# Forward/Backward Decomposition for Dispersive Wave Propagation Measurements
## 5 Conclusions and Future Work  
We have presented a method for decomposing wave propagation measurements into their forward- and backward-propagating components in a one-dimensional linear waveguide. The general theory and algorithm based on Fourier analysis were presented,  
Figure 10: Decomposed low-frequency (500 Hz) beam responses showing the forward- and backward-propagating wave components for sensor 39. The relative error between the reconstructed signal and the measured signal is 16.75%.  
Figure 9: Relative errors for the signal reconstructions at all 40 scan points. The average relative error for the high-frequency case (1500 Hz) is 17.37%, and the average relative error for the low-frequency case (500 Hz) is 17.71%.  
with specific algorithmic implementation details for the case of dispersive vibration waves propagating in a Timoshenko beam. While existing methods in the literature typically require isolating incident waves from reflected waves, this requirement can be relaxed to identifying forward- and backward-propagating components using the proposed approach. This permits the use of lower-frequency signals, longer pulses with more localized frequency content, and measurement locations closer to reflection sources. The method was validated on experimental data from a clamped-clamped Timoshenko beam, for which the algorithm successfully identified the forward- and backward-propagating waves.  
Direct applications for this work include non-destructive testing and SHM. By including the wavenumber \(k_{1}\) as an unknown, the system of Equation (8) can be solved for the dispersion relation in addition to the forward- and backward-propagating waves, enabling characterization of material and geometry properties, such as stress. While the algorithm currently satisfactorily identifies the incident wave and its primary reflections, the current time sampling criteria are a known area of improvement. It is possible that introducing some artificial damping into the measurements to give more weight to the incident wave could improve the results. It is also possible that the technique could be built upon with additional information in order to detect reflection sources (i.e., damage localization) or further isolate the true incident wave from other forward-propagating components, although this has not been investigated.  
Conceptualization, N.A.C. and P.T.; methodology, N.A.C. and P.T.; software, N.A.C.; validation, N.A.C.; formal analysis, N.A.C.; investigation, N.A.C.; resources, P.T.; data curation, N.A.C.; writing--original draft preparation, N.A.C.; writing--review and editing, N.A.C. and P.T.; visualization, N.A.C.; supervision, P.T.; project administration, P.T.; funding acquisition, P.T. All authors have read and agreed to the published version of the manuscript.  
The authors would like to acknowledge the Federal Railroad Administration for providing funding for this work under Grant 693 JJ618C000013.  
The original data presented in the study are openly available in the repository at [[https://doi.org/10.18738/T8/JQSOVI](https://doi.org/10.18738/T8/JQSOVI)]([https://doi.org/10.18738/T8/JQSOVI](https://doi.org/10.18738/T8/JQSOVI)) (accessed on 11 November 2024).  
The authors acknowledge that this work was performed during their time at Virginia Polytechnic Institute and State University in the Mechanical Engineering and Biomedical Engineering and Mechanics Departments.  
The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} SHM & Structural Health Monitoring \\ RNT & Rail Neutral Temperature \\ FFT & Fast Fourier Transform \\ MFC & Macro Fiber Composite \\ PSV & Polytec Scanning Vibrometer \\ \end{tabular}