## 2 Mathematical Models
### Guided Waves Excitation, Sensing, and Scattering  
Since both of the homogenized anisotropic waveguides are orthotropic with principal axes coinciding with directions \(x_{1}\) and \(x_{2}\), they, as well as the initial cross-ply laminate, support propagation of in-plane EGWs along these directions. Therefore, for simplicity's sake, the problem of EGW excitation, sensing, and scattering is considered further on within the plane-strain assumption (Figure 1b,c). In the case of two rectangular surface-mounted PWTs acting as an actuator and as a sensor, two corresponding domains (\(\hat{V}^{\text{(a)}}\) and \(\hat{V}^{\text{(s)}}\), upper index \(a\) stands for the actuator, while \(s\) stands for sensor) are included in the mathematical model. Two PWTs are assumed of the same length \(a\) and thickness \(d\), being situated at the upper surface of the waveguide \(\hat{S}^{+}\) with centers located at \(s=50\) mm from each other (see Figure 3). PWTs are assumed to be made of the same piezoelectric material with mass density \(\hat{\rho}\) and the tensors of elastic, piezoelectric, and dielectric constants \(\hat{C}_{ijkl}\), \(\hat{e}_{kl}\), \(\hat{e}_{ik}\) respectively. \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline
**Material** & \(C_{\text{11}}\) & \(C_{\text{22}}\) & \(C_{\text{33}}\) & \(C_{\text{12}}\) & \(C_{\text{13}}\) & \(C_{\text{23}}\) & \(C_{\text{44}}\) & \(C_{\text{55}}\) & \(C_{\text{66}}\) \\ \hline \([0^{\circ},90^{\circ}]_{2\text{s}}\) & 122.4 & 11.5 & 11.5 & 4.13 & 4.13 & 6.47 & 2.51 & 4.85 & 4.85 \\ \hline Model 1, & & & & & & & & & \\ SLWH & 67.0 & 67.0 & 11.5 & 4.13 & 5.30 & 5.30 & 3.31 & 3.31 & 4.85 \\ \hline Model 2, & & & & & & & & & \\ LWHM & 64.8 & 64.8 & 11.1 & 7.99 & 3.63 & 3.90 & 2.85 & 3.39 & 4.92 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Elastic properties of a single ply and of a homogenized waveguide [GPa]. The governing equations for the time-harmonic motion of the piezoelectric media with the tensors of elastic, piezoelectric, dielectric constants \(\hat{C}_{ijkl}\), \(\hat{e}_{kij}\), \(\hat{e}_{ik}\) and mass density \(\hat{\rho}\) can be written in terms of the displacement vector \(\hat{\mathbf{u}}\) and the electric potential \(\hat{\phi}\)[47]:  
\[\hat{C}_{ijkl}\frac{\partial^{2}\hat{u}_{k}(\mathbf{x})}{\partial x_{l}\partial x_ {j}}+\hat{e}_{kij}\frac{\partial^{2}\hat{\phi}(\mathbf{x})}{\partial x_{k}\partial x _{j}}+\hat{\rho}\omega^{2}\hat{u}_{i}(\mathbf{x})=0, \tag{8}\]  
\[\hat{e}_{ikl}\frac{\partial^{2}\hat{u}_{k}(\mathbf{x})}{\partial x_{l}\partial x_ {i}}-\hat{e}_{ik}\frac{\partial^{2}\hat{\phi}(\mathbf{x})}{\partial x_{k}\partial x _{i}}=0. \tag{9}\]  
The components of the electric field vector are the derivatives of the electric potential \(\phi\):  
\[E_{k}=-\frac{\partial\phi}{\partial x_{k}}.\]  
Figure 3: Geometry sketch of the considered plain-strain problem of EGW excitation, sensing, and scattering. Figure 2: Wavenumbers \(k_{n}(\gamma)\) for \([0^{\circ},90^{\circ}]_{28}\): multi-layered waveguide (thick solid lines), homogenized waveguides (thin solid and dashed lines for Models 1 and 2, respectively). For the PWTs \(\hat{V}^{(j)}\), the outer boundary \(\partial\hat{V}^{(j)}\) is split into three parts \(\partial\hat{V}^{(j)}=\hat{S}^{(j)}_{\mathrm{D}}\cup\hat{S}^{(j)}_{0}\cup\hat{ S}^{(j)}_{\phi}\cup\hat{S}^{(j)}_{\phi}\) in accordance with the boundary conditions.