# Calculation of Guided Wave Dispersion Characteristics Using a Three-Transducer Measurement System
## 3 Methodology  
In this section, the methodology for determining the phase shift and hence the phase velocity is presented. This general methodology is complemented by a description of a specific experiment setup described in Section 6.1. The technique is based on the analysis of two signals acquired at different distances in the direction of wave propagation. One of them will be treated as the baseline and the other signal will be modified in time and phase in order to match with the baseline. The methodology is composed of different signal processing techniques which are presented in a block diagram in Figure 3.  
The experimental setup is composed of one transmitter and two receivers along the propagation direction of the wave to be measured, as shown in Figure 4. For isotropic systems the wave velocity is independent of direction. The technique analyses the effect of propagation between the first receiver and the second receiver. Spacing between receivers of a few centimeters is selected in order to avoid large phase shifts and waveform deformations due to dispersion. The first step in analyzing the data is the isolation of the desired wavepacket by removing the rest of the signal. The boundaries of each wavepacket are determined by studying the slope of its envelope. The boundaries are set at the closest zero slopes before and after each peak, see example of real signals in Figure 5. Once the region containing the wavepacket is identified, the remainder of the signal is discarded. Both signals are truncated with the same boundaries in order to maintain the time difference between the wavepackets, \(w_{1}\) and \(w_{2}\). The boundaries used are the left limit of the signal of the closest receiver to the transmitter and the second limit of the signal of the further receiver, as can be seen in Figure 6. Afterwards, the Hilbert Transform (H(...)) [28] is applied to both wavepackets and the magnitude extracted in order to determine the envelope of each wavepacket:  
\[e_{1}(t) =|H(w_{1}(t))|, \tag{10}\] \[e_{2}(t) =|H(w_{2}(t))|. \tag{11}\]  
Figure 4: Schematic of the test setup. Figure 3: Methodology’s block diagram to extract the optimum phase shift and time delay. These two envelopes are then cross-correlated to determine the time delay between them:  
\[\Delta t=\underset{\tau}{arg\ max}((e_{1}\star e_{2})(\tau)), \tag{12}\]  
where \(\tau\) is the applied time delay and \(\Delta t\) the value that maximizes the correlation. Once this time delay is known, the phase delay to the harmonic part of \(w_{2}(t)\) that maximizes the cross-correlation between the delayed phase-shifted wavepacket \(w_{2}^{*}(t-\Delta t)\) and \(w_{1}(t)\) is calculated. This process consists of the modification in phase of the signal from the furthest receiver by a phase angle, \(\phi\), and finding the phase shift that maximizes the cross correlation. The phase shift, \(\phi\), is applied to \(w_{2}^{*}\); then, the cross correlation between \(w_{1}\) and \(w_{2}^{*}\) is calculated. The optimum phase shift is the phase angle when the correlation coefficient with time delay, \(\Delta t\), is highest. \[\phi_{opt}=\underset{\phi}{arg\ max}((w_{1}\star w_{2}^{*})(\Delta t)(\phi)) \tag{13}\]  
Figure 4.0 Wavepacket detection and boundary determination established by the algorithm for two signals acquired at: (**a**) 30 cm and (**b**) 35 cm from the transmitter using a d\({}_{33}\)-type transducer. Figure 4.1 Shortening of the signals to reduce computational time. The same limits are used to truncate each signal in order to maintain the time difference between wavepackets. Signals acquired at: (**a**) 30 cm and (**b**) 35 cm from the transmitter. This methodology enables the calculation of the lag time and phase shift between the two signals acquired at different distances. By knowing these three variables (distance, time and phase) and also the excitation frequency, (2) and (9) can be used to determine the group and phase velocity, respectively.