## 2 Theoretical Aspects of Lamb Waves
### Dispersion Curves of Lamb Waves  
The dispersion curves of Lamb waves refer to the relationship between the longitudinal and transverse waves of Lamb waves in symmetric and antisymmetric modes as a function of frequency; that is, the wave speed-frequency curve. Since the wavelength and frequency of Lamb waves are influenced by factors such as the thickness of the structural plate and material properties, the dispersion curves of Lamb waves are closely related to the geometric dimensions and material parameters of the structural plate.  
Figure 1: In a 1 mm thick aluminum plate, the wave speed–frequency dependency relationships of axial waves, bending waves, Lamb waves, and Rayleigh waves are considered.  
Figure 2: Structural plate and coordinate system.  
To obtain the dispersion curves of Lamb waves, it is necessary to construct the dispersion equations of Lamb waves. For a plate structure with an assumed thickness of \(2\,h\), the Lamb dispersion curve equation is  
\[\frac{\tan(k_{s}h)}{\tan(k_{l}h)}=-\left[\frac{4k^{2}k_{l}k_{s}}{(k^{2}-k_{s}^{2 })}\right]^{\pm 1} \tag{3}\]  
In the equation \(k=\omega/c_{p},k_{l}^{2}=\omega^{2}/v_{l}^{2}-k^{2},k_{s}^{2}=\omega^{2}/v_{s} ^{2}-k^{2}\), \(k\) represents the angular wave speed, \(\omega\) represents the angular frequency, and \(c_{p}\) represents the phase velocity of the material plate, while \(v_{l}\) and \(v_{s}\) are the longitudinal and transverse wave speeds of the plate, respectively. When the superscript in Equation (3) is \(+1\), it represents the symmetric mode; when it is \(-1\), it represents the antisymmetric mode. Solving the aforementioned Equation (3) yields the phase velocity \(c_{p}\) dispersion curves and group velocity \(c_{g}\) dispersion curves, as shown in Figure 3.  
From Figure 3, it can be seen that in the group velocity dispersion curve, within the low-frequency range, there are \(A_{0}\) and \(S_{0}\) modes, and at 3600 kHz, \(A_{1}\) and \(S_{1}\) modes begin to appear; in the phase velocity dispersion curve, \(S_{0}\) and \(A_{0}\) modes appear at low frequencies, and at 3600 kHz, \(A_{1}\) and \(S_{1}\) modes appear. This indicates that for a three-dimensional aluminum plate, only waves of \(S_{0}\) and \(A_{0}\) exist at lower frequencies, while at higher frequencies, different modes of guided waves will appear, leading to mutual interference. In practical applications of structural damage monitoring, to reduce the dispersion and multimode effects of guided waves and simplify signal processing, Lamb waves of a single mode are usually selectively excited in the low-frequency range.  
Typically, the \(S_{0}\) mode is chosen as it helps alleviate the impact of wave dispersion and multimode characteristics. This selection facilitates easier signal analysis and processing.  
For different modes of Lamb waves, the relationship between phase velocity and group velocity also varies. In the low-frequency range, the phase and group velocities \(A_{0}\) and \(S_{0}\) are relatively close so they can be used simultaneously for SHM. However, as the frequency increases, the difference between the phase and group velocities of higher-order modes becomes more pronounced, so the appropriate mode must be selected for monitoring and analysis according to the actual situation.