## 3 Solution Procedure
### Spectral Element Method
Further, with scanning the frequency, if the imaginary part of \(q\) is vanished, the incident wave can propagate freely, which indicates a pass band; if the imaginary part of \(q\) is present, the incident wave will be attenuated along the pipe, with absorption of the vibration energy by the PC structure, which indicates a stop band, i.e., a BG. The BG regions of the present periodic pipe can thus be measured. **Finally**, for the whole periodic pipe with \(n\) cells, the stiffness assembling procedure used above is also adopted, and the spectral stiffness matrix for such composite structure is ultimately achieved as  
\[\mathbf{S}_{n}=\left[ \begin{array}{cccccc}\mathbf{S}1&\mathbf{S}2&0&\cdots &\cdots&0\\ \mathbf{S}3&\mathbf{S}4+\mathbf{S}1&\mathbf{S}2&0&\vdots&\vdots\\ 0&\mathbf{S}3&\mathbf{S}4+\mathbf{S}1&\ddots&0&\vdots\\ \vdots&0&\ddots&\ddots&\mathbf{S}2&0\\ \vdots&\vdots&0&\mathbf{S}3&\mathbf{S}4+\mathbf{S}1&\mathbf{S}2\\ 0&\cdots&\cdots&0&\mathbf{S}3&\mathbf{S}4\\ \end{array} \right], \tag{25}\]  
which is a \((2n+2)\)\(\times\)\((2n+2)\) matrix. The relationship between the general force and displacement at both ends of all cells along the whole pipe can then be obtained as  
\[\left[ \begin{array}{cccccc}Q_{\mathrm{L}}&M_{\mathrm{L}}&\cdots&Q_{\mathrm{ R}}&M_{\mathrm{R}}\\ \end{array} \right]^{\mathrm{T}}=\mathbf{S}_{n}\cdot\left[ \begin{array}{cccccc} w_{\mathrm{L}}&\theta_{\mathrm{L}}&\cdots&w_{\mathrm{R}}&\theta_{\mathrm{R}}\\ \end{array} \right]^{\mathrm{T}}. \tag{26}\]  
Using Equation (26), for an arbitrary incident wave, the edge displacement of each cell, namely, the node displacement, can be calculated, and consequently the frequency response and vibration shape can be attained. In addition, based on Equation (26), the natural frequency and vibration mode of the periodic pipe can be determined. Since these properties are associated with the end condition, we first give the general displacement and force at the left and right ends of the pinned-pinned pipe and those at both ends of all cells as  
\[ \begin{array}{ccccc}w_{\mathrm{L}}=w_{\mathrm{R}}=0,\ \theta_{\mathrm{L}}\
eq 0,\ \theta_{\mathrm{R}}\
eq 0,\ \mathrm{others}\
eq 0,\\ Q_{\mathrm{L}}\
eq 0,\ Q_{\mathrm{R}}\
eq 0,\ M_{\mathrm{L}}=M_{\mathrm{R}}=0,\ \mathrm{other\ loads}=0.\end{array}  \tag{27}\]  
Thus, Equation (26) can be written as  
\[\left[ \begin{array}{cccccc}Q_{\mathrm{L}}&0&\cdots&Q_{\mathrm{R}}&0\\ \end{array} \right]^{\mathrm{T}}=\mathbf{S}_{n}\cdot\left[ \begin{array}{cccccc }0&\theta_{\mathrm{L}}&\cdots&0&\theta_{\mathrm{R}}\\ \end{array} \right]^{\mathrm{T}}. \tag{28}\]  
A nontrivial solution of Equation (28) requires a simplified determinant of the coefficient matrix equal to zero, which yields the characteristic equation of the system as  
\[\det(\mathbf{S}_{n,s})=0, \tag{29}\]  
where  
\[\mathbf{S}_{n,s}=\left[ \begin{array}{cccccc}\mathbf{S}1(2,2)&\mathbf{S}2(2, 1)&\mathbf{S}2(2,2)&0&\cdots&\cdots&0\\ \mathbf{S}3(1,2)&\mathbf{S}41(1,1)&\mathbf{S}41(1,2)&\ddots&0&\vdots&\vdots\\ \mathbf{S}3(2,2)&\mathbf{S}41(2,1)&\mathbf{S}41(2,2)&\ddots&0&0&\vdots\\ 0&\ddots&\ddots&\ddots&\ddots&\ddots&0\\ \vdots&0&0&\ddots&\mathbf{S}41(1,1)&\mathbf{S}41(1,2)&\mathbf{S}2(1,2)\\ \vdots&\vdots&0&\ddots&\mathbf{S}41(2,1)&\mathbf{S}41(2,2)&\mathbf{S}2(2,2)\\ 0&\cdots&\cdots&0&\mathbf{S}3(2,1)&\mathbf{S}3(2,2)&\mathbf{S}4(2,2)\\ \end{array} \right], \tag{30}\]and \(\mathbf{S}41=\mathbf{S}4+\mathbf{S}1\). The natural frequency can then be determined by Equation (29), and the mode shape can be simulated in terms of the node displacements obtained by Equation (28).