# Numerical Study of Mode Conversion between Fundamental Lamb and Rayleigh Waves
## 4 Discussion  
One of the objectives of the present work is to understand the physical mode conversion phenomenon using numerical analysis rather than an analytical expression, which cannot capture the complete scattering behavior. For the case of Lamb to Rayleigh conversion, the energy exiting the plate can be treated at the discontinuity plane as a set of Huygens' sources as shown in Figure 7a. These can act as line or point sources depending on several factors such as the number of sources, the width of sources, the frequency, etc. Due to the complexity of deriving an analytical solution for buried sources in a quarter space, for simplicity, let us neglect the vertical edge of the quarter space and treat it as a half space. This gives rise to two hypotheses: (a) the sources closest to the surface will result in grazing incidence of bulk modes at the top stress-free boundary, and (b) the other sources will result in diffracted bulk waves whose incident angles and phase will result in a \"generation length\" for the Rayleigh wave. Grazing incidenceFirst, we test the hypothesis of grazing incidence to generate Rayleigh waves. [PERSON] and [PERSON] [23] showed that for the case of grazing incidence of bulk waves at a stress-free boundary, the nature of the bulk mode (P vs. SV) will dictate the mode conversion phenomenon. For the case of a grazing incident SV wave, a wave with an exponentially decaying component in the depth direction and an SV wave with a linearly increasing amplitude will be generated. For the case of a grazing incident P wave, the plane wave solutions show that an SV wave is generated along with a P wave travelling with a linearly increasing amplitude. The derivations of these conditions can be found in [3; 23]. The exponentially decaying wave forms the basis for Rayleigh wave generation and the linearly increasing amplitude waves are not physically admissible. To place this in the context of the present work, it is necessary to study the energy exiting the plate by  
Figure 6: Rayleigh to Lamb mode conversion: (**a**) snapshot showing the vector plot of the total displacement in the quarter-space configuration. P and SV refer to the longitudinal and shear waves, respectively, and R is the Rayleigh wave. (**b**) Bar plot for percentages of converted modes. Figure 7: Schematic showing the hypotheses of (**a**) Lamb to Rayleigh mode conversions and (**b**) Rayleigh to Lamb mode conversion. The diffraction will be similar to a line load source with a set of Huygens’ sources. This may give rise to two hypotheses: mode conversion of Lamb to Rayleigh via grazing incidence of bulk modes on the half-space and direct diffraction of bulk modes. capturing the bulk mode conversions. Instead of using a free-space model, a symmetric half space, as shown in Figure 2c, can be used to capture the radiation. The energy exiting the plate at \(90^{\circ}\) to the discontinuity into the half space was captured at the receiver point \(X\) as shown in Figure 8 and used to calculate the mode conversion efficiencies. For an incident \(A_{0}\) mode as shown in Figure 8a, the conversion to P waves was <0.001%, whereas 33.23% was converted into SV waves. In the case of an incident \(S_{0}\) mode as shown in Figure 8b, 3.43% was converted into P waves and <0.001% was converted into SV waves. Combing the analytical solution from [PERSON] (1973) with the above-measured conversion efficiencies, it can be observed that an incident \(A_{0}\) mode will mode convert predominantly into SV waves, as shown in Figure 7b, which will further mode convert into Rayleigh waves. The \(A_{0}\rightarrow\) SV and \(A_{0}\rightarrow\) R was \(\approx\) 34% (Figure 4b), which suggests that the grazing incidence of the SV mode could be the main factor for mode conversion between \(A_{0}\) and R modes. For the case of the \(S_{0}\) mode, as shown in Figure 7b, the \(S_{0}\) will mode convert into P waves and further into Rayleigh waves. However, the \(S_{0}\rightarrow\) P is 3.43% and \(S_{0}\rightarrow\) R is only 1% (Figure 5b).