# Forward/Backward Decomposition for Dispersive Wave Propagation Measurements
## 2 Theory  
The typical experimental setup for wave propagation measurements consists of an actuator and a set of sensors providing measurements \(\mathbf{u}_{i}\) at locations \(\mathbf{x}_{i}\) as a function of time \(t\), such as in Figure 1. Depending on the sensor type, the measurements \(\mathbf{u}_{i}\) may be displacement, velocity, or acceleration measurements. In this work, we will consider one-dimensional wave propagation, such that the wave measurements take the form \(u(x_{i},t)\). Furthermore, we consider systems for which a measurement of the response consisting of \(N\) time samples over a period of \(T\) can be decomposed using spectral analysis using the discrete Fourier transform as  
Figure 1: Diagram showing sensors along waveguide. The orange rectangle represents an actuator, and the black circles represent the sensors. \[u(x,t)=\sum_{n=0}^{N-1}\hat{u}_{n}(x)e^{i\omega_{n}t}, \tag{1}\]  
where \(\hat{u}_{n}\) is the \(n^{th}\) Fourier coefficient at frequency \(\omega_{n}=\frac{2\pi n}{T}\). Generally, the Fourier coefficients \(\hat{u}_{n}\) can be further decomposed as  
\[\hat{u}_{n}(x)=\hat{U}_{n}\hat{G}(k_{n}x), \tag{2}\]  
where \(\hat{U}_{n}\) is called the _amplitude spectrum_ and \(\hat{G}(k_{n}x)\) is called the _phase spectrum_ for a particular wave mode. The phase spectrum \(\hat{G}(k_{n}x)\) represents a transfer function defining how the wave mode specified by the amplitude spectrum \(\hat{U}_{n}\) manifests at different locations \(x\). The quantity \(k_{n}\) is known as the wavenumber corresponding to the frequency \(\omega_{n}\). As written, Equation (2) represents the response containing a single wave mode; in reality, several wave modes may exist concurrently, which can be accounted for by introducing additional modes to the right-hand side of Equation (2) to form a superposition solution. This representation of a signal is valid for a wide variety of systems which are governed by linear, separable PDEs [5]. In this paper in particular, we consider elementary beam theory for modeling the response of a linear elastic beam with constant cross-sectional and material properties, in which case the transfer function simplifies to \(\hat{G}(k_{n}x)=e^{-ik_{n}x}\); however, the formulation presented herein is also valid for other transfer functions, so long as the transfer function is given by a model. Nonetheless, for common structural elements such as beams (flexural waves) and bars (longitudinal or torsional waves), the simplification \(\hat{G}(k_{n}x)=e^{-ik_{n}x}\) is sufficient. With this, the general solution is given by the so-called spectral solution  
\[u(x,t)=\hat{U}e^{-i(kx-\omega t)}, \tag{3}\]  
where for notational simplicity we have omitted the summations over \(n\) (corresponding to the different frequency components of the signal) and over the various wave modes which may exist, which we will define next. Since the governing equations are linear, several solutions (wave modes) may exist which can be linearly superimposed to form the total response. In elementary beam theories such as the Euler-Bernoulli or Timoshenko beam theories, four solutions exist. The four wavenumber solutions are obtained by inserting the spectral solution into the governing equations and rearranging for \(k\); in the Timoshenko beam theory, this yields  
\[k_{1}=\pm\sqrt{\frac{-b+\sqrt{b^{2}-4 ac}}{2a}}\quad\text{and}\quad k_{2}=\mp \sqrt{\frac{-b-\sqrt{b^{2}-4 ac}}{2a}}, \tag{4}\]  
with \(a=EI\), \(b=F_{0}-\rho\omega^{2}\frac{EI}{G\kappa}-\rho\omega^{2}I\), and \(c=\rho^{2}\omega^{4}\frac{I}{G\kappa}-\rho A\omega^{2}\). For the low frequency ranges considered in this work, the positive and negative solutions for \(k_{1}\), which are real-valued, represent forward- and backward-propagating waves, respectively, whereas the positive and negative solutions for \(k_{2}\), which are complex-valued, represent forward- and backward-evanescent waves, respectively.