## 2 Principle of Time-Reverse Algorithms
For single-frequency or narrowband Lamb waves, Equation (12) can be expressed as follows:  
\[k_{S_{0}}=\frac{\omega_{0}}{V_{p\omega_{0}S_{0}}} \tag{2}\]0  
At lower frequencies of Lamb waves, when the \(A_{0}\) mode predominates in propagation, Equation (13) can be approximately simplified as follows:  
\[k_{S_{0}}=\frac{\omega_{0}}{V_{p\omega_{0}S_{0}}} \tag{2}\]1  
Equation (14) represents the simplified expression of the frequency response function of the transfer function for the single-frequency or low-frequency narrowband Lamb waves propagating in a plate structure in the \(A_{0}\) mode. As depicted in Figure 1, with \(n+1\) piezoelectric sensors affixed to the plate structure, assuming the frequency response of the transfer function from the source location \(0\) to the \(i\)th piezoelectric sensor for propagating Lamb waves is denoted as \(H_{0i}\), and the excitation signal emitted by the source is represented as \(E_{0}\), the response signal received by each piezoelectric sensor is expressed as follows:  
\[k_{S_{0}}=\frac{\omega_{0}}{V_{p\omega_{0}S_{0}}} \tag{2}\]2  
By time-reversing the response signals received at each receiving end (by taking the complex conjugate in the frequency domain) and subsequently using the corresponding receiving end as the excitation point, each piezoelectric sensor re-excites the time-reversed response signal of the Lamb wave it originally received back into the structure. Consequently, the signal received at the original source's piezoelectric sensor is as follows:  
\[k_{S_{0}}=\frac{\omega_{0}}{V_{p\omega_{0}S_{0}}} \tag{2}\]3  
\[k_{S_{0}}=\frac{\omega_{0}}{V_{p\omega_{0}S_{0}}} \tag{2}\]4 Schematic diagram of single-source time-reversed focusing method. where \(E_{i}^{*}\) represents the conjugate form of \(E_{0}\), indicating the time-reversed signal of the source signal, and \(H_{i0}\) represents the conjugate form of \(H_{0i}\). According to the reciprocity principle of acoustic wave propagation, exchanging the positions of the excitation end and the receiving end results in the transfer function between the sensors and the source being independent of the propagation direction and remaining unchanged, denoted as \(H_{i0}=H_{0i}\). Therefore, Equation (16) can be expressed as follows:  
\[k_{S_{0}}=\frac{\omega_{0}}{V_{p\omega_{0}S_{0}}} \tag{2}\]5  
\(\sum\limits_{i=1}^{n}H_{i0}H_{i0}^{*}\) in Equation (17) can be transformed into matrix form:  
\[k_{S_{0}}=\frac{\omega_{0}}{V_{p\omega_{0}S_{0}}} \tag{2}\]6  
Among these terms, \(H\) represents the transmission function matrix of signal communication, \(\left|H\right|^{2}\) denotes the proportion of the propagation function of the sound wave in the medium, which is the accumulation of the common function, and its value must be a positive real function. According to Equation (18), if the sound source incentive signal is a low-frequency narrowband Lamb wave signal, the term \(\left|H\right|^{2}\) can be expressed by \(\pi^{2}|A(\omega)|^{2}\). Equation (18) can be expressed as follows:  
\[k_{S_{0}}=\frac{\omega_{0}}{V_{p\omega_{0}S_{0}}} \tag{2}\]7  
Taking the Fourier transform of Equation (19) yields the following:  
\[k_{S_{0}}=\frac{\omega_{0}}{V_{p\omega_{0}S_{0}}} \tag{2}\]8  
It can be observed that the Lamb wave signal excited from the source is re-excited to the corresponding piezoelectric sensors after time reversal. Moreover, the focused signal \(e_{0}^{\prime}(t)\) at the original source is amplified by a certain factor compared to the source excitation signal \(e_{0}(t)\). This process represents signal focusing and amplification. Equations (19) and (20) are derived under the assumption of low-frequency narrowband Lamb wave excitation, where the \(A_{0}\) mode predominates as the dominant propagating mode in the structure. They are not applicable to structures where the dominant mode of propagation is the \(S_{0}\) mode or for wideband Lamb waves. When the Lamb wave excited in the structure is not primarily the \(A_{0}\) mode, the \(\left|H\right|^{2}\) term can be expressed using Equation (21).