## 2 Multimode and Multipath Time Reversal in a Rod
The reconstructed signal in the time domain \(V_{R}(t)\) can be obtained by taking the inverse Fourier transform to \(V_{R}(\omega)\):  
\[V_{R}(t)=\sum_{p=1}^{n_{s}}\sum_{q=1}^{n_{s}}\frac{1}{2\pi}\int_{-\infty}^{+ \infty}\overline{A}_{pq}(\omega)K_{TR}(\omega)V_{A}^{*}(\omega)\exp\big{\{}i[ \overline{\theta}_{pq}(\omega)+\omega t]\big{\}}d\omega \tag{4}\]  
when an original input signal is a narrowband tone burst with a center frequency \(\omega_{c}\), then  
\[\overline{A}_{pq}(\omega)K_{TR}(\omega)\approx\overline{A}_{pq}(\omega_{c})K_ {TR}(\omega_{c}) \tag{5}\]  
when \(p=q\), \(V_{R}(t)\) in Equation (4) can be rewritten as follows:  
\[V_{R}(t)=\sum_{p=1}^{n_{s}}\overline{A}_{pp}(\omega_{c})K_{TR}(\omega_{c})V_{A }(T-t) \tag{6}\]  
reverse \(V_{R}(t)\) in Equation (6) in time domain, then the reconstructed input signal \(V_{AR}(t)\) is represented as follows:  
\[V_{AR}(t)=\sum_{p=1}^{n_{s}}\overline{A}_{pp}(\omega_{c})K_{TR}(\omega_{c})V_{ A}(t) \tag{7}\]  
when \(p\
eq q\), the \(V_{R}(t)\) in Equation (4) can be rewritten alternatively as follows:  
\[V_{R}(t)=\sum_{p=1}^{n_{s}}\sum_{q=1}^{n_{s}}\frac{1}{2\pi}\int_{-\infty}^{+ \infty}\overline{A}_{pq}(\omega_{c})K_{TR}(\omega_{c})V_{A}^{*}\exp\big{\{}i[ \omega t+\overline{\theta}_{pq}(\omega)]\big{\}}d\omega \tag{8}\]  
\(\overline{\theta}_{pq}(\omega)\) is expanded using a Taylor series near the center frequency \(\omega_{c}\) up to the first order term [21]  
\[G_{p}(\omega)=\prod_{i=1}^{k}A_{pi}(\omega)\exp[-ik_{pi}(\omega)l_{i}]= \overline{A}_{p}(\omega)\exp[-i\overline{k}_{p}(\omega)l] \tag{1}\]0  
By using the relationships among the equivalent wave number \(\overline{k}_{p}(\omega)\), the equivalent group velocity \(\overline{w}_{p}(\omega)\), and the equivalent phase velocity \(\overline{v}_{p}(\omega)\) of the \(p\)th propagating process, Equation (10) is expressed as follows:  
\[G_{p}(\omega)=\prod_{i=1}^{k}A_{pi}(\omega)\exp[-ik_{pi}(\omega)l_{i}]= \overline{A}_{p}(\omega)\exp[-i\overline{k}_{p}(\omega)l] \tag{1}\]1  
take Equation (10) into Equation (9), then  
\[G_{p}(\omega)=\prod_{i=1}^{k}A_{pi}(\omega)\exp[-ik_{pi}(\omega)l_{i}]= \overline{A}_{p}(\omega)\exp[-i\overline{k}_{p}(\omega)l] \tag{1}\]2  
where, \(\overline{v}_{pq}(\omega_{c})=\Big{\{}\Big{[}\frac{l}{\overline{v}_{p}(\omega _{c})}-\frac{l}{\overline{w}_{p}(\omega_{c})}\Big{]}-\Big{[}\frac{l}{\overline {v}_{q}(\omega_{c})}-\frac{l}{\overline{v}_{q}(\omega_{c})}\Big{]}\Big{\}}\), \(\overline{t}_{pq}(\omega_{c})=\Big{[}\frac{l}{\overline{w}_{p}(\omega_{c})}- \frac{l}{\overline{w}_{q}(\omega_{c})}\Big{]}\). Using Equation (11), Equation (8) is expressed as follows:  
\[G_{p}(\omega)=\prod_{i=1}^{k}A_{pi}(\omega)\exp[-ik_{pi}(\omega)l_{i}]= \overline{A}_{p}(\omega)\exp[-i\overline{k}_{p}(\omega)l] \tag{1}\]3  
then, the reconstructed input signal \(V_{AR}(t)\) is represented as follows:  
\[G_{p}(\omega)=\prod_{i=1}^{k}A_{pi}(\omega)\exp[-ik_{pi}(\omega)l_{i}]= \overline{A}_{p}(\omega)\exp[-i\overline{k}_{p}(\omega)l] \tag{1}\]4when \(p=q\), that means the propagating processes are identical in both forward and backward of the specified transmission route. The scaling factor \(\sum\limits_{p=1}^{n_{s}}\overline{A}_{pp}(\omega_{c})K_{TR}(\omega_{c})\) in Equation (7) is a real number, which indicates that multiple modes are fully compensated and converge to a reconstructed main band. Alternatively, when \(p\
eq q\), that means the propagating processes are different in both forward and backward of the specified transmission route. The scaling factor \(\overline{A}_{pq}(\omega_{c})K_{TR}(\omega_{c})\exp\bigl{[}i\omega_{c} \overline{\tau}_{pq}(\omega_{c})\bigr{]}\) in Equation (13) is no longer a real number because of the item \(\exp\bigl{[}i\omega_{c}\overline{\tau}_{pq}(\omega_{c})\bigr{]}\) representing phase shift. As a result, multiple modes are partially compensated and form side bands which are shifted from \(V_{A}(t)\) by \(-\overline{t}_{pq}(\omega_{c})\) in time domain.