## 2 Multimode and Multipath Time Reversal in a Rod  
Supposing that a rod is separated into parts by defects, mode reflection or conversion occurs at boundaries or structural defects, and so guided waves will transmit through various routes in different modes from an actuator to a sensor, which results in multimode and multipath natures of guided waves propagation in a rod. As to a specified transmission route from an actuator to a sensor, the response signal is a composition of modes with their individual propagating process defined specially for the aims of this study. The explanation to the multiple propagating ways is schematically shown in Figure 1. For brevity, assume that a rod is divided into two transmission parts by one defect, and the direct path among multiple transmission routes is the only one considered here. Mode 1 is excited by transducer A and then propagates right through the Part 1. Supposing that the energy of Mode 1 is separated into sections at the defect, one will still transmit through the Part 2 as Mode 1, but another one converts to Mode 2 and propagates along the remaining path. Therefore, the response signals captured by transducer B on the other side are composed of Mode 1 and Mode 2 which derive from the identical direct path but with different propagating processes. In general, as to a rod with multiple defects, the transfer function \(G_{p}(\omega)\) of the \(p\)th propagating process in a specified transmission route can be expressed in the frequency domain as follows:  
\[G_{p}(\omega)=\prod_{i=1}^{k}A_{pi}(\omega)\exp[-ik_{pi}(\omega)l_{i}]= \overline{A}_{p}(\omega)\exp[-i\overline{k}_{p}(\omega)l] \tag{1}\]  
where, \(\omega\) is the angular frequency, \(k\) is the number of disturbance due to discontinuities (including defects and boundaries), \(A_{pi}(\omega)\) is the amplitude dispersion function after the \(i\)th disturbance, \(k_{pi}(\omega)\) and \(l_{i}\) denote the wave number of the mode after the \(i\)th disturbance and length of transmission part between the \(i\)th and the \((i+1)\)th disturbance respectively. For simplified expression, \(l\) is the total length of the specified transmission route, \(\overline{A}_{p}(\omega)\) and \(\overline{k}_{p}(\omega)\) indicate the equivalent amplitude dispersion function and the equivalent wave number of the \(p\)th propagating process respectively. When an excitation signal \(V_{A}(\omega)\) is applied to the actuator, the response signal \(V_{S}(\omega)\) at the sensor can be represented in the frequency domain as follows:  
\[V_{S}(\omega)=\sum_{p=1}^{n_{s}}V_{A}(\omega)G_{p}(\omega)K_{A}(\omega)K_{S}(\omega) \tag{2}\]  
\(K_{A}(\omega)\) and \(K_{S}(\omega)\) are the electro-mechanical efficiency coefficients of the actuator and the sensor respectively, and \(n_{s}\) can be defined as the total number of propagating processes in the specified transmission route. We truncate \(V_{S}(\omega)\) at time \(t=T\), reverse the acquired signal in the time domain, and then reload it to the sensor rather than the actuator. Noticing that reversion in time domain is equivalent to taking a complex conjugate of the signal in frequency domain, the reconstructed signal \(V_{R}(\omega)\) captured by the transducer which is acted previously as an actuator is shown below:  
\[V_{R}(\omega)=\sum_{q=1}^{n_{s}}V_{S}^{*}(\omega)G_{q}(\omega)K_{A}(\omega)K_ {S}(\omega)=\sum_{p=1}^{n_{s}}\sum_{q=1}^{n_{s}}\overline{A}_{pq}(\omega)\exp [i\theta_{pq}(\omega)]K_{TR}(\omega)V_{A}^{*}(\omega) \tag{3}\]  
Figure 1: Illustration of two propagating ways in the direct transmission route. where, \(\overline{A}_{pq}(\omega)=\overline{A}_{p}^{*}(\omega)\overline{A}_{q}(\omega),\)\(K_{TR}(\omega)\)\(=\)\(K_{A}^{*}(\omega)K_{S}^{*}(\omega)K_{A}(\omega)K_{S}(\omega)\) and \(\overline{\theta}_{pq}(\omega)=l[\overline{k}_{p}(\omega)-\overline{k}_{q}( \omega)]\).