# Calculation of Guided Wave Dispersion Characteristics Using a Three-Transducer Measurement System
## 2 Theoretical Basis  
Assuming that the propagating pulse is sufficiently narrow-band that dispersion does not significantly distort the wavepacket over the propagation distance, the signal can be approximated as [27],  
\[\psi(x,t)=G\bigg{(}t-\frac{x}{v_{g}}\bigg{)}e^{i(kx-\omega t)}, \tag{1}\]  
where \(x\) is propagation distance, \(t\) is time, \(\omega=2\pi f\) is the angular frequency of the harmonic wave at frequency \(f\) and \(k=2\pi/\lambda\) is the wavenumber. \(G(t)\) is a function defining the temporal pulse shape. \(G(t)\) is defined such that the pulse has a peak at \(t=0\). \(v_{g}\) is the group velocity at which the packet propagates, and so if the trajectory of the wavepacket peak is defined as the point whereby \(G\big{(}t_{0}-x_{0}/v_{g}\big{)}=G(0)\), then  
\[v_{g}=\frac{x_{0}}{t_{0}}. \tag{2}\]  
The wavepacket is shown schematically in Figure 1. The harmonic part of the signal has a phase of  
\[\phi=kx-\omega t, \tag{3}\]  
and propagates at the phase velocity [6; 27].  
\[v_{p}=\frac{\omega}{k} \tag{4}\]  
The phase of the harmonic signal at the centre of the wavepacket \(x_{0}\), \(t_{0}\) is  
\[\phi_{0}=kx_{0}-\omega t_{0}. \tag{5}\]  
Rearranging and substituting for \(v_{p}\) and \(v_{g}\) gives  
\[\phi_{0}=kt_{0}\bigg{(}\frac{x_{0}}{t_{0}}-\frac{\omega}{k}\bigg{)}, \tag{6}\]\[\phi_{0}=kt_{0}\big{(}v_{g}-v_{p}\big{)}. \tag{7}\]  
For nondispersive waves, \(v_{p}=v_{g}\) and Equation (7) gives the expected result that the phase at the wavepacket center is \(\phi_{0}=0\) for all propagation times. Where the system is dispersive there is a finite phase difference between the harmonic part of the signal and the wavepacket, \(\phi_{0}\), which increases linearly with propagation time and is positive if \(v_{g}>v_{p}\), negative if \(v_{g}<v_{p}\). Assuming that this phase difference can be measured experimentally, Equations (4) and (5) further yield two useful equations for evaluating the phase velocity:  
\[v_{p}=\frac{x_{0}-\frac{\phi_{0}}{k}}{t_{0}}, \tag{8}\]  
\[v_{p}=\frac{x_{0}}{t_{0}+\frac{\phi_{0}}{\omega}}. \tag{9}\]  
In Figure 2, an illustrative example of a S\({}_{0}\)-mode wavepacket at 150 kHz is shown. The solid line is the S\({}_{0}\) mode after 1.2-m propagation including the dispersion. The dashed line represents the S\({}_{0}\) mode after the propagation without dispersion, namely, the wavepacket was calculated using the same phase velocity for all frequencies. In this example, the phase shift is negative (-1.26 radians), which means the phase velocity is higher than the group velocity. Using (9), the calculation of the phase velocity would be 1.2 m divided by the sum of 223 us, which is the time of the wavepacket to travel 1.2 m at group velocity, plus -1.34 us which is the time that the signal at an angular frequency, \(\omega=2\pi\):150000, takes to cover -1.26 radians. Group velocity and phase velocity are calculated using (2) and (9), respectively: \(v_{g}=5381\) ms\({}^{-1}\) and \(v_{p}=5414\) ms \({}^{-1}\).  
Figure 1: Wavepacket arriving at \(x_{0}\) at time \(t_{0}\) with phase \(\phi_{0}\) propagating at group velocity \(v_{g}\). \(G(t)\) is represented as the envelope of the wavepacket.  
Figure 2: S\({}_{0}\)-mode wavepacket of 5 cycles at 150 kHz after propagating 1.2 m, with dispersion and without dispersion.