## 2 Propagation on Stems with Rectangular Section
### Analytical Models of Wave Propagation in Waveguides  
[PERSON] [14], in 1917, developed a solution for wave propagation in plates whose response relates to angular frequency (\(\omega\)) with the wavelength (\(\lambda\)). This solution is known as the transcendental equations of Rayleigh-Lamb [6; 8]. In addition to the analytical solution, the graphical representation of these equations is known as the plate dispersion curves. In the dispersion curves, each curve represents a propagation mode [15], which can be written in terms either of frequency (\(\omega=2\pi/\)T, f = 1/T) vs. the wavenumber (k = 2\(\pi\)/\(\lambda\) k = 1/\(\lambda\)), or phase velocity (c) vs. frequency, remembering that c = \(\omega\)/k. Moreover, it is possible to represent the dispersion curves in terms of group velocity vs. frequency since the group velocity is defined as: \(c_{\mathbf{g}}=\mathrm{d}\omega/\mathrm{d}k\)[6]. This way to present the solution of the waveguide propagation (curves that represent the propagation modes and their frequency and wavenumber relations) is usual in acoustics, acting as a map of the waveguide propagation. There are a lot of publications on the computational methods on which research was based on to build the dispersion curves; among others, we cited the work presented in [16] where the key works in this topic are commented. In the particular case of a rectangular waveguide the dispersion curves obtained using an analytical approach presents two options: to consider the waveguide as a plate or as a beam, both with an \"infinite\" length. Next, the most relevant computational methods due to their historical relevance and for their practicality are presented:  
Considering the rectangular rod as a plate, [PERSON] and [PERSON] [10] developed an adaptation of the Lamb plate theory applied to rectangular sections. Mode wave families propagating on plates are called: Sn (symmetrical to the plate), An (antisymmetric) and SHn (horizontal shear, perpendicular to the propagation direction). In [PERSON] [8], an analogous model is presented, where two fundamental modes, S\({}_{0}\) and A\({}_{0}\) are defined by the expressions \(\omega=kcs\) and \(\omega=kcp\). The family modes Sn and An tend to these fundamental modes when the frequencies decrease. In Figure 2 the modes Sn and An are presented for the rectangular rod studied and modes A\({}_{0}\) and S\({}_{0}\) are highlighted. If the rectangular rod is considered as a beam, it is possible to generate the dispersion curves from the balance of forces applied over an infinitesimal portion of the waveguide, and considering the hypotheses that allow the solution of the differential equation of equilibrium in terms of \(\omega\) and \(k\)[11; 17]. In Figure 3, four modes from three known movements of beams: axial tension, torsion, and bending in two main directions are shown. In Table 1, the expression that defines these curves are presented and described. The expressions above (Table 1) present the dispersion function of each mode, where \(\rho\) is the density, \(C_{t}\) the modulus of torsional stiffness, \(J\) the polar moment of inertia, \(A\) the transversal area, \(I\) the moment of inertia with respect to the bending axis (Axis 2 or 3, depicted in Figure 5b) and \(E\) the axial elastic modulus. Some corrections can be applied to the analytical expressions obtained to represent the dispersion curves either for the formulations from the plate or beam model. For example: Love correction for longitudinal waves in beams [11]; modification of the dispersion relation for the longitudinal waves based on [PERSON]'s formulation applied in Plates presented by [PERSON] [12]; [PERSON]'s correction in the transverse waves of the beam model [18]. In all cases, the corrections are made aiming to improve the analytical solutions.