# Development of Post Processing for Wave Propagation Problem: Response Filtering Method
## 4 Numerical Examples  
This section illustrates examples to demonstrate the benefits of the RFM in terms of accuracy as well as its limitations. Figure 7: Changes to the numerical solution ([PERSON]’s method) in frequency domain after filtering with different filtering ratio and comparison with the exact solution for benchmark problem in Figure 2. (**a**) Numerical solution without RFM, (**b**) with RFM (\(FR=0.5\%\)), and (**c**) with RFM (\(FR=2\%\)). **Example 1**. : _Bar with three different segments_  
For the first example, a one-dimensional bar with different cross-section areas is considered. The problem geometry is composed of three different segments, as shown in Figure 8a, and each segment has a different cross-sectional area:\(0.5\times 1\), \(1.5\times 1\), and \(0.5\times 1\) in\({}^{2}\), respectively. A step axial load of 100 psi in the \(x\)-direction is applied on the left side of the bar, and the example structure is discretized using \(0.5\) in \(\times\) 0.5 in Q4 element meshes. The detailed geometry, boundary conditions, and material properties are illustrated in Figure 8.  
Different from the previous benchmark example, the impedance mismatches at \(x=8\) and \(11\) in cause the occurrence of reflection and transmission waves owing to the change in the cross-sectional area. It should be noted that when a wave that propagates through a medium encounters the boundary of another medium having a different impedance, it produces reflection and transmission waves, as shown in Figure 9 [27]. The wave propagation impedance, \(\mathbf{Z}\), transmission coefficient, \(\mathbf{T}\), and reflection coefficient, \(\mathbf{R}\), can be defined by the following equations:  
\[\mathbf{Z}=\frac{\mathbf{p}}{\mathbf{v}}, \tag{35}\]  
\[\mathbf{T}=\frac{2\mathbf{Z}_{2}}{\mathbf{Z}_{2}+\mathbf{Z}_{1}}=\frac{2}{1+ \mathbf{Z}_{1}/\mathbf{Z}_{2}}, \tag{36}\]  
\[\mathbf{R}=\frac{\mathbf{Z}_{2}-\mathbf{Z}_{1}}{\mathbf{Z}_{2}+\mathbf{Z}_{1 }}=\frac{1-\mathbf{Z}_{1}/\mathbf{Z}_{2}}{1+\mathbf{Z}_{1}/\mathbf{Z}_{2}}, \tag{37}\]  
where \(\mathbf{p}\) and \(\mathbf{v}\) denote the pressure (stress) and velocity vectors of the wave propagation, respectively. The transmission coefficient, \(\mathbf{T}\), and reflection coefficient, \(\mathbf{R}\), have the following relationship:  
\[1+\mathbf{R}=\mathbf{T}. \tag{38}\]  
Figure 8: Geometric parameters of Example 1 **(a)** The problem definition (\(L_{1}=8\) in, \(L_{2}=3\) in, \(L_{3}=9\) in, \(h_{1}=0.5\) in, \(h_{2}=1.5\) in, \(h_{3}=0.5\) in, and Thickness \(=1\) in), and **(b)** the force condition. Figure 9: Transmission wave (subscript \(t\)) and reflection wave (subscript \(r\)) of an incident wave (subscript \(i\)) perpendicular to the boundary between two media. Based on Equations (35)-(38), the values for each coefficient can be obtained in this example. First, the impedance of Domain 1 and Domain 3 is three times greater than that of Domain 2. The transmission coefficients of the \((+)\)\(x\)\(dir\) wave are 1/2 and 3/2 at \(x=8\) in and \(x=11\) in, respectively, and the reflection coefficient can be obtained from (37) accordingly (\(\mathbf{R}_{x=8\text{ in}}=-1/2\), \(\mathbf{R}_{x=11\text{ in}}=1/2\)). In the same way, the transmission and reflection coefficients for the \((-)\)_xdir_ wave can also be obtained. The analytical stress value at \(x=9.75\) in can be obtained through a step-by-step procedure, as shown in Figure 10.  
Figure 11a shows the differences between the numerical solution and the analytical solution before applying the RFM, while Figure 11b-d shows the shape of the filtered numerical solution as the filtering length of the RFM changes. The shape of the filtered numerical solution changes according to the size of the filtering length of the RFM. The filtered numerical solution gradually becomes increasingly similar to the analytical solution as the filtering length increases. In addition, it can be confirmed that, in this example, the oscillation of the numerical solution decreases or disappears with the application of the RFM.