## 3 Study on Instantaneous Phase of an Active Lamb Wave Monitoring Signal
### Analysis of the Instantaneous Phase of the Monitoring Signal
According to Figure 4c,dIII, the direction and deformation of the instantaneous phase after damage could be obtained. In Figure 4c, the instantaneous phase after the damage was smaller  
Figure 4: The sensing signal and the instantaneous phase in the active Lamb wave-based monitoring of the epoxy glass fiber plate structure. (**a**) Experimental setup for the active Lamb wave-based monitoring of the epoxy glass fiber plate. (**b**) Sensing signal of the active Lamb wave-based monitoring, and subfigures (**I,II**) show the response of the plate structure excited by the signal with the center frequency of 100 kHz and 150 kHz, respectively; (**c**) The monitoring signal with a center frequency of 100 kHz and the instantaneous phase, and subfigure (**I**) shows the instantaneous phase change, subfigure (**II**) denotes a locally enlarged view of (**II**), and subfigure (**III**) shows some information for the monitoring signal. (**d**) The monitoring signal with a center frequency of 150 kHz and the instantaneous phase. The explanation of the subfigures (**I–III**) is the same as (**c**). than the original instantaneous phase. According to the relationship between the superposition site and the direction of the instantaneous phase change obtained in Section 3.2, the superposition site of the scattering wave should be between the maximum and minimum of the direct wave. The real scattering wave and direct wave were used to verify this inference. In Figure 4 cIII, the Wave packet of the direct wave is labeled \"Wave packet 1\", and the damage scattering wave is labeled as \"Scattering wave.\" According to the definition of the superposition site in Section 3.2, the maximum point of the \"Scattering wave\" is marked as \"**, and the intersection point between the projection of this maximum \"** point on the abscissa, and the direct wave is the superposition site, which is marked as \"A\". The local magnification in Figure 4 cIII shows that \"A\" is indeed between the maximum and minimum of the direct wave. In Figure 4 dIII, based on the same processing and analysis, the instantaneous phase after the damage is larger than the original phase, and the superposition position of scattering wave should be between the minimum and maximum. The local magnification in Figure 4 dIII shows that \"A\" is indeed between the minimum and maximum of the direct wave. The instantaneous phase analysis results of the two experiments are consistent with the relation between the instantaneous phase change direction and the scattering wave superposition summarized in Section 3.2. The instantaneous phases are shown in Figure 4(cII,dII), and their deformation degree is different from the original instantaneous phase shape. From the analysis of the instantaneous phase change in Section 3.2, when the superposition site makes the phase difference between the scattering wave and the direct wave tend to be in antiphase, the greater the deformation degree is, and vice versa. Compared with the superposition site in Figure 4 dIII, the superposition site in Figure 4 cIII made the scattering wave closer to the antiphase of the direct wave, and thus the local deformation of its instantaneous phase should be larger. Large deformation is beneficial for improving the recognition of instantaneous phase change. Moreover, the influence of scattering wave energy on the local deformation range of the instantaneous phase was discussed in Section 3.2. Although the scattering wave energy ratio (A1/A0 = 0.1) in Figure 4 dIII is higher than that in Figure 4 cIII (A1/A0 = 0.06), there is no significant difference between their instantaneous phase deformation ranges. It is considered that the damage scattering wave is the rescattering of the wave, and its energy is already very small; thus, its effect on the instantaneous phase change is limited. That is to say, in the case of the same or close energy, the effect of phase difference on the instantaneous phase change is higher than that of energy.