## 2 Overview of the TMM
### The Theory of the MVTMM
#### 2.3.1 The Construction of the Mixed Variable Transfer Matrix
\begin{bmatrix}\exp(\lambda^{U}(l-l_{0}))&0\\ 0&\exp(\lambda^{D}(l-l_{1}))\end{bmatrix}\begin{bmatrix}A^{*,U}\\ A^{*,D}\end{bmatrix}  \tag{21}\]  
where \(l_{0}\) and \(l_{L}\) denote the initial and final positions of the pipeline system, respectively, while \(A^{*,U}\) and \(A^{*,D}\) represent the amplitudes of the incident and reflected mechanical waves within the pipeline. Therefore, the following is true:  
\[ \begin{bmatrix}\textbf{Var}(l,s)\\ \textbf{Var}_{\textbf{c}}(l,s)\end{bmatrix} = \begin{bmatrix}\textbf{S}(s)^{UU} &\textbf{S}(s)^{UD}\\ \textbf{S}(s)^{DU}&\textbf{S}(s)^{DD}\end{bmatrix}\begin{bmatrix}\exp( \lambda^{U}(l-l_{0}))&0\\ 0&\exp(\lambda^{D}(l-l_{1}))\end{bmatrix}\begin{bmatrix}A^{*,U}\\ A^{*,D}\end{bmatrix}  \tag{22}\]  
Let \(\mathbf{\Lambda}(s)^{U}=\exp(\lambda(s)^{U}L)\) and \(\mathbf{\Lambda}(s)^{D}=\exp(-\lambda(s)^{D}L)\), where \(L=l_{1}-l_{0}\) represents the length of the pipeline. This is in conjunction with the boundary conditions established at both ends of the pipeline:  
\[ \begin{bmatrix}\textbf{Var}(l_{0},s)\\ \textbf{Var}_{\textbf{c}}(l_{0},s)\end{bmatrix} = \begin{bmatrix}\textbf{S}(s)^ {UU}&\textbf{S}(s)^{UD}\\ \textbf{S}(s)^{DU}&\textbf{S}(s)^{DD}\end{bmatrix}\begin{bmatrix}\textbf{I}&0 \\ 0&\mathbf{\Lambda}(s)^{D}\end{bmatrix}\begin{bmatrix}A^{*,U}\\ A^{*,D}\end{bmatrix} = \begin{bmatrix}\textbf{S}(s)^{UU}&\textbf{S}(s)^{UD} \mathbf{\Lambda}(s)^{D}\\ \textbf{S}(s)^{DU}&\textbf{S}(s)^{DD}\mathbf{\Lambda}(s)^{D}\end{bmatrix}\begin{bmatrix} A^{*,U}\\ A^{*,D}\end{bmatrix}  \tag{23}\]  
\[ \begin{bmatrix}\textbf{Var}(l_{1},s)\\ \textbf{Var}_{\textbf{c}}(l_{1},s)\end{bmatrix} = \begin{bmatrix}\textbf{S}(s)^ {UU}&\textbf{S}(s)^{UD}\\ \textbf{S}(s)^{DU}&\textbf{S}(s)^{DD}\end{bmatrix}\begin{bmatrix}\mathbf{\Lambda} (s)^{U}&0\\ 0&\textbf{I}\end{bmatrix}\begin{bmatrix}A^{*,U}\\ A^{*,D}\end{bmatrix} = \begin{bmatrix}\textbf{S}(s)^{UU}\mathbf{\Lambda}^{I}& \textbf{S}(s)^{UD}\\ \textbf{S}(s)^{DU}\mathbf{\Lambda}^{I}&\textbf{S}(s)^{DD}\end{bmatrix}\begin{bmatrix} A^{*,U}\\ A^{*,D}\end{bmatrix}  \tag{24}\]  
Based on Equations (23) and (24), we can conclude the following:  
\[ \begin{bmatrix}\textbf{Var}(l_{1},s)\\ \textbf{Var}_{\textbf{c}}(l_{0},s)\end{bmatrix} = \begin{bmatrix}\textbf{X}_{l_{1},l_{0}}&-\textbf{Z}_{l_{1},l_{0}}\\ \textbf{Y}_{l_{1},l_{0}}&\textbf{W}_{l_{1},l_{0}}\end{bmatrix}\begin{bmatrix} \textbf{Var}(l_{0},s)\\ \textbf{Var}_{\textbf{c}}(l_{1},s)\end{bmatrix}  \tag{25}\]  
\[ \begin{bmatrix}\textbf{X}_{l_{1},l_{0}}&-\textbf{Z}_{l_{1},l_{0}}\\ \textbf{Y}_{l_{1},l_{0}}&\textbf{W}_{l_{1},l_{0}}\end{bmatrix} = \begin{bmatrix} \textbf{S}^{UU}(s)\mathbf{\Lambda}(s)^{U}&\textbf{S}(s)^{UD}\\ \textbf{S}(s)^{DU}&\textbf{S}(s)^{DD}\mathbf{\Lambda}(s)^{D}\end{bmatrix}\begin{bmatrix} \textbf{S}(s)^{UU}&\textbf{S}(s)^{UD}\mathbf{\Lambda}(s)^{D}\\ \textbf{S}(s)^{DU}\mathbf{\Lambda}(s)^{U}&\textbf{S}(s)^{DD}\end{bmatrix} ^{-1} \tag{26}\]  
\(\textbf{Z}_{l_{1},l_{0}}\) and \(\textbf{Y}_{l_{1},l_{0}}\) can be interpreted as the impedance matrix and the admittance matrix of the pipeline, respectively. As indicated in Equation (25), the state vectors on both sides of the equation consist of variables at both ends of the pipeline; thus, they are referred to as mixed variable vectors. The corresponding transfer matrix is termed the mixed variable transfer matrix. From Equation (26), it is evident that the diagonal elements of the inverse matrix on its right side do not include exponential terms. Consequently, when the length of thepipeline is substantial, this matrix remains non-singular. Conversely, for shorter pipelines, this matrix corresponds to an eigenvector matrix derived from the Hamiltonian matrix, which retains invertibility due to inherent properties associated with Hamiltonian matrices. Furthermore, by applying the recursive algorithm pertaining to mixed variable transfer matrices, we can derive a global transfer matrix for chain systems without necessitating additional operations involving inverse matrices.