# 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 \(Qes_{0}\)
#### 2.2.2 Mode Shapes  
To investigate the effect of the curvature of a waveguide on mode shapes of the \(QES_{0}\) wave mode, the normal displacement of nodes along lines 1 (\(U_{z}\)) and 2 (\(U_{y}\)) was extracted (see Figure 5). Figure 5a,b illustrate the location of lines 1 and 2, respectively, where line 1 is located on the top of the surface of the edge and line 2 is on the outer surface of the specimen. The simulation results from the present models were compared with the corresponding simulation results for \(ES_{0}\), which were validated with analytical results [12]. Figure 6a-c show the normal displacement (\(U_{z}\)) of the top edge extracted from line 1. The data were normalised with the corresponding maximum value. The mode shape of \(ES_{0}\) (marked by the green line) can be seen to be the baseline. It can be observed that the mode shape of \(ES_{0}\) was symmetric about the mid-plane of the top edge surface, and that the amplitudes increased from the mid-plane toward both the outer and inner sides. Meanwhile, it can also be observed that the symmetricity of the \(QES_{0}\) was largely affected by the waveguide curvature. With the decrease in the R/T ratio, the difference in amplitudes between the outer and inner sides of the curved edge increased. In addition, it  
Figure 4: Coefficient of attenuation against R/T ratio. can also be noted that the mode shape of \(QES_{0}\) was more affected by the change in the R/T ratio in the case of high frequencies. Figure 6(a)-c show the normal displacements (\(U_{y}\)) of the side surface, which are extracted from line 2. It can be observed that the amplitudes of \(ES_{0}\) and \(QES_{0}\) waves rapidly decayed with the depth from the top edge surface. With the increase in excitation frequency, the effective depths of \(ES_{0}\) and \(QES_{0}\) were closer to the surface of the top edge. This behaviour is similar to the classical Rayleigh wave mode, where energy is more concentrated on the surface with the increase in the excitation frequency [25]. In addition, it can also be observed that the amplitudes of leaky waves from \(QES_{0}\) is highly correlated to the waveguide curvature. With the decrease in the R/T ratio (larger curvature), more energy was transferred from \(QES_{0}\) to leaky waves that propagated on the side surfaces of the edge. The results from this numerical case study of \(QES_{0}\), propagated along curved edges, provide potential guidelines for optimising sensor numbers and locations to maximise the signal-to-noise ratio. From a practical application perspective, when inspecting a relatively thick curved edge of a structural component, a higher wave magnitude can be measured from a location on the top edge surface that is close to the outer side, compared to the inner surface or mid-plane (see Figure 7(a)-c). Figure 5: Schematic of the location of measurement points for extracting mode shapes: (**a**) top view and (**b**) zoom-in view. Figure 6: Normal displacement (\(U_{z}\)) of line 1 with different R/T ratios: (**a**) FTV = 2.0, (**b**) 2.5, and (**c**) 3.0. A 3D FE Model for Investigating Scattering Characteristics of the \(Qes_{0}\) Wave Mode at an Edge Defect  
In this section, the R/T ratio of the 3D FE model is fixed at 20. The modelling approach is largely the same as that which was described in Section 2.1. To investigate the scattering characteristic of \(QES_{0}\) at edge defects, a narrow notch-like edge defect was modelled by removing elements from the model as shown in Figure 8. The location of the edge defect is in 45\({}^{\circ}\) direction, as illustrated in Figure 8a. The edge defect was modelled as 1 mm thick (along the wave propagation direction), which avoids the possible contact between the surfaces of the edge defect. To investigate the dependence of amplitudes of scattered waves and the size of the edge defect, the depth of the through-thickness edge defect was modelled, ranging from 2 to 14 mm with increments of 3 mm. Figure 8b presents a snapshot of an 8 mm long through-thickness edge defect in the FE model.