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Time domain modelling of a Helmholtz resonator analogue for water waves

Published online by Cambridge University Press:  10 June 2021

Leo-Paul Euvé ,
Kim Pham ,
Philippe Petitjeans ,
Vincent Pagneux  and
Agnès Maurel Open the ORCID record for Agnès Maurel [Opens in a new window]
Leo-Paul Euvé*
Affiliation:
PMMH, ESPCI, Sorbonne Université, Université PSL, 1 rue Jussieu, 75005Paris, France Bluerium, Av. L. Philibert, 13100Aix-en-Provence, France
Kim Pham
Affiliation:
IMSIA, CNRS, EDF, CEA, ENSTA Paris, Institut Polytechnique de Paris, 828 Bd des Maréchaux, 91732Palaiseau, France
Philippe Petitjeans
Affiliation:
PMMH, ESPCI, Sorbonne Université, Université PSL, 1 rue Jussieu, 75005Paris, France
Vincent Pagneux
Affiliation:
Lab. d'Acoustique de l'Université du Mans (LAUM), Av. O. Messiaen, 72085Le Mans
Agnès Maurel
Affiliation:
Institut Langevin, ESPCI Paris, Université PSL, CNRS, 1 rue Jussieu, 75005Paris, France
*
†Email address for correspondence: leo-paul.euve@espci.fr
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Abstract

In the context of water waves, we consider a resonator with deep subwavelength resonance, analogue to the Helmholtz resonator in acoustics. In the shallow water regime, using asymptotic analysis, a one-dimensional model is derived in which the effect of the resonator is reduced to effective transmission conditions. These conditions clearly highlight two contributions. The first is associated with the dock on its own and it is responsible for a jump of the potential at the free surface. The second is due to the resonant cavity and it is responsible for a jump in the horizontal velocity. It involves as well the uniform amplitude within the resonant cavity with a transient dynamics explicitly given by the equation of a damped oscillator forced by the incident waves. The one-dimensional model is validated in the harmonic regime by comparison to direct two-dimensional numerics. It is shown to reproduce accurately the scattering coefficients and the amplitude within the resonator; interestingly, this remains broadly true for finite water depths. We further inspect the spatio-temporal behaviour of different types of wave packets interacting with the resonating and radiating cavity.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave scattering
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 920 , 10 August 2021 , A22
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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