## 5 Results, Analysis, and Discussion
### Reflection and Transmission Coefficients  
Simulations have been performed for each case with different excitation frequencies (\(f_{\rm c}\)) and different stiffener heights (\(h\)), and the scattering coefficient in each case can be computed using Equations (3) and (5). The obtained scattering coefficients of the incident L(0,1) mode are reported in Figure 13 for different frequencies and stiffener heights, which are calculated by the scattering coefficients presented in Section 2.2. It can be observed that the reflection and transmission coefficients are displayed as a function of the excitation frequency and the height of the stiffener. Figure 14 extracts the scattering coefficients calculated at different frequencies, including 110 kHz, 130 kHz, 150 kHz, and 170 kHz, to further study the underlying characteristics. From Figure 14, it should be noted that the reflection coefficient shows a special trend with respect to \(f_{\rm c}\) and \(h\), while the trend of the transmission coefficient is roughly the opposite. The reflection coefficient for the incident A\({}_{0}\) mode reaches the maximum value of around 0.4 when h is about 5 mm, while the transmission coefficient acquires the minimum value of around 0.2 for the same h. Regarding the stiffener with larger heights, the effect of stiffeners on the coefficients decreases gradually, and both coefficien  
Figure 12: The mechanism of the L(0,1) mode propagates in the cylindrical shell with the stiffener. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Symbol** & **Description** & **Tc/μs** & **Ts/μs** & **\(\Delta\)T=1**T**s\(-\)**Tc**/μs** \\ \hline \(t_{\rm L(0,1)}\) & Incident L(0,1) mode & 20.0 & 19.8 & 0.2 \\ \(t_{\rm L(0,2)}\) & Reflected L(0,2) mode & 48.6 & 44.2 & 4.4 \\ \(t_{\rm L(0,1)}\) & Reflected L(0,1) mode & 58.6 & 56.2 & 2.4 \\ \(t_{\rm L(0,2)}\) & Transmitted L(0,2) mode & 57.9 & 57.2 & 0.7 \\ \(t_{\rm L(0,1)}\) & Transmitted L(0,1) mode & 77.9 & 72.4 & 5.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The comparison about the arrival time of each wave with \(f_{\rm c}\) = 150 kHz and \(h\) = 32 mm. Moreover, the reflection coefficient and the transmission coefficient can be considered in three portions in terms of the rate of change versus \(h\), namely the sensitive part, the fluctuation part, and the stable part, as shown in Figure 14. The corresponding regions are region I, region II, and region III. It is a noticeable phenomenon that the reflection coefficient and the transmission coefficient in region I present linear variations with the increase in \(h\). The difference is that the reflection coefficient increases monotonously to the maximum as \(h\) increases, but the transmission coefficient, which monotonously reduces to the minimum value, shows the opposite in this region. With the increase in \(h\) in region II, the reflection coefficients fluctuate and show a downward trend. Conversely, the transmission coefficient fluctuates oppositely but has a slight upward trend, and the rate of change is relatively high. This intuitively indicates that both the reflection coefficient and the transmission coefficient change with respect to \(h\) are nonlinear. Both the reflection coefficient and the transmission coefficient remain basically unchanged in region III. It can be reasonably deduced that the reflection coefficient and transmission coefficient are not very sensitive to the height of the stiffener increasing after \(h\) exceeds a certain value. Additionally, for different \(f_{\rm c}\), the rate of variation of the reflection coefficient and transmission coefficient versus \(h\) differs a little. If the frequency increases, critical points between the three parts shift to smaller \(h\) regions, that is, the sensitivity to the presence of \(h\) increases. Maximum values of the reflection coefficient under different \(f_{\rm c}\) all appear at smaller \(h\) regions (around \(h\) = 5), but minimum transmission coefficients appear at the same \(h\).