## 2 Theoretical Analysis  
### Eigenequation in the Pipeline  
It should first be noted that although calculations in pipe acoustics have generally been performed under the assumption of absolute soft or absolute hard boundary conditions, the characteristic impedance of a liquid is large compared with that of air, and cannot be ignored, and so ideal boundary conditions are no longer applicable. The infinitely long straight pipeline considered here has outer diameter \(a\) and inner diameter \(b\), as shown in Figure 1[31]. In the following, the displacement scalar potential function is denoted by \(\phi\), the vector potential function by \(\overrightarrow{\Psi}\), the longitudinal-wave velocity in the wall by \(c_{l}\), the shear-wave velocity by \(c_{s}\), the shear modulus of the pipe wall material by \(\mu\), the wave velocity in the liquid in the pipe by \(c_{0}\), and the liquid density in the pipe by \(\rho_{1}\). Some previous studies have assumed an axisymmetric source in a plate [32] or in a pipe, using a cylindrical coordinate system [33], while some have assumed a point source [34; 35]. In the present study, the assumption of axisymmetric excitation is an important one. Under axisymmetric excitation, with \(\overrightarrow{\Psi}=(0,\psi_{\theta},0)\), the wave equation in the wall can be represented by the following equations for two scalar potential functions:  
\[ \begin{array}{c}\
abla^{2}\phi=\frac{1}{c_{l}^{2}}\frac{\partial^{2}\phi}{ \partial t^{2}},\\ \Big{(}\
abla^{2}-\frac{1}{r^{2}}\Big{)}\psi_{\theta}=\frac{1}{c_{l}^{2}} \frac{\partial^{2}\psi_{\theta}}{\partial t^{2}},\end{array}  \tag{1}\]  
where  
\[\
abla^{2}=\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{ \partial r}+\frac{\partial^{2}}{\partial z^{2}}.\]  
The radial and axial components of the displacement are  
\[ \begin{array}{c}u_{r}=\frac{\partial\phi}{\partial r}-\frac{\partial\psi_{ \theta}}{\partial z},\\ u_{z}=\frac{\partial\phi}{\partial z}+\frac{1}{r}\frac{\partial(r\psi_{\theta })}{\partial r},\end{array}  \tag{2}\]  
where \(u_{r}\) is the radial components of the displacement in the wall, \(u_{z}\) is the axial components of the displacement in the wall, and the normal and tangential components of the stress are  
\[ \begin{array}{c}\delta_{rr}=\lambda\Delta+2\mu\frac{\partial u_{r}}{ \partial r},\\ \delta_{rz}=\mu\Big{(}\frac{\partial u_{r}}{\partial z}+\frac{\partial u_{z}}{ \partial r}\Big{)},\end{array}  \tag{3}\]  
where \(\delta_{rr}\) is the normal component of the stress in the wall, \(\delta_{rz}\) is the tangential components of the stressin the wall, \(\lambda\) and \(\mu\) are the Lame coefficients. Figure 1: Infinitely long straight fluid filled pipeline model. Under the assumption of a simple harmonic vibration in the \(z\) direction, the displacement potential function can be expressed as  
\[\phi=\Phi e^{i(k_{z}z-\omega t)},\psi_{\theta}=\Psi e^{i(k_{z}z-\omega t)}.\]  
With the time dependence ignored, by substitution of these expressions into the wave equation, the relationship between the displacement potential function and the displacement and stress in the pipe wall can be obtained. When \(b\leq r\leq a\), the formal solution for the potential function in the wall is  
\[ \begin{array}{l}\phi(r,z)=\left[AJ_{0}(k_{t}r)+BY_{0}(k_{t}r)\right]e^{ik_{z }z},\ k_{t}^{2}+k_{z}^{2}=\left(\omega/c_{t}\right)^{2},\\ \psi_{\theta}(r,z)=\left[CJ_{0}(k_{t}r)+DY_{0}(k_{t}r)\right]e^{ik_{z}z},\ k_{t}^{2}+k_{z}^{2}= \left(\omega/c_{s}\right)^{2},\end{array}  \tag{4}\]  
where \(A\), \(B\), \(C\), and \(D\) are constants. The wave equation satisfied by the water potential function is  
\[\
abla^{2}\phi_{1}=\frac{1}{c_{0}^{2}}\frac{\partial^{2}\phi_{1}}{\partial t^ {2}}. \tag{5}\]  
The radial and axial components of the displacement in the water are  
\[ \begin{array}{l}u_{rf}=\frac{\partial\phi_{1}}{\partial r},\\ u_{zf}=\frac{\partial\phi_{1}}{\partial z},\end{array}