## Appendix A  
The displacement components \(u_{r}\), \(u_{\theta}\) and \(u_{z}\) in Equation (5) can be expanded by a series of Legendre polynomials:  
\[\mathbf{u}=\sum_{n=0}^{N-1}\Psi_{I}^{n}P_{n}(\chi)\ l=r,\theta,z \tag{1}\]  
where \(P_{n}(\chi)\) is the n-th Legendre polynomial and \(\Psi_{I}^{n}\) are three component quantities of the expansion coefficients at a provided polynomial order. Considering that the domain of radius \(r\) associated with the displacement component is \([a,b]\), while the domain of Legendre polynomial is \([-1,1]\), the argument must be converted as:  
\[\chi=\frac{2r-(a+b)}{b-a} \tag{2}\]  
then the displacement components \(u_{i}(i=r,\theta,z)\) can be expressed as:  
\[u_{i}=\sum_{n=0}^{N-1}\Psi_{i}^{n}P_{n}(\chi) \tag{3}\]  
In addition, the orthogonality of Legendre polynomial can be expressed as:  
\[ \begin{split}\int_{-1}^{1}P_{n}(x)P_{m}(x)dx&=\frac {2}{2m+1}=\frac{2}{2n+1}& n=m\\ \int_{-1}^{1}P_{n}(x)P_{m}(x)dx&=0& n \
eq m\end{split}  \tag{4}\]It can be derived by Equation (A4) that:  
\[\sum_{n=0}^{N-1}\int_{-1}^{1}P_{n}(\chi)P_{m}(\chi)d\chi=\frac{2}{2m+1}\] (A5)  
Combining Equations (A3) and (A5), an equation of the displacement component can be derived as:  
\[\int_{-1}^{1}u_{i}P_{m}(x)dx=\sum_{n=0}^{N-1}\int_{-1}^{1}\Psi_{i}^{n}P_{n}( \chi)P_{m}(x)dx=\sum_{n=0}^{N-1}\frac{2}{2n+1}\Psi_{i}^{n}\] (A6)  
There are several derivative terms in Equation (5), which interferes with the calculation of analytical solution. Therefore, it is necessary to transform it into forms that can be simplified by Equation (A6). First, an axially propagating wave can be expressed as:  
\[u_{i}=u_{i}(r)e^{i[kz+n\theta-\omega t]}\] (A7)  
where \(k\) is the axial wave number, \(n\) is the circumferential wave number and \(\omega\) is the angular frequency. The result of its partial derivatives are as follows:  
\[ \begin{array}{c}\frac{\partial u_{i}}{\partial\theta}=jnu_{i}\\ \frac{\partial u_{i}}{\partial z}=jku_{i}\\ \frac{\partial u_{i}}{\partial t}=-j\omega u_{i}\end{array} \] (A8)  
By using Equation (A8) to simplify the partial derivative term in Equation (5), we can obtain:  
\[ \begin{array}{c}\lambda\frac{\partial^{2}u_{r}}{\partial r^{2}}+\frac{(jn)^ {2}}{r^{2}}Gu_{r}+(jk)^{2}Gu_{r}+\frac{(jn)}{r}(\lambda+3G)\frac{\partial u_{ \theta}}{\partial r}+(jk)(\lambda+3G)\frac{\partial u_{r}}{\partial r}\\ +\frac{1}{r}\lambda\frac{\partial u_{r}}{\partial r}-\frac{(jn)}{r^{2}}( \lambda+G)u_{\theta}-\frac{1}{r^{2}}\lambda u_{r}=(j\omega)^{2}\rho u_{r}\\ G\frac{\partial^{2}u_{\theta}}{\partial r^{2}}+\frac{(jn)^{2}}{r^{2}}\lambda u _{\theta}+(jk)^{2}Gu_{\theta}+\frac{(jn)}{r}(\lambda+3G)\frac{\partial u_{r}} {\partial r}+\frac{(jn)(jk)}{r}(\lambda+3G)u_{z}\\ +\frac{1}{r}G\frac{\partial u_{\theta}}{\partial r}+\frac{(jn)}{r^{2}}( \lambda+G)u_{r}-\frac{1}{r^{2}}Gu_{\theta}=(j\omega)^{2}\rho u_{\theta}\\ G\frac{\partial^{2}u_{r}}{\partial r^{2}}+\frac{(jn)^{2}}{r^{2}}Gu_{z}+(jk)^{2 }\lambda u_{z}+(jk)(\lambda+3G)\frac{\partial^{2}u_{r}}{\partial r}+\frac{(jn )(jk)}{r}(\lambda+3G)u_{\theta}\\ +\frac{(jk)}{r}(\lambda+3G)u_{r}+\frac{1}{r}G\frac{\partial u_{r}}{\partial r }=(j\omega)^{2}\rho u_{z}\end{array} \] (A9)  
After using Equation (A1) to perform Legendre transformation on Equation (A9), we can use Equation (A6) to simplify some terms, and the remaining terms are \(\chi P_{n}(\chi)\), \(\chi^{2}P_{n}(\chi)\), \(\frac{\partial P_{n}(\chi)}{\partial\chi}\), \(\chi\frac{\partial P_{n}(\chi)}{\partial\chi}\), \(\chi^{2}\frac{\partial P_{n}(\chi)}{\partial\chi}\), \(\frac{\partial^{2}P_{n}(\chi)}{\partial\chi^{2}}\), \(\chi\frac{\partial^{2}P_{n}(\chi)}{\partial\chi^{2}}\) and \(\chi^{2}\frac{\The simplification of the above eight terms must be carried out based on the definition of Legendre polynomial, Equation (A1). The simplified equations are shown as:  
\[