# Forward/Backward Decomposition for Dispersive Wave Propagation Measurements
## 4 Experimental Validation
### Experimental Results and Discussion
Consider the first pulse, between 0.002 and 0.004 s. As the forward-propagating incident wave reflects from the boundary, it will immediately turn into a backward-propagating reflection and return over the sensor near the boundary. Hence, in this first group, we see the incident wave and a first reflection. Then, at around 0.006 to 0.008 s, we see the same phenomenon, but with the initially backward-propagating incident wave. This wave has at this point traveled backwards toward the rear boundary, and then forward past the sensors where it is measured for the first time, and then reflected back from the front boundary, at which time it is registered for a second time. Figure 4: Beam responses to the two-cycle toneburst with measurements at locations 1 and 40. **(a)** 1500 Hz center frequency toneburst. **(b)** 500 Hz center frequency toneburst. Figure 5: Decomposed high-frequency (1500 Hz) beam responses showing the forward- and backward-propagating wave components for sensor 1. The relative error between the reconstructed signal and the measured signal is 35.37%. An important note here is that every time a reflection is generated, it changes between forward- and backward-propagating, but the algorithm groups all of the signals into these two categories. Hence, the algorithm does not truly isolate the incident wave from all other signals; rather, it isolates forward-propagating components from backward-propagating ones. Further processing involving information about boundary conditions would be required to truly isolate the incident wave from reflections, which is outside the scope of this work. Next, consider the low-frequency case in Figures 7 and 8. Here, the incident and reflected waves are not clearly identifiable in the unprocessed time domain measurements plotted as the dotted gray lines. We emphasize that this undesirable characteristic typically prompts users to increase the excitation frequency in order to obtain separation between the incident wave and subsequent reflections, and that the motivation of this work is to provide an alternative. As in Figure 5, we can observe that the algorithm correctly identifies the incident wave at the location of sensor 1 shown in Figure 7, where it is mostly uncontaminated, whereas at the location of sensor 40 depicted in Figure 6 we see that there is a reflection from the end of the beam, which immediately contaminates the signal. Figure 6: Decomposed high-frequency (1500 Hz) beam responses showing the forward- and backward-propagating wave components for sensor 40. The relative error between the reconstructed signal and the measured signal is 21.27%. Figure 7: Decomposed low-frequency (500 Hz) beam responses showing the forward- and backward-propagating wave components for sensor 1. The relative error between the reconstructed signal and the measured signal is 34.68%. In the low-frequency case, because the wavelengths are so long, the separation in the time domain between \"incident waves\" and \"reflected waves\" even in the decomposed signals is diminished. Nonetheless, by going into the frequency domain, the algorithm is able to identify the portion of the signal corresponding to forward-propagating waves and backward-propagating waves. Based on the intuition of the expected propagating signals, the algorithm results match our expectations. To more quantitatively assess the performance of the algorithm, we can consider the error between the reconstructed signals obtained by combining the identified forward- and backward-propagating components and the measured signals. We consider the relative error, defined as the norm of the absolute error between the reconstructed signal and the measured signal divided by the norm of the measured signal at a particular scan point located at \(x_{i}\), as follows:  
\[E_{\text{rel}}(x_{i})=\frac{||u_{\text{reconstructed}}(x_{i},t)-u_{\text{ measured}}(x_{i},t)||_{2}}{||u_{\text{measured}}(x_{i},t)||_{2}}.\]  
Figure 9 depicts the relative errors for both the high- and low-frequency cases at all 40 scan points.