# Partial Energy Transfer Model of Lamb Waves Scattering in Materially Isotropic Waveguides
## 2 Mode Matching Technique
### Scattering of Lamb Waves at Vertical Notch
\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{J}_{1,m}&[0]_{NxN}\\ \left[0\right]_{NxN}&\mathbf{J}_{1,m}\end{array} \Bigg{]},\]  
\[{}_{m}\mathbf{M}_{5}=\Bigg{[} \begin{array}{cc}\mathbf{G}_{1,m}&[0]_{NxN}\\ \left[0\right]_{NxN}&\mathbf{G}_{1,m}\end{array} \Bigg{]}\Bigg{[} \begin{array} []{cc}_{2m}\mathbf{\Lambda}&[0]_{NxN}\\ \left[0\right]_{NxN}&2_{m}\mathbf{\Lambda}\end{array} \Bigg{]}^{-1}\Bigg{[} \begin{array}{cc}\mathbf{J}_{1,m}&[0]_{NxN}\\ \left[0\right]_{NxN}&\mathbf{J}_{1,m}\end{array} \Bigg{]},\]  
\[{}_{m}\mathbf{M}_{6}=\Bigg{[} \begin{array}{cc}\mathbf{G}_{1,m}&[0]_{NxN}\\ \left[0\right]_{NxN}&\mathbf{G}_{1,m}\end{array} \Bigg{]}\Bigg{[} \begin{array} []{cc}_{2m}\mathbf{\Lambda}&[0]_{NxN}\\ \left[0\right]_{NxN}&2_{m}\mathbf{\Lambda}\end{array} \Bigg{]}^{-1}\Bigg{[} \begin{array}{cc}\mathbf{J}_{1,m}&[0]_{NxN}\\ \left[0\right]_{NxN}&\mathbf{J}_{1,m}\end{array} \Bigg{]},\]  
\[{}_{m}\mathbf{M}_{7}=\Bigg{[} \begin{array}{cc}_{2m}\mathbf{M}_{7}&[0]_{NxN} \\ \left[0\right]_{NxN}&2_{m}\mathbf{\Lambda}\end{array} \Bigg{]}^{-1}\Bigg{[} \begin{array}{cc}_{2m}\mathbf{M}_{7}&[0]_{NxN}\\ \left[0\right]_{NxN}&\mathbf{J}_{1,m}\end{array} \Bigg{]}, \tag{24}\]  
\[{}_{m}\mathbf{M}_{8}=\Bigg{[} \begin{array}{cc}_{2m}\mathbf{R}^{+-}(x_{1})& {}_{m}\mathbf{T}^{-}(x_{2},x_{1})\\ {}_{m}\mathbf{T}^{+}(x_{1},x_{2})&{}_{m}\mathbf{R}^{-+}(x_{2})\end{array}  \Bigg{]}, \tag{25}\]  
where \({}_{m}\mathbf{R}^{+-}(x_{1})\) represents the reflection matrix affiliated to the \(m\)-th waveguide and, at the same time, containing the complex amplitudes of left-moving reflected waves at interface \(x=x_{1,\ m}\mathbf{R}^{-+}(x_{2})\) refers to reflection matrix containing the complex amplitudes of right-moving reflected waves at interface \(x=x_{2,\ m}\mathbf{T}^{+}\)\((x_{1},x_{2})\) represents the transmission matrix containing the complex amplitudes of right-moving waves transmitted through the waveguide section defined by coordinates \(x_{1},x_{2}\) and \({}_{m}\mathbf{T}^{-}\)\((x_{2},x_{1})\) refers to a transmission matrix containing the complex amplitudes of left-moving waves transmitted through the waveguide section defined by coordinates \(x_{2},x_{1}\) (see Figure 2). The local scattering matrices for sub-waveguides 2-4 are defined in the following way:  
\[{}_{m}\mathbf{S}(x_{1},x_{2})=\Bigg{[} \begin{array}{cc}_{m}\mathbf{R}^{+-}(x _{1})&{}_{m}\mathbf{T}^{-}(x_{2},x_{1})\\ {}_{m}\mathbf{T}^{+}(x_{1},x_{2})&{}_{m}\mathbf{R}^{-+}(x_{2})\end{array}  \Bigg{]}. \tag{26}\]  
In the case of sub-waveguide 2, the following applies:  
\[{}_{2}\mathbf{S}(x_{1},x_{2})=\Bigg{[} \begin{array}{cc}_{2}\mathbf{R}^{+-}(x _{1})&[0]_{NxN}\\ \left[0\right]_{NxN}&{}_{2}\mathbf{R}^{-+}(x_{2})\end{array} \Bigg{]}, \tag{27}\]  
where, due to the symmetry of the vertical notch with respect to the axis \(y\), the reflection matrices \({}_{2}\mathbf{R}^{+-}(x_{1})\) and \({}_{2}\mathbf{R}^{-+}\mathbf{R}^{-+}(x_{2})\) are identical. The resulting expression of the scattering matrix is as follows [27]:  
\[{}_{m}\mathbf{R}^{+-}(x_{1})={}_{m}\mathbf{R}^{-+}(x_{2})= \Big{(}\mathbf{L}+\mathbf{V}^{T}-2\mathbf{\Lambda}\Big{)}^{-1}\Big{(} \mathbf{V}^{T}-\mathbf{L}\Big{)}, \tag{28}\]  
with:  
\[\mathbf{V}(n,w)=(-\widetilde{\mathbf{u}}^{n}|\widetilde{\boldsymbol{ \sigma}}_{xx}^{w})|_{z=z=z_{1}}^{z=z_{2}}, \tag{29}\]\[\mathbf{L}(n,w)=(\widetilde{\sigma}_{xz}^{n}|\widetilde{w}^{w})|_{z=z_{1}}^{z=z_{2}}, \tag{32}\]  
\(\boldsymbol{u}/\sigma\) denotes a vector form of a given displacement vector/stress tensor component and \(z_{1}/z_{2}\) refers to the given integration limits. The local scattering matrices of sub-waveguides 3 and 4 will be in following form:  
\[{}_{m}\mathbf{S}(x_{1},x_{2})=\left[ \begin{array}{cc}[0]_{NxN}&diag\Big{(}e ^{i\ k_{n}|\Delta x|}\Big{)}\\ diag\Big{(}e^{i\ k_{n}|\Delta x|}\Big{)}&[0]_{NxN}\end{array}