## 2 Theoretical Analyses
\right. \tag{4}\]  
Introducing this into the constitutive equations yields stress-free boundary conditions at \(r=a\) and \(r=b\) with \(\sigma_{rr}=\sigma_{rz}=\sigma_{t\theta}=0\):  
\[\left\{ \begin{array}{l}\sigma_{\theta\theta}\\ \sigma_{zz}\\ \sigma_{rr}\\ \sigma_{r\theta}\\ \sigma_{z\theta}\end{array} \right\}=\left[ \begin{array}{cccc}c_{11}&c_{12} &c_{13}&0&0&0\\ &c_{22}&c_{23}&0&0&0\\ &&c_{33}&0&0&0\\ &&c_{44}&0&0\\ &symmetry&&c_{55}&0\\ &&c_{66}\end{array} \right]\left\{ \begin{array}{l}\varepsilon_{\theta\theta} \\ \varepsilon_{zz}\\ \varepsilon_{rr}\\ 2\varepsilon_{rz}\\ 2\varepsilon_{z\theta}\end{array} \right\}\pi(r)-\left\{ \begin{array}{l} \beta_{1}\\ \beta_{2}\\ \beta_{3}\\ 0\\ 0\\ 0\end{array} \right\}\left\{T\right\}\pi(r) \tag{5}\]  
For a free harmonic wave propagating along the longitudinal direction in an infinitely long hollow cylinder, let its displacement and the component of temperature change be:  
\[ \begin{array}{l}u_{r}(r,\theta,z,t)=\exp(ikz+in\theta-i\omega t)U(r)\\ u_{\theta}(r,\theta,z,t)=\exp(ikz+in\theta-i\omega t)V(r)\\ u_{z}(r,\theta,z,t)=\exp(ikz+in\theta-i\omega t)W(r)\\ T(r,\theta,z,t)=\exp(ikz+in\theta-i\omega t)X(r)\end{array}  \tag{6}\]  
where \(U(r)\), \(V(r)\), and \(W(r)\) denote the displacement amplitude along the three directions \(r\), \(\theta\), and \(z\), respectively, \(X(r)\) denotes the temperature amplitude along the \(r\) direction, \(k\) denotes the wave number, \(\omega\) is the angular frequency, and \(n\) denotes the number of circumferential orders. Substituting Equations (2), (5) and (6) into Equation (3) to eliminate the strain, the control equation can be obtained as follows:  
\[ \begin{array}{l}\left\{inc_{31}V^{\prime}+ikr^{2}c_{32}W^{\prime}+r^{2}c_{ 33}U^{\prime\prime}-r^{2}\beta_{3}X^{\prime}-n^{2}c_{55}U+inc_{55}V^{\prime}- inc_{55}V-k^{2}r^{2}c_{44}U\\ +ikr^{2}c_{44}W^{\prime}+rikc_{32}W+rc_{33}U^{\prime}-r\beta_{3}X-inc_{11}V- c_{11}U-ikrc_{12}W+r\beta_{1}X\right\}\pi(r)\\ +\left\{inc_{31}V+rc_{31}U+ikr^{2}c_{32}W+r^{2}c_{33}U^{\prime}-r^{2}\beta_{3}X \right\}[\delta(r-a)-\delta(r-b)]=-r^{2}\rho\omega^{2}U\\ \\ \left\{inc_{55}U^{\prime}+r^{2}c_{55}V^{\prime\prime}-nc^{2}c_{11}V+inc_{11}U- nkrc_{12}W+inc_{13}U^{\prime}-int\beta_{1}X\right.\\ \left.-k^{2}r^{2}c_{66}V-nkrc_{66}W+inc_{55}U+rc_{55}V^{\prime}-c_{55}V\right\} \pi(r)+\left\{inc_{55}U\right.\\ \left.+r^{2}c_{55}V^{\prime}-rc_{55}V\right\}\left[\delta(r-a)-\delta(r-b) \right]=-r^{2}\rho\omega^{2}V\end{array} \right. \tag{7a}\]\[ \begin{array}{l}\{ikr^{2}c_{44}U^{\prime}+r^{2}c_{44}W^{\prime\prime}-nkrc_{66}V- n^{2}c_{66}W-nkrc_{21}V+ikrc_{21}U \tag{7c}\] \[-k^{2}r^{2}c_{22}W+ikr^{2}c_{23}U^{\prime}-ikr^{2}\beta_{2}X+ikrc_{44}U+rc_{44}W^ {\prime}\}\pi(r)+\{ikr^{2}c_{44}U\] \[+r^{2}c_{44}W^{\prime}\}[\delta(r-a)-\delta(r-b)]=-r^{2}\rho\omega^{2}W\\ \\ r^{2}K_{3}X^{\prime\prime}+rK_{1}X^{\prime}-n^{2}K_{1}X-k^{2}r^{2}K_{2}X+ikr^{ 2}T_{0}\omega^{2}\beta_{2}W+r^{2}T_{0}\omega^{2}\beta_{3}U^{\prime}\\ +inrT_{0}\omega^{2}\beta_{1}V+rT_{0}\omega^{2}\beta_{1}U=-r^{2}\rho C_{\epsilon }\omega^{2}X\end{array}  \tag{7d}\]  
Since \(r^{2}T_{0}\omega^{2}\beta_{3}U^{\prime}\) exists in Equation (7d), we must now address this equation:  
\[\begin{array}{l}\rho r\big{(}r^{2}K_{3}X^{\prime\prime}+rK_{1}X^{\prime}-n^{ 2}K_{1}X-k^{2}r^{2}K_{2}X\big{)}+ikT_{0}\beta_{2}r\cdot\rho r^{2}\omega^{2}W+T _{0}\beta_{3}r\cdot\rho r^{2}\omega^{2}U^{\prime}\\ +in\beta_{1}T_{0}\rho r^{2  
The 4(M + 1) equations can be obtained by multiplying both sides of Equations (7a)-(7d) concurrently by \(Q_{j}^{\prime}(r)\) picked from 0 to M. Then, using [PERSON]'s polynomials' orthogonality feature, one can integrate the radius \(r\) from \(a\) to \(b\):  
\[\left\{