## 3 Numerical Simulation and Verification
### Method Comparison and Factor Analysis  
The results of the proposed synthetic aperture focusing technology method for circumferential scanning (CSAFT) are compared with the B-scan image and the results by the DAS method [29]. The sound field of the MST with the length 2L = 70 mm is used to modulate the signals from the simulation. The time signals are turned into distance signals with the group velocity \(\hat{c}_{g}\) by \(R=t\ \times\ \hat{c}_{g}\); all the distance signals are stacked to become the B-scan image in the order of the angle positions of the receiving points. By the DAS method, the image region is first divided into pixels; then, the flight time of each pixel is calculated with the group velocity and the distance between the pixel and the transducer in the scanning. The amplitude of each pixel is the superposition of the amplitudes extracted from the time signals of the transducer at different positions according to the flight times. It should be noticed that, since only the procedure of receiving is simulated, the group velocity \(\hat{c}_{g}\) will be the same as the group velocity \(c_{g}=3208\) m/s. The results in Figures 9-11 are all calculated with the excitation signal of three cycles. The results for the angular step 0.5\({}^{\circ}\) are shown in Figure 9. In the B-scan image of Figure 9a, extra waves existing in front of the defect waves are Lamb modes generated by the defect, which travel faster than the SH0 mode. The reflection waves from the boundaries appear after the defect waves. As in Figure 6, the divergence angle of the transducer cannot be zero, and so the defect can also be detected even if it is not directly in front of the transducer. That is the reason for the tailings of the defect waves in the B-scan image, which significantly lower the angular resolution. The image is  
Figure 10: Comparisons of three methods with different factors: (**a**) the angular bandwidth \(\theta_{w}\) by the transducer with different lengths; (**b**) Calculation times with different numbers of signals. The excitation signal is a tone-burst signal with three cycles. Figure 9: The results for the angular step 0.5\({}^{\circ}\) by different methods: (**a**) B-scan, (**b**) delay and sum (DAS), (**c**) circumferential synthetic aperture focusing technique (CSAFT). The excitation signal is a tone-burst signal with three cycles. improved by the DAS method, as in Figure 9b. However, over adjustments exist as a result, leading to some residual trailing. The result of the CSAFT method is shown in Figure 9c. The best angular resolution of the defect spot with the least trailing can be obtained compared with those by other methods. The defect locations (0.76-0.77 m) in Figure 9 are calculated with the peaks of the defect spots, which results in the localization error of the distances (6.7%) for the width of the defect waves in the time signals. The denoted angular positions (9.5-11.5\({}^{\circ}\)) have little error compared with the actual value. The effect of the length of the transducer is investigated in the simulation. The angular bandwidth of the defect spot \(\theta_{w}\), defined as the largest angle range of the region in which the amplitudes are no less than half of the peak, is used to denote the size of the defect spot in the angular direction. The divergence angle \(\theta_{d}\) of the transducer decreases with the increment of the length of the transducer length 2L as in Figure 6. The angular bandwidth of the defect spot in the B-scan image has the same tendency of decrement as the divergence angle of the transducer, as in Figure 10a, which is 12-29.5\({}^{\circ}\) for a length of the transducer between 2 and 150 mm. The DAS method provides a significant improvement of the angular bandwidth (3.4-3.8\({}^{\circ}\)), which is basically not affected by the variations of the transducer lengths. The angular bandwidths by the CSAFT method (1.5-2.5\({}^{\circ}\)) are slightly superior to those by the DAS method.