## 2 Mathematical Models
### Guided Waves Excitation, Sensing, and Scattering
Lower surfaces of PWTs (\(\hat{S}^{(j)}_{0}=\hat{V}^{(j)}\bigcap V,\quad j=\{\mathrm{a},\mathrm{s}\}\)) are assumed to be grounded, which means that electric potentials are equal zero  
\[\hat{\phi}(\mathbf{x},t)=0,\qquad\mathbf{x}\in\hat{S}^{(j)}_{0},\quad j=\{\mathrm{a}, \mathrm{s}\}. \tag{10}\]  
An electric input impulse \(p(t)\) with a given voltage \(\varphi_{0}\) is applied at the second electroded surface \(\hat{S}^{\mathrm{a}}_{\phi}\) of the piezoelectric actuator \(\hat{V}^{(\mathrm{a})}\), is assumed to be known:  
\[\hat{\phi}(\mathbf{x},t)=\varphi_{0}\cdot p(t),\qquad\mathbf{x}\in\hat{S}^{\mathrm{a} }_{\phi},\quad t\geq t_{0}. \tag{11}\]  
Zero electric displacements are assumed at the boundaries \(\hat{S}^{(j)}_{\mathrm{D}}\) without electrodes  
\[\hat{D}^{(j)}_{1}(\mathbf{x})=0,\quad\mathbf{x}\in\hat{S}^{(j)}_{\mathrm{D}},\quad j= \{\mathrm{a},\mathrm{s}\} \tag{12}\]  
whereas electric potential at the grounded electrodes is assumed to be zero  
\[\hat{\phi}^{(j)}(\mathbf{x})=0,\quad\mathbf{x}\in\hat{S}^{(j)}_{0}, \tag{13}\]  
Electric potentials at electroded surfaces of sensor denoted is to be determined using two additional boundary conditions  
\[ \begin{split}&\hat{\phi}(\mathbf{x})=\bar{\varphi},\qquad\mathbf{x}\in \hat{S}^{\mathrm{s}}_{\phi},\\ &\hat{Q}(\hat{\mathbf{u}},\hat{\phi})=\int\limits_{\hat{S}^{\mathrm{s }}_{\phi}}\hat{D}_{2}(\mathbf{x})\mathrm{d}S=0,\end{split}  \tag{14}\]  
where \(\hat{Q}(\mathbf{u},\phi)\) is the electric charge and the constant electric potential \(\bar{\varphi}\) is assumed to be unknown. The continuity boundary conditions for the displacement and the traction vectors are also assumed in the contact area  
\[u^{(L)}_{i}\Big{|}_{z=H}=\left.\hat{u}^{(n)}_{i}\right|_{z=H},\quad\mathbf{\tau}^ {(L)}_{i}\Big{|}_{z=H}=\left.\hat{\mathbf{\tau}}^{(n)}_{i}\right|_{z=H},\quad n=\{ \mathrm{a},\mathrm{s}\} \tag{15}\]  
and the stress free boundary conditions in \(\hat{S}^{(n)}_{\mathrm{D}}\cup\hat{S}^{(n)}_{\phi}\). In the case of the stringer being glued at the surface of the laminate \(\hat{V}\), two adjacent domains \(V^{(\mathrm{g})}\) and \(V^{(\mathrm{st})}\) describing respectively the glue layer and the T-shaped stringer are introduced as depicted in Figure 3. Since the stringer and the adhesive are elastic and isotropic, governing equation (2) reduced for isotropy are employed. The outer boundaries of \(V^{(\mathrm{g})}\) and \(V^{(\mathrm{st})}\) are stress-free, whereas the displacement and the traction vectors are assumed continuous at the interfaces \(V\cap V^{(\mathrm{g})}\) and \(V^{(\mathrm{g})}\cap V^{(\mathrm{st})}\) so that relations identical to (15) hold there. To solve the formulated in-plane boundary value problem (1)-(3) together with (8)-(15), a recently developed SAHA has been employed. Its starting point is the integral representation describing wave fields induced due to the presence of PWTs and stringer, which is obtained form Equation (5) after its reduction to the plain-strain case:  
\[\mathbf{u}^{(j)}(\mathbf{x})=\frac{1}{2\pi}\int\mathbf{K}^{(j)}(\alpha,x_{3})\mathbf{Q}^{( j)}(\alpha)\mathrm{e}^{-\mathrm{i}\alpha x_{1}}\mathrm{d}\alpha,\quad j=\{ \mathrm{a},\mathrm{s},\mathrm{st}\} \tag{16}\]  
Here \(\hat{\mathbf{K}}^{(j)}(\alpha,x_{3})\) is the Fourier transform of the waveguide Green's matrix, \(\mathbf{Q}^{(j)}(\alpha)\) is the Fourier transform of the unknown traction vector \(\mathbf{q}^{(n)}(x_{1})\) occurring in the second equality of (15) with respect to \(x_{1}\) coordinate, while \(\Gamma\) is the integration contour in the complex plane \(\alpha\) along the real axis surrounding poles of the integrand in accordance with the limiting absorption principle [39; 40]. The surface-mounted PWTs are discretized using the frequency domain spectral element method (FDSEM) [48; 49] using Lagrange interpolation polynomials at the Gauss-Lobatto-Legendre point.