## 3 The Spectral Method Algorithm for a Fluid-Filled Elastic Tube
### The Spectral Method Algorithm
 \tag{56}\]  
The corresponding coefficients are  
\[\text{rs1} =-\lambda\tfrac{\omega^{2}}{c_{1}^{2}}\text{I}+2\mu D^{(2)} \tag{57}\] \[\text{rs2} =-\mu D^{(1)}\] \[\text{rs3} =\mu D^{(1)}\] \[\text{rs4} =2\mu n\Big{[}\text{diag}\Big{(}\tfrac{1}{r}\Big{)}D^{(1)}-\text{ diag}\Big{(}\tfrac{1}{r^{2}}\Big{)}\Big{]}\]\[ \begin{split}\text{rts1}&=-2\mu n\Big{[}\text{diag} \Big{(}\frac{1}{r}\Big{)}D^{(1)}-\text{diag}\Big{(}\frac{1}{r^{2}}\Big{)} \Big{]}\\ \text{rts2}&=\frac{\mu}{2}\Big{[}D^{(1)}+(n-1) \text{diag}\Big{(}\frac{1}{r}\Big{)}\Big{]}\\ \text{rts3}&=\frac{\mu}{2}\Big{[}D^{(1)}-(n+1) \text{diag}\Big{(}\frac{1}{r}\Big{)}\Big{]}\\ \text{rts4}&=\mu\Big{[}-D^{(2)}+\text{diag}\Big{(} \frac{1}{r}\Big{)}D^{(1)}-n^{2}\text{diag}\Big{(}\frac{1}{r^{2}}\Big{)}\Big{]} \\ \text{rzs1}&=-2\mu\Bigg{[}\begin{array}{c}D^{(3)}+ \text{diag}\Big{(}\frac{1}{r}\Big{)}D^{(2)}-\Big{(}n^{2}+1\Big{)}\text{diag} \Big{(}\frac{1}{r^{2}}\Big{)}D^{(1)}\\ +2n^{2}\text{diag}\Big{(}\frac{1}{r^{3}}\Big{)}+\frac{\omega^{2}}{c_{1}^{2}}D ^{(1)}\end{array}\Bigg{]}\\ \text{rzs2}&=\frac{\mu}{2}\Big{[}D^{(2)}-(n-1)\text{diag}\Big{(} \frac{1}{r}\Big{)}D^{(1)}+(n-1)\text{diag}\Big{(}\frac{1}{r^{2}}\Big{)}+\text {L}_{2}\Big{]}\\ \text{rzs3}&=-\frac{\mu}{2}\Big{[}D^{(2)}+(n+1)\text{diag}\Big{(} \frac{1}{r}\Big{)}D^{(1)}-(n+1)\text{diag}\Big{(}\frac{1}{r^{2}}\Big{)}+\text {L}_{3}\Big{]}\\ \text{rzs4}&=-\mu n\Big{[}\text{diag}\Big{(}\frac{1}{r}\Big{)}D^{(2)}+\text{ diag}\Big{(}\frac{1}{r^{2}}\Big{)}D^{(1)}-n^{2}\text{diag}\Big{(}\frac{1}{r^{3}} \Big{)}+\frac{\omega^{2}}{c_{2}^{2}}\text{diag}\Big{(}\frac{1}{r}\Big{)} \Big{]}\end{split}  \tag{59}\]  
The differential matrix expression of Equation (34) is  
\[ik_{z}\
abla\cdot\overset{\rightarrow}{\Psi}=\big{(}\text{gs1}\quad\text{gs2 }\quad\text{gs3}\quad\text{gs4}\big{)} \begin{pmatrix}\phi\\ \hat{h}_{1}\\ \hat{h}_{2}\\ \end{pmatrix}  \tag{60}\]  
The coefficients can be given as  
\[ \begin{split}\text{gs1}&=0\cdot\text{I}\\ \text{gs2}&=\frac{1}{2}\Big{[}D^{(1)}+(1-n)\text{ diag}\Big{(}\frac{1}{r}\Big{)}\Big{]}\\ \text{gs3}&=\frac{1}{2}\Big{[}D^{(1)}+(1+n)\text{ diag}\Big{(}\frac{1}{r}\Big{)}\Big{]}\\ \text{gs4}&=-D^{(2)}-\text{diag}\Big{(}\frac{1}{r}\Big{)}D^{(1)}+n^{2} \text{diag}\Big{(}\frac{1}{r^{2}}\Big{)}-\bigg{(}\frac{\omega^{2}}{c_{2}^{2}} \Big{)}\text{I}\end{split}  \tag{61}\]  
For the fluid, the differential matrix expressions of the displacement vector components are as follows:  
\[ \begin{split}\begin{pmatrix}u_{rf}\\ u_{\theta f}\\ \hat{u}_{zf}\end{pmatrix}&=\begin{pmatrix}\text{ufd1}\\ \text{ufd2}\\ \text{ufd3}\end{pmatrix}\phi_{fl}\\ \text{ufd1}&=D^{(1)}\\ \text{ufd2}&=-n\text{diag}\bigg{(}\frac{1}{r}\Big{)}\end{split}  \tag{62}\]  
\[\text{ufd3}=-\text{L}_{f} \tag{63}\]  
The differential matrix expression of the stress tensor component \(\tau_{rrf}\) of the fluid is  
\[\tau_{rrf}=(\text{rs1})\phi_{fl} \tag{64}\]\[\text{rs1}=-\lambda\frac{\omega^{2}}{c_{fl}^{2}}\cdot\text{I} \tag{67}\]  
Taking the fluid-filled elastic tube with a single-layer wall as an example, Figure 2 shows the appropriate position of the boundary conditions in the L matrix. This can be performed by introducing a matrix \(Q\) on the right-hand side of Equation (51), as shown in Equation (68):  
\[\widetilde{L}\Theta=k_{2}^{2}Q\Theta \tag{68}\]  
where \(\Theta\) is a matrix consisting of potential functions:  
\[\Theta=\left(\phi_{fl}\quad\phi\quad\hat{h}_{1}\quad\hat{h}_{2}\quad h_{3} \right)^{T} \tag{69}\]  
\(Q\) is a \(5N\times 5N\) matrix and can be defined as follows:  
\[Q= \begin{pmatrix}M&&&&\\ &M&&&\\ &&M&&\\ &&&M&\\ &&&&M&\\ \end{pmatrix}  \tag{70}\]  
where \(M\) is an \(N\times N\) diagonal matrix with the following form:  
\[M= \begin{pmatrix}0&&&&\\ &1&&&\\ &&\ddots&&\\ &&&1&\\ &&&0\\ \end{pmatrix}  \tag{71}\]  
Equation (68) is a generalized eigenvalue problem that can be solved using the MATLAB routine \(\text{EIG}\left(\widetilde{L},Q\right)\)[24]. Figure 2: The \(\widetilde{L}\) matrix and boundary condition distribution of a single-layer fluid-filled elastic tube.