Modeling Surface Wave Propagation over Meniscus and Scattering by a Surface-piercing Barrier

Abstract

Recent experiments have revealed that the meniscus formed near a surface-piercing barrier can significantly alter the propagation and scattering of capillary-gravity surface waves, beyond what classic flat-surface models predict. In particular, wave transmission increases as the barrier pulls up the meniscus, then drops sharply if the barrier is raised further, overturning the meniscus. Motivated by these findings, this paper develops two linearized theoretical frameworks that incorporate meniscus geometry and pinned contact line conditions into capillary-gravity wave propagation and scattering. Model~1 assumes a vertical wave perturbation of the unperturbed meniscus, thereby extending classic flat-surface boundary conditions in a relatively straightforward manner. Model~2 takes a more comprehensive approach, defining surface wave perturbations normal to the curved meniscus and reparameterizing boundary conditions in terms of arc length. While Model~1 proves convenient mathematically, it is restricted to single-valued meniscus shapes. By contrast, Model~2 is capable of describing multi-valued or overturning free surfaces, thereby capturing a wider range of physically realistic scenarios. Numerical simulations based on both models reproduce the experimental observations on how wave transmission varies with changes in meniscus height and contact angles as the barrier is lifted, underscoring the critical influence of meniscus curvature in small-scale wave-structure interactions. These results establish a robust theoretical foundation for predicting and optimizing capillary-gravity wave scattering in microfluidic, industrial, and scientific applications.