# Calculation of Guided Wave Dispersion Characteristics Using a Three-Transducer Measurement System
## 4 Synthetic Signal Analysis  
The first validation was carried out using synthesized signals. The signal synthesis is based on the adjustment in frequency and wavenumber of the input signal in the frequency domain. Knowing the waveform at one point, in this case the input signal at the transmitting point, and the dispersion characteristics of each wave mode, the wave modes can be reconstructed in the time domain at any distance [29]. If \(u(x,t)\) is the reconstructed wave mode at a distance \(x\) from the transmitting point, \(t\) is the time, \(F(\omega)\) is the Fourier transform of the input signal and \(k(\omega)\) is the wavenumber as a function of the angular frequency, then  
\[u(x,t)=\int_{-\infty}^{\infty}F(\omega)e^{i(k(\omega)x-\omega t)}d\omega \tag{14}\]  
The relation between wavenumber and angular frequency is extracted directly from the dispersion curves provided by Disperse(r) and introduced in the integral. As an initial test of the signal processing method, developed signals are synthesized that are highly pure, without noise and overlapping between modes. The higher the sampling frequency is set, the better accuracy the technique provides. The sampling frequency was set at 10 MHz, since it has been observed that it provides a good balance between computational time and accuracy. A set of signals for symmetric (S\({}_{0}\)), antisymmetric (A\({}_{0}\)) and shear horizontal (SH\({}_{0}\)) wave modes were created by evaluating equation (14) in MATLAB for two propagation distances, 30 cm and 35 cm: the analysed propagated distance (\(\Delta d\)) in this case is 5 cm. The analysed frequencies are from 60 to 370 kHz with steps of 10 kHz. The results after applying the signal processing algorithm described in Section 3 to the synthesized signals are presented in Figure 7. The results (black circles) accurately reconstruct the theoretical values from Disperse(r) (grey lines). In Figure 8, the extracted phase shift has been represented against the frequency for the three fundamental modes. For the S\({}_{0}\) mode, the phase values have negative values decaying from -2 degrees at 60 kHz to -52 degrees at 370 kHz. For the A\({}_{0}\) mode, conversely, the phase is positive, increasing along the frequency, since the phase velocity is lower than the group velocity; from 377 degrees at 60 kHz to 673 degrees at 370 kHz. In this case, the phase values are much higher than the S\({}_{0}\) and SH\({}_{0}\) ones, because the A\({}_{0}\) mode is highly dispersive at these frequencies. For the SH\({}_{0}\) mode, the extracted phase values are 0 degrees at 60 kHz and 1 degree at 370 kHz. The SH\({}_{0}\) is a nondispersive mode so the phase value remains at near 0 degrees. The sampling frequency is an important factor, since it determines the degree of resolution of the group velocity. As these examples are performed for short propagation distances, the time that the wave modes take to cover that distance is small. Thus, low sampling frequencies are not able to determine the group velocity accurately. Faster modes require higher sampling frequencies to get the same resolution as slower modes. In this case of 5 cm spacing, the propagation time for the S\({}_{0}\) mode at 200 kHz is 9.3 \(\upmu\)s; a sampling frequency of 1 MHz will have a time resolution of 1 \(\upmu\)s which yields a velocity resolution for this particular example of 556 m/s; with a sampling frequency of 10 MHz, the velocity resolution will be 58.4 m/s; and with a sampling frequency of 100 MHz, the velocity resolution will be 5.8 m/s. This resolution issue can be observed in Figure 7; at lower velocities the curves are smoother, and at higher velocities (S\({}_{0}\) mode) the curves have poorer velocity resolution. Figure 8: Comparison of the phase shift of S\({}_{0}\), A\({}_{0}\) and SH\({}_{0}\) between the results from the synthesized signals (black circles) and theoretical values (grey lines).