## 3 Study on Instantaneous Phase of an Active Lamb Wave Monitoring Signal
### Experiment of the Instantaneous Phase of Monitoring Signal Model  
The monitoring model was used to analyze the instantaneous phase change by the damage scattering wave. The experiment required the assignment of \(\alpha\) and \(\Delta\mathrm{t}\). According to the above description of the monitoring signal model, \(\alpha\) is defined as the ratio of the amplitude of the scattering wave to the direct wave. Generally, the energy of scattering wave is far less than that of the direct wave. According to the two experiments in this paper, the ratio of the maximum amplitude of scattering wave(A1) to that of the direct wave(A0) was 0.06 and 0.1 (as shown in the figures in Section 3.3). Therefore, the range of the \(\alpha\) value was set to 0.01-0.1, and the change step was 0.01. \(\Delta\mathrm{t}\) should be given different values to simulate the scattering wave coming from different damage locations. Its range is equal to the time range of the direct wave (0-1 s), and the change step was 0.01 s. There is no necessary correlation between \(\Delta\mathrm{tanalyzing the influence of one variable on the instantaneous phase, another variable is set as a constant. The advantage of this approach is that the influence of variables (\(\alpha\) or \(\Delta\)t) on the instantaneous phase change can be demonstrated. In the following experiments, when analyzing the influence of the superposition site on instantaneous phase change, \(\alpha\) was set to a constant, which was between 0.01 and 0.1, and the step of \(\Delta\)t increased gradually by 0.01 s. When analyzing the influence of attenuation on instantaneous phase change, the typical superposition site was chosen as the research object, and the step of \(\alpha\) increased gradually by 0.01. The experiment is shown in Figure 3. Equations (6) and (7) were used to generate a direct wave and a scattering wave. The attenuation coefficient was set to 0.05, and the simulated scattering wave is shown in Figure 3 aI. The maximum point of the scattering wave (indicated by the triangle (\"\(\Delta\)\") in Figure 3 aI) was taken as the observation point of scattering signal moving right, and the superposition site was defined as the intersection point between the projection of the maximum point (\"\(\Delta\)\") on the abscissa and the direct wave (indicated by the asterisk (\"*\") in Figure 3 aI). The initial time coordinate of the observation point was t\({}_{0}\), and the \(\Delta\)t in the monitoring signal model was the time difference between the superposition site and t\({}_{0}\). The instantaneous phase of the direct wave is shown in Figure 3 aII. In Figure 3b-d, the instantaneous phase of change of the monitoring signal after scattering wave superposition is shown. The asterisks in subfigure I represent the superposition site; in subfigure II, the instantaneous phase of the signal after the superposition with the scattering wave is demonstrated; and subfigure III denotes a locally enlarged view of subfigure II. In Figure 3b-d, the arrow in subfigure I indicates the moving direction of the superposition site (the moving interval is 0.01 s), while the arrow in subfigure III indicates the corresponding change in the direction of the instantaneous phase. The effect of the attenuation coefficient \(\alpha\) is shown in Figure 3b-d. The red triangles in Figure 3 eI-f1 are the two superposition sites. When \(\alpha\) increased from 0.01 to 0.1 (the step size was 0.01), the 10 scattering waves are shown in Figure 3(eI,f1). The change of the instantaneous phase with the attenuation coefficient increasing is shown in Figure 3(eI,f1). The subfigures III denote a locally enlarged view of subfigure II, and the arrow indicates the change direction on the instantaneous phase when the attenuation coefficients increased. The experimental scheme constructed in this paper shows the effect of the scattering signal on the instantaneous phase of the signal.