## 4 Identification of Surface Crack Location
### Location of Surface Crack on Cylinders with Different Diameters  
On the basis of the surface wave velocity equation for the cylindrical surface [53], [PERSON] separated the phase term of the surface wave as a part of the dispersion equation, and obtained a relationship between the phase, velocity, frequency, radius, and elastic constants. The formula is as follows [22]:  
\[ \begin{array}{c}(V_{1}p^{4}+V_{2}p^{3})+(V_{4}p^{3}+V_{5}p^{2}+V_{6}p)\Phi i+ \\ (V_{7}p^{2}+V_{8}p+V_{9})\Phi^{2}+(V_{10}p+V_{11})\Phi^{3}i+V_{12}\Phi^{4}=0 \end{array}  \tag{13}\]  
where \(V_{1}\)... \(V_{12}\) represent the different functions of \(v_{L}\), \(v_{S}\), and \(v_{R}\) and have the units of velocity squared; \(p=kR\), \(R\) is the radius of the cylinder; \(k\) is the wavenumber; \(\Phi\) is the phase of wave propagation. Equation (13) indicates that the phase of the cylindrical surface wave is not only related to the material of the cylinder but also closely related to its diameter. Therefore, this section explores the location of the surface crack on cylindrical structures with different diameters using the laser ultrasonic SLLS method. Figure 11 shows the peak-to-peak variation of SLLS signals at detection positions of \(\theta=30^{\circ}\), \(60^{\circ}\), \(90^{\circ}\), \(120^{\circ}\), and \(150^{\circ}\) for a cylinder with a diameter of 10 mm. There is a signal enhancement near the surface crack edge and a rapid drop in the peak-to-peak amplitudes after the crack passes the laser source at any \(\theta\). Comparing Figure 11a-e, the SLLS peaks are still different at different positions of detection. It is observed that the SLLS peaks of the 10 mm diameter cylinder at \(\theta=120^{\circ}\) and \(150^{\circ}\) are significantly higher than those of the other detection angles, while the SLLS peaks of the 20 mm diameter cylinder are significantly higher at \(\theta=90^{\circ}\) and \(120^{\circ}\). For the unipolar position of the surface waves of the intact cylinder (Figure 11f), the SLLS peak is significantly higher than that for the bipolar positions (see Figure 11a-e), which is the same trend for the lager cylinder. Figure 12 shows the phase shift of the surface waves on the intact cylinder with a diameter of 10 mm and of RWs on the cracked cylinder. The fit curve of the phase shift on the intact cylinder is \(P=1.34\theta-6.65\). When \(P=0\) or \(180^{\circ}\), the unipolar position of the surface waves on the intact cylinder is \(\theta=5^{\circ}\)or \(139^{\circ}\). The fit curve for the phase shift of the RWs on the cracked cylinder is \(P=1.56\theta-194.7\). The unipolar position of the RWs is \(\theta=125^{\circ}\). This suggests that the SLLS peaks at \(\theta=120^{\circ}\) and \(150^{\circ}\), which are near the unipolar position, are higher than those at other detection angles because of the superimposition of the unipolar waves of the surface waves and the RWs. Furthermore, as can been seen from the initial phase of the surface waves, the surface waves and the RWs are unipolar and the polarities are opposite at \(\theta=0\). Figure 10: Phase shifts of cylindrical surface waves on the intact cylinder and the RWs on the cracked cylinder, both with a 20 mm diameter. The solid lines represent fit curves for the phase shift of the cylindrical surface waves and the RWs. Likewise, the unipolar position of the surface waves of an intact cylinder with different diameters can be recognized as the approximate optimal position for crack detection using the SLLS method. It is clear from Figures 10 and 12 that the phase shift rule of surface waves is different for different diameters. In order to describe the difference of the phase shift in cylinders of different diameters, the phase shift rate (PSR) here is defined as the variation of the phase per unit distance along the circumferential surface. The PSR of different diameters is investigated using the FEM of laser ultrasonics, as described in the previous section.