## 4 Identification of Surface Crack Location
### Optimal Detection Position in the SLLS Method  
In [22], there appeared to be strong shift effects as the wave propagated along the circumferential path of the cylindrical specimens. The phase shift effect was observed even under conditions where  
Figure 7: Peak-to-peak amplitudes of the cylindrical surface waves. dispersion is not detected. The phase shift causes the polarity of the surface wave to be different at different positions from the excitation source [22; 25], as shown in Figure 8f. When the scattered waves superimpose with the surface waves, which have a different polarity, the peak-to-peak of the cylindrical surface waves may increase or decrease. Figure 8a-e show the variations in the peak-to-peak amplitudes of surface waves at different positions of detection, when the SLLS method is used. It is clear that the crack at \(\alpha=90^{\circ}\) can be identified by the peak-to-peak amplitude enhancement. However, the SLLS peaks are different at different positions of detection. The SLLS peaks at \(\theta=90^{\circ}\) and \(120^{\circ}\) is higher than that of \(\theta=30^{\circ}\), \(60^{\circ}\), and \(120^{\circ}\). The SLLS peaks of the cracked cylinder (see Figure 8a-e) at the unipolar position of the intact surface waves (see Figure 8f) are significantly higher than that of the cracked cylinder at the bipolar position. In fact, higher SLLS peaks are more conducive to the identification of the surface crack. To explore the inducement of the phenomenon shown in Figure 8, we define the remaining waves (RWs) as the difference between the waves propagating on the cracked cylinder and the waves propagating on the intact cylinder. According to the analysis in the previous section, the RWs are the superposition of LTR waves and reflection waves on the surface waves, although the LTR waves constitute the main part, as shown in Figures 6 and 9. It is observed from Figure 9 that there is also a regular change in the polarity of the RWs at different positions of detection, which indicates that the phase shift also exists in the RWs. Comparing Figure 9a-e, it is found that, when the surface waves of the intact cylinder or the RWs are unipolar, the superposition of the RWs and the surface waves increase the peak-to-peak amplitudes. When the surface waves and the RWs are bipolar, there is a partial wave cancellation as the RWs superimpose with the surface waves, as shown in Figure 9b. In order to further study the signal enhancement at different positions of detection, the phase shift of the surface waves on the intact cylinder and the RWs on the cracked cylinder with a diameter of 20 mm are shown in Figure 10. The calculation method of the instantaneous phase of surface waves is obtained from the method described in [22]. The fit curve for the phase shift of the surface waves on the intact cylinder is \(P~{}=~{}1.44\theta~{}-~{}1.84\), and the fit curve for the phase shift of the RWs on the cracked cylinder is \(P=1.58\theta-151.57\). This indicates that the phase shift of the surface waves on the intact cylinder at the unit circumferential angle is 1.44, while that of the RWs on the cracked cylinder is 1.58. As shown in Figure 10, the change in the phase shift for these two situations is similar. The main reason for the difference is that the RWs are not composed of one type of wave but, rather, are a superposition of the Rr and the LTR waves. The initial phase of the surface wave on the intact cylinder is \(1.84^{\circ}\) when the circumferential position is \(0^{\circ}\). It is known that the surface wave excited by the laser line source is a unipolar wave when the detection position is near the laser source. The initial phase of the RWs on the cracked cylinder is \(-151.57^{\circ}\), which indicates that the polarity of the RWs is opposite to the surface wave and the RWs are almost unipolar waves at \(\theta=0^{\circ}\). According to the fit curve of the phase shift displayed in Figure 10, we obtain the unipolar position of the surface waves and the RWs.