## 2 Theoretical Analyses on the Torsional Wave-Damage Interaction
### Characterization of T(0,1) Mode Wave  
In light of the characteristics of the T(0,1) mode wave, it can be naturally characterized with a shaft torsional wave whose wave equation writes [34],  
\[\rho I_{p}\frac{\partial^{2}\varphi}{\partial^{2}t}=\frac{\partial}{\partial z }\left(GI_{p}\frac{\partial\varphi}{\partial z}\right) \tag{3}\]  
Figure 1: (**a**) Dispersion curves of a solid steel rod with a 3.5 mm radius; (**b**) the mode shape of the T(0,1) mode wave at 50 kHz.  
Equation (10), \(R_{1}\) should be equal to \(-R_{2}\). Therefore, \(R_{1}\) is replaced by \(R\) in the subsequent discussions.  
The reflected torsional wave from the defect at the receiving point is the result of the superposition of the waves associated with different reflection paths as sketched in Figure 3. For the first reflection path, the corresponding reflected torsional wave \(\varphi_{d1}\) is obtained as,  
\[\varphi_{d1}=R\hat{\varphi}_{i}e^{i(\theta_{0}-\omega t)} \tag{12}\]  
where \(\theta_{0}\) is the phase difference with respect to the incident wave. By synthesizing the wave transmission and reflection at the interfaces and wave propagation distances, the torsional wave associated with the \(N\)th reflection (\(N\geq 2\)) path writes,  
\[\varphi_{dN}=-S_{1}R^{2N-3}S_{2}\hat{\varphi}_{i}e^{i(\theta_{0}+(2N-2)kw- \omega t)}=-S_{1}R^{2N-3}S_{2}e^{i(\frac{(N-4)\pi w}{\delta})}\hat{\varphi}_{i }e^{i(\theta_{0}-\omega t)} \tag{13}\]  
where \(\delta\) is the wavelength of the torsional wave in the damaged segment. As a result, the reflected torsional waves from the damage \(\varphi_{d}\) is calculated as,  
\[\varphi_{d}=\sum_{N=1}^{\infty}\varphi_{dN}=\left(1-S_{1}S_{2}\left(\lim_{N \rightarrow\infty}e^{i(\frac{4\pi w}{\delta})}\frac{1-R^{2N}e^{i(\frac{4N\pi w }{\delta})}}{1-R^{2}e^{i(\frac{4\pi w}{\delta})}}\right)\right)\,R\hat{ \varphi}_{i}e^{i(\theta_{0}-\omega t)} \tag{14}\]  
According to Equation (10), the absolute value of the reflection coefficient \(R\) should be smaller than \(1\) so that Equation (14) further reduces to,  
\[\varphi_{d}=\left(1-S_{1}S_{2}\frac{e^{i(\frac{4\pi w}{\delta})}}{1-R^{2}e^{i (\frac{4\pi w}{\delta})}}\right)\,R\hat{\varphi}_{i}e^{i(\theta_{0}-\omega t )}=\hat{\varphi}_{d}e^{i(\theta_{0}-\omega t)} \tag{15}\]  
Figure 3: Sketch of the torsional wave reflections in a damaged cylindrical structure.  
Following the same procedure, the reflected wave at the receiving point from the end \(\varphi_{e}\) is calculated as,  
\[\begin{array}{c}\varphi_{e}=\sum\limits_{N=1}^{\infty}\varphi_{eN}=\bigg{(}1+ \bigg{(}\lim\limits_{N\to\infty}R^{2}e^{i(\frac{4\pi\omega}{2\omega})}\frac{1-R^ {2N}e^{i(\frac{4\pi\omega}{2\omega})}}{1-R^{2}e^{i(\frac{4\pi\omega}{2\omega})} }\bigg{)}\bigg{)}S_{1}{}^{2}S_{2}{}^{2}\hat{\varphi}_{i}e^{i(\theta_{1}- \omega t)}\\ \hskip 14.226378 pt=\bigg{(}1+\frac{R^{2}e^{i(\frac{4\pi\omega}{2\omega})}}{1-R^ {2}e^{i(\frac{4\pi\omega}{2\omega})}}\bigg{)}S_{1}{}^{2}S_{2}{}^{2}\hat{\varphi }_{i}e^{i(\theta_{1}-\omega t)}=\hat{\varphi}_{e}e^{i(\theta_{1}-\omega t)}\end{array} \tag{16}\]  
where \(\theta_{1}\) is also a certain phase difference with respect to the incident wave. Finally, the reflection coefficients induced by the defect \(\varnothing_{d}\) and end \(\varnothing_{e}\) are, respectively, defined as,  
\[\varnothing_{d}=\frac{\hat{\varphi}_{d}}{\hat{\varphi}_{i}},\;\varnothing_{e} =\frac{\hat{\varphi}_{e}}{\hat{\varphi}_{i}} \tag{17}\]  
So far, the reflections from the defect and end in a damaged solid rod have been ascertained. Since the governing equations of the torsional waves in pipes are identical to those in solid rods, the derived reflection coefficients should also be applicable for pipes. In addition, as the cross-section of the damage zone is characterized with the polar moment of inertia in the derivations, the above theoretical analyses, in principle, allow assessing the reflections from defects with complex cross-sections.