## 2 Admittance-Based Immission Prediction
### How to Determine the Transfer Admittance
#### 2.1.1 Practical Aspects--Approximations of Line Transfer Admittance  
In practice, determining line transfer admittances through the integration of multiple point transfer admittances--especially when dealing with numerous measurement points or a long section of track--can be very time-consuming. Based on theoretical considerations, the authors created a simplified method to estimate the line transfer admittance (Y\({}_{line}\)) at an immission point from the maximum point transfer admittance (\(Y_{point,\ max}\)), as shown in Equation (4). This approach measures the transfer admittance only at the shortest distance between the track and immission point (assuming this yields \(Y_{point,\ max}\)), which is multiplied by factor \(\kappa\) that depends on the geology (which affects material damping), the integration length from \(a\) to \(b\) (train length \(L\)) and the distance between the emission and immission points (\(D\)). The values obtained for factor \(\kappa(\beta,\ L,\ D)\) are shown in Figure 4 and are based on the following theoretical considerations:  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Vibration Generator** & **Impact Tester** \\ \hline Peak excitation & 225 kN & 600 J \\ Frequency bands & 1–250 Hz & 4–315 Hz \\ Baseplate size & 2 m\({}^{2}\) & 0.03 m\({}^{2}\) \\ Actuator mass & 3505 kg & 50 kg \\ \hline \hline \end{tabular}
\end{table}
Table 2: Technical specifications of the artificial vibration sources used in this study.  
Figure 3: Decay of peak vibration amplitudes with distance at (**a**) 16 Hz and (**b**) 32 Hz when measuring line sources (trains) and artificial point sources.  
In a first step, additional point transfer admittances are calculated along the train length using Equation (5) [14], where \(v_{0}\) and \(v\) are the vibration amplitudes at distance \(r_{0}\) and \(r\) from the source, respectively, and \(d\) is the damping. The exponent \(n\) accounts for the amplitude reduction due geometric damping. The exponent \(\frac{\omega d}{c}(r-r_{0})\) describes the amplitude reduction due to material damping. As a result, material damping increases with frequency, leading to lower frequencies becoming more dominant with increasing distance, while higher frequencies diminish more quickly. In practice, since the contributions of geometric and material damping cannot be determined independently, the simpler form of the equation is typically used, shown in Equation (6). This approach encompasses both the geometric and the material damping within a single frequency-dependent parameter \(\beta(w)\).  
\[Y_{line}=\kappa\ Y_{point,\ max} \tag{4}\]  
\[v=v_{0}\left(\frac{r_{0}}{r}\right)^{n}exp^{(\frac{\omega d}{c}(r-r_{0}))} \tag{5}\]  
\[v=v_{0}\left(\frac{r_{0}}{r}\right)^{\beta(\omega)} \tag{6}\]  
In a second step, these calculated point transfer admittances are summed up according to Equation (3), giving a theoretical estimate of \(Y_{line}\). Lastly, \(\kappa\) was then computed as the inverse of Equation (4) from the estimate of \(Y_{line}\).  
Practical experience has shown that the influence of the attenuation exponent \(\beta\) diminishes with increasing distance from the track (see Figure 4, in particular, subplot (a)). The authors therefore recommend using \(\beta=1\) for impression points at larger distances in settings where the exponent is not known, as shown in Figure 5. This pragmatic approach simplifies the modeling of \(Y_{line}\) at every single emission point along a railway track in projects with budgetary constraints.  
Figure 4: Factor \(\kappa\) to estimate the line transfer admittance from the maximum point transfer admittance at an impression point depending on the attenuation coefficient, \(\beta\), the distance from the track, \(D\), and train length, \(L\). (**a**) When assuming a constant \(L=200\) m and (**b**) assuming a constant \(D=10\) m.