## 2 The Model for Propagation
### Determination of the Optimal Position of the Sensor
\[\theta_{i}=\theta_{r} \tag{8}\]  
Figure 3: The propagation model for the cylindrical shell with the stiffener: (**a**) several reflection paths; (**b**) several transmission paths. Hence, for the propagation paths \(\rm{r_{5}}\), \(\rm{r_{6}}\), \(\rm{t_{2}}\), and \(\rm{t_{5}}\), the signal is reflected vertically at the edge. Then, ES and R should satisfy following equations:  
\[\left\{ \begin{array}{c}T(\rm{D_{r}})=T(\rm{s_{1}})=l(\rm{s_{1}})/\upsilon_{g} (\rm{L}(0,1))\\ T(\rm{RS})=T(\rm{s_{2}})=l(\rm{s_{2}})/\upsilon_{g}(\rm{L}(0,1))\\ T(circle_{1})=T(\rm{s_{3}})=T(\rm{s_{4}})=l(\rm{t_{3}})/\upsilon_{g}(\rm{F}(1,1 ))\\ T(\rm{RE_{1}})=T(\rm{s_{5}})=l(\rm{s_{5}})/\upsilon_{g}(\rm{L}(0,1))\\ T(\rm{RE_{2}})=T(\rm{s_{6}})=l(\rm{s_{6}})/\upsilon_{g}(\rm{L}(0,1))\\ \end{array} \right. \tag{9}\]  
where \(T(\rm{D_{r}})\) is the time as the direct signal propagating acquired in R; \(T(\rm{RS})\) is the propagating time of the signal reflected from the stiffer acquired in R; \(T(circle_{1})\), \(T(\rm{RE_{1}})\), and \(T(\rm{RE_{2}})\) are the propagating times acquired in R of the signal propagating around the circumference and signals reflected from edges in the cylindrical shell; \(L(\rm{s_{i}})\) is the length of the path \(\rm{s_{j}}\), where \(\rm{j}\) = 1, 2, 3, 4, 5, 6; and \(\upsilon_{g}(\rm{L}(0,1))\) and \(\upsilon_{g}(\rm{F}(1,1))\) are the group velocities of L(0,1) mode and F(1,1) mode in the cylindrical shell, respectively. The parameters in Figure 3 are obtained by the following:  
\[\left\{ \begin{array}{c}l(\rm{s_{1}})=d_{1}\\ l(\rm{s_{2}})=d_{1}+2d_{2}\\ l(\rm{s_{3}})=l(\rm{s_{4}})=\sqrt{\left(d_{1}\right)^{2}+\left(2\pi a\right)^{ 2}}\\ l(\rm{s_{5}})=2d_{4}+d_{1}\\ l(\rm{s_{6}})=d_{1}+2d_{2}+2d_{3}+2d_{5}\end{array} \right. \tag{10}\]  
The constraint conditions in the time domain for separating the reflected signal from the direct signal and other waves are as follows:  
\[\left\{ \begin{array}{c}\Delta T_{1}=T(\rm{RS})-T(\rm{D_{r}})\geq B/f_{c}\\ \Delta T_{2}=T(circle_{1})-T(\rm{RS})\geq B/f_{c}\\ \Delta T_{3}=T(\rm{RE_{1}})-T(\rm{RS})\geq B/f_{c}\\ \Delta T_{4}=T(\rm{RE_{2}})-T(\rm{RS})\geq B/f_{c}\end{array} \right. \tag{11}\]  
Substituting Equations (9) and (10) into Equation (11), the specific separation conditions for the reflection are obtained:  
\[\left\{\begin{array}{c}d_{2}\geq\frac{B\times\upsilon_{g}(\rm{L}(0,1))}{2f_{ c}}\\ \frac{\sqrt{\left(d_{1}\right)^{2}+\left(2\pi a\right)^{2}}}{\upsilon_{g}(\rm{F}( 1,1))}-\frac{d_{3}+2d_{3}}{\upsilon_{g}(\rm{L}(0,1))}\geq\frac{B}{f_{c}}\\ d_{4}\geq\frac{B\times\upsilon_{g}(\rm{L}(0,1))}{2f_{c}}+d_{The specific separation conditions for the transmission can be inferred as follows:  
\[\left\{ \begin{array}{l}\Delta T_{\rm{5}}=T(circle_{2})-T^{\prime}(\rm{D}_{t}) \geq B/f_{\rm{c}}\\ \Delta T_{\rm{6}}=T^{\prime}(\rm{RE}_{1})-T(\rm{D}_{t})\geq B/f_{\rm{c}}\\ \Delta T_{\rm{7}}=T^{\prime}(\rm{RE}_{2})-T(\rm{D}_{t})\geq B/f_{\rm{c}}\\ \end{array} \right. \tag{15}\]  
Substituting Equations (13) and (14) into Equation (15), a system of equations that enable us to separate the transmission signals from other signals is obtained. \[\left\{ \begin{array}{l}\frac{\sqrt{(d_{1}+d_{2}+d_{3})^{2}+(2\pi a)^{2}}}{v_ {g}(\rm{F}(\rm{I},1))}-\frac{d_{1}+d_{2}+d_{3}}{v_{g}(\rm{I}(\rm{I},0))}\geq \frac{B}{f_{\rm{c}}}\\ d_{5}\geq\frac{B\times v_{g}(\rm{I}(\rm{O},1))}{2f_{\rm{c}}}\\ d_{4}\geq\frac{B\times v_{g}(\rm{I}(\rm{O},1))}{2f_{\rm{c}}}\\ \end{array} \right. \tag{16}\]