# Numerical Study of Mode Conversion between Fundamental Lamb and Rayleigh Waves
## 1 Introduction  
Mode conversion between different propagating wave types is of interest in several academic communities such as seismology, elastodynamics, structural health monitoring and nondestructive evaluation. Specifically, the conversion of one mode type to another based on interaction with geometric features has been explored extensively. This is of specific interest to the nondestructive evaluation and testing community since geometric discontinuities are potential sources of defects, which eventually need to be characterized. Several researchers have explored the mode conversion between P and SV waves [1; 2; 3], guided Lamb waves and P and SV waves [4; 5; 6] and P and SV and Rayleigh waves [7; 8; 9; 10; 11; 12; 13]. However, analyses for the conversion between Rayleigh- and Lamb-type waves are limited and will be the main focus of this article. Although they both are generated from bulk mode interaction with geometric features, their particle displacement, mode velocity, etc., are different, which make them interesting and of importance. Mode conversion between these waves has been studied typically in the specific case of a Lamb wave interaction with a delamination. This was first shown analytically [14], followed by experimental and numerical results for composite and metallic structures [15; 16]. In the case of Lamb mode interaction with a delamination, a large portion of the scattered bulk modes will contribute towards the formation of Lamb modes since the energy is trapped within the plate. A similar Lamb wave interaction with a step will result in energy being trapped within the plate. However, in the case of a plate attached to a half-space or quarter-space, the incident Lamb (in the plate) or Rayleigh (in the half-space) energy can be scattered through the halfor quarter spaces as bulk modes, which has not been explored in the literature. In different but related work, [PERSON] et al. [17] showed that an incident Rayleigh wave will be scattered into bulk modes when interacting with a sub-surface horizontal crack. However, the analysis was limited to bulk modes and generation of Lamb modes was not explored.  
The main objective of the present work is to understand the mode conversion between Rayleigh and Lamb waves in terms of bulk mode interactions. This will be accomplished using numerical models of examples to unify previous understanding of Lamb and Rayleigh mode conversion into bulk modes and bulk mode conversion into Lamb and Rayleigh waves to explain the conversion between Lamb and Rayleigh waves. Mode conversion efficiencies between \(\mathrm{L}\rightarrow\mathrm{R}\) and \(\mathrm{R}\rightarrow\mathrm{L}\) modes were calculated numerically for a plate attached to a quarter-space. To understand this conversion, the Lamb energy exiting out of the plate was captured using a half-space model. The calculated bulk mode conversion efficiency was then used to explain the \(\mathrm{L}\rightarrow\mathrm{R}\) conversion. Furthermore, numerical time-distance plots were used to investigate the difference between the two conversions.  
Most practical applications of Lamb waves in NDT/E and SHM utilize the fundamental symmetric and anti-symmetric modes since they can be generated at lower frequency-thickness products. Therefore, to keep the analysis simple, only the fundamental symmetric and anti-symmetric Lamb modes were considered in this study. It is well known that Lamb waves exhibit dispersion characteristics [1; 2; 3], and different modes can be generated due to the fundamental relationships between the excitation frequency and thickness of the plate. As an example, a thin plate (1 mm) was chosen and the frequency of excitation was chosen such that only the fundamental symmetric (\(5_{0}\)) and anti-symmetric (\(A_{0}\)) Lamb modes can be generated, as shown in Figure 1 at 1 MHz-mm for aluminum. The material properties will be presented in the next section.