# Calculation of Guided Wave Dispersion Characteristics Using a Three-Transducer Measurement System
## 1 Introduction
The technique requires precise control of the incidence angle of the ultrasonic pressure from the ACT upon the surface. By relating the frequency to the incidence angle at which the wave amplitude is maximized, it is possible to calculate the phase velocity using [PERSON]'s law. However, the technique has limited applicability outside of the laboratory. In works by [PERSON] et al. [18; 19], a zero-crossing technique is used to calculate the phase velocity based on the measurement of the position of constant phase points of the pulse corresponding to zero amplitude as a function of time. A similar approach is taken in the phase velocity method in [20], tracking the peaks of the pulse rather than the zero crossings. To produce convincing results, both techniques require the acquisition of a large number of signals at different distances, making these techniques time-consuming. For the sake of reducing time and sampling points in space, [PERSON] et al. [21] presented a method using an array probe from which a time-space matrix can be created. A 2D filter is applied to the matrix to extract the phase velocities. The technique was demonstrated to be valid in simulation using FE analysis. However, for realistic probe sizes, the technique has limited experimental applications. In [22], sparse wavenumber analysis was used to experimentally recover the dispersion curves from an aluminum plate; however, seventeen sensors were needed to deploy the technique. An alternative approach to getting dispersion curves from experimental signals is to use time-frequency representations [23], where group velocity can be directly determined. However, the calculation of phase velocity involves the integration of group velocity requiring precise values of \(\omega\) and k at the lower limit, which are not easy to obtain experimentally. For medical purposes, two techniques, SVD and LRT, were evaluated for extracting the dispersion curves of cortical bones in [24]. Both techniques require a large number of receivers and transmitters. Experimental calculation of bulk wave velocities in dispersive materials has been studied [25]. [PERSON] and [PERSON] used the method of phase spectral analysis, where the phase delay is calculated using the real and imaginary parts of the Fourier transform of the pulse, and then obtaining the group delay through the differentiation of the phase characteristics of the pulse with respect to frequency. [PERSON] et al. presented the amplitude spectrum technique overcoming the requirement of the phase spectrum technique for the separation of successive pulses in the time domain for bulk waves [26]. However, the phase spectral technique provided better results when pulses can be separated. In this paper, an experimental technique for straightforward calculations of phase and group velocities of guided waves at frequencies of interest is presented. The technique requires just the acquisition of two signals spaced a few cm using conventional transducers, enabling its application on in-service structures located in poorly controlled environments. There is no requirement for prior knowledge of the material properties or thickness of the inspected structure. In Section 2, the formulation on which the experimental technique relies is presented. Section 3 describes in detail the methodology to extract the group delay and phase shift from experimental signals. Successive sections present validation tests using synthesized signals and simulated signals from FE analysis. Finally, experimental signals from a 3 mm-thick aluminium plate are used to validate the technique as a solution to the problem presented in the introduction.