# Forward/Backward Decomposition for Dispersive Wave Propagation Measurements
## 1 Introduction  
Propagating waves appear in a variety of engineering and science applications, ranging from radar and communications [1] to cosmology [2], seismology [3], acoustics [4], and structural mechanics [5]. In particular, measuring the propagation of _vibration_ waves is popular in structural engineering applications. Contrasting with the widely used modal analysis approach to vibration testing, which is highly dependent on the boundary conditions of a structure or test article, wave propagation in principle is agnostic to boundary conditions: a propagating wave is unaffected by boundaries until it reaches them. Applications in which the boundary conditions are not of interest are therefore often prime candidates for wave propagation-based testing methods. Such applications in structural engineering include damage localization, material characterization, and parameter estimation, which are all examples seen in structural health monitoring (SHM). Very similar techniques can be applied in other settings as well, such as for footstep localization in smart structures [6] or target detection in radar [1]. Generally, a wide variety of different types of waves are used in various applications, ranging from electromagnetic waves to acoustic waves and ultrasonic waves, each having different characteristics. Nonetheless, the mathematical basis for analyzing propagating waves is often similar despite the different applications and wave properties. One frequent limitation of existing analysis techniques is the requirement to isolate the initial propagating wave signal, known as the incident wave. Often, reflections of the incident wave (from boundaries or other objects in the propagation path) are the primary hindrance in isolating the incident wave. Consequently, wave propagation techniques typically use short burst signals and/or high-frequency signals with short wavelengths to aidin distinguishing the incident wave. Furthermore, the most common analysis techniques are based on leading or trailing edge detection and are only suitable for nondispersive waves which maintain their shape as they propagate (dispersive waves feature frequency-dependent propagation velocities; this causes multi-frequency signals such as pulses to not retain their shape as they propagate). Since some types of propagating waves become dispersive in certain frequency ranges, this imposes a further restriction on the frequencies which can be used in testing. Flexural waves in beams, for example, become dispersive at low frequencies, precluding the possibility of using low frequencies for non-destructive testing with traditional methods. However, for certain applications, such as rail neutral temperature (RNT) and stress measurement in rails [7], it is desirable to use lower frequencies given their enhanced sensitivity to the parameters of interest. Even in ultrasonic applications, there are examples in the literature where dispersion and reflections push researchers to use higher frequencies than they might otherwise use [8]. For nondispersive signals, it is relatively straightforward to compensate for reflections by using time-shifts inversely proportional to the propagation velocity to subtract the reflected signal components. Such methods are not applicable to dispersive waves though, which have frequency-dependent propagation velocities. On the other hand, dispersion compensation in the absence of reflections [7; 9] has been addressed in the literature, but to the authors' knowledge, the issue of compensating for reflections in dispersive waves has until now been unaddressed. In this work, we present a signal processing technique for separating forward- and backward-propagating signal components in a set of dispersive wave measurements. Of particular interest is the case where the incident wave is not separable from reflections in the time domain in all of the measurements.