# Forward/Backward Decomposition for Dispersive Wave Propagation Measurements
## 4 Experimental Validation
### Experimental Results and Discussion  
We begin this section by showing some representative examples of the raw measurements. The 1500 Hz test case is shown on the left of Figure 4, and the 500 Hz test case is  
Figure 3: Time and frequency domain representations of the excitation signal. shown on the right. In each plot, the dark line shows the signal at the first measurement location, whereas the light gray signal is the measurement at the 40 th scan point approximately 80 cm away. In both cases, the dispersion effect is apparent, as the toneburst changes shape as it propagates. In the 500 Hz case, the measurement at sensor location 1 exhibits relatively low dispersion, and the incident two-cycle Hanning windowed toneburst is even identifiable in the first sensor measurement. This is because the sensor spacing is relatively small compared to the long wavelength for the low-frequency waves; thus, the wave has had very little time to disperse. In contrast, although it technically is less-dispersive, the high-frequency case exhibits more dispersion already in the first measurement (note the qualitative difference in the pulse shape compared with the original toneburst). This is because, relative to the wavelengths of the frequency components of the signal, the first sensor's location is already relatively far from the actuator. As a result, the wave has already had time to disperse, even if the incident wave is still identifiable. In both cases, the incident wave is not identifiable in the gray signal representing the last measurement location due to reflections from the boundary of the beam. For techniques which rely on incident wave isolation, these measurements would not be acceptable. The results of executing the forward/backward decomposition proposed in this paper are shown in Figures 5-8. Recall that the process performed by Algorithm 1 involves solving an inverse vector addition problem at each frequency to identify the forward- and backward-propagating components in each of the signals from the 40 scan points. Consider first the high-frequency case, in which we can clearly identify the incident wave and reflected waves in the unprocessed signals. The unprocessed signal from Figure 4 is plotted as the dotted gray line in Figure 5; the purple line is the forward-propagating component identified by the algorithm, whereas the yellow line is the backward-propagating component identified by the algorithm. Finally, the solid black line is the reconstructed signal obtained by combining the identified forward- and backward-propagating signals; ideally, the reconstructed signal should match the measured signal. The algorithm successfully decomposes the measurements. First, we see that the incident pulse between 0 and 0.002 s is completely categorized as forward-propagating as desired. However, we also see other forward-propagating components, which we will proceed to identify. Consider that the actuator actually sends a pulse symmetrically forward along the structure and backward along the structure; hence, while we only measure the region in front of the actuator and only consider one of these pulses as the incident wave, we should expect to see two reflections (one from each boundary) of these two initial waves. At the location of sensor 1 depicted in Figure 5, which is near the center of the beam, these two reflections arrive at almost the same time (around 0.004 s), so in the time domain measurement it is not obvious that there are actually two signals present. Based on the measurements at other sensor locations though, the algorithm is able to identify and decompose these two reflections accordingly, with one being forward- and the other backward-propagating. At the location of sensor 40 depicted in Figure 6, which is near the boundary, we again see that what initially appear to be individual signal packets are actually various pulses superimposed and traveling in opposite directions.