## 2 Mathematical Models
### General Problem Statement
#### 2.1.2 Homogenized Waveguide  
Employing Voigt notation and assumptions regarding the transverse isotropy of unidirectional and their orientation (either 0 or 90 degrees), stiffness tensor \(C_{ijkl}\) could be reduced to a \(6\times 6\) stiffness matrix \(C_{ij}\) with 12 non-zero elements [42]. Taking these matrices \(C_{ij}^{(l)},l=1,\dots,L\) as a starting point, two approaches for deriving the stiffness matrix of the homogenized single layer waveguide \(\bar{C}_{ij}\) are considered. Within the first model (Model 1--static long-wave homogenization (SLWH)), following relations for the components of \(\bar{C}_{ij}\) are employed [31] taking into account that all the prepregs have identical thickness \(h^{(l)}=H/L\):  
\[\bar{C}_{ij}=\sum_{l=1}^{L}C_{ij}^{(l)},\quad i,j=1,2,3,6 \tag{6}\] \[\bar{C}_{pq}=\left(\sum_{l=1}^{L}C_{pq}^{(l)}/\Delta_{l}\right)/ \Delta,\quad p,q=4,5,\]  
\[\Delta_{l}=C_{44}^{(l)}C_{55}^{(l)},\quad\Delta=\left(\sum_{l=1}^{L}C_{44}^{( l)}/\Delta_{l}\right)\left(\sum_{l=1}^{L}C_{55}^{(l)}/\Delta_{l}\right)\]  
From Equation (6), it is clear that if a balanced cross-ply laminate is considered, i.e., the one where \(\eta=\theta\) in the stacking sequence, equalities \(\bar{C}_{11}=\bar{C}_{22}\) and \(\bar{C}_{44}=\bar{C}_{55}\) hold. The second model (Model 2--Lamb wave homogenization method (LWHM)) is inspired by the approach proposed in Ref. [33]. In LWHM, the homogenization is achieved by equating at a set of given frequencies angular-dependent wavenumbers \(k_{n}\) of EGWs (quasi-Lamb waves), (whereas slowness surfaces of Floquet waves are employed in Ref. [33]) in an actual multi-layered composite medium to the corresponding dispersion curves of EGWs propagating in the effective anisotropic homogeneous single layer. Effective elastic constants \(\bar{C}_{ij}\) in the class of orthotropic materials are obtained from the optimization of the discrepancy between these angular dispersion curves of \(k_{n}\). For this purpose, an objective function is constructed from the components of the Fourier transform of the [PERSON]'s matrix for the considered homogenized anisotropic waveguide [43; 44]  
\[F(\bar{C}_{ij})=\sum_{r=1}^{N_{f}}\sum_{n=1}^{N_{nr}}\sum_{m=1}^{N_{nr}}|K_{3 3}^{-1}(k_{n}(\gamma_{m}),f_{r})|. \tag{7}\]Here \(K_{33}\) is the \((3,3)\) component of the [PERSON]'s matrix \(K(\alpha,\gamma,f)\), of which the evaluation algorithm is taken from Ref. [41]; \(N_{f}\), \(N_{w}\) and \(N_{\gamma}\) are the numbers of frequencies, normal modes of EGWs and propagation directions for which angular and frequency-dependent wavenumbers \(k_{n}(\tau_{m},f_{r})\) for the initial multi-layered composite are preliminarily evaluated by employing the approach described in Ref. [41]. Since poles of the [PERSON]'s matrix \(K(\alpha,\gamma,f)\) in the complex plane are equal to the wavenumbers \(k_{n}\) of normal modes propagating in the direction \(\gamma\)[41], appropriate values of \(\hat{C}_{ij}\) would result in higher magnitudes of \(|K_{33}|\). The minimization of (7) is achieved with a micro-genetic algorithm (\(\mu\)-GA) [45]. To illustrate the performance and peculiarities of the implemented models, the 8-layered cross-ply laminate with the stacking sequence \([0^{\circ},90^{\circ}]_{2\text{s}}\) is considered with a total thickness of \(H=2.24\) mm and prepreg's density \(\rho=1528\) kg/m\({}^{3}\). Ply elastic properties [46] as well as the non-zero values of \(\hat{C}_{ij}\) obtained with both of the described models are summarized in Table 1. For Model 2, the following parameters were considered: \(N_{f}=2\) (\(f=50\) and \(200\) kHz), \(N_{w}=3\), which stands for fundamental quasi-antisymmetric, symmetric, and SH-modes \(A0\), \(S0\) and \(SH0\) respectively, and \(N_{\gamma}=50\) resulting in \(50\)\(\gamma_{m}\) points evenly distributed along the segment \(\gamma\in[0,\pi/2]\). Due to the stochastic nature of \(\mu\)-GA, corresponding data in Table 1 represent averaged values after 10 runs of the optimization algorithm.