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Experimental study on water-wave trapped modes

  • P. J. COBELLI (a1), V. PAGNEUX (a2), A. MAUREL (a3) and P. PETITJEANS (a1)
  • Please note a correction has been issued for this article.

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.

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Corresponding author

Email address for correspondence: cobelli@pmmh.espci.fr

References

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JFM classification

Type Description Title
VIDEO
Movies

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.

 Video (417 KB)
417 KB
VIDEO
Movies

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.

 Video (210 KB)
210 KB
VIDEO
Movies

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.

 Video (497 KB)
497 KB
VIDEO
Movies

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.

 Video (161 KB)
161 KB

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