# Numerical and Experimental Investigation of Guided Wave Propagation in a Multi-Wire Cable
## 3 Transient Finite Element Simulations
### Single Wire  
In order to analyze multimodal and dispersive guided wave signals, discrete wave signals in time and space were recorded at the receiving nodes of the FE model. Signals were mapped to dispersion data in the frequency and wavenumber domains applying the two-dimensional (2D) fast Fourier transform (FFT) method [34]. The essence of the 2D FFT is to perform two fast Fourier transforms on the signal, temporally and spatially. A guided wave signal is harmonical in time and space from the perspective of wave propagation. Therefore, the time and space domain signals \(f(z,\ t)\) can be mapped to angular frequency and wavenumber domain signals \(F(k,\ \omega)\) as:  
\[F(k,\omega)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(z,t)e^{-i(kz+ \omega t)}\mathrm{d}z\mathrm{d}t \tag{3}\]  
\[c_{ph}=\omega/k \tag{4}\]  
where \(f(z,\ t)\) is the acceleration signal matrix recorded at the receiving nodes along the \(z\) axis of the wire, \(c_{ph}\) is the phase velocity, \(\omega\) is the angular frequency, and \(k\) is the wavenumber. In this case, the accelerations of 500 series with a spatial interval of 1 mm excited by a triangular pulse were used to extract the 2D FFT spectrum, as shown in Figure 7.  
The theoretical phase velocity dispersion curves overlaid on the 2D FFT spectra in Figure 7 agreed with the transient simulation results for the longitudinal and flexural modes. As a result of the radial vibration displacement component \(u_{r}\) and the \(z\)-axial vibration displacement component \(u_{z}\), excited by the oblique imposed forces along the \(x\), \(y\), and \(z\) directions, all longitudinal (i.e., L(0,1)) and flexural modes (i.e., F(1,1), F(1,2), and F(1,3)) below 500 kHz were generated in the single wire. On the other, the only the wave mode L(0,1) was excited by using the perpendicular triangular pulse along the \(z\)-direction.  
In order to minimize the dispersion of guided waves during propagation in the structure, signals with a narrow bandwidth are commonly employed to excite waves for defect detection in structures. The tone burst consisting of several cycles of Hann-windowed sine waves is a typical narrow bandwidth signal, with the feature of: the larger the number of wave cycles, the narrower the bandwidth of the signal. Figure 8 shows the 2D FFT spectrum extracted from the nodal accelerations data recorded using the single wire FE model. The guided wave in the wire was excited with a Hann-windowed, five-cycle, 60 kHz sinusoidal tone burst perpendicular to the wire end face.  
Figure 7: The phase velocity dispersion spectra obtained from nodal accelerations with different excitation methods: (**a**) an oblique triangular pulse signal with three equal components, and (**b**) a perpendicular triangular pulse signal with a \(z\)-axial component. The theoretical phase velocity dispersion curves obtained in Figure 1a are represented by the black dotted lines.  
The dispersion nephogram shows a section of the L(0,1) mode with bandwidth of about 40 kHz, which is the same as that of the excitation signal (see Figure 5b).