## 2 Theory
### Frequency Response Function (FRF) and Transmissibility Function  
FRFs and transmissibility functions, which are well established in SHM [72], allow for evaluating a system's response for various time history inputs. For the case of Lamb wave propagation investigated in this paper, both functions can be derived starting from the discretized form of the equilibrium equations of a Linear Time Invariant (LTI) Multi Degree of Freedom (MDOF) system:  
\[\mathbf{M}\ddot{\mathbf{u}}(t)+\mathbf{C}\dot{\mathbf{u}}(t)+\mathbf{K}\mathbf{u}(t)=\mathbf{f}_{\mathbf{s}} \cdot f(t) \tag{2}\]  
where \(\mathbf{M}\),\(\mathbf{C}\), and \(\mathbf{K}\in\mathbb{R}^{n\times n}\) describe the discretized mass, damping, and stiffness matrices, respectively, and vectors \(\mathbf{u}\), \(\dot{\mathbf{u}}\), and \(\ddot{\mathbf{u}}\in\mathbb{R}^{n\times 1}\) denote the deformation vector and its first and second derivative with respect to time \(t\). The right hand side of Equation (2) describes the excitation of the system, where \(\mathbf{f}_{\mathbf{s}}\) denotes the spatial distribution of the excitation and \(f(t)\) its time history. This split of the excitation is possible, when the spatial distribution of the excitation is time invariant, which is usually the case for Lamb wave excitation [73]. Applying the Fourier transform to Equation (2), one obtains:  
\[(\mathbf{K}+i\omega\mathbf{C}-\omega^{2}\mathbf{M})\mathbf{u}(\omega)=\mathbf{f}_{\mathbf{s}}\cdot f(\omega) \tag{3}\]  
In the above, the same symbols are used for displacement \(\mathbf{u}\) and excitation \(f\), regardless of whether they refer to frequency or time domain, since the corresponding domain is evident from their dependence on time or frequency, respectively. Reorganizing Equation (3) yields the FRF [73]:  
\[FRF(\omega)=\frac{\mathbf{u}(\omega)}{f(\omega)}=(\mathbf{K}+i\omega\mathbf{C}-\omega^{2} \mathbf{M})^{-1}\mathbf{f}_{\mathbf{s}} \tag{4}\]  
For discrete frequencies \(\omega_{i}\), the FRF can be evaluated numerically:  
\[FRF(\omega_{i})=\frac{\mathbf{u}(\omega_{i})}{f(\omega_{i})} \tag{5}\]  
Dimilarly, if the FRF is available, the response for any given excitation can be computed as follows:  
\[\mathbf{u}(\omega_{i})=FRF(\omega_{i})f(\omega_{i}) \tag{6}\]  
Next, the transmissibility function is derived for discrete frequencies from Equation (3). First, the equation is simplified by substituting \(\mathbf{\hat{K}}(\omega)=\mathbf{K}+i\omega\mathbf{C}-\omega^{2}\mathbf{M}\):  
\[\mathbf{\hat{K}}(\omega)\mathbf{u}(\omega)=\mathbf{f}_{\mathbf{s}}\cdot f(\omega) \tag{7}\]  
Figure 2: Signal shifted in time and its periodic response in frequency domain. Next, it is assumed that the response is known at a single Degree of Freedom (DOF) and should be calculated for all remaining locations. To this end, the system is partitioned in two sets of equations:  
\[ \begin{bmatrix}\hat{K}_{uu}(\omega)&\hat{K}_{uk}(\omega)\\ \hat{K}_{ku}(\omega)&\hat{K}_{kk}(\omega)\end{bmatrix}\begin{bmatrix}\mathbf{u}_{u} (\omega)\\ u_{k}(\omega)\end{bmatrix} = \begin{bmatrix}\mathbf{f}_{s,u}\\ f_{s,k}\end{bmatrix} \cdot f(\omega) \tag{8}\]  
where subindex \(k\) refers to the DOF, where the response is known, and the subindex \(u\) to the unknown DOFs. This means \(u_{k}(\omega)\in\mathbb{R}^{1\times 1}\) and \(\mathbf{u}_{u}(\omega)\in\mathbb{R}^{m\times 1}\), where \(m=n-1\).