## 8 The Right Reflection Matrix. The Spectral Asymptotics
\begin{array}{ll}a(\lambda)=\lambda^{3}S\rho^{2}+\lambda\rho EI\Lambda(\lambda),&\widetilde{a}(\lambda)=\lambda^{3}S\rho^{2}+\lambda\rho EI\widetilde{\Lambda} (\lambda),\\ b(\lambda)=i\lambda^{2}\alpha S\rho\sqrt{\Lambda(\lambda)}+i\alpha EI\Lambda^ {3/2}(\lambda),&\widetilde{b}(\lambda)=i\lambda^{2}\alpha S\rho\sqrt{ \widetilde{\Lambda}(\lambda)}+i\alpha EI\widetilde{\Lambda}^{3/2}(\lambda). \end{array}  \tag{208}\]  
Rewriting Equation (207) in terms of Equation (208), we have  
\[ \begin{array}{ll}C_{1}(\lambda)\epsilon(\lambda)\Big{[}a(\lambda)-b(\lambda) \Big{]}+D_{1}(\lambda)e^{-1}(\lambda)\Big{[}a(\lambda)+b(\lambda)\Big{]}+\\ C_{2}(\lambda)\widetilde{e}(\lambda)\Big{[}\widetilde{a}(\lambda)- \widetilde{b}(\lambda)\Big{]}+D_{2}(\lambda)\widehat{e}^{\,-1}(\lambda) \Big{[}\widetilde{a}(\lambda)+\widetilde{b}(\lambda)\Big{]}=0.\end{array}  \tag{209}\]  
Now, we turn to the second boundary condition in Equation (204). Let  
\[\Theta(\lambda)=\frac{\lambda\theta^{2}}{C\Big{(}\frac{1}{C\mathcal{R}}+i \lambda\Big{)}}. \tag{210}\]  
Then, Equation (204) becomes  
\[EI\varphi^{\prime}(\lambda,L)=\Theta(\lambda)\varphi(\lambda,L). \tag{211}\]  
The second boundary condition generates the second relation between \(C_{i}(\lambda)\) and \(D_{i}(\lambda)\), \(i=1,2\)  
\[ \begin{array}{ll}C_{1}(\lambda)e(\lambda)\Big{[}EI\sqrt{\Lambda(\lambda)}-i \Theta(\lambda)\Big{]}-D_{1}(\lambda)\big{(}e(\lambda)\big{)}^{-1}\Big{[}EI \sqrt{\Lambda(\lambda)}+i\Theta(\lambda)\Big{]}\\ +C_{1}(\lambda)\widetilde{e}(\lambda)\Big{[}EI\sqrt{\widetilde{\Lambda}( \lambda)}-i\Theta(\lambda)\Big{]}-D_{1}(\lambda)\big{(}\widetilde{e}(\lambda) \big{)}^{-1}\Big{[}EI\sqrt{\widetilde{\Lambda}(\lambda)}+i\Theta(\lambda) \Big{]}=0.\end{array}  \tag{212}\]  
Let us introduce two two-vectors:  
\[\mathbb{C}(\lambda)=\left[ \begin{array}{c}C_{1}(\lambda)\\ C_{2}(\lambda)\end{array} \right],\qquad\mathbb{D}(\lambda)=\left[ \begin{array} []{c}D_{1}(\lambda)\\ D_{2}(\lambda)\end{array} \right]. \tag{213}\]  
Equations (209) and (212) can be written in a matrix form as  
\[\mathbb{A}(\lambda)\mathbb{E}(\lambda)\left[ \begin{array}{c}C_{1}(\lambda) \\ C_{2}(\lambda)\end{array} \right]=\mathbb{B}(\lambda)\mathbb{E}^{-1}(\lambda) \left[ \begin{array}{c}D_{1}(\lambda)\\ D_{2}(\lambda)\end{array} \right], \tag{214}\]  
where  
\[\mathbb{A}(\lambda)=\left[ \begin{array}{ll}a(\lambda)-b(\lambda)&\widetilde{ a}(\lambda)-\widetilde{b}(\lambda)\\ EI\sqrt{\Lambda(\lambda)}-i\Theta(\lambda)&EI\sqrt{\widetilde{\Lambda}( \lambda)}-i\Theta(\lambda)\end{array} \right], \tag{215}\]  
\[\mathbb{B}(\lambda)=\left[ \begin{array}{ll}-a(\lambda)+b(\lambda)&- \widetilde{a}(\lambda)+\widetilde{b}(\lambda)\\ EI\sqrt{\Lambda(\lambda)}-i\Theta(\lambda)&EI\sqrt{\widetilde{\Lambda}( \lambda)}-i\Theta(\lambda)\end{array} \right], \tag{216}\]  
and \(\mathbb{E}(\lambda)\) is a diagonal matrix: \(\mathbb{E}(\lambda)=\text{diag}\Big{\{}e(\lambda),\widehat{e}(\lambda)\Big{\}}\). The matrix that relates vectors \(\mathbb{C}(\lambda)\) and \(\mathbb{D}(\lambda)\) of Equation (213), given by \(\mathfrak{R}_{r}(\lambda)=\mathbb{E}^{-1}(\lambda)\mathbb{A}^{-1}(\lambda) \mathbb{B}(\lambda)\mathbb{E}^{-1}(\lambda)\), is called the right reflection matrix. The next statement is the main result of this Section. **Theorem 6**. : _(1) The initial-boundary value problem in Equations (161)-(164) has a countable set of complex vibrational modes, which belongs to a strip parallel to the real axis and can accumulate only at infinity. There may be only a finite number of multiple modes of a finite algebraic multiplicity each._  
_(2) The entire set of modes asymptotically splits into two disjoint subsets: we call them the \(\alpha\)-branch and the \(\theta\)-branch and denote them by \(\left\{\lambda_{n}^{\alpha}\right\}_{n\in\mathbb{Z}^{\prime}}\) and \(\left\{\lambda_{m}^{\theta}\right\}_{m\in\mathbb{Z}^{\prime}}\), \(\mathbb{Z}^{\prime}=\mathbb{Z}\diagdown\{0\}\).