## 2 Theory
### Propagation of Bending Waves  
The simplest type of partition for which it is possible to determine the sound transmission loss is a homogeneous wall. In this case, a simplified theory can be applied, once the bending stiffness of the structure is obtained either theoretically or experimentally. Before discussing the model related to the estimation of the sound transmission loss, the background on the propagation of the bending waves in thin plates must be established. The complete presentation can be found in [20] and will be summarized here. When the sound impinges one side of a plate, vibrations are induced in the partition, and the partition develops bending waves. Let us consider a straight and thin beam bending in the \((x,z)\) plane as shown in Figure 2, where \(\zeta\) is the lateral displacement due to bending. It is possible to derive a differential equation to describe the propagation of the bending waves starting from the equation of motion applied to a small portion of beam having a length \(\Delta x\). When the beam bends, the different fiber layers and the axis undergo a bending motion, as well. Consequently, some shear and traction/compression forces are induced in the beam. Figure 1: Breakdown of the incident power: (1) into the main components: (2) reflected; (3) transmitted to adjacent structures; (4) dissipated; (5) transmitted across the partition. Assuming that plane sections remain plain and perpendicular to the axis, thus neglecting the deformations produced by the shearing forces, the normal stress over the cross-section can be written as \(\sigma=E\times\varepsilon\left(z\right)\), where \(E\) is the Young's modulus of the material, and the classic Euler-Bernoulli theory of 'pure bending waves' is derived. This assumption can generally be accepted when the bending wavelength is large if compared with the cross-sectional dimensions. In this condition, the bending moment \(\mathbf{M}_{y}\) can be obtained as:  
\[\mathbf{M}_{y}=\int_{S}\sigma z\,\mathrm{d}S=\int_{S}E\times\varepsilon\left( z\right)z\,\mathrm{d}S\approx-E\frac{\partial^{2}\zeta}{\partial x^{2}}\int_{S}z^{2} \,\mathrm{d}S=-El_{b}\frac{\partial^{2}\zeta}{\partial x^{2}} \tag{4}\]  
where \(l_{b}\) is the moment of inertia relative to the beam cross-section \(S\). Since the bending stiffness \(D\) is the proportionality constant in the relation between the bending moment \(\mathbf{M}_{y}\) and the curvature, \(\partial^{2}\zeta/\partial x^{2}\), it can be defined as:  
\[D=El_{b}. \tag{5}\]  
The sound insulation characteristics of the material are strongly related to its flexural behavior. The bending wave phase velocity in an isotropic panel, \(c_{B}\), is frequency-dependent and can be expressed as:  
\[c_{B}=\sqrt[4]{\frac{\left(2\pi f\right)^{2}D}{\mu}} \tag{6}\]  
where \(\mu\) is the mass per unit area of the wall. When the incident sound waves have the same trace wavelength as free bending waves, the so-called 'coincidence effect' occurs. In this region, the bending wave speed is equal to the speed of sound in air, \(c_{air}\), the wall oscillations are amplified and the acoustic waves are directly transmitted almost without attenuation. The critical frequency, \(f_{c}\), is related to \(D\) and is defined as the lowest frequency for which the coincidence effect occurs, that is the frequency at which the bending wave becomes supersonic:  
\[f_{c}=\left(\frac{c_{air}^{2}}{2\pi}\right)\sqrt{\frac{\mu}{D}}\,. \tag{7}\]  
Figure 2: Bending of a beam in the plane \(\left(x,y\right)\). [PERSON] [21] derived the sound transmission coefficient \(\tau\left(\varphi,\omega\right)\) for a infinite thin homogeneous plate as a function of the angle of incidence \(\varphi\) of the acoustic wave and of the angular frequency, \(\omega=2\pi f\):  
\[