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Scattering of gravity waves by a periodically structured ridge of finite extent

Published online by Cambridge University Press:  21 May 2019

Agnès Maurel Open the ORCID record for Agnès Maurel [Opens in a new window] ,
Kim Pham  and
Jean-Jacques Marigo
Agnès Maurel*
Affiliation:
Institut Langevin, ESPCI ParisTech, CNRS UMR 7587, 1 rue Jussieu, 75005 Paris, France
Kim Pham
Affiliation:
IMSIA, ENSTA ParisTech – CNRS – EDF – CEA, Université Paris-Saclay, 828 Bd des Maréchaux, 91732 Palaiseau, France
Jean-Jacques Marigo
Affiliation:
Lab. de Mécanique des Solides, Ecole Polytechnique, Route de Saclay, 91120 Palaiseau, France
*
†Email address for correspondence: agnes.maurel@espci.fr
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Abstract

We study the propagation of water waves over a ridge structured at the subwavelength scale using homogenization techniques able to account for its finite extent. The calculations are conducted in the time domain considering the full three-dimensional problem to capture the effects of the evanescent field in the water channel over the structured ridge and at its boundaries. This provides an effective two-dimensional wave equation which is a classical result but also non-intuitive transmission conditions between the region of the ridge and the surrounding regions of constant immersion depth. Numerical results provide evidence that the scattering properties of a structured ridge can be strongly influenced by the evanescent fields, a fact which is accurately captured by the homogenized model.

JFM classification

Geophysical and Geological Flows: Shallow water flows Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 871 , 25 July 2019 , pp. 350 - 376
Copyright
© 2019 Cambridge University Press 

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References

Berraquero, C., Maurel, A., Petitjeans, P. & Pagneux, V. 2013 Experimental realization of a water-wave metamaterial shifter. Phys. Rev. E 88 (5), 051002.Google Scholar
Bobinski, T., Eddi, A., Petitjeans, P., Maurel, A. & Pagneux, V. 2015 Experimental demonstration of epsilon-near-zero water waves focusing. Appl. Phys. Lett. 107 (1), 014101.Google Scholar
Cakoni, F., Guzina, B. & Moskow, S. 2016 On the homogenization of a scalar scattering problem for highly oscillating anisotropic media. SIAM J. Math. Anal. 48 (4), 2532–2560.Google Scholar
Chen, H., Yang, J., Zi, J. & Chan, C. T. 2009 Transformation media for linear liquid surface waves. Europhys. Lett. 85 (2), 24004.Google Scholar
Dupont, G., Guenneau, S., Kimmoun, O., Molin, B. & Enoch, S. 2016 Cloaking a vertical cylinder via homogenization in the mild-slope equation. J. Fluid Mech. 796, R1.Google Scholar
Dupont, G., Kimmoun, O., Molin, B., Guenneau, S. & Enoch, S. 2015 Numerical and experimental study of an invisibility carpet in a water channel. Phys. Rev. E 91 (2), 023010.Google Scholar
Farhat, M., Enoch, S., Guenneau, S. & Movchan, A. B. 2008 Broadband cylindrical acoustic cloak for linear surface waves in a fluid. Phys. Rev. Lett. 101 (13), 134501.Google Scholar
Guo, X., Wang, B., Mei, C. C. & Liu, H. 2017 Scattering of periodic surface waves by pile-group supported platform. Ocean Engng 146, 46–58.Google Scholar
Hu, X. & Chan, C. T. 2005 Refraction of water waves by periodic cylinder arrays. Phys. Rev. Lett. 95 (15), 154501.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Marigo, J.-J. & Maurel, A. 2017 Second order homogenization of subwavelength stratified media including finite size effect. SIAM J. Appl. Maths 77 (2), 721–743.Google Scholar
Maurel, A. & Marigo, J.-J. 2018 Sensitivity of a dielectric layered structure on a scale below the periodicity: a fully local homogenized model. Phys. Rev. B 98 (2), 024306.Google Scholar
Maurel, A., Marigo, J.-J., Cobelli, P., Petitjeans, P. & Pagneux, V. 2017 Revisiting the anisotropy of metamaterials for water waves. Phys. Rev. B 96 (13), 134310.Google Scholar
Mei, C. C., Chan, I.-C., Liu, P., Huang, Z. & Zhang, W. 2011 Long waves through emergent coastal vegetation. J. Fluid Mech. 687, 461–491.Google Scholar
Newman, J. N. 2014 Cloaking a circular cylinder in water waves. Eur. J. Mech. (B/Fluids) 47, 145–150.Google Scholar
Porter, R. 2017 Cloaking in water waves. In Handbook of Metamaterials Properties, vol. 2. World Scientific Publishing Company.Google Scholar
Porter, R.2019 An extended linear shallow water equation. J. Fluid Mech. (submitted)https://people.maths.bris.ac.uk/∼marp/abstracts/jfmcswe.pdf.Google Scholar
Porter, R. & Newman, J. N. 2014 Cloaking of a vertical cylinder in waves using variable bathymetry. J. Fluid Mech. 750, 124–143.Google Scholar
Rosales, R. R. & Papanicolaou, G. C. 1983 Gravity waves in a channel with a rough bottom. Stud. Appl. Maths 68 (2), 89–102.Google Scholar
Sheinfux, H. H., Kaminer, I., Plotnik, Y., Bartal, G. & Segev, M. 2014 Subwavelength multilayer dielectrics: ultrasensitive transmission and breakdown of effective-medium theory. Phys. Rev. Lett. 113 (24), 243901.Google Scholar
Tuck, E. O. 1976 Some classical water-wave problems in variable depth. In Waves on Water of Variable Depth, Lecture Notes in Physics, vol. 64, pp. 9–20. Springer.Google Scholar
Vinoles, V.2016 Problèmes d’interface en présence de métamatériaux: modélisation, analyse et simulations. PhD thesis, Université Paris-Saclay.Google Scholar
Wang, B., Guo, X. & Mei, C. C. 2015 Surface water waves over a shallow canopy. J. Fluid Mech. 768, 572–599.Google Scholar
Xu, J., Jiang, X., Fang, N., Georget, E., Abdeddaim, R., Geffrin, J.-M., Farhat, M., Sabouroux, P., Enoch, S. & Guenneau, S. 2015 Molding acoustic, electromagnetic and water waves with a single cloak. Sci. Rep. 5, 10678.Google Scholar
Zareei, A. & Alam, M.-R. 2015 Cloaking in shallow-water waves via nonlinear medium transformation. J. Fluid Mech. 778, 273–287.Google Scholar
Zhang, C., Chan, C.-T. & Hu, X. 2014 Broadband focusing and collimation of water waves by zero refractive index. Sci. Rep. 4, 6979.Google Scholar