## 2 Theory  
### Wave Equations  
In this section, using linear elastic theory, the governing wave propagation equations in the elastic tube wall and the fluid are derived, respectively. The corresponding boundary conditions at different interfaces are given. Figure 1 shows an infinite, homogeneous, and isotropic fluid-filled elastic tube model. The inner radius of the tube is \(r_{1}\), the outer radius is \(r_{2}\), and the cylindrical coordinate system \((r,\theta,z)\) is established with the axis of the cylindrical tube as the z-axis. Waves propagating along the axial direction in a fluid-filled elastic tube can be divided into  
longitudinal waves and torsional waves, both of which have axisymmetric modes and non-axisymmetric modes [27]. In some references, longitudinal and torsional waves are defined as waves with only axisymmetric modes, and waves with non-axisymmetric modes are collectively called flexural waves. In this paper, the positive integer \(n\) is used to represent the circumferential order of the mode, and the axisymmetric mode satisfies \(n=0\). For the elastic tube model in Figure 1, assuming that the outer surface satisfies the stress-free boundary condition, the wave equation governing the propagation of elastic waves [27] is  
\[\mu\
abla^{2}\overset{\rightarrow}{U}+(\lambda+\mu)\
abla\
abla\cdot\overset{ \rightarrow}{U}=\rho\frac{\partial^{2}\overset{\rightarrow}{U}}{\partial t^ {2}} \tag{1}\]  
where \(\overset{\rightarrow}{U}\) represents the displacement field, which is a function of the three cylindrical coordinates and time. The bulk wave velocities in the elastic material are determined by the density \(\rho\) and the Lame constants \(\lambda\) and \(\mu\). If \(c_{1}\) represents the dilatational longitudinal bulk wave velocity and \(c_{2}\) is the shear bulk wave velocity, then  
\[c_{1}=\sqrt{\frac{\lambda+2\mu}{\rho}} \tag{2}\]  
\[c_{2}=\sqrt{\frac{\mu}{\rho}} \tag{3}\]  
The displacement field \(\overset{\rightarrow}{U}\) can be described using the dilatational scalar potential \(\Phi\) and the equivoluminal vector potential \(\overset{\rightarrow}{\Psi}\):  
\[\overset{\rightarrow}{U}=\
abla\Phi+\
abla\times\overset{\rightarrow}{\Psi} \tag{4}\]  
Substituting Equation (4) into Equation (1) gives  
\[\
abla^{2}\Phi=\frac{1}{c_{1}^{2}}\frac{\partial^{2}\Phi}{\partial t^{2}} \tag{5}\]  
\[\
abla^{2}\overset{\rightarrow}{\Psi}=\frac{1}{c_{2}^{2}}\frac{\partial^{2} \overset{\rightarrow}{\Psi}}{\partial t^{2}} \tag{6}\]  
Using elasticity theory, the potentials \(\Phi\) and \(\overset{\rightarrow}{\Psi}\) have the following form in cylindrical coordinates:  
\[\
abla^{2}\Phi=\bigg{(}\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{ \partial}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}+ \frac{\partial^{2}}{\partial z^{2}}\bigg{)}\Phi \tag{7}\]  
Figure 1: The infinitely long, uniform, isotropic fluid-filled elastic tube model. \[ \begin{array}{rl}\
abla^{2}\overrightarrow{\Psi}&=\
abla^{2}\Big{(}\varphi_{r} \overrightarrow{e}_{r}+\varphi_{\theta}\overrightarrow{e}_{\theta}+\varphi_{ \theta}\overrightarrow{e}_{z}\Big{)}\\ &=\Big{(}\
abla^{2}\varphi_{r}-\frac{1}{r^{2}}\varphi_{r}-\frac{2}{r^{2}} \frac{\partial\varphi_{\theta}}{\partial\theta}\Big{)}\overrightarrow{e}_{r}+ \Big{(}\
abla^{2}\varphi_{\theta}-\frac{1}{r^{2}}\varphi_{\theta}+\frac{2}{r^ {2}}\frac{\partial\varphi_{r}}{\partial\theta}\Big{)}\overrightarrow{e}_{ \theta}+\
abla^{2}\varphi_{z}\overrightarrow{e}_{z}\end{array}  \tag{8}\]  
[PERSON] [3] presented expressions for the potentials satisfying Equations (5) and (6):  
\[ \begin{array}{rl}\Phi=\phi\cos n\theta e^{i(k_{z}z-\omega t)}\\ \varphi_{z}=\varphi_{1}\sin n\theta e^{i(k_{z}z-\omega t)}\\ \varphi_{r}=\varphi_{2}\sin n\theta e^{i(k_{z}z-\omega t)}\\ \varphi_{\theta}=\varphi_{3}\cos n\theta e^{i(k_{z}z-\omega t)}\end{array}  \tag{9}\]  
where the integer \(n\) is known as the circumferential order of a wave mode and \(k_{z}\) is the axial wavenumber.