## 2 Theoretical Background
### Sound Transmission Loss  
The Transmission Loss represents the ability of a structure to block the sound propagation in neighboring ambient and is defined as  
\[TL=10\cdot log\frac{1}{\tau} \tag{11}\]  
Materials characterized by a low transmission coefficient have a high \(TL\). The sound insulation properties depend mainly on the mass per unit area of the structure, the angle of incidence and the frequency of the impinging wave. Other factors influencing the transmission of the sound are the nature of the partition (single, double), the internal losses and the boundary conditions [29]. The acoustic insulation performance of a homogeneous wall can be divided into four regions as shown in Figure 4.  
In the low frequency region, the \(TL\) is governed by the material stiffness and it decreases 6 dB/oct. The behavior of the panel is then dominated by the modes of the specimen, which depend on the elastic and geometric properties of the wall. When the modal density is sufficiently high, the wall behaves according to the mass law [30]. In this region, the transmission loss has a linear trend and increases 6 dB/oct. The mass-law region is limited by the coincidence effect that occurs when the wavelength of the sound in the air is the same as the wavelength of the bending waves in the partition. In this region the partition does offer a weak opposition to sound propagation. The coincidence frequency \(f_{c}\) of a homogeneous board is related to its size, thickness, Young's modulus, and surface density:  
\[f_{c}=\frac{c^{2}}{2\pi}\sqrt{\frac{\mu^{\prime\prime}}{D}} \tag{12}\]  
where \(D\) is the bending stiffness calculated as  
\[D=E\cdot I_{b} \tag{13}\]  
\(E\) is the Young modulus, and \(I_{b}\) is the moment of inertia. The bending stiffness can be computed also for complex structures, once the Young's modulus and the moment of inertia of the elements are known [31]. The sound insulation of a wall can be significantly increased if it is built as a multiple structure. One common way to increase the sound insulation of a partition, without increasing the mass per unit area, is to build it with two or more layers separated by an air gap, possibly filled by sound absorbing material. In this case the wall behaves like a multiple mass-spring-mass system. When an acoustic wave passes through such a construction, the total transmission factor \(\tau_{tot}\) is:  
\[\tau_{tot_{n-1}}^{n=N}=\tau_{1}\cdot\tau_{2}\cdot\ldots\cdot\tau_{n} \tag{14}\]  
in which the assumption is that the \(N\) layers have a transmission factor \(\tau_{n}\). The equation holds at sufficiently high frequency (above the mass-spring-mass frequency \(f_{0}\)), for large enough distances between the layers and when the damping of the gap, in the form of sound absorbing material, is sufficiently high. Below the mass-spring-mass resonance standing waves between the layers modify the transmission factor. The most common case is the one featuring a double wall. If the mass law holds, then the following equation can be applied:  
\[R_{\perp}=20 log(\mu^{\prime\prime}\cdot f)-42\text{ for }f<f_{c} \tag{15}\]  
Figure 4: Sound transmission loss of a single panel: (1) stiffness controlled region, (2) resonances region, (3) mass law region, (4) coincidence region. In case of a finite double wall, the sound transmission loss can be computed as:  
\[R_{\perp}^{double}=20 log\left[\frac{2\cdot\pi\cdot f\cdot\mu_{1}^{\prime\prime}} {(2\rho_{0}\cdot c_{0})}\right]+20\cdot log\left[\frac{(2\cdot\pi\cdot f\cdot \mu_{2}^{\prime\prime})}{(2\rho_{0}\cdot c_{0})}\right]\text{ for }f<f_{c} \tag{16}\]  
The mass law has a lower bound given by the lower mechanical resonance of the system. This resonance corresponds to the mass-spring-mass resonance of the wall, where the air enclosed in the gap acts as a spring, while the walls act like two masses.