## 3 Signal Enhancement of Cylindrical Surface Waves
(**b**) The velocity magnitude map at 3.7 \(\mu\)s.  
According to the Huygens principle, when the ultrasonic waves act on the edge of a crack, the corner of the crack can be recognized as a new source of ultrasonic waves reflector. The surface skimming longitudinal wave (L) reaches the edge of the crack first, and interacts with the crack to generate the reflected surface skimming longitudinal wave (Lr) and the mode-converted longitudinal-to-Rayleigh wave (LTR). The surface wave (R) then interacts with the crack and generates the reflected Rayleigh wave (Rr), the mode-converted Rayleigh-to-longitudinal wave (RTL), and the mode-converted Rayleigh-to-shear wave (RTS) [52]. Since the RTS has less energy in the out-of-plane displacement, it is not obvious in the B-scan shown in Figure 4. At the edge of the crack, the R wave and any Rr or mode-converted waves interfere, and a significant amplitude enhancement can be observed. This region is marked in Figure 4. The signal enhancement in the near field of the crack can be explained by the wave superposition of RTL, R, and Rr. From the spatial distribution of the mode-converted bulk wave, the out-of-plane displacement has a relatively large contribution from the RTL that comes from the lower crack tip [38], as shown at Position 2 in Figure 3. It has been suggested that the signal enhancement near the surface crack can be used as a fingerprint of the presence of the surface crack, which is the same as that of the plate. Unfortunately, as the distance from the surface crack increases, the amplitude of RTL attenuates rapidly, hence the signal enhancement can only be detected in a small area near the edge of the surface crack. With the crack scanning close to the pulsed laser source, the counterclockwise propagating surface wave generated by the pulsed laser source interacts with the crack at position \(\alpha\), as shown at Position 1 in Figure 3. As mentioned above, Lr, LTR, Rr, RTL, and RTS are generated by the interaction between waves that propagate on the cylindrical surface and the crack. The arrival time of these waves, whose paths (\(0\leq\alpha\leq 90^{\circ}\)) are shown schematically at Position 1 in Figure 3, are given by  
\[t_{Rr}=\frac{\pi d(180^{\circ}-2\alpha+\theta)}{360^{\circ}v_{R}} \tag{8}\]  
Figure 1:0  
Figure 1:1  
Figure 1:2  
Figure 1:3  
Figure 1:4 B-scan showing the out-of-plane velocity from FEM when \(\alpha=150^{\circ}\), \(\theta=10^{\circ}\)\(\sim\)90\({}^{\circ}\). The dotted lines in the graph are used to assist with the identification of the types of scattered waves. where \(d\) is the diameter of the cylinder; \(v_{R}\), \(v_{L}\), and \(v_{S}\) are the propagation velocities of the Rayleigh, longitudinal, and shear waves in the solid cylinder, and they can be estimated at dominant frequency of laser excitation. \(t_{RTL}\) and \(t_{RTS}\) ignore the time of the wave propagation in the crack depth and the crack width. Figure 5a shows the out-of-plane displacement at \(\theta=30^{\circ}\) when the crack position \(\alpha\) is \(60^{\circ}\), \(65^{\circ}\), \(70^{\circ}\), \(75^{\circ}\), \(80^{\circ}\), and \(85^{\circ}\). According to Equations (8)-(12), the various wave modes can be identified and are thus labeled in Figure 5a. It is observed that, at the crack position \(\alpha=85^{\circ}\), a new wave appears at the position of LTR and superimposes with the surface wave. Since the six signals in Figure 5 are displayed at the same time, the amplitude of each signal will be scaled down. For a clearer analysis of this new wave, the out-of-plane displacements of \(\alpha=60^{\circ}\), \(80^{\circ}\), and \(85^{\circ}\) and \(\theta=30^{\circ}\) are selected and are presented in Figure 1:5 Compared with other modes of waves, the surface wave is dominant when the crack position \(\alpha\) is \(60^{\circ}\) and \(80^{\circ}\).