## 3 Experimental Analyses  
In the present section, experimental analyses of the elastic wave propagation in rectangular section waveguides are illustrated. The four fundamental modes will be excited; applying a tone burst force with main frequency bands between 10 to 100 kHz. As a response, the displacements were measured at equally distanced points arranged along the waveguide propagating direction. The information of every test was saved in a matrix ([s] \(\times\) [m]), where the colons indicate the spatial point where the sensor was fixed and the rows indicate the information stored in each time. Applying the double Fourier transform technique [35] in the data collected, the dispersion curve domain (frequency [1/s] \(\times\) wave number [1/m]) was written. Consequently, it is possible to identify the wavelength and frequency of the wave propagating in the waveguide. It is possible to induce a particular space mode applying excitation forces in different directions, with a tone burst function as a temporal variation. As a result, the dispersion curves for a frequency-wave number dominium were obtained. The dispersion curves obtained with this methodology were compared with the dispersion curves obtained by the SAFE methods already presented in [33]. Preliminary, experimental and theoretical results also were presented by the authors in [36]. The setup consists of:  
* A bar, made of ABNT 1020 steel [37], with dimensions of \(5\times 15\times 1500\) [mm] used as waveguide. * Two piezoelectric ceramics (PZT) Ferroperm(r), with dimensions \(1\times 3\times 13\) [mm] and longitudinal and perpendicular polarizations which were used as an excitation transducer. Its frequency vs. Impedance response is presented in Figure 8, where it is possible to perceive that in the interval of  
Figure 7: Results using the [PERSON] et al. [25] SAFE method, the dispersion curves and the displacement field of each mode. Notice that in this graph the wave number is defined as k = 2\(\pi\)/\(\Lambda\) [m\({}^{-1}\)]. frequency used [0, 100 KHz], these transductors present a response without resonances for the longitudinal and transversal polarization. * A function generator Agilent (Keysight) 33,521, used to create an excitation signal that applies a potential difference to the piezoelectric ceramics transducer; a voltage amplifier Krohn-Hite 7500 used to amplify the excitation signal emitted for the function generator. * Two vibrometers Polytec(r) OFV-505, linked with Polytec(r) OFV-5000 controller, used as sensors to measure the surface displacement of the specimen. * A National(r) NI PXIe-1062Q data acquisition system with the LabVIEW(r) software [38] to scan the vibrometer measures. Data is processed on PC with an Intel(r) 3.30 GHz processor and 8 GB of RAM (random access memory), equipped with MatLab(r) software [39]. The PZT ceramic was glued at the end of the specimen with an epoxy resin, as shown in Figure 8c. The setup is excited with a potential difference V(t) with a tone-burst function, built within Hanning window [6], with different frequencies in the range between 0, 100 [kHz], (5 cycles, with 100 [V] of decrement peak to peak). The ceramic vibration excites the surface of the specimen, and this perturbation propagates along the bar as a guided wave. The surface displacement is captured by the vibrometers. In Figure 9a,b the position of the vibrometers are shown. Notice that the vibrometer's arrangement is sensitive to the displacements directions. Figure 9e shows the scheme of how the components are arranged as well as the path of the electrical signals. The configuration considered uses two vibrometers. This scheme lets to know if the displacements are in the same or out of the propagation plane. Therefore, the phase between the signals of the two sensors must be determined. Equal phases indicate off-plane displacement, and opposite phases signals (phase difference of 180\({}^{\circ}\)) indicate displacement in the plane of the wave propagation.