## 2 Baseline-Free Damage Detection Using the Time Reversal Method of Lamb Wave  
The time reversal method originated from time reversal acoustics [23; 24]. Then the time reversal concept extended to Lamb wave propagation to reduce the effects of dispersion and to realize baseline-free damage detection [12; 13]. In accordance with the time reversal concept, an original input signal, \(s(t)\), can be reconstructed at the initial excitation position A, if the response signal received at position B is reversed in the time domain, and re-emitted from position B to position A. This process is referred to as the time reversibility of Lamb waves. Therefore, damages (e.g., fiber breakage and delamination, crack opening-and-closing) could be detected by comparing the waveform of the reconstructed signal with that of the original input signal [13]. The schematic diagram of the TRP is shown in Figure 1.  
The theoretical analysis of the TRP in the frequency domain is described as follows. Use piezoelectric (PZT) as transducer. When a voltage signal \(V_{A}(\omega)\) is excited at actuator PZT A, it then converts to mechanical strain \(\varepsilon_{A}(\omega)\) and activates a Lamb wave signal that propagates within the plate [25]. \[\varepsilon_{A}(\omega)=k_{a}(\omega)V_{A}(\omega), \tag{1}\]  
where, \(\omega\) is the angular frequency, and \(k_{a}(\omega)\) is the electro-mechanical coupling coefficient at PZT A. The response signal, \(V_{B}(\omega\),\(r)\), that captured by sensor PZT B satisfies,  
\[V_{B}(\omega,r)=k_{b}^{-1}(\omega)k_{a}(\omega)V_{A}(\omega)G(\omega,r), \tag{2}\]  
where, \(k_{b}(\omega)\) is the electro-mechanical coupling coefficient of PZT B, \(r\) is the propagation distance of Lamb waves from actuator to sensor, \(G(\omega\),\(r)\) is the transfer function of the sensing path. If the actuator and sensor are the same, \(k_{a}(\omega)=k_{b}(\omega)\)[16]. Thus, Equation (2) can be rewritten as,  
\[V_{B}(\omega,r)=V_{A}(\omega)G(\omega,r), \tag{3}\]  
The response signal inversed in the time domain is equivalent to take complex conjugate in the frequency domain. Thus, the time reverse operation of the response signal \(V_{B}(\omega\),\(r)\) represented in the frequency domain is,  
\[V_{B}^{*}(\omega,r)=V_{A}^{*}(\omega)G^{*}(\omega,r), \tag{4}\]  
where, the superscript \"*\" represents the complex conjugate operation. The reversed signal is then re-emitted at PZT B which now acts as an actuator. The corresponding response signal captured by PZT A is the reconstructed signal \(V_{R}(\omega\),\(r)\). It is defined as,  
\[V_{R}(\omega,r)=V_{B}^{*}(\omega)G(\omega,r), \tag{5}\]  
Figure 1: Schematic diagram of the time reversal process (TRP). PZT: Piezoelectric. Substituting Equation (4) into Equation (5), the reconstructed signal is rewritten as  
\[V_{R}(\omega,r)=V_{A}^{*}(\omega)|G(\omega,r)|^{2}, \tag{6}\]  
Here, \(\mid G(\omega,r)\mid^{2}\) is referred to as time reversal operator. It varies with frequency because the transfer function \(G(\omega,r)\) of Lamb waves propagating in composite laminates (including the effects of material damping) is frequency dependent. Therefore, the wave components at different frequencies will be non-uniformly attenuated and the initial input signal cannot be properly reconstructed. To alleviate this problem, narrowband signals are usually used as excitations [13; 19]. The TRM-based damage detection is accomplished by comparing the waveform of the reconstructed signal with that of the original input signal. A damage index (DI) is used to measure the discrepancy between the waveform of the original input signal and that of the reconstructed signal. It is defined as follows [14],  
\[DI=1-\int\limits_{t_{0}}^{t_{1}}V_{A}(t)V_{r}(t)\mathrm{dt}/\sqrt{\int\limits_ {t_{0}}^{t_{1}}V_{A}(t)^{2}\mathrm{dt}\int\limits_{t_{0}}^{t_{1}}V_{r}(t)^{2} \mathrm{dt}}\,, \tag{7}\]  
where, \(t_{0}\) and \(t_{1}\) define the time interval in which the two waveforms are compared, \(V_{A}(t)\) is the original input signal, \(V_{r}(t)\) is the reconstructed signal.