## 8 The Right Reflection Matrix. The Spectral Asymptotics  
In this section, we derive the constraints on the coefficients \(C_{i}(\lambda)\) and \(D_{i}(\lambda)\), \(i=1,2\), that follow from the fact that the general solution in Equation (179) has to satisfy two right-hand side boundary conditions. As we already know, the boundary conditions at the right end of the beam are  
\[K\big{(}\varphi(\lambda,L)-w^{\prime}(\lambda,L)\big{)}=i\lambda xw(\lambda,L ),\qquad\alpha>0, \tag{203}\]  
\[EI\varphi^{\prime}(\lambda,L)=\frac{i\lambda\theta}{C\Big{(}\frac{1}{C \mathbb{R}}+i\lambda\Big{)}}\varphi(\lambda,L),\qquad\theta,C,R>0. \tag{204}\]  
To rewrite the first condition in terms of \(\varphi\), we use Equations (180) for \(w\) and (166) for \(w^{\prime}\) to get the equation  
\[\lambda^{2}S\varphi q(\lambda,L)+EIq^{\prime\prime}(\lambda,L)=i\lambda \alpha S\varphi^{\prime}(\lambda,L)+\frac{i\alpha}{\lambda}\frac{EI}{\rho} \varphi^{\prime\prime\prime}(\lambda,L). \tag{205}\]  
Using Equations (177) and (178), we derive the general solution derivatives:  
\[ \begin{split}\Psi(\lambda,x)=& C_{1}(\lambda)\exp \bigg{(}\sqrt{\Lambda(\lambda)}x\bigg{)}+D_{1}(\lambda)\exp\bigg{(}-\sqrt{ \Lambda(\lambda)}x\bigg{)}+\\ & C_{2}(\lambda)\exp\bigg{(}\sqrt{\widetilde{\Lambda}(\lambda)} x\bigg{)}+D_{2}(\lambda)\exp\bigg{(}-\sqrt{\widetilde{\Lambda}(\lambda)}x \bigg{)},\\ \Psi^{\prime}(\lambda,x)=&\sqrt{\Lambda(\lambda)} \leavevmode\
obreak\ \leavevmode\
obreak\ C_{1}(\lambda)\exp\bigg{(}\sqrt{\Lambda(\lambda)}x \bigg{)}-\sqrt{\Lambda(\lambda)}\leavevmode\
obreak\ \leavevmode\
obreak\ D_{1}(\lambda)\exp\bigg{(}-\sqrt{\Lambda(\lambda)}x \bigg{)}+\\ &\sqrt{\widetilde{\Lambda}(\lambda)}\leavevmode\
obreak\ \leavevmode\
obreak\ C_{2}(\lambda)\exp\bigg{(}\sqrt{\widetilde{\Lambda}( \lambda)}x\bigg{)}-\sqrt{\widetilde{\Lambda}(\lambda)}\leavevmode\
obreak\ \leavevmode\
obreak\ D_{2}(\lambda)\exp\bigg{(}-\sqrt{\widetilde{\Lambda}( \lambda)}x\bigg{)},\\ \Psi^{\prime\prime\prime}(\lambda,x)=&\Lambda(\lambda)C_{1 }(\lambda)\exp\bigg{(}\sqrt{\Lambda(\lambda)}x\bigg{)}+\Lambda(\lambda)D_{1 }(\lambda)\exp\bigg{(}-\sqrt{\Lambda(\lambda)}x\bigg{)}+\\ &\widetilde{\Lambda}(\lambda)C_{2}(\lambda)\exp\bigg{(}\sqrt{ \widetilde{\Lambda}(\lambda)}x\bigg{)}+\widetilde{\Lambda}(\lambda)D_{2}( \lambda)\exp\bigg{(}-\sqrt{\widetilde{\Lambda}(\lambda)}x\bigg{)},\\ \Psi^{\prime\prime\prime\prime}(\lambda,x)=&\Big{(} \Lambda(\lambda)\Big{)}^{3/2}C_{1}(\lambda)\exp\bigg{(}\sqrt{\Lambda(\lambda)} x\bigg{)}-\Big{(}\Lambda(\lambda)\Big{)}^{3/2}D_{1}(\lambda)\exp\bigg{(}-\sqrt{ \Lambda(\lambda)}x\bigg{)}+\\ &\Big{(}\widetilde{\Lambda}(\lambda)\Big{)}^{3/2}C_{2}(\lambda) \exp\bigg{(}\sqrt{\widetilde{\Lambda}(\lambda)}x\bigg{)}-\Big{(}\widetilde{ \Lambda}(\lambda)\Big{)}^{3/2}D_{2}(\lambda)\exp\bigg{(}-\sqrt{\widetilde{ \Lambda}(\lambda)}x\bigg{)}.\end{split}  \tag{206}\]  
Substituting Equations (206) into the condition in Equation (205), we obtain  
\[ \begin{split}& C_{1}(\lambda)e(\lambda)\bigg{[}\lambda^{3}S \varphi^{2}+\lambda\rho EI\leavevmode\
obreak\ \leavevmode\
obreak\ \Lambda^{3/2}(\lambda)\bigg{]}+\\ & D_{1}(\lambda)e^{-1}(\lambda)\bigg{[}\lambda^{3}S\varphi^{2}+ \lambda\rho EI\leavevmode\
obreak\ \leavevmode\
obreak\ \Lambda^{3/2}(\lambda)\bigg{]}+\\ & C_{2}(\lambda)\widehat{e}(\lambda)\bigg{[}\lambda^{3}S\varphi^{2}+ \lambda\rho EI\leavevmode\
obreak\ \widetilde{\Lambda}(\lambda)-i\lambda^{2}\alpha S\rho\sqrt{\widetilde{ \Lambda}(\lambda)}-i\alpha EI\leavevmode\
obreak\ \leavevmode\
obreak\ \widetilde{\Lambda}^{3/2}(\lambda)\bigg{]}+\\ & D_{2}(\lambda)\widehat{e}^{-1}(\lambda)\bigg{[}\lambda^{3}S \varphi^{2}+\lambda\rho EI\leavevmode\
obreak\ \widetilde{\Lambda}(\lambda)+i\lambda^{2}\alpha S\rho\sqrt{\widetilde{\Lambda} (\lambda)}+i\alpha EI\leavevmode\
obreak\ \leavevmode\
obreak\ \widetilde{\Lambda}^{3/2}(\lambda)\bigg{]}=0.\end{split}  \tag{207}\]It is convenient to introduce new notations. Namely, let  
\[