# Partial Energy Transfer Model of Lamb Waves Scattering in Materially Isotropic Waveguides
## 2 Mode Matching Technique
### Scattering of Lamb Waves at Vertical Notch
Using the biorthogonality relation (Equation (7)) and the related systematic projections of the individual vectors \(\mathbf{M},\mathbf{M}_{n}\), \(\mathbf{N},\mathbf{N}_{n}\), it is possible to separate the \(m_{n}\), \(n_{n}\) variables, which are affiliated to the main waveguide (section number 1--see Figure 2):  
\[\left.({}_{1}\mathbf{M}|_{1}\mathbf{N}_{w})\right|_{t_{\text{fl}}}=\left.({}_{2}\mathbf{M} |_{1}\mathbf{N}_{w})\right|_{t_{\text{fl}}}+\left.({}_{3}\mathbf{M}|_{1}\mathbf{N}_{w}) \right|_{t_{\text{fl}}}+\left.({}_{4}\mathbf{M}|_{1}\mathbf{N}_{w})\right|_{t_{\text{fl }}}={}_{1}m_{w1}\Lambda_{w}, \tag{17}\]  
\[\left.({}_{1}\mathbf{M}_{n}|_{1}\mathbf{N})\right|_{t_{\text{fl}}}=\left.({}_{1}\mathbf{M} _{n}|_{2}\mathbf{N})\right|_{t_{\text{fl}}}+\left.({}_{1}\mathbf{M}_{n}|_{3}\mathbf{N}) \right|_{t_{\text{fl}}}+\left.({}_{1}\mathbf{M}_{n}|_{4}\mathbf{N})\right|_{t_{\text{fl }}}={}_{1}n_{1}\Lambda_{n}, \tag{18}\]  
and for the present sub-waveguides denoted by section numbers 2-4 (see Figure 2):  
\[\left.({}_{1}\mathbf{M}|_{m}\mathbf{N}_{n})\right|_{t_{\text{fl}}}=\sum\limits_{w=1}^{ N}{}_{1}m_{w}\left.({}_{1}\mathbf{M}_{w}|_{m}\mathbf{N}_{n})\right|_{t_{\text{fl}}}={}_{m}m_{ nm}\Lambda_{n}, \tag{19}\]  
\[\left.({}_{m}\mathbf{M}_{w}|_{1}\mathbf{N})\right|_{t_{\text{fl}}}=\sum\limits_{n=1}^{ N}{}_{1}n_{n}\left.({}_{m}\mathbf{M}_{w}|_{1}\mathbf{N}_{n})\right|_{t_{\text{fl}}}={}_{m}n_{ wm}\Lambda_{w}, \tag{20}\]  
where \(\left|{}_{t_{\text{fl}}}\right.\) denotes the integration over the specified sub-waveguide thickness in the form of the given inner product. After the rearrangement of Equations (17)-(20) [43, 44], we can derive the explicit formulation of the global scattering matrix, a linear operator, which links the vectors containing the complex amplitudes of the reflected waves \(v_{R}=\left[ \begin{array}{cc}1\mathbf{\alpha}^{-}(x_{1})&{}_{1}\mathbf{\alpha}^{+}(x_ {2})\end{array} \right]^{T}\) with the vector containing the complex amplitudes of the incident ones--\(v_{IN}=\left[ \begin{array}{cc}1\mathbf{\alpha}^{+}(x_{1})&{}_{1}\mathbf{\alpha}^{-}(x_ {2})\end{array} \right]^{T}\):  
\[\left[ \begin{array}{c}1\mathbf{\alpha}^{-}(x_{1})\\ 1\mathbf{\alpha}^{+}(x_{2})\end{array} \right]={}_{1}\mathbf{S}\left. (x_{1},x_{2})\left[ \begin{array}{c}1\mathbf{\alpha}^{+}(x_{1})\\ 1\mathbf{\alpha}^{-}(x_{2})\end{array} \right], \tag{21}\]where:  
\[{}_{1}\mathbf{S}\ (x_{1},x_{2})=\Bigg{\{}_{1}\mathbf{M}_{1}-\sum_{m=2}^{N_{\rm{D}}=3} \ _{m}\mathbf{M}_{2}+_{m}\mathbf{M}_{3}\Bigg{\}}^{-1}\Bigg{\{}\sum_{m=2}^{N_{\rm{D}}= 3}\ _{m}\mathbf{M}_{4}+_{m}\mathbf{M}_{5}\Bigg{\}}, \tag{22}\]  
with [43]:  
\[{}_{1}\mathbf{M}_{1}=\Bigg{[} \begin{array}{cc}2_{1}\mathbf{\Lambda}&[0]_{NxN }\\ \left[0\right]_{NxN}&2_{1}\mathbf{\Lambda}\end{array} \Bigg{]}\]  
\[{}_{m}\mathbf{M}_{2}=\Bigg{[} \begin{array}{cc}\mathbf{H}_{1,m}&[0]_{NxN}\\ \left[0\right]_{NxN}&\mathbf{H}_{1,m}\end{array} \Bigg{]}\Bigg{[} \begin{array} []{cc}_{m}\mathbf{R}^{+-}(x_{1})&{}_{m}\mathbf{T}^{-}(x_{2},x_{1})\\ {}_{m}\mathbf{T}^{+}(x_{1},x_{2})&{}_{m}\mathbf{R}^{-+}(x_{2})\end{array}  \Bigg{]}\Bigg{[} \begin{array}{cc}_{2m}\mathbf{\Lambda}&[0]_{NxN}\\ \left[0\right]_{NxN}&2_{m}\mathbf{\Lambda}\end{array} \Bigg{]}^{-1}\Bigg{[} \begin{array}{cc}\mathbf{K}_{1,m}&[0]_{NxN}\\ \left[0\right]_{NxN}&\mathbf{K}_{1,m}\end{array} \Bigg{]},\]  
\[{}_{m}\mathbf{M}_{3}=\Bigg{[} \begin{array}{cc}\mathbf{G}_{1,m}&[0]_{NxN}\\ \left[0\right]_{NxN}&\mathbf{G}_{1,m}\end{array} \Bigg{]}\Bigg{[} \begin{array} []{cc}_{2m}\mathbf{\Lambda}&[0]_{NxN}\\ \left[0\right]_{NxN}&2_{m}\mathbf{\Lambda}\end{array} \Bigg{]}^{-1}\Bigg{[} \begin{array}{cc}\mathbf{K}_{1,m}&[0]_{NxN}\\ \left[0\right]_{NxN}&\mathbf{K}_{1,m}\end{array} \Bigg{]}, \tag{23}\]  
\[{}_{m}\mathbf{M}_{4}=\Bigg{[} \begin{array}{cc}\mathbf{H}_{1,m}&[0]_{NxN}\\ \left[0\right]_{NxN}&\mathbf{H}_{1,m}\end{array} \Bigg{]}\Bigg{[} \begin{array} []{cc}_{m}\mathbf{R}^{+-}(x_{1})&{}_{m}\mathbf{T}^{-}(x_{2},x_{1})\\ {}_{m}\mathbf{T}^{+}(x_{1},x_{2})&{}_{m}\mathbf{R}^{-+}(x_{2})\end{array}  \Bigg{]}\Bigg{[}