# Propagation, Scattering and Defect Detection in a Circular Edge with Quasi-Edge Waves
## 4 Directivity Pattern of Scattered Waves
### Influence of Location and Size of Partially Through-Thickness Edge Defect  
Figure 15a-d show the comparison between the 8 mm deep (defect depth-to-wavelength ratios = 0.32) types I (located on the inner side) and II (located on the outer side) of partially through-thickness edge defects with widths of 1, 2, 3 and 4 mm, respectively. All the SDPs in Figure 15 show that the energy of scattered waves are concentrated in forward and normal directions relative to the incident wave propagation direction, which is similar to the case of the through-thickness edge defect. In Figure 15, the SDPs of type I are similar to the SDPs of type II in the corresponding dimension. Compared to type I, the amplitudes of scattered waves of type II in the \(-135^{\circ}\) directio  
Figure 14: Polar plot of normalised maximum amplitudes of scattered waves at the through-thickness edge defects with depths of (a) 2, (b) 8 and (c) 14 mm. Figure 16a,b present the comparison of SDPs for the edge defects of types I and II with different widths, respectively. The black solid line can be taken as a benchmark, which represents the scattered waves for an 8 mm deep through-thickness edge defect. It can be observed that the amplitude of scattered waves increased with the width of defects. The energy of scattered waves at the type I edge defect is concentrated in the forward and normal directions relative to the incident wave propagation direction. For type II, the energy of scattered waves not only increased in forward and normal directions but also in the \(-135^{\circ}\) direction, as shown in Figure 16b. However, the amplitudes of scattered waves significantly decreased at the dimension of 5 mm, where type II converts to a through-thickness edge defect. To further elucidate the correlation between defect width and the amplitudes of scattered waves, Figure 17 presents the dependences of the amplitudes of scattered waves in various directions as a function of the defect width. Figure 16: Polar plot of the normalised maximum amplitudes of scattered waves of (**a**) type I (located on the inner side) and (**b**) type II (located on the outer side) partially through-thickness edge defects with various widths. Figure 15: Polar plot of the normalised maximum amplitudes of scattered waves of type I (blue solid line) and type II (red dashed line) edge defects with widths of (**a**) 1, (**b**) 2, (**c**) 3 and (**d**) 4 mm. Figure 17a-c show the normalised amplitudes of forward, normal and backward scattered waves at the type I edge defect (see Figure 8c) with various widths. Figure 17a shows the normalised amplitudes of forward scattered waves in the directions of \(-180^{\circ}\), \(-157.5^{\circ}\), \(-135^{\circ}\) and \(-112.5^{\circ}\) (with respect to the incident wave direction), respectively. It can be observed that the amplitudes of scattered waves in the directions of \(-180^{\circ}\), \(-157.5^{\circ}\) and \(-135^{\circ}\) show an increasing trend with the defect width, while the change in amplitudes of scattered waves in the direction of \(-180^{\circ}\) are relatively stronger than  
Figure 17: Normalised amplitudes of scattered waves of type I (located on the inner side) in **(a)** forward, **(b)** normal and **(c)** backward direction; normalised amplitudes of scattered waves of type II (located on the inner side) partially through-thickness edge defect in **(d)** forward, **(e)** normal, and **(f)** backward direction. the other two directions. For forward scattered waves in the direction of \(-112.5^{\circ}\), the amplitude increases with the defect width in the range from 1 to 4 mm. However, when the defect geometry approaches a through-thickness geometry, the amplitude of scattered waves in the direction of \(-112.5^{\circ}\) decreases. Figure 17b shows the scattered wave amplitude in the normal direction (\(-90^{\circ}\)) relative to the incident wave propagation direction. The results indicate that the amplitude of scattered waves increases proportionally with the width of the type I edge defect in the range between 1 and 4 mm.