## 4 Discussion
### Comparison between the Spectral Method Algorithm and the COMSOL Simulation  
The parameters of the water-filled PMMA tube adopted by [PERSON] [26] are shown in Table 1, where \(b\) is the inner radius of the tube, \(h\) is the wall thickness, \(v_{p}\), \(v_{s}\) are the longitudinal and shear sound speed of PMMA, respectively, \(c_{1}\) is the sound speed in water, \(\rho_{s}\) is the density of PMMA, and \(\rho_{1}\) is the density of water. The two-dimensional model of the water-filled PMMA tube with the above parameters was established in COMSOL, as shown in Figure 4. \begin{table}
\begin{tabular}{c c c c c c c} \hline \(b\) (cm) & \(h\) (cm) & \(v_{p}\) (m/s) & \(v_{s}\) (m/s) & \(c_{1}\) (m/s) & \(\rho_{s}\)(kg/m\({}^{3}\)) & \(\rho_{1}\) (kg/m\({}^{3}\)) \\ \hline
4.445 & 0.5 & 2690 & 1340 & 1479 & 1190 & 1000 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of the water-filled PMMA tube [26]. Figure 3: The flowchart of the proposed spectral method algorithm. The inner part was water, and its physical field was set to be a \"Pressure Acoustics\" field. The wave equation for time-harmonic acoustic waves in 2D geometry is Equation (72),  
\[\
abla\cdot\left(-\frac{1}{\rho_{c}}\Big{(}\
abla p_{t}-\overset{\rightarrow}{q} _{d}\Big{)}\right)-\frac{k_{eq}^{2}p_{t}}{\rho_{c}}=Q_{m} \tag{72}\]  
\[p=p(x,y)e^{-ik_{z}z} \tag{73}\]  
Where \(p_{t}\) means total pressure, which is the sum of \(p\) and the possible background pressure field \(p_{b}\), as shown in Equation (74):  
\[p_{t}=p+p_{b} \tag{74}\]  
\(k_{eq}^{2}\) is shown as Equation (75), where \(k_{z}\) is the out-of-plane wave number:  
\[k_{eq}^{2}=\left(\frac{\omega}{c_{c}}\right)^{2}-k_{z}^{2} \tag{75}\]  
The subscript \"\(c\)\" in \(\rho_{c}\) and \(c_{c}\) means that they may be complex-valued. Lossy media, like porous materials or highly viscous fluids, can be modeled by using the complex-valued speed of sound and density. In Equation (72), \(\overset{\rightarrow}{q}_{d}\) is the dipole domain source, which corresponds to a domain force source on the right-hand side of the momentum equation, and \(Q_{m}\) is the monopole domain source, which corresponds to a mass source on the right-hand side of the continuity equation. In the mode analysis study of COMSOL, \(-ik_{z}\) is used as the eigenvalue. The outer part is the PMMA tube wall, and its physical field, is set to be a \"Solid Mechanics\" field. The governing equation used here is  
\[-\rho\omega^{2}\overset{\rightarrow}{u}=\
abla\cdot\left(C:\
abla\cdot \overset{\rightarrow}{u}\right) \tag{76}\]  
In Equation (76), \(\overset{\rightarrow}{u}\) is the structural displacement vector, and \(C\) is the elasticity tensor. The outer tube wall is the free boundary, which satisfies the stress-free boundary condition, and the inner tube wall is set to be the \"Acoustic-Structure Boundary\", which is used to couple a Pressure Acoustics model to any structural component. The coupling includes the fluid load on the structure and the structural acceleration as experienced by the fluid. The condition on exterior boundaries is shown as Equations (77) and (78):  
\[-\overset{\rightarrow}{n}\cdot\left(-\frac{1}{\rho_{c}}\Big{(}\
abla p_{t}- \overset{\rightarrow}{q}_{d}\Big{)}\right)=-\overset{\rightarrow}{n}\cdot u _{tt} \tag{77}\]  
\[F_{A}=p_{t}\overset{\rightarrow}{n} \tag{78}\]  
where \(u_{tt}\) is the structural acceleration, \(\overset{\rightarrow}{n}\) is the surface normal, \(p_{t}\) is the total acoustic pressure and \(F_{A}\) is the load (force per unit area) experienced by the structure. Figure 4: A two-dimensional model of the water-filled PMMA tube in COMSOL. The condition on interior boundaries is shown as follows:  
Figure 3:0  
Figure 3:1  
Figure 3:2  
such that the acoustic load is given by the pressure drop across the thin structure. The up and down subscripts refer to the two sides of the interior boundary.