Analytical study to control phase gradient in elastic meta-interface for wave mode conversions

We present an analytical and design framework for achieving efficient elastic wave mode conversion across meta-interfaces governed by Generalized Snell's Law (GSL), which prescribes wave motion based on a spatial phase gradient along the interface. In contrast to conventional optimization-based approaches, our method provides clear physical insight into the mechanism of pressure-to-shear wave conversion through a simplified one-dimensional axial wave model with the transfer matrix approach. This model yields analytical expressions for geometric conditions and identifies the relevant design parameter space. Based on this framework, we propose a compact chiral-pattern subunit with a frequency-scalable geometry, enabling straightforward implementation across a range of wave conditions. Full-scale numerical simulations confirm that the resulting meta-interface achieves strong mode conversion performance and accurately reproduces key phenomena including transmitted angles. Additionally, we demonstrate symmetric transmission by introducing mirrored phase gradients, further validating the flexibility of the GSL-based design. While the conversion efficiency is constrained by angular limits inherent to the material's Poisson ratio, the framework provides a foundation for future improvements. This work bridges analytical modeling and practical design, offering an interpretable and scalable approach to engineered wave manipulation.