## 2 Basic Theory of Acoustic-Solid Coupling
### Timoshenko Beam Model  
The flow transport pipeline model is usually constructed as a beam model or a shell model. Popular beam models include the Euler Bernoulli beam and the Timoshenko beam, which have different conditions for application. The former does not consider the shear effect of the pipe, while the DLLFP has a thick wall, and its shear deformation is not negligible. The Timoshenko beam model, however, can be used to construct the model of DLLFP and achieve calculation results in good alignment with reality. Therefore, in this paper, the DLLFP is calculated using the Timoshenko beam model, and the solid elastic waves are classified into shear, longitudinal, and bending waves based on the relative position between the axial direction of the pipe and the direction of wave motion, as shown in a study by [46]. In the Timoshenko theoretical model, the bending wavenumber analytic equation, the longitudinal wavenumber analytic equation, and the shear wavenumber analytic equation are expressed as follows [47]. The model of a Timoshenko beam subjected to transverse vibrations caused by transverse loads in the x-y plane is shown in Figure 1, and the equations of motion are shown below [48]. \[EI\frac{\partial^{4}y}{\partial x^{4}}-(q-\rho A\frac{\partial^{2}y}{\partial t ^{2}})-\rho I\frac{\partial^{4}y}{\partial x^{2}\partial t^{2}}+\frac{EI}{AG \mu}\frac{\partial^{2}}{\partial x^{2}}(q-\rho A\frac{\partial^{2}y}{\partial t ^{2}})-\frac{\rho I}{AG\mu}\frac{\partial^{2}}{\partial t^{2}}(q-\rho A\frac{ \partial^{2}y}{\partial t^{2}})=0 \tag{18}\]  
where \(E\) is the modulus of elasticity, \(G\) is the shear modulus, \(\mu\) is the cross-sectional shape factor, \(q\) indicates the external force, \(\rho\) is the density of the beam material, \(A\) is the cross-sectional area of the beam, and \(I\) is the moment of inertia of the beam cross-section to the neutral axis. Since the pipe has a circular cross-section, the shape factor \(\mu\) calculated from shear is set at 0.9 based on the strain energy theory. If it is assumed that the external force is 0, that is, \(q=0\); then, the basic solution of the equation is \(y=Be^{i(c+t-tx)}\), and the number of waves \(k=\omega/c;B\) is the wave amplitude, \(\omega\) is the angular frequency, \(c\) is the wave speed. Substitution of these variables into Equation (18) results in the following equation:  
\[\frac{EI}{\rho A}k^{4}-k^{2}c^{2}-\frac{I}{A}(1+\frac{E}{G\mu})k^{4}c^{2}+\frac {\rho I}{AG\mu}k^{4}c^{4}=0 \tag{19}\]  
The following equations are used to simplify calculation:  
\[b=\sqrt{\frac{I}{A}(1+\frac{E}{G\mu})},\ \ \ s=\sqrt{\frac{\rho I}{AG\mu}},\ \ a= \sqrt{\frac{EI}{\rho A}}\]  
Equation (19) can be simplified into  
\[s^{2}k^{4}c^{4}-k^{2}c^{2}-b^{2}k^{4}c^{2}+a^{2}k^{4}=0 \tag{20}\]  
Let \(\omega=2\pi f\), where \(f\) is the frequency, and the bending wavenumber of the Timoshenko beam model can be obtained as  
\[k_{\rm T}=\sqrt{\frac{b^{2}(2\pi f)^{2}\pm\sqrt{b^{4}(2\pi f)^{4}-4a^{2}[s^{2 }(2\pi f)^{4}-(2\pi f)^{2}]}}{2a^{2}}} \tag{21}\]  
The longitudinal vibration model of the pipe is shown in Figure 2, where \(u\) denotes the longitudinal displacement at a point on the cross-section at \(x\), \(S\) denotes the combined axial stress at the cross-section at \(x\), \(A\) is the cross-sectional area, and \(\rho\) is the mass density of the material. According to [PERSON] theorem, the axial force on each pipe section during longitudinal vibration of the pipeline is derived, and then the longitudinal wavenumber of the pipeline Timoshenko beam model is obtained from the relationship between wave number and frequency combined with [PERSON]'s law as follows:  
\[k_{\rm L}=\omega\sqrt{\frac{\rho}{E}} \tag{22}\]  
Figure 1: [PERSON]’s theoretical model. Figure 2: Longitudinal vibration model of pipeline.