## 2 Analytical Model for Guided Waves in a Hollow Cylinder  
To study the axial propagation characteristics of guided waves in the arterial wall, we consider the vessel as a hollow cylinder structure with an inner diameter \(a\) and outer diameter \(b\). As shown in Figure 1, a cylindrical coordinate system with radial direction \(r\), circumferential direction \(\theta\) and axial direction \(z\) is established, and the coordinate axes are aligned with the center line of the cylinder. Assuming that the arterial wall is uniform, isotropic, and elastic, the equilibrium equation in the column coordinate system [23] can be expressed as:  
\[ \begin{split}\frac{\partial\sigma_{rr}}{\partial r}+\frac{1}{r} \frac{\partial\sigma_{\theta}}{\partial\theta}+\frac{\partial\sigma_{zz}}{ \partial z}+\frac{\sigma_{rr}-\sigma_{\theta\theta}}{r}=\rho\frac{\partial^{2 }u_{z}}{\partial t^{2}}\\ \frac{\partial\sigma_{\theta\theta}}{\partial r}+\frac{1}{r} \frac{\partial\sigma_{\theta\theta}}{\partial\theta}+\frac{\partial\sigma_{ \theta z}}{\partial z}+\frac{2\sigma_{\theta\theta}}{r}=\rho\frac{\partial^{2 }u_{\theta}}{\partial t^{2}}\\ \frac{\partial\sigma_{zz}}{\partial r}+\frac{1}{r}\frac{\partial \sigma_{yz}}{\partial\theta}+\frac{\partial\sigma_{zz}}{\partial z}+\frac{ \sigma_{zz}}{r}=\rho\frac{\partial^{2}u_{\theta}}{\partial t^{2}}\end{split}  \tag{1}\]  
where \(\sigma_{ij}\) is the component of the stress tensor in the blood vessel (\(i,j=r,\theta,z\)), \(u_{r}\), \(u_{\theta}\) and \(u_{z}\) is the radial, circumferential and axial displacement component, respectively, and \(\rho\) is the density of the vessel. For an isotropic material, the stress-strain relationship in the cylindrical coordinate system can be described as [24]:  
\[\sigma_{ij}=2G\varepsilon_{ij}+\lambda\varepsilon_{kk}\delta_{ij} \tag{2}\]  
where \(\varepsilon_{ij}\) is the strain tensor, \(\lambda\) is Lame constant, \(G\) is the shear modulus, \(\delta_{ij}\) is the Kronecker delta, and \(\varepsilon_{kk}=\varepsilon_{ii}+\varepsilon_{\theta\theta}+\varepsilon_{zz}\) is the strain tensor trace. \(\lambda\) and \(G\) are determined by vascular mechanical properties; it can also be expressed by [PERSON]'s modulus \(E\) and [PERSON]'s ratio \(\
u\) as [24]:  
\[ \begin{split}\lambda&=\frac{E_{\
u}}{(1+\
u)(1-2\
u )}\\ G&=\frac{E}{2(1+\
u)}\end{split}  \tag{3}\]  
Figure 1: Hollow cylinder model of arterial vessel. In the arterial model, the mechanical properties of representative blood vessels are shown in Table 1.  
In addition, in the column coordinate system, each strain component can be represented by displacement as:  
\[ \begin{array}{l}\varepsilon_{rr}=\frac{\partial u_{r}}{\partial r}\ \varepsilon_{\theta \theta}=\frac{u_{r}}{r}+\frac{1}{r}\frac{\partial u_{\theta}}{\partial\theta} \ \varepsilon_{zz}=\frac{\partial u_{z}}{\partial z}\\ \varepsilon_{rz}=\frac{\partial u_{z}}{\partial r}+\frac{\partial u_{r}}{ \partial z}\ \varepsilon_{\theta z}=\frac{1}{r}\frac{\partial u_{z}}{ \partial\theta}+\frac{\partial u_{\theta}}{\partial z}\\ \varepsilon_{r\theta}=\frac{1}{r}\frac{\partial u_{\theta}}{\partial\theta}+ \frac{\partial u_{\theta}}{\partial r}-\frac{u_{\theta}}{r}\\ \end{array}  \tag{4}\]  
By combining Equations (1)-(4), and eliminating the stress and strain tensors, it can be derived that:  
\[