## 2 Mathematical Models  
### General Problem Statement  
#### 2.1.1 Multi-Layered Waveguide  
As a starting point, the three-dimensional elastic multi-layered waveguide is considered occupying the domain \(\hat{V}=\{-\infty<x_{1},x_{2}<\infty,0\leq x_{3}\leq H\}\) in the Cartesian coordinates \(\mathbf{x}=\{x_{1},x_{2},x_{3}\}\) introduced in such a way that the plane \(x_{1}Ox_{2}\) is parallel to the surfaces \(S^{-}=\{x_{3}=0\}\) and \(S^{+}=\{x_{3}=H\}\) or the outer boundaries of \(\hat{V}\) (\(\partial\hat{V}=S^{-}\cup S^{+}\)). The waveguide \(\hat{V}\) is a union of \(L\) homogeneous elastic transversally isotropic layers\(V^{(l)}=\{-\infty<x_{1},x_{2}<\infty,z^{(l-1)}\leq x_{3}\leq z^{(l)}\}\) of thicknesses \(h^{(l)}=|z^{(l)}-z^{(l-1)}|\) with stiffness tensor \(C^{(l)}_{ijkl}\) and mass densities \(\rho^{(l)}\), \(l=1,2,...L+1\) and \(z^{(1)}=0\), \(z^{(L+1)}=H\) (see Figure 1a). Below, where it does not lead to misunderstanding, the superscript is omitted. Constitutive equations  
\[\sigma_{ij}=C_{ijkl}s_{kl}. \tag{1}\]  
relate components of stress tensor \(\sigma_{ij}\) and displacement vector \(u_{i}\), where  
\[s_{kl}=\frac{1}{2}(u_{k,l}+u_{l,k}).\]  
Therefore, governing equations for time-harmonic motion \(u_{i}\mathrm{e}^{\mathrm{i}\omega t}\) (\(\omega=2\pi f\) is angular frequency, and \(f\) stands for dimensional frequency) in the absence of body forces. \[C_{ijkl}\frac{\partial^{2}u_{k}(\mathbf{x})}{\partial x_{j}\partial x_{j}}+\rho \omega^{2}u_{i}(\mathbf{x})=0 \tag{2}\]  
are valid in each sub-layer \(V^{(l)}\). Continuity of displacements and traction vector \(\mathbf{\tau}=(\sigma_{13},\sigma_{23},\sigma_{33})\) at the interfaces between sub-layers is assumed, i.e.,  
\[u_{i}^{(l)}\Big{|}_{z=z^{(l)}}=\left.u_{i}^{(l+1)}\right|_{z=z^{(l)}},\quad \left.\mathbf{\tau}_{i}^{(l)}\right|_{z=z^{(l)}}=\left.\mathbf{\tau}_{i}^{(l+1)}\right| _{z=z^{(l)}},\,l=2,...,L. \tag{3}\]  
The lower boundary of the waveguide is stress free while the excitation of EGWs is achieved by applying certain traction \(\mathbf{q}\) in a finite area \(\Omega\) of its upper boundary:  
\[\mathbf{\tau}|_{z=0}=0,\quad\left.\mathbf{\tau}\right|_{z=H}=\mathbf{q}, \tag{4}\]  
and no other external loads are considered. Since an infinite waveguide is considered, to close the problem statement, the limiting absorption principle [39] is used as a radiation condition. The geometry of the considered boundary value problem (2)-(4) allows to apply the two-dimensional Fourier transform \(F_{xy}\) over the horizontal spatial variables \(x_{1}\), \(x_{2}\) and to derive its explicit solution in terms of the inverse Fourier two-fold path integral [40; 41]:  
\[\mathbf{u}(\mathbf{x},\omega)=\frac{1}{4\pi^{2}}\int\limits_{\Gamma^{+}}\int\limits_{ 0}^{2\pi}K(\mathbf{\alpha},\gamma,x_{3})\mathbf{Q}(\mathbf{\alpha},\gamma)e^{-\mathrm{i} \pi\omega\cos(\gamma-\varphi)}d\gamma\alpha d\mathbf{\alpha}, \tag{5}\]  
Figure 1: Multi-layered composite with a piezoelectric transducer. where \(K=F_{xy}[k]\) and \(\mathbf{Q}=F_{xy}[\mathbf{q}]\) are Fourier symbols of the \(3\times 3\) [PERSON]'s matrix \(k(\mathbf{x})\) and the contact stress vector \(\mathbf{q}(x_{1},x_{2})\) (the notations of Ref. [41] are employed); polar coordinates \((r,\varphi)\) and \((\alpha,\gamma)\) are introduced in spatial and Fourier domains, respectively. Integration contour \(\Gamma^{+}\) goes in the complex plane \(\alpha\) along the real semi-axis \(\text{Re}\mathbf{x}\geq 0\), \(\text{Im}\mathbf{\alpha}=0\) from the real axis bypassing real poles of the Green's matrix \(K\) either from below or above according to the limiting absorption principle [39; 41]. Onwards, an important case of multi-layered composite structures, namely, cross-ply laminates fabricated from identical unidirectional transversely isotropic prepregs with the general stacking sequence \([0^{\circ}_{\eta},90^{\circ}_{\beta}]_{p\{s\}}\) are considered.