# The Rapid Detection Technology of Lamb Wave for Microcracks in Thin-Walled Tubes
## 2 Theoretical Principle
### Mode Selection  
Through the dispersion curves of the Lamb waves, we determined that the optional modes were \(A_{0}\) or \(S_{0}\). However, how to distinguish between them required further study of the direction of particle vibrations and the distributions of particle displacements for the \(A_{0}\) and \(S_{0}\) modes in plates. The particle displacement-amplitude curve of the Lamb waves reflects the variation and distribution of Lamb wave energy in the plate, which is one of the main bases for mode selection. The in-plane and out-of-plane displacement amplitudes of the particle vibrations are denoted as \(U\) and \(V\), respectively. If the vertical distance between the particle and waveguide axis is \(x\), their mathematical expressions are as follows [44]:  
Symmetric mode:  
\[U_{s}=\frac{(k_{z}^{2}-s^{2})\sin(0.5 sd)}{2q^{\prime}\sin(0.5q^{\prime}d)}\cos (q^{\prime}x)B+s\cos(sx)B, \tag{3}\]  
\[V_{s}=\frac{(k_{z}^{2}-s^{2})\sin(0.5 sd)}{2k_{z}\sin(0.5q^{\prime}d)}\sin(q^{ \prime}x)B+k_{z}\sin(sx)B, \tag{4}\]  
Antisymmetric mode:  
\[U_{a}=\frac{(k_{z}^{2}-s^{2})\cos(0.5 sd)}{2q^{\prime}\cos(0.5q^{\prime}d)}\sin (q^{\prime}x)D-s\sin(sx)D,\ V_{a}=\frac{(k_{z}^{2}-s^{2})\cos(0.5 sd)}{2k_{z} \cos(0.5q^{\prime}d)}\cos(q^{\prime}x)D+k_{z}\cos(sx)D, \tag{5}\]  
where \(k_{z}=\frac{2\pi f}{c_{p}}\), \(s=k_{z}\sqrt{\left(\frac{c_{p}}{c_{T}}\right)^{2}-1}\), \(q^{\prime}=k_{z}\sqrt{\left(\frac{c_{p}}{c_{L}}\right)^{2}-1}\), and \(c_{p}\) is the phase velocity of Lamb wave.  
In the symmetric mode, the vibration of a particle in a plate is symmetrical in the center of the plate, and its in-plane displacement is symmetric along the thickness of the plate, while the out-of-plane displacement is antisymmetric. Under the antisymmetric mode, the vibration phase of the particle in the plate is the same on the upper and lower surfaces, and the out-of-plane displacement is symmetric along the thickness of the plate, while the in-plane displacement is antisymmetric along the thickness of the plate. For the steel plate with 1-mm thickness, the particle displacement-amplitude curves of the \(A_{0}\) and \(S_{0}\) modes are shown in Figure 3.  
Figure 2: Relationship between the phase velocity \(c_{p}\) and frequency-times-thickness \(fd\) for Lamb waves in a steel plate (the S-wave velocity was 5970 m/s, the P-wave velocity was 3200 m/s, and Poisson’s ratio was 0.28).  
In Figure 3, the dashed line is the in-plane displacement along the thickness direction; that is, the vibration direction of the particle is parallel to the plate surface, while the real line is the out-of-plane displacement, that is, the vibration direction of the particle is perpendicular to the plate surface. The figure shows that the in-plane displacement of mode \(S_{0}\) had a larger component in the plate, which is suitable for detecting the defects perpendicular to the plate surface, and the out-of-plane displacement of mode \(A_{0}\) had a larger component in the plate, which is suitable for detecting defects on the inner and outer surfaces of the plate. Therefore, the \(A_{0}\) mode, which is sensitive to the surface defects of the inner and outer walls of thin-walled tubes, was selected as the detection mode. This is consistent with the conclusion of Ref. [37].