## 2 Principle of Time-Reverse Algorithms  
Damages occurring in a structure can act as excitation signals generated by wave sources within the structure, conveying information about the damages. When an excitation signal is generated at a specific point as a wave source within the structure, it propagates in various directions through vibrations. Observation points are positioned at other locations within the structure to capture response signals. By conducting time reversal in the time domain on the received response signals and employing them as new excitation signals at the observation points, a reconstructed signal is obtained at the original wave source. The reconstructed signal demonstrates focusing characteristics. Reverse focusing of the signal is accomplished based on the principles of acoustic reciprocity and the time reversal invariance of the linear elastodynamic wave equation. Utilizing the piezoelectric effect of piezoelectric components and the propagation characteristics of Lamb waves in plate structures, analytical models were developed for the \(S_{0}\) and \(A_{0}\) modes of Lamb waves propagated in isotropic plate structures. The expression for the Lamb wave unit impulse response in a plate structure is given as follows:  
\[\varepsilon(x,t)|_{y=d}=-j\frac{\pi\tau_{0}}{G}\sin\bigl{(}k_{A_{0}}a\bigr{)} \frac{N_{A}\bigl{(}k_{A_{0}}\bigr{)}}{D^{\prime}_{A}\bigl{(}k_{A_{0}}\bigr{)} }\times e^{-j\bigl{(}k_{A_{0}}x-\omega_{0}t\bigr{)}}-j\frac{\pi\tau_{0}}{G} \sin\bigl{(}k_{s_{0}}a\bigr{)}\frac{N_{S}\bigl{(}k_{S_{0}}\bigr{)}}{D^{\prime}_ {S}\bigl{(}k_{S_{0}}\bigr{)}}\times e^{-j\bigl{(}k_{s_{0}}x-\omega_{0}t\bigr{)}} \tag{1}\]  
Among the terms, \(\varepsilon(x)\) represents the strain solution for Lamb waves, where \(x\) denotes the distance of Lamb wave propagation. \(a\) represents the radius of the piezoelectric sensor, and \(\tau_{0}\) represents the distributed shear force at the edge of the piezoelectric sensor. \(k_{S_{0}}\) and \(k_{A_{0}}\) represent the wave numbers for the \(S_{0}\) and \(A_{0}\) mode, respectively, as obtained from Equations (2) and (3). The coefficients \(N_{s}\), \(N_{A}\), \(D_{S}\), and \(D_{A}\) are determined from Equations (4)-(7). \[k_{S_{0}}=\frac{\omega_{0}}{V_{p\omega_{0}S_{0}}} \tag{2}\]  
\[k_{A_{0}}=\frac{\omega_{0}}{V_{p\omega_{0}A_{0}}} \tag{3}\]  
\[N_{S}=kq\Bigl{(}k^{2}+q^{2}\Bigr{)}\cos(pd)\cos(qd) \tag{4}\]  
\[N_{A}=kq\Bigl{(}k^{2}+q^{2}\Bigr{)}\sin(pd)\sin(qd) \tag{5}\]  
\[D_{S}=\Bigl{(}k^{2}-q^{2}\Bigr{)}^{2}\cos(pd)\sin(qd)+4k^{2}qp\sin(pd)\cos(qd) \tag{6}\]  
\[D_{A}=\Bigl{(}k^{2}-q^{2}\Bigr{)}^{2}\sin(pd)\cos(qd)+4k^{2}qp\cos(pd)\sin(qd) \tag{7}\]  
where \(V_{p\omega_{0}S_{0}}\) and \(V_{p\omega_{0}A_{0}}\) represent the phase velocities of their respective modes. Equation (1) can be interpreted as the sum of the amplitude terms associated with the \(S_{0}\) mode and the \(A_{0}\) mode, along with the phase term modulation, which can be simplified as follows:  
\[\varepsilon(x,t)=A(\omega_{0})\times e^{-j\bigl{(}k_{A_{0}}x-\omega_{0}t \bigr{)}}+S(\omega  
\[S(\omega_{0})=-j\frac{\pi\upsilon_{0}}{G}\frac{\sin(k_{S_{0}}a)}{k_{S_{0}}}\frac{N _{S}(k_{S_{0}})}{D_{S}^{\prime}(k_{S_{0}})} \tag{10}\]  
Performing a Fourier transform on Equation (8) yields the following:  
\[H(x,\omega)=\int_{-\infty}^{+\infty}A(\omega_{0})\times e^{-j(k_{A_{0}}x- \omega_{0}t)}e^{-j\omega t}dt+\int_{-\infty}^{+\infty}S(\omega_{0})\times e^{- j(k_{0}x-\omega_{0}t)}e^{-j\omega t}dt \tag{11}\]  
The frequency response function of Equation (11) is the following:  
\[H(x,\omega)=\left[\pi A(\omega_{0})\times e^{-j\omega_{0}x}+\pi S(\omega_{0}) \times e^{-jk_{0}x}\right]\times\left[\delta(\omega-\omega_{0})+\delta(\omega +\omega_{0})\right] \tag{12}\]  
where \(\delta\) is the unit impulse function.