## 3 Principles of Image Method  
Based on Equation (14), it is known that for low-frequency narrowband Lamb wave signal propagation, the dominant mode of propagation is the \(A_{0}\) mode. The transfer function can be divided into two components: the magnitude component \(\pi A(\omega_{c})\) and the phase component \(e^{-jk_{A0}x}\). In the principle of time reversal, these two components exert distinct influences on signal focalization. When the time-reversed signal propagates back to the original source location, Equations (19) and (20) indicate that the signals undergo constructive interference. During signal focalization, the phase component of the transfer function is compensated. The magnitude component of the transfer function represents the magnitude of the reconstructed signal waveform after focalization, and the phase component plays a crucial role in determining whether the signal can achieve focalization after time reversal. Therefore, phase modulation can be applied to Lamb wave response signals to achieve a similar effect of time-reversed focalization, thus enabling localization. The key to phase modulation lies in acquiring the propagation velocity of Lamb waves in the structure, which is utilized to modulate the phase of the signal. This eliminates the need to acquire the transfer function along the propagation path, thereby reducing the demand for experimental or modeled transfer function determination and facilitating the implementation of time reversal in software. As depicted in Figure 4, \(n\) piezoelectric sensors are arranged on the structure. At position \(s(x,y)\), there exists an unknown sound source. The response signals that can be received by each piezoelectric sensor are extracted using the Shannon continuous complex wavelet transform to obtain narrowband signals with the desired center frequency. Let \(E_{i}(\omega)\) denote the frequency response of the narrowband signal, with a center frequency of \(\omega_{c}\). The distances from position \(s(x,y)\) to each piezoelectric sensor are denoted as \(r_{1}\), \(r_{2}\), \(\ldots\), \(r_{n}\). The phase of the narrowband signal \(E_{i}(\omega)\) is delayed by \(e^{jk_{A0}(-r_{i})}\) and then synthesized. The synthesized signal is expressed as follows:  
\[V_{S}=\underset{i=1}{\overset{n}{\sum}}E_{i}(\omega)^{*}e^{-jk_{A_{0}}(\omega )r_{i}} \tag{30}\]  
By performing the inverse Fourier transform on the synthesized signal \(V_{S}\) and taking its modulus, the following can be obtained:  
\[|v_{s}(t,x,y)|=\left|\frac{1}{2\pi}\underset{i=1}{\overset{n}{\sum}}\underset {-\infty}{\overset{n}{\sum}}E_{i}(\omega)^{*}e^{-jk_{\Phi_{0}}(\omega)r_{i}}e ^{j\omega t}d\omega\right| \tag{31}\]  
In this context, \(v_{s}(t,x,y)\) signifies the phase-synthesized signal, \(e^{-jk_{\Phi_{0}}(\omega)r_{i}}=e^{-jr_{i}\omega/c_{pA0}}\), and \(c_{pA0}\) denotes the phase velocity of Lamb wave propagation. According to the dispersion characteristics of narrowband Lamb waves, for the narrowband signal \(E_{i}(\omega)\), Equation (31) can be expressed as follows [18]:  
\[ \begin{split}&|v_{s}(t,x,y)|=|\frac{1}{2\pi}\sum\limits_{i=1}^{n} \int_{-\infty}^{+\infty}\left(\sum\limits_{\omega_{0}\in[\omega_{l},\omega_{h} ]}E_{i}(\omega_{0})^{*}e^{-j\mu_{i}\omega_{0}/c_{\mu_{0}\
u_{0}\
u_{0}}}\right) e^{i\omega t}d\omega|\\ &=\frac{1}{2\pi}|\sum\limits_{i=1}^{n}\left[\sum\limits_{\omega_ {0}\in[\omega_{l},\omega_{h}]}e_{i}^{\prime}\big{(}\tau-t-r_{i}/c_{\mu_{0}\
u_ {0}A_{0}}\big{)}\right]|\end{split}  \tag{32}\]  
In this context, \(E_{i}(\omega_{0})\) represents the frequency component of \(E_{i}(\omega)\) at frequency \(\omega_{0}\), and \(e_{i}^{\prime}\) denotes the temporal representation of \(E_{i}(\omega_{0})\). \(c_{\mu_{0}\
u_{0}A_{0}}\) is the phase velocity corresponding to the frequency signal with a frequency of \(\omega_{0}\). The center frequency of the narrowband signal \(E_{i}(\omega)\) is \(\omega_{c}\), with a frequency range of \([\omega_{l},\omega_{h}]\).