## 7 Conclusions  
The propagation of mechanical waves in rectangular section waveguides (\(15\times 5\) [mm]) was studied within the frequency range from 0 to 100 [kHz]. This geometry could be assimilated with an element of structural armor of a flexible riser commonly used in the off shore oil industry. Different ways to build the dispersion curves of the studied waveguide were verified. Two dispersion curves were created analytically. One of them, based on the development made in plates, presented consistent results, but it was not possible to represent the torsion mode. The other set of analytical curves, based on the beam theory, showed to be consistent in a small range of wave numbers in which the wavelength used allows to consider as valid the Euler-Bernoulli beam theory hypothesis. Two SAFE methods were used, the decomposition method in harmonics proposed originally by [PERSON] et al. [28] and the method proposed by [PERSON] et al. [25]. On one hand, the first method is easier to implement but does not allow to obtain the wave tridimensional mode configuration directly. On the other hand, the second method requires a more significant effort in the assembly of the dispersion curves, but it presents as a bonus the possibility to present the mode of propagation in a tridimensional display. Both methods presented identical precision in the definition of the dispersion curves for the structure analyzed. Experimental results allowed to capture the spatial and time displacement distribution when a specific excitation is applied. The tone burst function is applied with a piezoelectric actuator in one of the rod ends in an axial and transversal rod direction. The responses using a laser transducer displacement over a scan line were acquired. These data allowed us to determine the spatial and temporal distribution of the wave displacement mapped in a frequency vs. wavenumber domain. The experimental test was also simulated with a finite element model using an explicit scheme of integration in the time domain showing a consistency with the experimental results. By combining the experimental information acquired, it was also possible to build points over the dispersion curves. The comparison between the experimental results and different theoretical solutions show the consistency of the SAFE approaches implemented and also help to understand the propagation which is characteristic of the guided waves in rectangular rods provided subsidies to implement applications related to NDT techniques. The parametric study shows how the propagation conditions change as a function of the base/height ratio of the rectangular cross-section waveguide. Finally, in the present work, two applications linked to the knowledge of the dispersion curves in this kind of structure were presented. As regards these applications, it is possible to observe that the non-conventional measurement of wave displacement performed by using fiber grating Bragg cell (FGB), was consistent. This method is an excellent option when a low level of intervention on the structure analyzed is an essential factor since only one point of access used to emit and receive the wave is enough to prospect the region of interest. Considering the simulation of AE events, the numerical results obtained with the numerical approach are also consistent. Notice that the knowledge about the way the wave emitted as AE signal is polarized in a specific form, in the present case, a waveguide with a rectangular cross-section is investigated and the possibility to design a specific AE device is pointed out. This specific device could be used to minimize the noise and maximizes the capture of EA events information. It is imperative to highlight that an experimental verification over the obtained results in the AE simulation is mandatory to calibrate the model studied with the kind of information that could be captured in the real problem.