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Determination of the bottom deformation from space- and time-resolved water wave measurements

Published online by Cambridge University Press:  27 November 2017

Pablo J. Cobelli Open the ORCID record for Pablo J. Cobelli [Opens in a new window] ,
Philippe Petitjeans ,
Agnès Maurel Open the ORCID record for Agnès Maurel [Opens in a new window]  and
Vincent Pagneux
Pablo J. Cobelli*
Affiliation:
Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IFIBA, CONICET, Ciudad Universitaria, Buenos Aires 1428, Argentina Laboratoire de Physique et Mécanique des Milieux Hétérogènes, UMR CNRS 7636, Ecole Supérieure de Physique et de Chimie Industrielles, ESPCI-ParisTech, 10 rue Vauquelin, 75231 Paris CEDEX 5, France
Philippe Petitjeans
Affiliation:
Laboratoire de Physique et Mécanique des Milieux Hétérogènes, UMR CNRS 7636, Ecole Supérieure de Physique et de Chimie Industrielles, ESPCI-ParisTech, 10 rue Vauquelin, 75231 Paris CEDEX 5, France
Agnès Maurel
Affiliation:
Institut Langevin, UMR CNRS 7587, Paris, France
Vincent Pagneux
Affiliation:
Laboratoire d’Acoustique de l’Université du Maine, UMR CNRS 6613, Avenue Olivier Messiaen, 72085 Le Mans CEDEX 9, France
*
†Email address for correspondence: [email protected]
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Abstract

In this paper we study both theoretically and experimentally the inverse problem of indirectly measuring the shape of a localized bottom deformation with a non-instantaneous time evolution, from either an instantaneous global state (space-based inversion) or a local time-history record (time-based inversion) of the free-surface evolution. Firstly, the mathematical inversion problem is explicitly defined and uniqueness of its solution is established. We then show that this problem is ill-posed in the sense of Hadamard, rendering its solution unstable. In order to overcome this difficulty, we introduce a regularization scheme as well as a strategy for choosing the optimal value of the associated regularization parameter. We then conduct a series of laboratory experiments in which an axisymmetric three-dimensional bottom deformation of controlled shape and time evolution is imposed on a layer of water of constant depth, initially at rest. The detailed evolution of the air–liquid interface is measured by means of a free-surface profilometry technique providing space- and time-resolved data. Based on these experimental data and employing our regularization scheme, we are able to show that it is indeed possible to reconstruct the seabed profile responsible for the linear free-surface dynamics either by space- or time-based inversions. Furthermore, we discuss the different relative advantages of each type of reconstruction, their associated errors and the limitations of the inverse determination.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 835 , 25 January 2018 , pp. 301 - 326
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Copyright
© 2017 Cambridge University Press 

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