## 5 Analysis of the Relationship between the Instantaneous Phase and Marginal Spectrum  
In Figure 7, by comparing the change of marginal spectrum amplitude before and after damage, it can be seen that the marginal spectrum of the Lamb wave with a center frequency of 100 kHz diffused to low frequencies after the damage of the plate structure while that of the Lamb wave with a center frequency of 150 kHz diffused to high frequencies. Such results could not be explained only by the change in the marginal spectrum. However, by considering the change in the instantaneous phase, these two different directions of diffusion could be explained.  
As presented in Figure 4c, when the center frequency of the monitoring signal was 100 kHz, the damage scattering wave reduced the instantaneous phase of the Lamb wave, that is, the slope of the relationship curve between the time and instantaneous phase decreased. According to Equation (3), the slope denoted the instantaneous frequency. Therefore, the active Lamb wave monitoring signal with a center frequency of 100 kHz, after superimposing the damage scattering signal, decreased and the corresponding frequency to the marginal spectrum also decreased. This explains why the marginal spectrum of the Lamb wave diffused to low frequencies after the plate structure damage, as shown in Figure 7a. When the center frequency of the monitoring signal was 150 kHz, the damage scattering wave increased the instantaneous phase of the Lamb wave. As shown in Figure 4c, the instantaneous frequency of the monitoring signal increased. Therefore, the frequency corresponding to the marginal spectrum also increased, which explains why the marginal spectrum of the Lamb wave diffused to high frequencies after plate structure damage, as shown in Figure 7b. In Figure 4c,d, the instantaneous phase change is smooth, and thus the instantaneous frequency change is continuous. It has been shown that instantaneous frequency change caused the marginal spectrum to diffuse towards low or high frequencies. The marginal spectrum diffusion after the damage was in the range around the center frequency only because the instantaneous frequency change was continuous, which showsthat the marginal spectrum changed in multiple continuous local frequency bands, and the marginal spectrum amplitude increased and decreased in some local frequency bands.  
The above analysis of the monitoring signal shows that, in the active Lamb wave based monitoring of the plate structural health, the change in the signal instantaneous phase could indicate the change in the signal marginal spectrum and vice versa. The damage scattering wave caused the change of the instantaneous phase of the monitoring signal, and the change of the instantaneous phase also showed the existence of the damage scattering wave. The monitoring method based on the difference between the monitoring signals before and after damage illustrated the change of the signals after damage from the time domain. In this paper, the monitoring methods based on instantaneous phase change and marginal spectrum change of monitoring signal described the change of the signal after damage from the frequency domain and energy. These three monitoring methods are all based on the direct wave changes caused by damage scattered signals. They are different explanations of the same physical phenomenon. The research content and analysis results also show that they are interrelated. Therefore, it is reliable and feasible to use the instantaneous parameter change of active Lamb wave to realize plate structure health monitoring.