## 3 Study on Instantaneous Phase of an Active Lamb Wave Monitoring Signal
### Establishment of the Monitoring Signal Model  
The active Lamb wave based monitoring of a plate structure utilizes the excitation source to generate a Lamb wave in a plate structure, and the Lamb wave scatters when it encounters the damage. In this way, a Lamb wave that is arriving directly at a sensor is superposed with the scattering wave, and the sensing signal contains the information on the plate structure health. This sensing signal is called a monitoring signal, and it can be defined as the sum of a direct wave and a scattering wave. A suitable monitoring signal model is needed for the following simulation experiment. According to the characteristics of the active Lamb wave based monitoring technology and the experimental data, a direct wave and a scattering wave were constructed to establish an approximate monitoring signal model. As introduced in Section 2.2, the tone burst signal, which is used as an excitation signal, generates a Lamb wave that propagates in the plate and has a waveform that is almost consistent with the excitation signal (see Wave packet 2 in Figure 2b). Therefore, a tone burst signal (see Equation (6)) was used as a direct wave in the monitoring signal model. The scattering wave is the scattering of the Lamb wave  
Figure 2: Lamb wave excited in the plate structure and the associated wavelet transform. (**a**) Excitation signal; (**b**) Plate structure response signal; (**c**) Time-frequency image of the wavelet transform of the excitation signal; (**d**) Time-frequency image of the wavelet transform of the plate structure response signal. in propagating caused by damage. This means that the scattering wave and the direct wave come from the same excitation source, and their waveforms are similar. The monitoring signal model was defined as the sum of the direct wave and the scattering wave. The direct wave remains unchanged before and after the damage, and thus the scattering signal is the dominant factor of the signal model. That is to say, the expression of the signal model is determined by the waveform and superposition location of the scattering wave. The damage conditions, such as the location and size, directly affect the waveform and superposition location of the scattering waves, and it is impractical to analyze the instantaneous phase change of the damage signal model directly according to the actual damage conditions. This requires a reasonable design so that the experimental process not only simulates the actual situation but also the instantaneous phase change analysis is feasible. For the monitoring signal, the influence of the damage condition on the scattering wave is ultimately reflected in the waveform and superposition location. Generally, the larger the damage size is, the higher the energy of the scattering wave is; the longer the propagation path of the scattering wave is, the more energy the scattering wave consumes. Other factors, such as scattering angle and plate material, also affect the energy of scattering waves [27; 28; 29]. However, these factors will not change the center frequency of the scattering wave, and the scattering wave and the direct wave come from the same excitation source. Thus, the attenuation of the scattering wave energy can be approximately the attenuation of the wave amplitude. Therefore, the attenuation coefficient \(\alpha\) of the scattering signal is simplified as the ratio of the amplitude of the scattering wave and the direct wave. In the damage signal model, the damage scattering wave is the product of direct wave and attenuation coefficient \(\alpha\). By changing \(\alpha\), the signal model can simulate the scattering waves with different energy. The propagation distance of the direct wave is constant before and after the damage, and thus the superposition site is determined by the propagation distance of the damage scattering wave, in other words, by the location of the damage.