# Numerical and Experimental Investigation of Guided Wave Propagation in a Multi-Wire Cable
## 2 UGW Propagation in Multi-Wire Waveguides
### Dispersion Analysis of a Single Rod  
A single wire can be considered to be the same as an individual cylindrical rod, whose dispersion properties can be analytically obtained using the Pochhammer-Chree equations. UGWs tend to travel with different velocities in a structure depending on the center excitation frequency. There are two ways to define the propagation velocities of UGWs: group and phase velocity. Group velocity refers to the propagation velocity of the entire wave packet. Phase velocity is the propagation velocity at which the phase of any one frequency component of the wave travels. For a cylindrical rod, group and phase velocity dispersion curves of a 7-mm-diameter steel wire (commonly used in a bridge cable), and a steel and aluminum wire with a diameter of 2.5 mm and 3.2 mm, respectively (commonly used in ACSR cable), were acquired using an open source software package named PCDISP developed in the Matlab (MathWorks Inc., Natick, MA, USA) environment. PCDISP is described in more detail in references [28; 29]. The material and geometric properties of wires used in the dispersion calculation models are summarized in Table 1. The results are shown in Figure 1.  
Figure 1 shows UGW modes for three different types of wires with frequencies up to 1000 kHz. The phase and group velocities of the fundamental order flexural mode F(1,1) are slow and display severe dispersive properties in the low-frequency range (below 200 kHz). L(0,1) and T(0,1) seem to be the applicable modes for wire breakage inspection of steel and aluminum wires, though the velocity of the L(0,1) mode is faster than that of the T(0,1) mode at low frequencies, yet is slightly dispersive. Therefore, the L(0,1) mode seems ideal for wire breakage detection due to its velocity and relatively low dispersive behavior. Figure 2a-c show the vibration displacement distribution in a cross-section of a single 7-mm-diameter steel wire carrying these three types of fundamental modes: L(0,1), T(0,1), and F(1,1), at a frequency of 60 kHz. The arrows indicate the particle vibration displacements in the radial direction (in-plane). The color indicates the magnitude of the particle displacements in the \(z\)-axial direction (out-of-plane).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & **Diameter** & **Elastic Modulus** & **Density** & **Poisson Ratio** \\ \hline Steel & 7 mm & 209 GPa & 7800 kg/m\({}^{3}\) & 0.3 \\ Steel & 2.5 mm & 209 GPa & 7800 kg/m\({}^{3}\) & 0.3 \\ Aluminum & 3.2 mm & 72 GPa & 2700 kg/m\({}^{3}\) & 0.35 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The material and geometric properties of wires used in the dispersion models.  
Figure 1: Phase and group velocity dispersion curves for (**a**) a single 7-mm-diameter steel wire and (**b**) a single steel and aluminum wire with a diameter of 2.5 mm and 3.2 mm, respectively.  
Both longitudinal and flexural guided waves create radial and axial vibration displacement. The coupling extent among the adjacent contact wires in the cable is determined by the radial vibration displacement at the wire surface (i.e., maximum radius). The larger the radial vibration displacement amplitude, the better the coupling. Therefore, the frequency of UGW wave modes with large radial vibration displacement at the wire surface become particularly critical during energy transfer. In Figure 2d,e, radial and axial vibration displacement distribution curves (wave structure) of the different radii of a 7-mm-diameter steel wire were calculated using Pochhammer-Chree equations for the L(0,1) and F(1,1) wave modes in the 20-180 kHz frequency range. As the wire radius increases, the radial displacements decrease but the axial displacements increase. This trend was significant with the increase in wave frequency below 200 kHz.