## 2 Theoretical Analyses  
As is shown in Figure 1, a hollow cylinder is represented in the column coordinate system, with the axial direction extending indefinitely along the z-axis and the inner and outer surfaces satisfying the stress-free boundary condition. It is assumed that the inner radius of the cylinder is a, the outer radius is b, and the outer-radius-to-wall-thickness ratio is \(\eta=\text{b}/(\text{b}-\text{a})\). The column coordinate system uses r, \(\theta\), and z to indicate the cylinder's three directions: radial, circumferential, and axial. The thermoelastic constitutive equation for orthotropic anisotropic materials in the column coordinate system can be expressed as:  
\[\left\{ \begin{array}{c}\sigma_{\theta\theta}\\ \sigma_{zz}\\ \sigma_{rr}\\ \sigma_{r\varepsilon}\\ \sigma_{r\theta}\\ \sigma_{z\theta}\end{array} \right\}=\left[ \begin{matrix}c_{11}&c_{12}&c_{13 }&0&0&0\\ &c_{22}&c_{23}&0&0&0\\ &&c_{33}&0&0&0\\ &&&c_{44}&0&0\\ &symmetry&&c_{55}&0\\ &&&c_{66}\end{matrix} \right]\left\{ \begin{array}{c}\varepsilon_{\theta \theta}\\ \varepsilon_{zz}\\ \varepsilon_{rr}\\ 2\varepsilon_{r\theta}\\ 2\varepsilon_{z\theta}\end{array} \right\}-\left\{ \begin{array}{c}\beta_{1} \\ \beta_{2}\\ \beta_{3}\\ 0\\ 0\\ 0\end{array} \right\}\{T\} \tag{1}\]  
where \(T\) is the temperature change above the uniform reference temperature \(T_{0}\). The geometric relations under the column coordinate system in the small deformation assumption are:  
\[ \begin{array}{l}\varepsilon_{\theta\theta}=\frac{1}{r}\frac{\partial u_{ \theta}}{\partial\theta}+\frac{\mu}{r},\varepsilon_{zz}=\frac{\partial u_{z}}{ \partial\varepsilon},\varepsilon_{rr}=\frac{\partial u_{r}}{\partial r}\,\\ \varepsilon_{r\theta}=\frac{1}{2}\left(\frac{\partial u_{r}}{\partial\theta}+ \frac{\partial u_{z}}{\partial r}-\frac{u_{z}}{r}\right),\varepsilon_{rz}= 2\left(\frac{\partial u_{r}}{\partial z}+\frac{\partial u_{z}}{\partial r} \right),\\ \varepsilon_{\theta z}=\frac{1}{2}\left(\frac{\partial u_{z}}{\partial z}+ \frac{1}{\partial u_{z}}\right)\end{array}  \tag{2}\]  
Figure 1: Hollow cylinder in the cylindrical coordinate system. When the volume force and heat source are both zero, the wave governing equation can be calculated using the GN generalized thermoelastic theory without energy dissipation combined with the equation of motion:  
\[\left\{ \begin{array}{l}\frac{\partial\sigma_{r\theta}}{\partial r}+\frac{1}{r} \frac{\partial\sigma_{r\theta}}{\partial\theta}+\frac{\partial\sigma_{r\theta} }{\partial z}+\frac{\sigma_{r\theta}-\sigma_{\theta\theta}}{r}=\rho\frac{ \partial^{2}u_{r}}{\partial l^{2}}\\ \frac{\partial\sigma_{r\theta}}{\partial r}+\frac{1}{r}\frac{\partial\sigma_{ \theta\theta}}{\partial\theta}+\frac{\partial\sigma_{\theta\theta}}{\partial z }+\frac{2\sigma_{r\theta}}{r}=\rho\frac{\partial^{2}u_{r}}{\partial l^{2}}\\ \frac{\partial\sigma_{r\theta}}{\partial r}+\frac{1}{r}\frac{\partial\sigma_{ r\theta}}{\partial\theta}+\frac{\partial\sigma_{rz}}{\partial\theta}+\frac{\sigma_{rz}}{ \sigma_{r\theta}}=\rho\frac{\partial^{2}u_{r}}{\partial l^{2}}\\ K_{3}\frac{\partial^{2}T}{\partial r^{2}}+K_{1}\left(\frac{1}{r}\frac{1}{r} \frac{1}{r}\frac{1}{r^{2}}\frac{\partial^{2}T}{\partial l^{2}}\right)+K_{2} \frac{\partial^{2}T}{\partial z^{2}}-T_{0}\frac{\partial^{2}}{\partial l^{2}} (\beta_{2}\varepsilon_{zz}+\beta_{3}\varepsilon_{rr}+\beta_{1}\varepsilon_{ \theta\theta})=\rho C_{e}\frac{\partial^{2}T}{\partial l^{2}}\end{array} \right. \tag{3}\]  
In the above Equations (1)-(3), \(\sigma_{ij}\), \(u_{i}\), and \(\varepsilon_{ij}\) represent stress, displacement, and strain, respectively; \(c_{ij}\) represents the elastic constant of the material; \(K_{i}\) is the constant property of the theoretical material; \(\beta_{i}\) represents the coefficient of volumetric expansion; and \(C_{e}\) represents the specific heat at a constant strain. In order to satisfy the stress boundary conditions for the hollow cylinder, a rectangular window function is introduced:  
\[\pi(r)=\left\{ \begin{array}{l}1,a\leq r\leq b\\ 0\ \ \textit{elsewhere}\end{array}