Modelling the influence of equilibrium menisci on capillary–gravity wave propagation and transmission through a surface-piercing barrier

Classical capillary–gravity wave theories incorporate surface tension through the free-surface dynamic boundary condition but assume a flat equilibrium interface, thereby excluding the spatially varying geometry of equilibrium menisci. Motivated by experiments on meniscus-modified wave transmission past surface-piercing barriers with pinned contact lines, we develop a linearised two-dimensional framework incorporating the equilibrium meniscus profile into wave propagation and scattering. Linearisation about a prescribed meniscus yields modified kinematic and dynamic boundary conditions with spatially varying coefficients determined by the local meniscus height, slope and curvature. Solved via the finite-element method, the model captures observed variations in transmission as a function of meniscus height and contact angle. For a 15 Hz wave in deep water, the model shows that the transmission coefficient varies on the order of 0.1 for each 1 mm increase in capillary rise, provided the meniscus remains single valued. Computed energy-flux fields provide a physical interpretation: depressed menisci redirect energy away from the barrier, while raised menisci guide energy through the elevated fluid region to enhance coupling. Local dispersion analysis further clarifies how meniscus geometry influences phase evolution and wave structure. This framework establishes a consistent free-surface formulation for waves on spatially varying interfaces, providing insight into wave–meniscus coupling and enabling future extensions to nonlinear, three-dimensional and multi-valued configurations.