## 3 Results
### Assessment by Vibrometry Approach  
Using vibrometry, by analyzing operational deflection shapes (ODSs), the radiated sound power (\(P\) (\(W\))) and the radiation efficiency (\(\sigma\) (-)), additional information can be obtained about the factors that affect the sound insulation of the considered walls. Some of the ODSs, determined as frequency response functions between accelerometer signals obtained on measurement grid and the reference signal, are shown in Figure 3. Figure 3 shows operational deflection shapes of the two structures at the lowest structural resonance frequencies, as well as the spectrum of the velocity level \(L_{v}\) (dB re ms\({}^{-1}\); \(v_{0}=5\times 10^{-8}\) m/s). The peaks in the spectrum of the velocity level \(L_{v}\) indicates resonances occurrence. By evaluating the response for each point of the wall surface, the ODS was plotted. The lowest natural frequency \(f_{(1;1)}\) is about 67 Hz. Although the phenomenon of coincidence is not related to a structural resonance, the velocity spectrum shows a peak at \(f_{c}\approx 160\) Hz, as a result of the increased coupling with the vibration exciting airborne waves at and above that frequency. The ETICS resonance appears as a peak in the \(L_{v}\) spectrum at about\(f_{0}\approx 119\) Hz. At \(f\approx 230\) Hz, the \(L_{v}\) spectra of Wall 1 and Wall 2 are equal, which is consistent with the equality at that frequency of the sound insulation spectra. \begin{table}
\begin{tabular}{c c c c} \hline
**Symbol** & **Wall 1**so & **Wall 2**iso & \(\Delta R_{w,\,\text{ISO}}\) \\ \hline \(R_{w}\) (dB) & \(59\) & \(58\) & \(-1\) \\ \(C\) (dB) & \(-2\) & \(-3\) & \(-1\) \\ \(C_{tr}\) (dB) & \(-6\) & \(-9\) & \(-3\) \\ \(C_{50-3150}\) (dB) & \(-3\) & \(-4\) & \(-1\) \\ \(C_{50-5000}\) (dB) & \(-2\) & \(-3\) & \(-1\) \\ \(C_{100-5000}\) (dB) & \(-1\) & \(-3\) & \(-2\) \\ \(C_{tr,50-3150}\) (dB) & \(-10\) & \(-12\) & \(-2\) \\ \(C_{tr,50-5000}\) (dB) & \(-10\) & \(-12\) & \(-2\) \\ \(C_{tr,100-5000}\) (dB) & \(-6\) & \(-9\) & \(-3\) \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison between ISO 10140-2 based single number ratings. Figure 3: \(Cont.\)  
Exploiting the rectangular symmetry of the wall, it was sufficient to measure the vibrational response of the wall in only one quadrant by means of accelerometers. The response of the other quadrants was determined by mirroring the maps of the measured quadrant along the horizontal and vertical axis and along the center point. In this manner, the measurement effort could be reduced, with the drawback that this approach is only valid for the lowest eigenmodes (natural frequencies) of the wall. Having measured on a grid of 9 \(\times\) 11 measurement points in one quadrant, a full dataset of 18 \(\times\) 22 was obtained in that manner. The operational deflection shapes thus obtained are shown in Figure 3. Vibration amplitudes \(a_{n}(x,y,\omega)\) are mapped along a grid consisting of 18 \(\times\) 22 points, with an interspacing of 15 cm between the measurement points. For every frequency component of the measured responses, a two-dimensional spatial Fourier transform was taken, to arrive at the wavenumber domain (\(k_{x},k_{y}\)):  
\[a_{n}\big{(}k_{x},k_{y},\omega\big{)}\ =\ \mathcal{F}\mathcal{F}\{a_{n}(x,y, \omega)\} \tag{12}\]  
where \(\mathcal{F}_{x}\) and \(\mathcal{F}_{x}\) represent a spatial Fourier transform in x- and y-direction, respectively. Although other methods exist, such as Prony's method or the wavenumber fit method [56], the spatial Fourier transform is the easiest approach to assess the wavenumbers present in the vibrating wall at each angular frequency \(\omega\). Spatial windowing techniques are usually employed to reduce possible leakage of the data [57]. However, in the present case, the wall was mounted in a heavy test-setup, which enforced the vibrations at the edges of the spatial domain to zero. This in itself reduced the possible occurrence of leakage, and thus a spatial window was not needed in this case.