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 derive a consistent model that explicitly describes this boundary layer for a small step.
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BibTeX
@misc{paulsen2026shallowwater,
title = {{The shallow--water limit of linear water waves over a discontinuous bottom}},
author = {Martin Oen Paulsen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10170},
eprint = {2609.10170},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10170},
}
RIS
TY - UNPB
TI - The shallow--water limit of linear water waves over a discontinuous bottom
AU - Martin Oen Paulsen
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.10170
UR - https://arxiv.org/abs/2609.10170
AB - 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 derive a consistent model that explicitly describes this boundary layer for a small step.
ER -