## 2 Mechanical Model and Dispersion Equation  
The mechanical schematic of a fluid-conveying PC pipe with periodically varying cross-section is shown in Figure 1. The pipe is composed of \(n\) identical cells, and each cell consists of two segments with different geometrical cross-sections. The inner diameter is uniform along the pipe, while the outer diameter is different for the two segments. The lengths of pipe A and pipe B are \(l_{\text{A}}\) and \(l_{\text{B}}\), respectively, and the lattice constant is \(l_{\text{PC}}=l_{\text{A}}+l_{\text{B}}\). \(r_{i}\), \(r_{\text{o,A}}\), and \(r_{\text{o,B}}\) are the uniform inner radius and outer radii of pipe A and pipe B, respectively. An inertial frame \(oxy\) is taken with the origin \(o\) at the left end of the pipe. The \(x\) and \(y\) axes coincide with the pipe axis and the direction of lateral deflection, respectively. Based on the Euler-Bernoulli beam model, and neglecting the gravity, imposed tension and pressurization effect, the relationship between the general force and displacement of the uniform pipe and fluid is derived as  
\[\left\{ \begin{array}{l}M=EI\frac{\partial\theta}{\partial x}\\ \theta=\frac{\partial w}{\partial x}\\ EI\frac{\partial^{2}w}{\partial x^{2}}+Q=0\\ \frac{\partial Q}{\partial x}=m_{\text{f}}\left(\frac{\partial}{\partial t}+U \frac{\partial}{\partial x}\right)^{2}w+m_{\text{p}}\frac{\partial^{2}w}{ \partial t^{2}}\end{array} \right. \tag{1}\]  
where \(E\), \(I\), \(m_{\text{p}}\) and \(m_{\text{f}}\) denote the Young's modulus, moment inertia, mass per unit length of the pipe and fluid, respectively, and \(U\) the flow velocity. \(Q\), \(M\), \(w\) and \(\theta\) are the shear force, bending moment, lateral deflection and slope of the pipe, respectively. The over dot and prime denote differentiations with respect to time \(t\) and the axial coordinate \(x\), respectively. It should be noted that the sound velocity of the fluid is not considered in this study, since it is far higher compared to the flow velocity, which will lead to a trivial influence of the sound velocity on the FSI of the system. Such hypothesis will lose effectiveness when the flow velocity is increased close to the sound velocity of the fluid. Eliminating \(\theta\) from Equation (1), the well-known governing equation of fluid-conveying pipes can be obtained as  
\[EI\frac{\partial^{4}w}{\partial x^{4}}+\left(m_{\text{f}}+m_{\text{p}}\right) \frac{\partial^{2}w}{\partial t^{2}}+2m_{\text{f}}U\frac{\partial^{2}w}{ \partial x\partial t}+m_{\text{f}}U^{2}\frac{\partial^{2}w}{\partial x^{2}}=0. \tag{2}\]  
Since the traveling wave generally exists in axially moving structures, herein, the harmonic solution with traveling wave form is first introduced as  
\[w(x,\ t)=C\text{e}^{\text{i}(\omega t+kx)}, \tag{3}\]  
where \(C\) and \(\omega\) (\(\omega=2\pi f\)) are the amplitude and angular frequency of the wave, respectively, \(k\) is the wavenumber and i the imaginary unit. Substitution of Equation (3) into Equation (2) leads to the dispersion equation as  
\[EIk^{4}-m_{\text{f}}U^{2}k^{2}-2m_{\text{f}}U\omega k-\left(m_{\text{f}}+m_{ \text{p}}\right)\omega^{2}=0. \tag{4}\]  
For a given frequency \(\omega\), the wavenumber \(k\) can be generally solved as four roots from Equation (4): two real ones and a conjugate pair of imaginary ones, denoted by \(k_{n}\)\(n=1\), 2, 3, 4. The real wavenumbers indicate propagating waves, while the imaginary ones describe evanescent waves. Once \(k\) is determined from Equation (4), the deflection and slope of the pipe can be expanded as  
\[\left\{ \begin{array}{l}w(x,\ t)=\sum\limits_{j=1}^{4}C_{j}\mathrm{e}^{ \mathrm{i}(\omega t+k_{j}x)}\\ \theta(x,\ t)=\sum\limits_{j=1}^{4}\mathrm{i}k_{j}C_{j}\mathrm{e}^{\mathrm{i} (\omega t+k_{j}x)}\end{array} \right. \tag{5}\]  
Substituting Equation (5) into Equation (1), the shear force and bending moment can be further obtained as  
\[\left\{