## 3 Methods
### Measurement of Sonic Velocity  
#### 3.2.1 Theoretical  
Various types of elastic waves can be produced in the matter, depending on the mode of excitation used. In the analysis of any type of wave, the material consisting of particles at rest, i.e., in a state of equilibrium, is considered. The average distances between the particles are on average constant, representing a condition of equilibrium between attractive and repulsive forces at the atomic or molecular level. If for some reason the particles are subject to moving away, the attractive forces prevail, so when the cause ceases, they return to their initial position; if they are subject to moving closer, repulsive forces prevail. Therefore, it can be said that, following the stresses, the particles undergo a shift from their position of equilibrium, and because of the elastic reaction of the material they tend to return to their primitive position. This applies if the stresses are contained within the elastic limits of the material. In solids, the particles can oscillate in the direction of propagation as longitudinal waves or perpendicular to the direction of propagation as transverse waves. Longitudinal waves are commonly used in the field of non-destructive diagnostics [56; 57], and are also used to estimate the mechanical parameters of the material in the context of this study. The general equation of a longitudinal wave that propagates along a bar of section small with respect to the wavelength, and with section, density, and modulus of elasticity that remain constant along the longitudinal development of the bar itself can be written as  
\[\partial^{2}\mathrm{u}/\partial\mathrm{x}^{2}\ -\ \rho/\mathrm{E}\,\partial^{2} \mathrm{u}/\partial\mathrm{t}=0 \tag{1}\]  
where x is the axis of the bar, u(x,t) is the temporary displacement that particles undergo under the action of the wave, t is the generic time, \(\rho\) is the density of the material, and E is the longitudinal modulus of elasticity that binds the longitudinal deformation \(\varepsilon\) to the pressure p induced by the passage of the wave. Assuming  
\[\rho/\mathrm{E}=1/\mathrm{V}^{2}, \tag{2}\]  
with V wave propagation velocity in the longitudinal direction, equation 1 takes the form  
\[\partial^{2}\mathrm{u}/\partial\mathrm{x}^{2}\ -\ 1/\mathrm{V}^{2}\, \partial^{2}\mathrm{u}/\partial\mathrm{t}=0 \tag{3}\]  
This equation defines the longitudinal motion u(x,t) of the considered bar. The differential equation has a general solution of the form  
\[\mathrm{u}(\mathrm{x},t)=\mathrm{f}_{1}(\mathrm{x}-\mathrm{V}t)+\mathrm{f}_{2 }(\mathrm{x}+\mathrm{V}t), \tag{4}\]  
where \(\mathrm{f}_{1}(\mathrm{x}-\mathrm{V}t)\) is a wave that propagates in a positive direction with a velocity \(\mathrm{V}=\sqrt{\left(\frac{E}{\rho}\right)}\) and \(\mathrm{f}_{2}(\mathrm{x}+\mathrm{V}t)\) is a wave that propagates in a negative direction with a velocity \(\mathrm{V}=-\sqrt{\left(\frac{E}{\rho}\right)}\). Wood is an orthotropic material, which means that its properties vary according to three mutually orthogonal directions, conventionally assumed to coincide with the longitudinal, radial, and tangential axes of a cylindrical coordinate system in which the longitudinal direction coincides with the axis of the trunk and then with the direction of the fibers of the material. Therefore, the velocity of an acoustic impulse that propagates in the wood depends on the direction of propagation too. The velocity in the longitudinal direction is the highest, being about five times greater than in the radial one, whereas the one in the tangential direction is the lowest [53; 58].