## 2 Method
### Synthetic Focusing Imaging for Circumferential Scanning
\right.. \tag{11}\]  
The guided waves generated by the defect propagate away from the original point to the transducer; thus, the transfer function for the guided wave fields at different radii can be derived as  
\[P(R-\Delta R,\ k_{\alpha},\ \omega)=P(R,\ k_{\alpha},\ \omega)\frac{H_{k_{ \alpha}}^{(1)}(k_{R}(R-\Delta R))}{H_{k_{\alpha}}^{(1)}(k_{R}R)} \tag{12}\]  
Equation (12) can be simplified using the Rayleigh-Sommerfeld diffraction formula [43]:  
\[P(R-\Delta R,\ k_{\alpha},\ \omega)\approx P(R,\ k_{\alpha},\ \omega)\sqrt{\frac{R- \Delta R}{R}}e^{-i\Delta R\sqrt{\frac{\omega^{2}}{\xi_{\delta}^{2}}-\frac{{k_{ \alpha}}^{2}}{R(R-\Delta R)}}} \tag{13}\]  
This function is valid if \(\cos(\alpha_{1}-\alpha_{2})\) is approximately equal to \(1-\frac{(\alpha_{1}-\alpha_{2})^{2}}{2}\) With a given step \(\Delta R\), the two-dimensional spectrums of the guided wave fields at the radii \((R-\Delta R,\ R-2\Delta R,R-3\Delta R,\ldots,R-m\Delta R)\) can be derived with those collected by the transducer at the radius \(R\) with Equation (13). The focused image can be obtained with the inverse Fourier transforms of the guided wave fields at different radii. The main steps of the proposed algorithm are described below:  
1. A data matrix is constructed with scanning signals collected by the transducer at the radius \(R_{1}\) as column vectors. 2. The two-dimensional Fourier transform of the data matrix is performed to obtain two-dimensional spectrum \(P(R_{1},\ k_{\alpha},\ \omega)\). 3. The length \(\Delta R\) and number \(m\) of the radius step are determined; The imaging range of the radius is between \(R_{1}\) and \(R_{1}-m\Delta R\). The counter \(q\) is \(1\). 4. Two-dimensional spectrum \(P(R_{1}-q\Delta R,\ k_{\alpha},\ \omega)\) at the radius \(R_{1}-q\Delta R\) is calculated with Equation (13). Row vector \(R_{v}(R_{1}-q\Delta R)\) is constructed with the summation of each column of the two-dimensional spectrum. 5. The inverse Fourier transform of Row vector \(R_{v}(R_{1}-q\Delta R)\) is performed. The result is the \(q\)th row of the image matrix. The counter \(q\) is increased by one. 6. Steps 4 and 5 are executed repeatedly until \(q\) is larger than \(m\). The image matrix is outputted at last. The attainable angular resolution can be calculated with the similar derivation as in [41]. If the divergence angle \(\theta_{d}\) is used to determine the angular coverage of the transducer, the thresholds of the angular wave number \(k_{\alpha}\) are \(-\frac{4\pi}{\lambda}R_{1}\sin\left(\frac{\theta_{d}}{2}\right)\) and \(\frac{4\pi}{\lambda}R_{1}\sin\left(\frac{\theta_{d}}{2}\right)\), thus, the bandwidth of the angular wave number \(\Delta k_{\alpha}\) is \(\frac{2\pi}{\lambda}R_{1}\sin\left(\frac{\theta_{d}}{2}\right)\). Attainable angular resolution \(\theta_{ar}\) can be calculated as follows:  
\[\theta_{ar}=\frac{2\pi}{\Delta k_{\alpha}}=\frac{\lambda}{R_{1}\sin\left(\frac {\theta_{d}}{2}\right)} \tag{14}\]For the SH0 mode at a certain frequency, the wave length is constant, and the attainable angular resolutions are affected by the divergence angles, which are determined by the structures of the transducers. The proposed method is applicable for the circumferential scanning of isotropic plates with non-dispersive modes. For dispersive modes like the Lamb waves and higher-order SH waves, dispersion compensation can be achieved with the relationships between the group velocities \(\hat{c}_{g}\) and the angular frequencies \(\omega\) from the group velocity curves in the calculations of two-dimensional spectrums using Equation (13). For anisotropic plates, the wave equation in Equation (8) is not suitable, since the wave propagations along different directions are not the same. The proposed method may be not applicable. The detection ability of the circumferential scanning is determined by multiple reasons such as the excitation frequency and the characteristics of the defects (shapes and sizes).