# Forward/Backward Decomposition for Dispersive Wave Propagation Measurements
## 2 Theory
At very high frequencies, \(k_{2}\) switches from imaginary to real, converting from an evanescent mode into a propagating mode; however, this switch occurs at frequencies significantly higher than those considered in this work, so we will refer to the components corresponding to \(k_{2}\) as evanescent modes. For a more detailed discussion on the interesting phenomenon of evanescent waves, which represent spatially decaying (rather than time-decaying) oscillations, we again refer to [5]. The quantities represented here are the following: \(E\) the elastic modulus of the material, \(I\) the cross-sectional moment of inertia (i.e., the second moment of area), \(F_{0}\) the axial loading on the beam, \(\rho\) the density of the material, \(\omega\) the angular frequency, \(G\) the flexural modulus, \(\kappa\) the Timoshenko correction factor, and \(A\) the cross-sectional area. Note that the wavenumber solution depends on the angular frequency quadratically and not linearly, which results in dispersive wave behavior. The general response according to superposition is therefore  
\[u(x,t)=\left( \begin{array}{c}\frac{\hat{A}e^{-ik_{1}x}}{\text{forward}}+\frac{ \hat{B}e^{-ik_{2}x}}{\text{forward}}+\frac{\hat{C}e^{ik_{1}x}}{\text{backward}}+ \frac{\hat{D}e^{ik_{2}x}}{\text{backward}}+\underbrace{\hat{D}e^{ik_{2}x}}_{ \text{backward}}+\underbrace{\ldots}_{\text{hinger-order}}\end{array} \right)e^{i \omega t}. \tag{5}\]  
Again, recall that this solution exists for every frequency component, though we have neglected the summation over each frequency \(\omega_{n}\) for notational simplicity. Also note how Equation (5) corresponds to Equation (2) with four wave modes. For further reading on the spectral solution approach to wave propagation problems, we refer the reader to [5]. Neglecting noise and higher-order terms from nonlinear responses, each measurement \(u(x_{i},t)\) of the structure's response measured by a sensor at a particular location \(x_{i}\) is a snapshot of the general solution for the location \(x_{i}\):  
\[u(x_{i},t)=\left(\hat{A}e^{-ik_{1}x_{i}}+\hat{B}e^{-ik_{2}x_{i}}+\hat{C}e^{ik_ {1}x_{i}}+\hat{D}e^{ik_{2}x_{i}}\right)e^{i\omega t}. \tag{6}\]  
Thus, taking the Fourier transform of a single measurement at location \(x_{i}\) using the Fast Fourier Transform (FFT) algorithm gives (for every frequency) the quantity  
\[\hat{u}(x_{i})=\hat{A}e^{-ik_{1}x_{i}}+\hat{B}e^{-ik_{2}x_{i}}+\hat{C}e^{ik_{1 }x_{i}}+\hat{D}e^{ik_{2}x_{i}}. \tag{7}\]  
In practice, one receives a vector of all of the Fourier coefficients when using an FFT algorithm, and the corresponding frequencies can be constructed according to the time discretization parameters. In Equation (7), the wavenumber solutions \(k_{1},k_{2}\) are known based on a model incorporating known material and geometric properties of the waveguide (here, we use the Timoshenko beam theory solutions in Equation (4)). The location of the measurement \(x_{i}\) is also presumed to be measurable. Therefore, for each measurement, Equation (7) contains four unknown quantities: \(\hat{A},\hat{B},\hat{C},\hat{D}\in\mathbb{C}\); these represent the amplitude spectra for the various wave modes which can be present in the structure. To solve for these four waves, four equations are needed, which can be obtained by taking four measurements at distinct locations. The equations given by all of the measurements form a linear system \(\mathbf{Ax}=\mathbf{b}\), with  
\[\mathbf{A}= \begin{bmatrix}e^{-ik_{1}x_{1}}&e^{-ik_{2}x_{1}}&e^{ik_{1}x_{1}}&e ^{ik_{2}x_{1}}\\ e^{-ik_{1}x_{2}}&e^{-ik_{2}x_{2}}&e^{ik_{1}x_{2}}&e^{ik_{2}x_{2}}\\ e^{-ik_{1}x_{3}}&e^{-ik_{2}x_{3}}&e^{ik_{1}x_{3}}&e^{ik_{2}x_{3}}\\ e^{-ik_{1}x_{4}}&e^{-ik_{2}x_{4}}&e^{ik_{1}x_{4}}&e^{ik_{2}x_{4}}\end{bmatrix} , \quad\mathbf{x}= \begin{bmatrix}\hat{A}\\ \hat{B}\\ \hat{C}\\ \hat{D}\end{bmatrix} ,\quad\text{and}\quad\mathbf{b}= \begin{bmatrix}\hat{u}(x_ {1})\\ \hat{u}(x_{2})\\ \hat{u}(x_{3})\\ \hat{u}(x_{4})\end{bmatrix} .