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Steady water waves with arbitrary surface pressure: their recovery from bottom-pressure measurements

Published online by Cambridge University Press:  19 April 2024

Didier Clamond Open the ORCID record for Didier Clamond [Opens in a new window]  and
Joris Labarbe Open the ORCID record for Joris Labarbe [Opens in a new window]
Didier Clamond*
Affiliation:
Université Côte d'Azur, CNRS UMR 7351, Laboratoire J. A. Dieudonné, Parc Valrose, 06108 Nice cedex 2, France
Joris Labarbe
Affiliation:
Université Côte d'Azur, CNRS UMR 7351, Laboratoire J. A. Dieudonné, Parc Valrose, 06108 Nice cedex 2, France
*
†Email address for correspondence: didierc@unice.fr
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Abstract

Equations relating the pressure at a horizontal seabed, the free-surface profile and the surface pressure are derived for two-dimensional irrotational steady water waves with arbitrary pressure at the free surface. Special cases include gravity, capillary, flexural and wind waves. Without approximations, we show that the free-surface recovery from the bottom pressure requires the resolution of only one first-order ordinary differential equation independent of the surface pressure, thus providing a new general recovery method valid for a broad class of water waves. Another equation provides an explicit expression for the surface pressure as a function of the bottom pressure and of the free surface. Thus, if unknown, the surface pressure can also be recovered if one extra measurement is available. This new recovery procedure is illustrated analytically for the linear approximation of a flexural–capillary–gravity wave, and numerically for fully nonlinear capillary–gravity waves.

JFM classification

Mathematical Foundations: General fluid mechanics Geophysical and Geological Flows: Coastal engineering
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 985 , 25 April 2024 , R2
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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