## 4 Acoustic Finite Element Analysis (AFEA) of Acoustic Wave Propagation in a Straight Pipe
### Simulation Results  
#### 4.3.1 Verification of Results Using Time of Flight  
The FEA of acoustic wave propagation in a pipe of diameter 25.4 mm and of length 4.02 m was earlier conducted by [PERSON] and [PERSON] [11]. A 20-node quadratic acoustic brick (AC3D20) element type was employed for their analysis, as illustrated in Figure 4. As a starting point in these investigations, the acoustic wave propagation in the same pipe was analysed using the AC3D20 element type and the same mesh seed size of 9 mm. The responses obtained at six monitoring points, which are designated as'sensor' locations \(\mathrm{s}_{1}\) to \(\mathrm{s}_{6}\), as shown in Figure 5, are illustrated in Figure 6. These results are identical to those shown in the paper by [PERSON] and [PERSON] [11], and the Time of Flight approach was used to check their accuracy. Figure 6 presents stacked plots of the dynamic pressure versus time waveforms at the six monitoring nodes identified in the intact pipe model. It should be noted that all the plots start with a dynamic pressure amplitude of zero. Thus, in all cases, the maximum peak amplitudes are less than 1.0 Pa. The stacking of the plots enables a direct comparison of the forward-going and backward-going (reflected) wavefronts at the different sensor locations. From the figure, the first positive peaks seen in each of the graphs indicate the arrival of the pressure pulse to the individual sensors. The compression pulse propagates past the respective sensors (\(\mathrm{s}_{1}\)-\(\mathrm{s}_{6}\)) in the order of time seen in the second column of Table 1. On the other hand, the first negative peaks seen at the various sensor locations are the times taken by the wave to travel to the pipe outlet and back to the corresponding  
Figure 3: A half sine wave acoustic impulse. sensor locations. Thus, since s\({}_{6}\) is located close to the pipe outlet, the reflection of the wave from P\({}_{\text{out}}\) is first noticed by \(s_{6}\), then followed by \(s_{5}\)-s\({}_{3}\), and subsequently to \(s_{2}\) and \(s_{1}\), at times tabulated in the third column of Table 1.  
A magnified plot of the dynamic pressure versus time waveform predicted for sensor \(s_{1}\) is shown in Figure 7. Considering the predicted times of flight, the wave speed can be deduced from the arrival times predicted for sensor locations \(s_{1}\) to \(s_{6}\) using the following relation:  
\[c=\frac{\Delta\ell}{\Delta\text{t}}=\frac{2\times(\ell-l_{s})}{t_{p}-t_{s}} \tag{16}\]  
Figure 4: Air-filled pipe model meshed using the acoustic continuum 3D 20-node brick elements (AC3D20). Figure 5: Schematic pipe geometry in 2D. Figure 6: Stacked Plots of pressure vs. time predicted using FEA at sensor \(s_{1}\) to \(s_{6}\) on intact air–filled pipe using AC3D20 (Note: all plots start with zero amplitude; maximum peak amplitude of each plot is less than 1.0 Pa). where \(\ell\) is the pipe length (m), \(l_{s}\) is the distance from the pipe inlet to sensor location (m), \(c\) is the wave speed, \(t_{p}\) is the time it takes the wave to travel from the inlet to the sensor then to the pipe exit and back to sensor location (s) and \(t_{s}\) is the time it takes the wave to travel from the inlet to sensor location \(s_{n}\) (s). Substituting the respective data in Equation (16) for sensor \(s_{1}\) gives;  
Figure 6.0  
Similarly, the wave speed c, according to the time of flight for other sensor points (\(s_{1}\)-\(s_{6}\)) are computed and tabulated in Figure 6.1 Using the properties of air used for the AFE analysis, the actual speed of sound in air is approximately 345 ms\({}^{-1}\). Comparison of the derived sound speed (derived from the predicted arrival times) with the actual sound speed used in the AFEA gives the errors E associated with each sensor location, as shown in the fifth column of Figure 6.2 These errors are rather large and unacceptable. They can be attributed to the coarseness of the mesh used in the Abaqus AFEA. Therefore, there is a need for mesh refinement, in order to get a more accurate result.