# Numerical Study of Mode Conversion between Fundamental Lamb and Rayleigh Waves
## 4 Discussion
Based on the analytical solution for grazing waves, only SV can give rise to exponentially decaying waves. This suggests that for the case of the \(S_{0}\) mode, \(S_{0}\rightarrow\) P \(\rightarrow\) SV \(\rightarrow\) R will be the sequence of mode conversions, which agrees well with the low mode conversion efficiency of \(S_{0}\). Moving onto the case of Rayleigh to Lamb wave conversion, it is apparent that the Rayleigh wave will simply act like a line load on the side of the plate. However, the Rayleigh amplitudes decay exponentially with depth; therefore, the excitation of the line load will also vary with plate depth. For the cases considered here, the Rayleigh waves will mode convert to bulk waves in the plate, whose wavelength is significantly larger than the plate thickness. The generation of Lamb waves from bulk wave interactions is well documented in the literature ([PERSON] _et al._, 2016; [PERSON] and [PERSON], 2016; [PERSON] and [PERSON], 2016), particularly the partial wave technique. This technique uses the principle of superposition of three upward and three downward propagating waves to satisfy the stress-free boundary condition used to describe Lamb wave propagation. The reconstructions of the reflected waves or transverse resonances eventually give rise to a guided Lamb mode ([PERSON] _et al._, 2016). This applies very well to the present geometry, where the incident Rayleigh wave will be scattered as bulk modes into the plate structure and subsequently develop into a Lamb wave, as shown in Figure 7.  
_Generation length:_ To test the second hypothesis, i.e., generation length for Rayleigh and Lamb modes, a numerical line scan was carried out. The starting point of the line scan was chosen such that it was 10 mm from the discontinuity, and surface A-Scans (displacement as  
Figure 8: Snapshot showing vector plots of the total displacement in a half-space configuration: (**a**) for an incident \(A_{0}\) mode and (**b**) for an incident \(S_{0}\) mode. a function of time) were extracted at 0.2 mm increments for up to 40 mm from the starting point. The Hilbert transform was used to obtain an analysis of each A-Scan, and finally a Hilbert line scan was obtained as shown in Figure 9. The location of the discontinuity (10 mm from start point) is marked in dashed lines in Figure 9. The numerical Uy line scan of the \(A_{0}\to R\) conversion is shown in Figure 9a. It can be noticed that after the \(A_{0}\) mode converts into an R wave, the amplitude of the R wave is low until 17 mm, after which the amplitude increases and stabilizes. A distance of 7 mm from the discontinuity can be taken as the generation distance for the R wave. This agrees well with the hypothesis that the diffraction from the discontinuity will combine with the grazing incidence, resulting in a generation length for the Rayleigh wave. For the case of the incident R mode, the numerical Ux line scan of \(R\to A_{0}\) and \(S_{0}\) conversion is shown in Figure 9b. After the discontinuity, the mode undergoes a resonance effect in space which results in an increase and a decrease in the amplitudes. The resonance effect can be explained by the difference in velocity of the Lamb modes and the constructive and destructive interference that results from the phase difference between the \(A_{0}\) and \(S_{0}\) modes. While there is no distinct diffraction effect for the case of \(R\to A_{0}\) and \(S_{0}\), the resonance effect will still result in a generation length for the Lamb modes. The resonance effects stops when overlap/superposition of \(A_{0}\) and \(S_{0}\) modes stops. From Figure 9b, it is apparent that this corresponds to a length of 33 mm after the defect. This generation length will also be a function of frequency and more work is required to understand this effect. It is important to note that in both cases, any power coefficient measurement in the region before the full generation occurs will result in erroneous transmission coefficient values.