## 2 Experimental Approach
### Determination of Airborne Sound Insulation Based on Vibrometry  
In view of the effect ETICS application plays mainly at low frequencies, and of the reduced accuracy of microphone-based measurements below 100 Hz, we also assessed the sound insulation of the two test configurations by means of a vibrometry-based approach [44].  
The measuring equipment for the vibrometry tests consisted of 6 accelerometers of Bruel & Kjaer type 4370, of which one was used as a reference. The signals, amplified by Bruel & Kjaer type 2635 charge amplifiers, were simultaneously recorded with an 8-channel Sinus Soundbook analyzing system, with Samurai 1.7.14 software. For calibration of the complete measuring chain, an acceleration calibrator type MMF VC10 was used.  
Figure 1: (**a**) Composition of the measured walls. Wall 1, without (left) and Wall 2, with ETICS (right). (**b**) Schematic representation of accelerometer locations used for vibrometry: (1) lime plaster (15 mm); (2) hollow brick (220 mm); (3) concrete (200 mm); (4) adhesive mortar (1.5 mm); (5) mineral wool (35 mm); (6) lime plaster (25 mm); (7) B&K acceleration receivers 6\(\times\); (8) the measurement grid of \(22\times 18\) points with spacing of 150 mm; (9) the cross-section of the test wall; and (10) acoustic excitation.  
The sound power radiated by the test wall in the receiving room due to transmission of sound coming from the sending room can be calculated by an approach that was elaborated in [44; 49; 50; 51]. Based on the measured normal velocity \(v_{n}(\mathbf{r}_{s},\omega)\) of the vibrating surface at positions \(\mathbf{r}_{s}\) on the wall, the acoustic pressure \(p(\mathbf{r},\omega)\) at position \(\mathbf{r}\) in the receiving room (denoted by distance \(\underline{R}=|\mathbf{r}-\mathbf{r}_{s}\,|\)) can be calculated by  
\[p(\mathbf{r},\omega)\ =\ \frac{i\omega\rho}{2\pi}\iint\limits_{S}v_{n}(\mathbf{r}_ {S},\omega)\frac{e^{-ik\underline{R}}}{\underline{R}}dS \tag{2}\]  
where \(S\) is the area of the vibrating wall \(S,\rho\) is the density of air, \(\omega\) is the angular frequency and \(c\) is the speed of sound in air. Based on calculated pressure \(p(\mathbf{r}_{s},\omega)\) and the measured normal velocity \(v_{n}(\mathbf{r}_{s},\omega)\), the active acoustic intensity \(I\) along the surface and the total radiated active sound power \(P\) can be computed as:  
\[I(\mathbf{r}_{  
as computed by the Rayleigh integral, considering the measured reverberation time of the receiving room, using the expression  
\[L_{p2}\ =\ L_{w}-10\log\biggl{(}\frac{A}{A_{0}}\biggr{)}-4.34\frac{A}{S}-10\log \biggl{(}1+\frac{cS}{8 Vf}\biggr{)}-C_{1}-C_{2}+6 \tag{7}\]  
which may also be expressed as  
\[L_{p2}\ =\ L_{w}-10\log A-K_{a}-K_{w}-C_{1}-C_{2}+K_{d} \tag{8}\]  
where \(A_{0}\) = 1 m\({}^{2}\) is the reference area.  
Vibration measurements were made by scanning the positions of 5 accelerometers on a grid with spacing of 15 cm on the receiving side of the wall (which was built on the side of the sending room (Figure 1b), while recording simultaneously the signal of an accelerometer at a reference position in one of the corners of the wall. The accuracy of interpolating deflection shapes depends on the number of samples per wavelength. According to the Nyquist-Shannon sampling theorem, the used spacing of g = 15 cm allows to accurately sample structural waves with wavelengths up to 4 g = 60 cm, which, based on the velocity dispersion curves in Figure 4, corresponds with a frequency of 1250 Hz (bending wave velocity 955 m/s). Results obtained by airborne exciting the sending room with pink noise can thus be expected to be reliable till about 1250 Hz.