## 2 Principle of Time-Reverse Algorithms
\[\left|\mathbf{H}\right|^{2}=\pi^{2}\left[ \begin{array}{c}|A(\omega)|^{2}+|S( \omega)|^{2}+S(\omega)A^{*}(\omega)\times e^{-j(k_{0}-k_{0})x}\\ +A(\omega)S^{*}(\omega)\times e^{-j(k_{0}-k_{0})x}\end{array} \right] \tag{21}\]  
The focused signal \(E_{0}^{\prime}\), obtained by substituting Equation (21) into Equation (18), demonstrates wavefront focusing. However, interference signals in the form of side lobes appear on both sides of the wavefront, presenting a symmetric pattern as depicted in Figure 2. Analyzing the focused signal presents challenges. Figure 2: Schematic diagram of time-reversed focused Lamb wave signals. The synthesized signal of the excitation signals from each piezoelectric sensor propagating at the structural health location \(h\), i.e., the non-source location, can be represented as follows:  
\[E_{h}^{\prime}=\sum_{t=1}^{n}H_{ih}E_{i}^{*}=\sum_{t=1}^{n}H_{ih}H_{0i}^{*}E_{0} ^{*} \tag{22}\]  
The synthesized signal at the structural health location \(h\), obtained by substituting Equation (14) into Equation (22), is given as follows:  
\[E_{h}^{\prime}=\left[\sum_{i=1}^{n}\pi^{2}|A(\omega)|^{2}\times e^{-j(k_{A_{0} }x_{A_{0}}-k_{A_{0}}x_{A_{0}})}\right]E_{0}^{*} \tag{23}\]  
Based on Equations (19), (20), and (23), it can be observed that the synthesized signal \(E_{h}^{\prime}\) at the structural health location \(h\) is smaller than the focused signal \(E_{0}^{\prime}\) at the original source location. The time-reversed signals, when propagated through the structure, reach their maximum value at the original source location, where they are superimposed and focused, resulting in an increased signal magnitude. However, at non-source locations in the healthy region, no significant focusing occurs, leading to smaller signal magnitudes. This facilitates the identification of the source location. When there are two sources of excitation within the structure, as shown in Figure 3, \(n+1\) piezoelectric sensors are still attached to the plate structure. Assuming that the frequency response of the transfer function from the source positions 0 and 1 to the \(i_{th}\) piezoelectric sensor, denoted as \(H_{0i}\) and \(H_{1i}\) for Lamb wave propagation, respectively, and the excitation signals emitted from the source positions 0 and 1 are \(E_{0}\) and \(E_{1}\), respectively, the response signal received by each piezoelectric sensor can be expressed as follows:  
\[E_{i}=H_{0i}E_{0}+H_{1i}E_{1} \tag{24}\]  
Similarly, by time-reversing the response signals from each receiver and utilizing the corresponding receiver as the excitation source, the signal received on the piezoelectric sensor at the original source can be defined as follows:  
\[E_{0}^{\prime}=\sum_{i=2}^{n}H_{i0}E_{i}^{*} \tag{25}\]  
By substituting Equation (24) into Equation (25), where \(H_{0i}=H_{i0}\), we can obtain the following:  
\[E_{\text{C}_{1}}^{\prime}=\sum_{i=2}^{n}H_{i0}H_{0i}^{*}E_{0}^{*}+\sum_{i=2}^{ n}H_{i0}H_{1i}^{*}E_{1}^{*} \tag{26}\]  
\[E_{\text{C}_{2}}^{\prime}=\sum_{i=2}^{n}H_{i1}H_{1i}^{*}E_{1}^{*}+\sum_{i=2}^{ n}H_{i1}H_{0i}^{*}E_{0}^{*} \tag{27}\]  
Figure 3: Schematic diagram depicting time-reversal focusing for multiple sources. Based on Equation (14), Equations (26) and (27) can be expressed as follows [15; 16; 17]:  
\[E_{0}^{\prime}=n\pi^{2}|A(\omega_{c})|^{2}E_{0}^{*}+\pi^{2}|A(\omega_{c})|^{2} \underset{i=2}{\overset{n}{\sum}}e^{-jk_{A_{0}}(x_{i0}-x_{i1})}E_{2}^{*} \tag{28}\]  
Figure 2:0  
From Equations (28) and (29), it can be inferred that the signal demonstrates focalization at both source locations 0 and 1, while also introducing additional sidelobes. As the number of sources increases, the sidelobes gradually amplify and begin to influence the focalized signals, ultimately causing the source locations to become indistinguishable and rendering them unidentifiable.