# Partial Energy Transfer Model of Lamb Waves Scattering in Materially Isotropic Waveguides
## 2 Mode Matching Technique  
### General Background  
Let us now consider a waveguide of homogeneous thickness under the plane strain condition with stress-free boundaries (see Figure 1). By assuming the stress and displacement field time dependence in the form of \(e^{-i\omega t}\) and, at the same time, using the constitutive equations:  
\[\sigma_{ij}=\lambda\theta\delta_{ij}+2\mu\epsilon_{ij} \tag{1}\]  
the equation of motion can be presented:  
\[\
abla\times\overline{\overline{\sigma}}=\rho\tilde{\mathbf{u}} \tag{2}\]expressed purely in terms of the displacement vector \(u\):  
\[\mu\
abla^{2}\mathbf{u}+(\lambda+\mu)\
abla(\
abla\times\mathbf{u})=-\rho\omega^{2}\mathbf{u}, \tag{3}\]  
where \(\mathbf{u}(\mathbf{x},\mathbf{z})=\{\mathbf{u},\mathbf{w}\}^{\mathbf{T}}\), \(\mathbf{t}\) refers to time, \(\delta_{ij}\) is Kronecker delta, \(\mathbf{\theta}\) denotes dilatation, \(\mathbf{\rho}\) refers to density, \(\mathbf{\omega}\) is angular frequency, \(\varepsilon_{ij}\) refers to the symmetrical strain tensor component, and \(\sigma_{ij}\) represents component of the symmetrical stress tensor \(\overline{\overline{\sigma}}\):  
\[\overline{\overline{\sigma}}=\left[ \begin{array}{cc}\sigma_{xx}&\sigma_{xx }\\ \sigma_{xx}&\sigma_{zz}\end{array} \right] \tag{4}\]  
Using the Helmholtz decomposition theorem [41] and the given boundary conditions (see Figure 1), it is possible to derive the Rayleigh-Lamb dispersion equation, which is used to identify the real, complex, or pure imaginary wavenumbers for given material parameters and \(\mathbf{ft}_{h}\) product where \(\mathbf{t}_{h}\) is plate thickness and \(\mathbf{f}\) refers to the frequency. The Lamb wave problem may be treated as an eigenvalue problem for wavenumber \(k\) under the assumption of separation of variables \(\mathbf{x}\) and \(\mathbf{z}\) in the displacement and stress field functions [24; 25], which forming a discrete spectrum as the associated eigenfunctions:  
\[\left[ \begin{array}{c}u^{n}(x,z)\\ w^{n}(x,z)\\ \sigma_{xx}^{n}(x,z)\\ \sigma_{xx}^{n}(x,z)\end{array} \right]=\left[ \begin{array}{c}\tilde{u}^{n}( z)\\ \tilde{w}^{n}(z)\\ \tilde{\sigma}_{xx}^{n}(z)\\ \tilde{\sigma}_{xx}^{n}(z)\end{array} \right]e^{ik^{n}x} \tag{5}\]  
where subscript \(.^{n}\) denotes the \(n\)-th wavenumber \(k^{n}\), and subscript\(\ \widetilde{\ }\) expresses the functional dependence on the spatial coordinate \(z\). The already identified spectrum may be subsequently split into two parts [24], thus representing the wave propagation in opposite directions with respect to a given coordinate axis \(x\). Assuming the completeness of Lamb wave mode, as proved by [PERSON] [42], the resulting stress/displacement field can be represented as a sum of individual stress/displacement components, which are weighted by complex amplitudes \(\alpha_{n}^{+}/\alpha_{n}^{-}\) for the \(n\)-th mode with wavenumber \(k^{n}\):  
\[\left[ \begin{array}{c}u(x,z)\\ w(x,z)\\ \sigma_{xx}(x,z)\\ \sigma_{zx}(x,z)\end{array} \right]=\sum_{n\in\mathbb{N}}\left[ \begin{array}[ ]{c}m_{n}(x)\tilde{w}^{n+}(z)\\ n_{n}(x)\tilde{w}^{n+}(z)\\ n_{n}(x)\tilde{\sigma}_{xx}^{n+}(z)\\ m_{n}(x)\tilde{\sigma}_{xx}^{n+}(z)\end{array} \right]=\sum_{n\in\mathbb{N}} \alpha_{n}^{+}(x)\left[ \begin{array}{c}\tilde{u}^{n+}(z)\\ \tilde{w}^{n+}(z)\\ \tilde{\sigma}_{xx}^{n}(z)\\ \tilde{\sigma}_{xx}^{n+}(z)\end{array} \right]+\sum_{n\in\mathbb{N}}\alpha_{n} ^{-}(x)\left[ \begin{array}{c}\tilde{u}^{n-}(z)\\ \tilde{w}^{n-}(z)\\ \tilde{\sigma}_{xx}^{n+}(z)\\ \tilde{\sigma}_{xx}^{n-}(z)\end{array} \right] \tag{6}\]  
where subscript \(.^{-}\) refers to left-moving waves and subscript \(.^{+}\) refers to right-moving waves. Let us now introduce the biorthogonality relation [26]:  
\[\int_{\mathbb{R}}-\tilde{u}^{n}(z)\tilde{\sigma}_{xx}^{m}(z)+\tilde{w}^{m}(z )\tilde{\sigma}_{xx}^{n}(z)\mathrm{d}z=\Lambda_{n}\delta_{mn}, \tag{7}\]  
Figure 1: Geometry of the Lamb wave problem. which, using vectors:  
\[M_{n}(z)=\left[ \begin{array}{c}\widehat{u}^{n}(z)\\ \widetilde{\sigma}^{n}_{xx}(z)\end{array}