## Appendix A The Class TMM  
The fundamental principle of class TMM is exemplified through the application of the L-MOC.  
When the effect of the friction term is disregarded, Equations (1)-(14) can be uniformly expressed in the following matrix form:  
\[\mathbf{A}\frac{\partial}{\partial t}\mathbf{\Phi}(l,t)+\mathbf{B}\frac{ \partial}{\partial z}\mathbf{\Phi}(l,t)+\mathbf{C}\mathbf{\Phi}(l,t)=0 \tag{1}\]  
where \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\) represent the coefficient matrices, while \(\mathbf{\Phi}(l,t)\) denotes the state variable vector comprising 14 variables:  
\[\mathbf{\Phi}(l,t)=(v,p,u_{z},f_{z},u_{y},f_{y},\varphi_{x},m_{x},u_{x},f_{x}, \varphi_{y},m_{y},\varphi_{z},m_{z})\]  
Let \(\mathbf{\Phi}(l,0)=0\). We applied the transformation of Equation (1) to obtain the following:  
\[s\mathbf{A}^{*}\mathbf{\Phi}(l,s)+\mathbf{B}\frac{\partial}{\partial z} \mathbf{\Phi}(l,s)=0 \tag{2}\]  
where \(\mathbf{A}^{*}=\mathbf{A}+\mathbf{C}/s\), and \(\mathbf{\Phi}(l,s)=(V,P,U_{z},F_{z},U_{y},F_{y},\psi_{x},M_{x},U_{x},F_{x}, \psi_{y},M_{y},\psi_{z},M_{z})\) represents the state variable in the frequency domain.  
From Equations (1) and (3), it is evident that the impact of friction coupling manifests in the fluid shear force. [PERSON] [33] proposed a universal model for shear force in the frequency domain:  
\[\widetilde{\tau}(s)=\frac{\rho/r_{i}}{\vartheta_{1}(jr_{i}\sqrt{s/v_{f}})-2}s (V-U_{z}) \tag{3}\]  
Therefore, the friction coupling is manifested in the correction of matrix \(\mathbf{A}\) within the frequency domain. This adjustment can be achieved by substituting the variable coefficients from Equation (3) into their corresponding terms in the matrix.  
To decouple Equation (2), make the following transformations:  
\[\mathbf{\Phi}(l,s)=\mathbf{S}(s)\mathbf{\eta}(l,s) \tag{4}\]  
Substituting Equation (4) into Equation (2) and subsequently eliminating variable \(\mathbf{A}\) yields the following result:  
\[s\mathbf{\eta}(l,s)+\mathbf{\Lambda}(s)\frac{\partial\mathbf{\eta}(l,s)}{ \partial z}=0 \tag{5}\]  
where \(\mathbf{\Lambda}(s)=\mathbf{S}(s)^{-1}\mathbf{A}^{*-1}(s)\mathbf{B}\mathbf{S} (s)\) represents the eigenvalue matrix of \(\mathbf{A}^{*-1}(s)\mathbf{B}\), \(\mathbf{\Lambda}(s)\) is the diagonal matrix, and \(\mathbf{S}(s)\) corresponds to the associated eigenvector.  
The general solution of Equation (A5) is  
\[\eta(l,s)=\mathbf{E}(l,s)\eta_{0}(s)\] (A6)  
\(\eta_{0}(s)\) represents the constant term that is independent of position. \(\mathbf{E}(l,s)=diag(\exp(-sl/\lambda_{1}),\exp(-sl/\lambda_{2}),\ldots etc)\), while \(\lambda\) denotes the eigenvalue associated with \(\mathbf{A}^{*-1}(s)\mathbf{B}\).  
Substituting Equation (A6) into Equation (A5) and integrating the boundary conditions at both ends of the pipeline yields the following:  
\[\mathbf{\Phi}(l,s)=\mathbf{T}(l,s)\mathbf{\Phi}(l_{0},s)\] (A7)  
where \(\mathbf{T}(l,s)=\mathbf{S}(s)\mathbf{E}(l,s)\mathbf{S}(s)^{-1}\) denotes the field transfer matrix of the pipeline section, and \(l_{0}\) indicates the initial position of the pipeline.  
At both extremities of the pipeline, the boundary conditions are defined as follows:  
\[\mathbf{D}_{s}(s)\mathbf{\Phi}(l_{0},s)=\mathbf{Q}_{s}(s)\] (A8)  
\[\mathbf{D}_{e}(s)\mathbf{\Phi}(l,s)=\mathbf{Q}_{e}(s)\] (A9)  
where \(\mathbf{D}_{s}\) and \(\mathbf{D}_{e}\) represent the boundary matrices at each end of the pipeline, while \(\mathbf{Q}_{s}\) and \(\mathbf{Q}_{e}\) denote the excitation vectors corresponding to both ends of the pipeline.  
Equations (A7)-(A9) are combined to obtain the following:  
\[\mathbf{\Phi}(0,s)=\begin{bmatrix}\mathbf{D}_{s}(s)\\ \mathbf{D}_{e}(s)\mathbf{T}(l,s)\end{bmatrix}^{-1}\begin{bmatrix}\mathbf{Q}_{s }(s)\\ \mathbf{Q}_{e}(s)\end{bmatrix}\] (A10)  
By integrating Equations (A10) and (A7), the response of the state variable at any given position within the pipeline can be determined.