## 2 Theoretical Background  
### Propagation of Bending Waves  
In the present paper, the wave propagation approach, as presented in the work by [PERSON] and [PERSON] (1990) is used as a theoretical background for defining a test method to obtain the mechanical and dynamic properties of a sandwich material. The symmetry of the structure is assumed in obtaining the equations governing its transverse displacement (Figure 2), whose derivation is summarized in [PERSON] (1990). The external laminates have thickness \(h_{l}\). The effective shear stiffness of the core is \(G_{c}\), and its thickness \(h_{c}\). The Young's modulus of the laminates, \(E_{l}\), is usually much greater than the Young's modulus of the core, \(E_{c}\). The core is generally stiff in the \(z\)-direction perpendicular to the laminates, which therefore can be assumed to move in-phase within the frequency range used for the acoustic predictions. The bending stiffness of the entire structure can be computed as:  
\[{D^{\prime}}_{1}=b\left[\frac{E_{c}h_{c}^{3}}{12}+E_{l}\left(\frac{h_{c}^{2}h_ {l}}{2}+h_{c}h_{l}^{2}+2\frac{h_{l}^{3}}{3}\right)\right] \tag{1}\]  
[PERSON]'s principle states that the action integral of a mechanical system is stationary. The action is the difference between the total potential energy, which for a sandwich beam is due to pure bending of the beam, bending of both laminates and shear in the core, and its total kinetic energy, due to the vertical motion of the beam and to the rotation of one of its sections. This formulation is used  
Figure 2: Sandwich beam with foam core: geometric and mechanical characteristics. to derive the system of equations that governs the flexural behavior of the structure. A sixth-order differential equation is obtained, whose solution shows that the sandwich material behaves like a simple homogeneous beam whose bending stiffness depends upon frequency. An \"apparent\", frequency-dependent bending stiffness \(D^{\prime}{}_{a}\) can therefore be obtained as the solution to the equation:  
\[\left(\frac{G_{c}S}{\sqrt{m^{\prime}}\omega}\right)+\left[\frac{D^{\prime}{}_ {a}^{3/2}}{D^{\prime}{}_{1}}-D^{\prime}{}_{a}^{1/2}\right]+D^{\prime}{}_{a}-2D ^{\prime}{}_{2}=0 \tag{2}\]  
where \(D^{\prime}{}_{2}\) is the bending stiffness of one laminate and \(\omega\) is the angular frequency, \(2\pi\)_f_. In the expression, \(m^{\prime}\) is the mass per unit length of the beam, calculated with respect to the beam cross section. This equation can be written in a more general way as:  
\[\frac{A}{f}D^{\prime}{}_{a}^{3/2}-\frac{B}{f}D^{\prime}{}_{a}^{1/2}+D^{\prime} {}_{a}-C=0 \tag{3}\]  
provided that \(A\), \(B\) and \(C\) coefficients are defined as follows:  
\[A=\frac{G_{c}S}{\sqrt{m^{\prime}}2\pi D^{\prime}{}_{1}}B=\frac{G_{c}S}{\sqrt{ m^{\prime}}2\pi}C=2D^{\prime}{}_{2} \tag{4}\]  
For a beam made by a non-metallic material, the Young's modulus can be frequency-dependent as well [27]. However, within the frequency range of interest the parameters \(D^{\prime}{}_{1}\), \(D^{\prime}{}_{2}\), and \(G_{c}\) in Equation (4) can be reasonably assumed to have constant values for the structure under investigation. The literature presents many examples of successful use of this theory for different purposes, like estimating the sound transmission loss of a structure, as described in Section 2.2, evaluating new materials, and performing optimization analyses. The definition of \"sandwich\" is here quite broad. For example, [PERSON] et al. [28] use this approach to estimate the sound insulation performances of a coated brick wall, while [PERSON] et al. [29] study the vibro-acoustic behavior of wood plastic composite panels, which, given their inner structure, are treated as orthotropic sandwich elements with lightweight core, obtaining the bending stiffness according to the [PERSON] and [PERSON]'s model and feeding it to the transfer matrix method to optimize the structure in terms of sound transmission loss.