## 3 Method
### Offline Phase  
As a first step, the scattering behavior of an arbitrary feature is numerically investigated to extract the relevant information needed for the online phase. To extract this information, an idealized setup of an infinite plate with a single actuator, one feature, and one sensor (Figure 3) is chosen, where only a single mode is excited. In this setup, a direct wave packet from the actuator and one or several wave packets stemming from the feature can be measured at the sensor location. Several wave packets with their origin at the feature might be present, as mode conversion might occur at the feature--depending on the frequency and the feature's shape. In what follows, we investigate the different wave packets, which reach a sensor, separately. Firstly, a baseline simulation is performed using the pristine setup, i.e., one without defects. Through baseline subtraction, the waves stemming from the feature can be extracted, given that the direct wave is known from the baseline. To differentiate between the different scattering waves from the feature, the deformations or internal forces on the cross section need to be investigated. Although this task is challenging in an experimental setting, it can be easily accomplished in simulation. Using this approach, all possible cases of wave packets propagating from the actuator to the feature and then to the sensor should be investigated. This means that in a setup where the two fundamental modes (\(S_{0}\) and \(A_{0}\)) are present and mode conversion occurs at the feature, four cases must be investigated. These cases are described by mode pairs, where the first mode is the one excited at the actuator and the second mode is the one investigated at the sensor: \(S_{0}-S_{0}\), \(S_{0}-A_{0}\), \(A_{0}-S_{0}\), and \(A_{0}-A_{0}\). Due to the linearity assumption, these cases can simply be superposed in a later step to obtain the full sensor response. Next, several sensors are used in a radial line from the defect, as shown in Figure 4. The first sensor needs to have a certain distance from the feature to prevent evanescent waves from affecting the sensor response. A single mode wave packet, originating from the feature (due to either reflection of the same mode or mode conversion), changes its shape in between two sensors due to two effects, namely dispersion and geometric damping. Furthermore, the wave packets arrive later in time as the distance between sensor and feature grows. These three influences (dispersion, geometrical damping, time delay) on the sensor response are investigated next for discrete signals in frequency domain. Firstly, the geometric damping is known to be inversely proportional to \(\sqrt{d}\)[64; 74], where \(d\) is the distance to the origin of the wave, and independent of the frequency. In contrast,  
Figure 3: Wave packets reaching a sensor with one scattering feature (e.g., a defect or a rivet hole): single mode actuation, several modes scattered at the feature. Dashed rays have their origin at the actuator, solid rays have their origin at the feature. No boundary reflections are considered at this point. dispersion is frequency-dependent and does not occur if the wave is composed of a single frequency. In the described method, the signal is separately investigated for each discrete frequency, implying that the dispersion is inherently accounted for, as it is a purely frequency-dependent phenomenon. As for the differentiation in terms of the time of arrival, it is shown in Section 2.2 that the response will be periodic. The time shift \(\Delta t\) in Equation (1) thereby reflects the difference in distance \(\Delta d\). The relation between \(\Delta t\) and \(\Delta d\) is expressed as follows:  
\[\Delta t=\frac{\Delta d}{c(\omega)} \tag{13}\]  
where \(c(\omega)\) is the frequency-dependent wave speed for the investigated mode and frequency (\(\omega\)).