## 2 Propagation on Stems with Rectangular Section
### Semi-Analytical Models to Build the Dispersion Curves  
The semi-analytical finite element method (SAFE) is a technique that uses finite element analysis (FEM) for the dispersion curves confection of a given waveguide. They are called semi-analytic because these models require an analytical condition that by-pass the fact that the waveguide has at least one \"infinite\" dimension. The analytical condition allows relating the frequency with the wave number. This method allows building the dispersion curves of complex geometry waveguides, such as train rails [20], stem or plates composed by two or more materials [21], among a series of other possibilities. The central idea of this technique is to compute the natural vibration frequencies of models that incorporate, using an analytical restriction, the infinite dimensions of the waveguide. As mentioned in [22], the first SAFE method appear in 1973 with [PERSON] [23], who developed a procedure for working with waveguides of complex geometry with one face clamped, and with [PERSON] [24] that developed a method to compute dispersion curves for orthotropic rectangular waveguides. Since these pioneer works, SAFE methods have been constantly evolving and making progress. A proof of this fact is the great number of works in the scientific community, among others could be cited: [25], where the study of 3D finite element model combined with periodic boundary conditions in the infinite direction is implemented to compute the dispersion curves; [26], the study where the focus is to build the dispersion curves in anisotropic waveguides; [27], the study where the dispersion curves are applied in the context of nonlinear wave propagation; [28], the study where the dispersion curves are computed using the _COMSOL_/FEM package; [29], where the dispersion curves are implemented considering an harmonic function in the infinite dimension, [30], where the dispersion curves is used as tool in the area of geophysics; and finally [31; 32], where the dispersion curves are used in order to understand the wave propagation in textile composite with complex microstructure. In the present paper, two SAFE methods, [PERSON] et al. [28] and [PERSON] et al. [25] are used to make the dispersion curves of a waveguide with rectangular section (\(15\times 5\) [mm]), \(E=205\) GPa and \(\
u=0.29\). Next, the theoretical framework of both methodologies and the results for the rectangular rod studied are presented. _SAFE, proposed by [PERSON] et al. [28]_: In this method the displacement field in the infinite direction (1) according to Figure 5g, occurs through the imposition of the displacement field presented in Equation (1), on the problem described in Equations (2) and (3). \[\mathfrak{u}=\mathrm{U}(\mathfrak{x}_{2},\mathfrak{x}_{3})\mathrm{e}^{\sqrt{- 1}(\mathfrak{kx}_{1}-\omega\mathfrak{t})} \tag{1}\]  
where \(\mathfrak{u}\) represents the displacement vector, \(\mathfrak{k}\) the wavenumber, \(\omega\) the frequency, \(\mathfrak{x}\) the spatial coordinate, \(\mathfrak{t}\) the time and \(\mathrm{U}\) the amplitude of the displacement in the transverse direction of the waveguide. Equation (2) is the Cauchy equation, which governs the static equilibrium in a domain \(\Omega\) and Equation (3) represents the stresses \(\sigma_{i}\) on the contour \(\Gamma\). \[C_{ijkl}\frac{\partial^{2}u_{j}}{\partial\mathfrak{x}_{k}\partial\mathfrak{x} _{l}}+\rho\omega^{2}u_{i}=0\;\;em\;\Omega \tag{2}\]  
\[\sigma_{i}=C_{ijkl}\frac{\partial u_{j}}{\partial\mathfrak{x}_{l}}n_{k}\;\; \;\;\;\;\mathrm{em}\;\Gamma \tag{3}\]  
where \(C\) is the elastic constitutive tensor and \(n_{k}\) are the components of the normal versors with \(i,j,k\) and \(l=1,2,3\). Figure 5: (a) Dispersion curves computed using the [PERSON] et al. method presented in [28]. Displacement field of (**b**) second torsional, (**c**) longitudinal, (**d**) torsional, (**e**) flexural (around Axis (2)), (**f**) flexural wave mode (around Axis (**f**)). (**g**) The waveguide scheme with the reference axes indicated.