## Appendix A The Propagation of Elastic Waves in the Rod  
To study the dispersive propagation of elastic waves in an infinite rod, [PERSON] provided the motion equation of the element in a rod based on the Navier-Stokes equation in cylindrical coordinates [33].  
\[(\lambda+2\mu)\frac{\partial\phi}{\partial r}-\frac{2\mu}{r}\frac{\partial\omega z _{z}}{\partial\theta}+2\mu\frac{\partial\omega_{\theta}}{\partial z}=\rho\frac {\partial^{2}u_{r}}{\partial t^{2}} \tag{11}\]  
\[(\lambda+2\mu)\frac{\partial\phi}{\partial\theta}-2\mu\frac{\partial\omega_{r} }{\partial z}+2\mu\frac{\partial\omega_{z}}{\partial r}=\rho\frac{\partial^{2 }u_{\theta}}{\partial t^{2}} \tag{12}\]  
\[(\lambda+2\mu)\frac{\partial\phi}{\partial z}-\frac{2\mu}{r}\frac{\partial(r \omega_{\theta})}{\partial r}+\frac{2\mu}{r}\frac{\partial\omega_{r}}{ \partial\theta}=\rho\frac{\partial^{2}u_{z}}{\partial t^{2}} \tag{13}\]  
where, \(\mu\) is [PERSON]'s ratio, and \(u_{r}\), \(u_{\theta}\), and \(u_{z}\) are the radial, circumferential, and axial displacement components of the rod, respectively; \(\phi\) is the volume invariant in cylindrical coordinates, and \(\omega_{r}\), \(\omega_{\theta}\), and \(\omega_{z}\) are the three components of the rotation vector.  
According to the motion equation given by [PERSON], the vibration displacement in the rod conforms to the law of simple harmonic vibration.  
\[u_{r}=U(r)\cos n\theta e^{i(zz-\omega t)} \tag{14}\]  
\[u_{\theta}=V(r)\sin n\theta e^{i(zz-\omega t)} \tag{15}\]  
\[u_{z}=W(r)\cos n\theta e^{i(zz-\omega t)} \tag{16}\]  
where, \(U(r)\), \(V(r)\), and \(W(r)\) are the radial, circumferential, and axial displacement amplitude of the rod respectively, \(\theta\) is the circumference angle, \(\omega\) is the angular frequency; \(k\) is wave number, \(n=0,1,2\cdots\) is the order of the circumference direction.  
When \(n=0\), Equations (14)-(16) are the axisymmetric propagation of longitudinal waves in the rod. Figure 11 shows a schematic diagram of the longitudinal axisymmetric mode.  
The frequency equation of the longitudinal propagation mode is as follows [34]:  
\[\frac{2\alpha}{a}\left(\beta^{2}+k^{2}\right)J_{1}(\alpha a)J_{1}(\beta a)- \left(\beta^{2}-k^{2}\right)^{2}J_{0}(\alpha a)J_{1}(\beta a)-4k^{2}\alpha \beta J_{1}(\alpha a)J_{0}(\beta a)=0 \tag{17}\]  
where, \(\alpha^{2}=\frac{\omega^{2}}{c_{z}^{2}}-k^{2}\), \(\beta^{2}=\frac{\omega^{2}}{c_{z}^{2}}-k^{2}\), and \(a\) are the radius of the rod, \(k\) is the wave number, \(c_{L}\) is the longitudinal wave velocity, \(c_{T}\) is the transverse wave velocity, \(J_{0}()\) is the zero-order Bessel function, and \(J_{1}()\) is a first-order Bessel function.  
When the radial and axial displacement components of the rod are zero (\(u_{r}=0\), \(u_{z}=0\)), Equations (14)-(16) are the propagation of torsional waves in the rod. Figure 12 shows a schematic diagram of the torsional mode.  
The frequency equation of the torsional mode is as follows [35]:  
\[(\beta a)J_{0}(\beta a)-2J_{1}(\beta a)=0 \tag{18}\]where, \(a^{2}=\frac{\omega_{c}^{2}}{c_{L}^{2}}-k^{2}\), \(\beta^{2}=\frac{\omega_{c}^{2}}{c_{T}^{2}}-k^{2}\), and \(a\) are the radius of the rod, \(k\) is the wave number, \(c_{L}\) is the longitudinal wave velocity, \(c_{T}\) is the transverse wave velocity, \(J_{0}()\) is the zero-order Bessel function, and \(J_{1}()\) is a first-order Bessel function.  
From the frequency equation of torsional mode, it is known that torsional mode is the non-dispersive mode.  
\(n=1\) denotes the most important bending mode of the rod. Figure 11 shows the bending mode of the rod. For the frequency equation of bending mode, [PERSON] [36; 37] and [PERSON] [38] conducted in-depth research.