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Experimental study on water-wave trapped modes

Published online by Cambridge University Press:  06 January 2011

P. J. COBELLI ,
V. PAGNEUX ,
A. MAUREL  and
P. PETITJEANS
P. J. COBELLI*
Affiliation:
Laboratoire de Physique et Mécanique des Milieux Hétérogènes, UMR CNRS 7636, Ecole Supérieure de Physique et de Chimie Industrielles, 10 rue Vauquelin, 75231 Paris CEDEX 5, France
V. PAGNEUX
Affiliation:
Laboratoire d'Acoustique de l'Université du Maine, UMR CNRS 6613, Avenue Olivier Messiaen, 72085 Le Mans CEDEX 9, France
A. MAUREL
Affiliation:
Laboratoire Ondes et Acoustique, Institut Langevin UMR CNRS 7587, Ecole Supérieure de Physique et de Chimie Industrielles, 10 rue Vauquelin, 75231 Paris CEDEX 5, France
P. PETITJEANS
Affiliation:
Laboratoire de Physique et Mécanique des Milieux Hétérogènes, UMR CNRS 7636, Ecole Supérieure de Physique et de Chimie Industrielles, 10 rue Vauquelin, 75231 Paris CEDEX 5, France
*
†Email address for correspondence: [email protected]
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Abstract

We present an experimental study on the trapped modes occurring around a vertical surface-piercing circular cylinder of radius a placed symmetrically between the parallel walls of a long but finite water waveguide of width 2d. A wavemaker placed near the entrance of the waveguide is used to force an asymmetric perturbation into the guide, and the free-surface deformation field is measured using a global single-shot optical profilometric technique. In this configuration, several values of the aspect ratio a/d were explored for a range of driving frequencies below the waveguide's cutoff. Decomposition of the obtained fields in harmonics of the driving frequency allowed for the isolation of the linear contribution, which was subsequently separated according to the symmetries of the problem. For each of the aspect ratios considered, the spatial structure of the trapped mode was obtained and compared to the theoretical predictions given by a multipole expansion method. The waveguide–obstacle system was further characterized in terms of reflection and transmission coefficients, which led to the construction of resonance curves showing the presence of one or two trapped modes (depending on the value of a/d), a result that is consistent with the theoretical predictions available in the literature. The frequency dependency of the trapped modes with the geometrical parameter a/d was determined from these curves and successfully compared to the theoretical predictions available within the frame of linear wave theory.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave scattering
Type
Papers
Information
Journal of Fluid Mechanics , Volume 666 , 10 January 2011 , pp. 445 - 476
Copyright
Copyright © Cambridge University Press 2011

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References

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Cobelli supplementary material

Movie 1. Experimental results for the time-evolution of the free-surface deformation for a/d = 0.5 and f = 2.5 Hz, close to the experimentally determined resonant (trapped-mode) frequency. This movie is composed of a sequence of 100 instantaneous free-surface deformation fields, registered at an acquisition rate of 250 Hz. Configuration II was chosen because the cylinder being closer to the entrance makes the amplitudes larger, rendering the phenomenon more evident. Local height is linearly color-coded between red and blue, the former corresponding to elevations and the latter to depressions with respect to the free-surface at rest. With this convention, green is associated with undeformed regions. The scale of the colorbar is in mm.

Download Cobelli supplementary material(Video)
Video 416.9 KB

Cobelli supplementary material

Movie 1. Experimental results for the time-evolution of the free-surface deformation for a/d = 0.5 and f = 2.5 Hz, close to the experimentally determined resonant (trapped-mode) frequency. This movie is composed of a sequence of 100 instantaneous free-surface deformation fields, registered at an acquisition rate of 250 Hz. Configuration II was chosen because the cylinder being closer to the entrance makes the amplitudes larger, rendering the phenomenon more evident. Local height is linearly color-coded between red and blue, the former corresponding to elevations and the latter to depressions with respect to the free-surface at rest. With this convention, green is associated with undeformed regions. The scale of the colorbar is in mm.

Download Cobelli supplementary material(Video)
Video 210 KB

Cobelli supplementary material

Movie 2. Experimental results for the time-evolution of the linear component of the free-surface deformation around the phase singularity (whose position is made evident by a black point at the center of the frame). The corresponding parameters are: a/d = 0.5 and f = 2.5 Hz, close to the experimentally determined resonant (trapped-mode) frequency. This movie is composed of a sequence of 100 instantaneous free-surface deformation fields, registered at an acquisition rate of 250 Hz. Local height is linearly color-coded between red and blue, the former corresponding to elevations and the latter to depressions with respect to the free-surface at rest. With this convention, green is associated with undeformed regions. The scale of the colorbar is in mm.

Download Cobelli supplementary material(Video)
Video 497.2 KB

Cobelli supplementary material

Movie 2. Experimental results for the time-evolution of the linear component of the free-surface deformation around the phase singularity (whose position is made evident by a black point at the center of the frame). The corresponding parameters are: a/d = 0.5 and f = 2.5 Hz, close to the experimentally determined resonant (trapped-mode) frequency. This movie is composed of a sequence of 100 instantaneous free-surface deformation fields, registered at an acquisition rate of 250 Hz. Local height is linearly color-coded between red and blue, the former corresponding to elevations and the latter to depressions with respect to the free-surface at rest. With this convention, green is associated with undeformed regions. The scale of the colorbar is in mm.

Download Cobelli supplementary material(Video)
Video 161.1 KB