\partial^{2}}{\partial\alpha^{2}}-\frac{1}{\dot{c}_{g} }\frac{\partial^{2}}{\partial t^{2}}\right]p(R,\alpha,t)=0 \tag{8}\]  
where \(p(R,\alpha,t)\) is the wave field of the point \((R,\alpha)\) at time \(t\), which can be written in the form of an inverse Fourier transform:  
\[p(R,\alpha,t)=\int P(R,\ k_{\alpha},\ \omega)e^{j(k_{\alpha}\alpha+\ \omega t)}dk_{ \alpha}d\omega \tag{9}\]  
where \(k_{\alpha}\) and \(\omega\) are the angular wave number and angular frequency, respectively. \(P(R,\ k_{\alpha},\ \omega)\) is the two-dimensional spectrum of \(p(R,\alpha,t)\), which can be solved by substituting Equations (9) into (8):  
\[P(R,\ k_{\alpha},\ \omega)=A_{k_{\alpha}}H_{k_{\alpha}}^{(1)}(k_{R}R)+B_{k_{ \alpha}}H_{k_{\alpha}}^{(2)}(k_{R}R) \tag{10}\]  
Figure 6: The relationship between the divergence angles of the main lobes and the lengths of the MSTs. where \(A_{k_{\alpha}}H_{k_{\alpha}}^{(1)}(k_{R}R)\) and \(B_{k_{\alpha}}H_{k_{\alpha}}^{(2)}(k_{R}R)\) denote the guided wave propagating away from the original point and towards the original point, respectively. \(H_{k_{\alpha}}^{(p)}(p=1,\ 2)\) is the \(p\)th kind of Hank function of the order \(k_{\alpha}\ k_{R}\) is the wave number along the radius. \[k_{R}=\left\{ \begin{array}{cc}\sqrt{\frac{\omega^{2}}{\xi_{\delta}^{2}}-{k_{ \alpha}}^{2}},\sqrt{\frac{\omega^{2}}{\xi_{\delta}^{2}}-{k_{\alpha}}^{2}}\geq 0 \\ 0,\hskip 28.452756 pt\sqrt{\frac{\omega^{2}}{\xi_{\delta}^{2}}-{k_{\alpha}}^{2} }<0\end{array}