## 2 Overview of the TMM
### The Theory of the MVTMM
#### 2.3.1 The Construction of the Mixed Variable Transfer Matrix  
The 14 frequency-domain state variables of the hydraulic pipeline system are organized into pairs of conjugate variable groups based on symmetry, specifically \(\mathbf{Var}=\left\{P,F_{zz},U_{y},M_{x},F_{x},\psi_{y},M_{z}\right\}\) and \(\mathbf{Var_{c}}=\left\{V,U_{z},F_{y},\psi_{x},U_{x}M_{y},\psi_{z}\right\}\). The concept of symmetry is discussed in the literature [10]. Subsequently, Equations (1)-(14) can be uniformly expressed in the following matrix form after applying Laplace transformation:  
\[\frac{\partial}{\partial l} \begin{bmatrix}P\\ F_{z}\\ U_{y}\\ M_{x}\\ F_{x}\\ \psi_{y}\\ M_{z}\\ V\\ U_{z}\\ F_{y}\\ \psi_{x}\\ U_{x}\\ M_{y}\\ \psi_{z}\end{bmatrix} = \begin{bmatrix}0&0&0&0&0&0&0&D_{1}&D_{2}&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&D_{3}&D_{4}&\frac{-1}{R_{w}}&0&0&0&0&0\\ 0&0&0&0&0&0&0&\frac{1}{R_{w}}&\frac{-1}{R_{w}^{2}GA_{p}}&-1&0&0&0\\ 0&0&0&0&0&0&0&0&1&D_{5}&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&D_{6}&0&0\\ 0&0&0&0&0&0&0&0&0&0&\frac{sE_{A}}{R_{w}}&\frac{1}{R_{w}}\\ 0&0&0&0&0&0&0&0&0&0&0&\frac{-1}{R_{w}}&\frac{s\rho}{R_{w}}\\ -\frac{s}{K^{s}}&\frac{2 us}{EA_{p}}&\frac{-1}{R_{w}}&0&0&0&0&0&0&0&0&0&0\\ D_{7}&\frac{s}{EA_{p}}&\frac{-1}{R_{w}}&0&0&0&0&0&0&0&0&0&0\\ -\frac{-A_{f}}{R_{w}}&\frac{1}{R_{w}}&D_{6}&0&0&0&0&0&0&0&0&0\\ 0&0&0&sE_{A}&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&\frac{s}{K^{s}GA_{p}}&1&0&0&0&0&0&0&0\\ 0&0&0&0&-1&D_{5}&\frac{1}{R_{w}}&0&0&0&0&0&0\\ 0&0&0&0&0&\frac{-1}{R_{w}}&\frac{s}{G\rho}&0&0&0&0&0&0\\ \end{bmatrix}\begin{bmatrix}P\\ F_{z}\\ U_{y}\\ M_{x}\\ F_{x}\\ M_{y}\\ V\\ U_{z}\\ F_{y}\\ V_{x}\\ U_{x}\\ M_{y}\\ \psi_{z}\end{bmatrix}  \tag{16}\]  
where  
\[D_{1} = -(\rho_{f}s+\frac{2\rho_{f}s}{\rho_{1}(\rho_{1}\sqrt{s/\
u_{f}}) -2}),\ D_{2}=\frac{2\rho_{f}s}{\rho_{1}(\rho_{1}\sqrt{s/\
u_{f}})-2},\ D_{3}=- \frac{2\pi\rho_{f}\rho_{1}^{2}s}{\rho_{1}(\rho_{1}\sqrt{s/\
u_{f}})-2},\ D_{5}=s \rho_{p}I_{p}+s\rho_{f}I_{f},\] \[D_{4}=A_{p}\rho_{f}s+\frac{2\pi\rho_{f}\rho_{1}^{2}s}{\rho_{1}( \rho_{1}\sqrt{s/\
u_{f}})-2},\ D_{5}=s\rho_{p}I_{p}+s\rho_{f}I_{f},\]  
and \(D_{6}=s\rho_{f}A_{f}+s\rho_{p}A_{p}\), \(D_{7}=-\frac{sH\
u_{f}}{E}\). Equation (16) can be abbreviated as follows:  
\[\frac{\partial}{\partial l} \begin{aligned} \mathbf{-} \\ \mathbf{-}\\ \mathbf{\Phi}(l,s)=\begin{bmatrix}\mathbf{Var}(l,s)\\ \mathbf{Var}_{c}(l,s)\end{bmatrix},\mathbf{T}^{\star}(s)=\begin{bmatrix}0& \mathbf{T}_{12}(s)\\ \mathbf{T}_{21}(s)&0\end{bmatrix}.\end{aligned}  \tag{17}\]  
\[\left(\mathbf{J}\mathbf{T}^{\star}(s)\right)^{H}=\mathbf{J}\mathbf{T}^{\star}(s ),\mathbf{J}= \begin{bmatrix}0&\mathbf{I}\\ -\mathbf{I}&0\end{bmatrix}  \tag{18}\]  
**T** is a 14 \(\times\) 14 square matrix that encapsulates all parameter information of the FSI 14-equation model. Furthermore, **T** satisfies Equation (18), which implies that **T** is a Hamiltonian matrix. According to the properties of Hamiltonian matrices, half of its eigenvalues correspond to the mechanical wave propagation constants traveling forward along the pipeline element, while the other half represent those propagating backward along the same element. Consequently, **T** can be expressed as follows:  
\[\mathbf{\Lambda}(s)=\textbf{S}(s)^{-1}\textbf{T}(s)\textbf{S}(s) \tag{19}\]  
where \(\mathbf{\Lambda}(s)= \begin{bmatrix}\lambda(s)^{U}&0\\ 0&\lambda(s)^{D}\end{bmatrix} \) is a square matrix with eigenvalues, \(\textbf{S}(s)= \begin{bmatrix}\textbf{S}(s)^{UU}&\textbf{S}(s)^{UD}\\ \textbf{S}(s)^{DU}&\textbf{S}(s)^{DD}\end{bmatrix} \) is a square matrix composed of the corresponding eigenvectors, and the superscripts \(U\) and \(D\) denote forward and reverse directions along the pipeline, respectively. The process described is analogous to the decoupling procedure outlined in Equation (A4) and is derived through the transformation of Equation (17):  
\[\frac{\partial}{\partial l}\eta(l,s)=\textbf{S}(s)^{-1}\textbf{T}(s)\textbf{S }(s)\eta(l,s)=\mathbf{\Lambda}(s)\eta(l,s) \tag{20}\]  
The solution to Equation (20) can be articulated as follows:  
\[\eta(l,s)=