The shallow--water limit of linear water waves over a discontinuous bottom

Abstract: We rigorously justify the zero--dispersion limit of linear water waves over a sharp bottom step, recovering the shallow-water equations as a transmission problem, and describe the localized dispersive effects generated near the discontinuity. To obtain estimates uniform in the shallow-water parameter, we study an abstract functional framework for the linear water waves equations in two-dimensional fluid domains with a Lipschitz curved bottom. The framework is based on fractional powers of the Dirichlet--Neumann operator, and we characterize low-regularity energy spaces using adapted Rellich identities. At higher regularity, the associated velocity potential develops corner singularities. For the step geometry, we use an explicit conformal mapping to obtain a version of Grisvard's shift theorem, where we derive an explicit formula for the singular part of the potential. The dependence on the horizontal variable reveals a narrow transition scale generated by the step. The convergence toward the shallow-water transmission problem holds away from this transition region, while a localized dispersive correction remains near the step. We illustrate this discrepancy numerically and identify the resulting step-induced dispersive boundary layer. Building on these observations, we consider the strongly dispersive regime and rigorously derive a dispersive model that explicitly describes this boundary layer for a small step.