## 7 The Harvester Model with Constant Physical Parameters. The Left Reflection Matrix
\begin{split}&\sqrt{\Lambda(\lambda)}\Big{(}\lambda^{2}S\rho+ \Lambda(\lambda)EI\Big{)}C_{1}(\lambda)+\sqrt{\widetilde{\Lambda}(\lambda)} \Big{(}\lambda^{2}S\rho+\widetilde{\Lambda}(\lambda)EI\Big{)}C_{2}(\lambda)\\ -&\sqrt{\Lambda(\lambda)}\Big{(}\lambda^{2}S\rho+ \Lambda(\lambda)EI\Big{)}D_{1}(\lambda)-\sqrt{\widetilde{\Lambda}(\lambda)} \Big{(}\lambda^{2}S\rho+\widetilde{\Lambda}(\lambda)EI\Big{)}D_{2}(\lambda)=0.\end{split}  \tag{183}\]  
Introducing a new notation  
\[\mathbb{S}(\lambda)=\sqrt{\frac{\widetilde{\Lambda}(\lambda)}{\Lambda( \lambda)}}\;\;\frac{\lambda^{2}S\rho+\widetilde{\Lambda}(\lambda)EI}{\lambda^ {2}S\rho+\Lambda(\lambda)EI} \tag{184}\]  
we reduce the system of Equations (182) to the following form:  
\[ \begin{cases}C_{1}(\lambda)+C_{2}(\lambda)+D_{1}(\lambda)+D_{2}(\lambda)=0\\ C_{1}(\lambda)+\mathbb{S}(\lambda)C_{2}(\lambda)-D_{1}(\lambda)-\mathbb{S}( \lambda)D_{2}(\lambda)=0.\end{cases}  \tag{185}\]  
By subtracting the first equation from the second one we immediately obtain that \(C_{2}(\lambda)\) can be represented as a linear combination of the coefficients \(D_{1}(\lambda)\) and \(D_{2}(\lambda)\):  
\[C_{2}(\lambda)=\frac{2}{\mathbb{S}(\lambda)-1}\;\;D_{1}(\lambda)+\frac{\mathbb{ S}(\lambda)+1}{\mathbb{S}(\lambda)-1}\;\;D_{2}(\lambda). \tag{186}\]  
Rewriting Equations (185) in the form  
\[ \begin{cases}C_{1}(\lambda)+C_{2}(\lambda)+D_{1}(\lambda)+D_{2}(\lambda)=0\\ \frac{1}{\mathbb{S}(\lambda)}C_{1}(\lambda)+C_{2}(\lambda)-\frac{1}{\mathbb{S }(\lambda)}D_{1}(\lambda)-D_{2}(\lambda)=0\end{cases}  \tag{187}\]  
we obtain \(C_{1}(\lambda)\) as a linear combination of the coefficients \(D_{1}(\lambda)\) and \(D_{2}(\lambda)\)  
\[C_{1}(\lambda)=-\frac{\mathbb{S}(\lambda)+1}{\mathbb{S}(\lambda)-1}\;\;D_{1}( \lambda)-\frac{2\mathbb{S}(\lambda)}{\mathbb{S}(\lambda)-1}\;\;D_{2}(\lambda). \tag{188}\]Rewriting Equations (186) and (188) as as one matrix equation, we obtain  
\[\left[ \begin{array}{c}C_{1}(\lambda)\\ C_{2}(\lambda)\end{array} \right]=\left[ \begin{array}{cc}-\frac{\mathbb{S}( \lambda)+1}{\mathbb{S}(\lambda)-1}&-\frac{2\mathbb{S}(\lambda)}{\mathbb{S}( \lambda)-1}\\ \frac{2}{\mathbb{S}(\lambda)-1}&\frac{\mathbb{S}(\lambda)+1}{\mathbb{S}( \lambda)-1}\end{array} \right]\left[ \begin{array}{c}D_{1}(\lambda)\\ D_{2}(\lambda)\end{array} \right]. \tag{189}\]  
The matrix at the right of this equation we will call the _left reflection matrix_ and denote it by \(\mathfrak{H}_{l}(\lambda)\). **Asymptotic approximations for the entries of \(\mathfrak{H}_{l}(\lambda)\) as \(|\lambda|\to\infty\). **  
In the sequel, we need the asymptotic approximation for \(\mathbb{S}(\lambda)\) as \(|\lambda|\to\infty\). Using Equations (172), (175) and (177), we obtain  
\[ \begin{split}\sqrt{\Lambda(\lambda)}&=i\lambda\sqrt{\frac {\rho}{K}}\ \ \left(1+\frac{1}{\lambda^{2}\big{(}\rho/K\big{)}\left(EI/K-S \right)}+\mathcal{O}\bigg{(}\frac{1}{\lambda^{4}}\bigg{)}\right)^{1/2}\\ &=i\lambda\sqrt{\frac{\rho}{K}}\ \ \left(1+\frac{1}{2\lambda^{2} \big{(}\rho/K\big{)}\left(EI/K-S\right)}+\mathcal{O}\bigg{(}\frac{1}{\lambda^{ 4}}\bigg{)}\right).\end{split}  \tag{190}\]  
Using Equations (172), (176) and (178), we obtain  
\[ \begin{split}\sqrt{\widetilde{\Lambda}(\lambda)}&=i \lambda\sqrt{\frac{\rho}{EI}}\ \ \left(1-\frac{1}{\lambda^{2}\big{(}S\rho/EI\big{)}\left(EI/K-S \right)}+\mathcal{O}\bigg{(}\frac{1}{\lambda^{4}}\bigg{)}\right)^{1/2}\\ &=i\lambda\sqrt{\frac{\rho}{EI}}\ \ \left(1-\frac{1}{2\lambda^{2} \big{(}S\rho/K-S^{2}\rho/EI\big{)}}+\mathcal{O}\bigg{(}\frac{1}{\lambda^{4}} \bigg{)}\right).\end{split}  \tag{191}\]  
Equations (190) and (191) yield the following result:  
\[ \begin{split}\sqrt{\frac{\widetilde{\Lambda}(\lambda)}{\Lambda( \lambda)}}&=\sqrt{\frac{S\rho\cdot K}{EI\cdot\rho}}\left(1-\frac{ 1}{2\lambda^{2}\big{(}S\rho/K-S^{2}\rho/EI\big{)}}+\mathcal{O}\bigg{(}\frac{1 }{\lambda^{4}}\bigg{)}\right)\times\\ &\qquad\qquad\qquad\qquad\qquad\left(1+\frac{1}{2\lambda^{2} \big{(}\rho/K\big{)}\left(EI/K-s\right)}+\mathcal{O}\bigg{(}\frac{1}{\lambda^{ 4}}\bigg{)}\right)^{-1}.\end{split}