## 3 The Spectral Method Algorithm for a Fluid-Filled Elastic Tube
### The Spectral Method Algorithm  
Wave equations and the corresponding boundary conditions in Section 2 are expressed in terms of the differential matrix as follows. The wave Equation (11) in the elastic tube wall can be written as  
\[\underbrace{ \begin{pmatrix}\text{L}_{1}&0&0&0\\ 0&\text{L}_{2}&0&0\\ 0&0&\text{L}_{3}&0\\ 0&0&0&\text{L}_{4}\end{pmatrix} }_{\text{L}}\underbrace{ \begin{pmatrix}\phi\\ \hat{h}_{1}\\ \hat{h}_{2}\\ \hat{h}_{3}\end{pmatrix} }_{\Theta}=k_{2}^{2} \begin{pmatrix}\phi\\ \hat{h}_{1}\\ \hat{h}_{2}\\ \hat{h}_{3}\end{pmatrix}  \tag{45}\]  
where L is a \(4N\times 4N\) differential matrix and expressions for L\({}_{i}(i=1,2,3,4)\) are  
\[\text{L}_{1}=D^{(2)}+\text{diag}\bigg{(}\frac{1}{r}\bigg{)}D^{(1)}-n^{2}\cdot \text{diag}\bigg{(}\frac{1}{r^{2}}\bigg{)}+\bigg{(}\frac{\omega^{2}}{c_{1}^{2 }}\bigg{)}\text{I} \tag{46}\]  
\[\text{L}_{2}=D^{(2)}+\text{diag}\bigg{(}\frac{1}{r}\bigg{)}D^{(1)}-(n-1)^{2} \cdot\text{diag}\bigg{(}\frac{1}{r^{2}}\bigg{)}+\bigg{(}\frac{\omega^{2}}{c_{2 }^{2}}\bigg{)}\text{I} \tag{47}\]  
\[\text{L}_{3}=D^{(2)}+\text{diag}\bigg{(}\frac{1}{r}\bigg{)}D^{(1)}-(n+1)^{2} \cdot\text{diag}\bigg{(}\frac{1}{r^{2}}\bigg{)}+\bigg{(}\frac{\omega^{2}}{c_{2 }^{2}}\bigg{)}\text{I} \tag{48}\]  
\[\text{L}_{4}=D^{(2)}+\text{diag}\bigg{(}\frac{1}{r}\bigg{)}D^{(1)}-n^{2}\cdot \text{diag}\bigg{(}\frac{1}{r^{2}}\bigg{)}+\bigg{(}\frac{\omega^{2}}{c_{2}^{2 }}\bigg{)}\text{I} \tag{49}\]  
In Equations (46)-(49), diag denotes a diagonal matrix and I denotes an \(N\times N\) unit matrix. The wave Equation (14) in the fluid is expressed in the form of the differential matrix as  
\[\text{L}_{f}=D^{(2)}+\text{diag}\bigg{(}\frac{1}{r}\bigg{)}D^{(1)}-n^{2}\text {diag}\bigg{(}\frac{1}{r^{2}}\bigg{)}+\bigg{(}\frac{\omega^{2}}{c_{fl}^{2}} \bigg{)}\text{I} \tag{50}\]Thus, the differential matrix expression of equations for wave propagation in a fluid-filled elastic tube is as follows:  
\[\underbrace{ \begin{pmatrix}\text{L}_{f}&&&\\ &\text{L}_{1}&&\\ &&\text{L}_{2}&&\\ &&&\text{L}_{3}&\\ &&&\text{L}_{4}\end{pmatrix} }_{\text{L}} \begin{pmatrix}\phi_{fl}\\ \phi\\ \hat{h}_{1}\\ \hat{h}_{2}\\ h_{3}\end{pmatrix} =k_{z}^{2} \begin{pmatrix}\phi_{fl}\\ \phi\\ \hat{h}_{1}\\ \hat{h}_{2}\\ h_{3}\end{pmatrix}  \tag{51}\]  
Similarly, the differential matrix expressions for the displacement vector components in the elastic pipe wall are  
\[ \begin{pmatrix}u_{r}\\ u_{\theta}\\ \hat{u}_{z}\end{pmatrix} = \begin{pmatrix}\text{urs1}&\text{urs2}&\text{urs3}& \text{urs4}\\ \text{uts1}&\text{uts2}&\text{uts3}&\text{uts4}\\ u\text{uzs1}&\text{uzs2}&\text{uzs3}&\text{uzs4}\end{pmatrix}\begin{pmatrix} \phi\\ \hat{h}_{1}\\ \hat{h}_{2}\\ h_{3}\end{pmatrix}  \tag{52}\]  
The corresponding coefficients are  
\[\text{urs1} =D^{(1)} \tag{53}\] \[\text{urs2} =-\tfrac{1}{2}\text{I}\] (54) \[\text{urs3} =\tfrac{1}{2}\text{I}\] \[\text{urs4} =n\cdot\text{diag}\Big{(}\tfrac{1}{r}\Big{)}\] \[\text{uts1} =-n\cdot\text{diag}\Big{(}\tfrac{1}{r}\Big{)}\] \[\text{uts2} =\tfrac{1}{2}\text{I}\] \[\text{uts3} =\tfrac{1}{2}\text{I}\] \[\text{uts4} =-D^{(1)}\] \[\text{uzs1} =-D^{(2)}-\text{diag}\Big{(}\tfrac{1}{r}\Big{)}D^{(1)}+n^{2}\cdot \text{diag}\Big{(}\tfrac{1}{r^{2}}\Big{)}- \begin{pmatrix}\omega^{2}\\ \omega_{1}^{2}\end{pmatrix} \text{I}\] (55) \[\text{uzs2} =\tfrac{1}{2}\Big{[}D^{(1)}+(1-n)\cdot\text{diag}\Big{(}\tfrac{1 }{r}\Big{)}\Big{]}\] \[\text{uzs3} =-\tfrac{1}{2}\Big{[}D^{(1)}+(1+n)\cdot\text{diag}\Big{(}\tfrac{1 }{r}\Big{)}\Big{]}\] \[\text{uzs4} =0\cdot 1\]  
The differential matrix expressions for the stress tensor components in the elastic tube wall are  
\[ \begin{pmatrix}\tau_{rr}\\ \tau_{r\theta}\\ \tau_{r\zeta}\end{pmatrix}  = \begin{pmatrix}\text{rs1}&\text{rs2}&\text{rs3}&\text{rs4}\\ \text{rs1}&\text{rs2}&\text{rs3}&\text{rs4}\\ \text{rzs1}&\text{rzs2}&\text{rzs3}&\text{rzs4}\end{pmatrix}\begin{pmatrix} \phi\\ \hat{h}_{1}\\ \hat{h}_{2}\\ h_{3}\end{pmatrix}