# Propagation, Scattering and Defect Detection in a Circular Edge with Quasi-Edge Waves
## 2 Numerical Study
### Effect of the Curvature on the Propagation Characteristics of \(Qes_{0}\)  
#### 2.2.1 Attenuation  
In this section, the effect of the radius of curvature on the attenuation of the \(QES_{0}\) wave mode is discussed. To quantify the attenuation of \(QES_{0}\) for different curvatures, a coefficient of attenuation (\(\mu\)) was introduced and defined as follows:  
\[\mu=\frac{A_{MP1}-A_{MP2}}{d} \tag{1}\]  
where \(A_{MP1}\) and \(A_{MP1}\) are the amplitudes of measurement points (MP) 1 and 2, respectively, and \(d\) is the propagation distance that is half the perimeter of the corresponding model. It should be noted that the amplitudes were normalised by the corresponding \(A_{MP1}\) in the calculation of \(\mu\).  
Figure 4 presents the relationship between \(\mu\) and the R/T ratio. It can be observed that the value of \(\mu\) increases with the \(FTV\) in the case of the same R/T ratio. On the other hand, it can also be observed that the value of \(\mu\) is highly correlative to the R/T ratio. At the stage of the R/T ratio ranging from 100 to 20, the value of \(\mu\) is similar, while the value of \(\mu\) dramatically increases when the R/T ratio changes from 20 to 10. This phenomenon aligns with the corresponding 2D FFT results shown in Figure 3a,b, which indicate a lower energy concentration within the bandwidth of excitation frequency as the \(R/T\) ratio of the waveguide decreases.  
Figure 3: A 2D FFT plot for FE results of (**a**) plate and (**b**) R/T = 10 curved edge model.  
To better describe the correlation between \(\mu\) and the R/T ratio across various \(FTVs\), a curve-fitting analysis was conducted using the FE simulation results. The establishment of algebraic expression for fitting can help to predict the effect of waveguide curvature on the propagation of \(QES_{0}\). The procedure for establishing this algebraic expression of the curve-fitting analysis is as follows:  
1. By observing the relationship between \(\mu\) and R/T ratio in Figure 4, it was found that the function of \(y=A\cdot e^{(-Bx)}+C\) has a good correlation to the simulated data, where \(A\), \(B\) and \(C\) are parameters related to \(FTV\), and \(x\) is the ratio of R/T.
2. The nonlinear least-squares optimisation was employed to find the values of \(A\), \(B\) and \(C\) parameters using an iteration procedure, which minimises the residual sum of squares (RSSs) between predicted values and the corresponding FE simulation results.
3. To generalise the fitting between \(\mu\) and R/T ratio across different \(FTVs\), the parameters of \(A\), \(B\) and \(C\) in different \(FTV\) cases were presented as linear.  
Finally, the relationship between \(\mu\) and the R/T ratio for different \(FTVs\) can be expressed as  
\[\mu=Ae^{-B\cdot\frac{B}{T}}+C \tag{2}\]  
where \(A=0.0029 FTV+0.0004\), \(B=-0.0224 FTV-0.1955\) and \(C=0.0003 FTV-0.003\), respectively. Figure 4 also shows a comparison between the FE simulation results and the predicted value (solid line) using Equation (2). It can be observed that there is a good agreement between Equation (2) and the simulation results.