Skip to main content Accessibility help
×
Home

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

  • Agnès Maurel (a1), Kim Pham (a2) and Jean-Jacques Marigo (a3)

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.

Copyright

Corresponding author

Email address for correspondence: agnes.maurel@espci.fr

References

Hide All
Berraquero, C., Maurel, A., Petitjeans, P. & Pagneux, V. 2013 Experimental realization of a water-wave metamaterial shifter. Phys. Rev. E 88 (5), 051002.
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.
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), 25322560.
Chen, H., Yang, J., Zi, J. & Chan, C. T. 2009 Transformation media for linear liquid surface waves. Europhys. Lett. 85 (2), 24004.
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.
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.
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.
Guo, X., Wang, B., Mei, C. C. & Liu, H. 2017 Scattering of periodic surface waves by pile-group supported platform. Ocean Engng 146, 4658.
Hu, X. & Chan, C. T. 2005 Refraction of water waves by periodic cylinder arrays. Phys. Rev. Lett. 95 (15), 154501.
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Marigo, J.-J. & Maurel, A. 2017 Second order homogenization of subwavelength stratified media including finite size effect. SIAM J. Appl. Maths 77 (2), 721743.
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.
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.
Mei, C. C., Chan, I.-C., Liu, P., Huang, Z. & Zhang, W. 2011 Long waves through emergent coastal vegetation. J. Fluid Mech. 687, 461491.
Newman, J. N. 2014 Cloaking a circular cylinder in water waves. Eur. J. Mech. (B/Fluids) 47, 145150.
Porter, R. 2017 Cloaking in water waves. In Handbook of Metamaterials Properties, vol. 2. World Scientific Publishing Company.
Porter, R.2019 An extended linear shallow water equation. J. Fluid Mech. (submitted)https://people.maths.bris.ac.uk/∼marp/abstracts/jfmcswe.pdf.
Porter, R. & Newman, J. N. 2014 Cloaking of a vertical cylinder in waves using variable bathymetry. J. Fluid Mech. 750, 124143.
Rosales, R. R. & Papanicolaou, G. C. 1983 Gravity waves in a channel with a rough bottom. Stud. Appl. Maths 68 (2), 89102.
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.
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. 920. Springer.
Vinoles, V.2016 Problèmes d’interface en présence de métamatériaux: modélisation, analyse et simulations. PhD thesis, Université Paris-Saclay.
Wang, B., Guo, X. & Mei, C. C. 2015 Surface water waves over a shallow canopy. J. Fluid Mech. 768, 572599.
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.
Zareei, A. & Alam, M.-R. 2015 Cloaking in shallow-water waves via nonlinear medium transformation. J. Fluid Mech. 778, 273287.
Zhang, C., Chan, C.-T. & Hu, X. 2014 Broadband focusing and collimation of water waves by zero refractive index. Sci. Rep. 4, 6979.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed