## 2 The Model for Propagation
### Determination of the Optimal Position of the Sensor  
A key element of the excitation signal is the selection and exploitation of a single mode. In this study, the single frequency of the AE excitation signal is generated by applying a modulated sinusoidal function, which could reduce the bandwidth for the signal to minimize the dispersion. The excitation signal is exerted to the cylindrical shell, and the AE signal mainly in L(0,1) mode is obtained. The excitation equation \(Y(t)\) is written as follows:  
\[Y(t)=X(t)\cdot H(t)=\sin(2\pi f_{c}t)\times\left[0.5-0.5\times\cos(2\pi f_{c} t/B)\right] \tag{6}\]  
Figure 2: Diagram of the signal propagating in a stiffened cylindrical shell. where \(H(t)\) is the window function; \(f_{c}\) is the central frequency for the exciting signal; \(B\) refers to the amount of peak, here taking 3; and \(X(t)\) represents the carrier signal. The signal then spreads out in all directions based on the composite feature of the structure. The original signal and the signal scattered from edges and characteristic structures arrive at the sensor in order and overlap with each other with their respective phases. To distinguish the main echoes in the stiffened cylindrical shell in the time domain and to highlight the effect of the stiffener on the AE signal, the multipath propagation model for AE signals propagating in the stiffened cylindrical shell is established in this paper, as shown in Figure 3.  
Figure 3 illustrates multiple simplified shortest signal propagation paths, which include a shortest direct path (s\({}_{1}\) and g\({}_{1}\)) and several other paths (s\({}_{2}\), s\({}_{3}\), s\({}_{4}\), s\({}_{5}\), s\({}_{6}\), g\({}_{2}\), g\({}_{3}\), g\({}_{4}\), and g\({}_{5}\)). The direct path refers only the segment between the excitation point (EP) and measurement points for reflection (R) and transmission (T). Other paths not only include indirect paths involving the first segmentation between EP and structural features (such as the edge, stiffener, etc.) and the second segmentation between structural features and R or T but also involves paths where signals travel from EP to R or T after propagating around the circumference. Therefore, the relative positions of EP, R, T, and the stiffener are very important for distinguishing signals of interest. Calculating the arrival time of the mentioned wave in conformity with the corresponding propagation path can derive the installation constraints about ES, R, and T to separate the signal of interest from other unwanted signals in the time domain. In order to maximize the value of the proposed model, not only should EP, R, and T be co-linear but also the connection line should be perpendicular to the plane where the stiffener is located. As the AE signal is applied at EP in Figure 3, the excited signal propagates towards different directions guided by the features of the structure. The propagation directions of the signals reflected from edges of the cylindrical shell are dominated by [PERSON]'s law [31]:  
\[k_{i}\cdot\sin(\theta_{i})=k_{r}\cdot\sin(\theta_{r}) \tag{7}\]  
where \(k_{i}\) and \(\theta_{i}\) are the wavenumber and angle of the incident wave, respectively, and \(k_{r}\) and \(\theta_{r}\) represent the wavenumber and the angle of the reflected wave. Regarding the propagation paths such as r\({}_{5}\), r\({}_{6}\), t\({}_{2}\), and t\({}_{5}\), the signal is incident vertical to the edge of the cylindrical shell. In this situation, it is assumed that, for the edge of the plate, the incident A\({}_{0}\) mode reflects A\({}_{0}\) only and that the S\({}_{0}\) mode reflects S\({}_{0}\). Similarly, it can also be assumed that, at the edge of the cylindrical shell, the incident L(0,1) mode is reflected as only L(0,1), while the L(0,2) mode is reflected as only L(0,2). Due to no mode conversion occurring, the propagation velocity and the wave number of each mode remain the same as before. Therefore, the angle of incidence and that of reflection are also unchanged.