## 3 Materials and Methods
### Preliminary Tests on Concrete Materials
One of the points taken as a reference for the passage of R-waves is highlighted in the graph. The distance between the point of impact and the registration point is 20 cm. Figure 5: (**a**) The layout of the ten measurement point positions; (**b**) Portion of the tested concrete wall with no imperfections in the way of propagation of the R-wave; (**c**) Portion of the wall with some imperfections precisely in the way of waves propagation. In Table 1 the outcomes of the experimental tests are summarized in terms of mean and standard deviation of R-waves propagation velocity, of their intervals related to a confidence level of 95% (obtained by using the \(t\)-distribution due to the small number of samples) and of the derived elastic modulus range of the material, obtained by the third expression of the system Equation (23)  
\[E=\frac{2\,{V_{R}}^{2}(1+\
u)^{3}\rho}{(0.87+1.12\
u)^{2}} \tag{25}\]  
in which \(V_{R}\) is the R-waves propagation velocity, \(\
u\) is the Poisson ratio and \(\rho\) the material density. Regarding the Poisson ratio and the material density, the values reported in Table 1 are taken from the literature in the absence of specific information. Figure 7 illustrates the convergence trend of the Young modulus mean values (\(\overline{E}\)) and of the interval \(\overline{E}\pm\Delta E\) derived by the confidence intervals of the R-wave velocities, with the increase in length of the distance between the measurement points. This aspect makes evidence of the effectiveness of the proposed methodology also for the elastic parameters estimation to be assigned to the equivalent homogenized continuum of a heterogeneous media. Figure 6: Typical signals recorded on the CLS2 wall by a couple of accelerometers 0.20 m apart and placed at a distance of 1.60 m between the source and the sensor Acc 1. The red circle highlights the passage of the S-waves. Figure 7: Young modulus estimation of wall CLS2 with the increase of distance between the two accelerometers: mean values (blue circles) and range values derived by a confidence interval of 95% of the R-wave velocities obtained by indirect test (red and green circles). The results highlight that in the case of CLS1, characterized by a material in a good condition and also confirmed by small values of \(\sigma_{V_{R}}\) in relation to a distance between the accelerometers of 0.20 m, the estimation of the elastic modulus can be considered in agreement with the values associated to a standard concrete class around C20/25 and C25/30. According to Eurocode standard [46], the elastic modulus of concrete can be estimated through the following formulation  
\[E=22,000\Bigg{[}\bigg{(}\frac{f_{ck}+8}{10}\bigg{)}^{0.3}\Bigg{]} \tag{26}\]  
being \(f_{ck}\) the characteristic compressive strength expressed in MPa. The application of Equation (26) to C20/25 and C25/30 gives values of Young modulus equal to 29,962 MPa and 31,476 MPa, respectively. On the other hand, the superficial degradation which characterizes the external layer of CLS2 determines an estimation of the elastic modulus lower than the standard expected values, in agreement with the visual inspections.