Skip to main content Accessibility help
×
Home

Cloaking of a vertical cylinder in waves using variable bathymetry

  • R. Porter (a1) and J. N. Newman (a2)

Abstract

The paper describes a process which allows a vertical circular cylinder subject to plane monochromatic surface gravity waves to appear invisible to the far-field observer. This is achieved by surrounding the cylinder with an annular region of variable bathymetry. Two approaches are taken to investigate this effect. First a mild-slope approximation is applied to the governing linearised three-dimensional water wave equations to formulate a depth-averaged two-dimensional wave equation with varying wavenumber over the variable bathmetry. This is then solved by formulating a domain integral equation, solved numerically by discretisation. For a given set of geometrical and wave parameters, the bathymetry is selected by a numerical optimisation process and it is shown that the scattering cross-section is reduced towards zero with increasing refinement of the bathymetry. A fully three-dimensional boundary-element method, based on the WAMIT solver (see www.wamit.com) but adapted here to allow for depressions in the bed, is used to assess the accuracy of the mild-slope results and then further numerically optimise the bathymetry towards a cloaking structure. Numerical results provide strong evidence that perfect cloaking is possible for the fully three-dimensional problem. One practical application of the results is that cloaking implies a reduced mean drift force on the cylinder.

Copyright

Corresponding author

Email address for correspondence: richard.porter@bris.ac.uk

References

Hide All
Alam, M. 2012 Broadband cloaking in stratified seas. Phys. Rev. Lett. 108, 084502.
Alù, A. & Engheta, N. 2005 Achieving transparency with plasmonic and metamaterial coatings. Phys. Rev. E 72, 016623.
Belibassakis, K. A. 2008 A boundary element method for the hydrodynamic analysis of floating bodies in variable bathymetry regions. Engng Anal. Bound. Elem. 32, 796810.
Brent, R. 2002 Algorithms for Minimization with Derivatives. Dover.
Chamberlain, P. G. & Porter, D. 1995 The modified mild-slope equation. J. Fluid Mech. 291, 393407.
Chamberlain, P. G. & Porter, D. 1999 Scattering and near-trapping of water waves by axisymmetric topography. J. Fluid Mech. 388, 335354.
Chen, H. & Chan, C. T. 2007 Acoustic cloaking in three dimensions using acoustic metamaterials. Appl. Phys. Lett. 91 (19), 183518.
Cummer, S. A., Popa, B.-I., Schurig, D., Smith, D. R., Pendry, J. B., Rahm, M. & Starr, A. 2007 Scattering theory derivation of a 3D acoustic cloaking shell. Phys. Rev. Lett. 100, 024301.
Cummer, S. A. & Schurig, D. 2007 One path to acoustic cloaking. New J. Phys. 9, 45.
Ehrenmark, U. 2005 An alternative dispersion equation for water waves over an inclined bed. J. Fluid Mech. 543, 249266.
Farhat, M., Enoch, S., Guenneau, S. & Movchan, A. B. 2008 Broadbanded cylindrical acoustic cloak for linear surface waves in a fluid. Phys. Rev. Lett. 101, 134501.
Farhat, M., Guenneau, S., Enoch, S. & Movchan, A. B. 2009 Cloaking bending waves propagating in the elastic plates. Phys. Rev. B 79 (4), 033102.
Ferreira, M. D. & Newman, J. N.2009 Diffraction effects and ship motions on an artificial seabed. In Proceedings of 24th International Workshop on Water Waves and Floating Bodies, Zelenogorsk, Russia.
Griffiths, L. S. & Porter, R. 2012 Focusing of surface waves by variable bathymetry. Appl. Ocean Res. 34, 150163.
Kurniawan, A. & Moan, T.2012 Multi-objective optimization of a wave energy absorber geometry. In Proceedings of 27th International Workshop on Water Waves and Floating Bodies, Copenhagen, Denmark.
Lee, C.-H. & Newman, J. N. 2004 Computation of wave effects using the panel method. In Numerical Modeling in Fluid–Structure Interaction (ed. Chakrabarti, S.). WIT Press.
Leonhardt, U. 2006 Optical conformal mapping. Science 312, 17771780.
Liu, H.-W., Wang, Q.-Y. & Tang, G.-T. 2013 Exact solution to the modified mild-slope equation for wave scattering by a cylinder with an idealized scour pit. ASCE J. Waterway, Port, Coastal, Ocean Engng. 139, 413423.
Maruo, H. 1960 The drift force of a body floating in waves. J. Ship Res. 4 (3), 110.
Mei, C. C. 1983 The Applied Dynamics of Ocean Surface Waves. Wiley Interscience.
Milton, G. W., Briane, M. & Willis, J. R. 2006 On cloaking for elasticity and physical equations with a transformation invariant form. New J. Phys. 8, 248.
Newman, J. N.2012 Scattering by a cylinder with variable bathymetry. In Proceedings of 27th International Workshop on Water Waves and Floating Bodies, Copenhagen, Denmark.
Newman, J. N. 2013 Cloaking a circular cylinder in water waves. Eur. J. Mech. B (in press); doi: 10.1016/j.euromechflu.2013.11.005.
Pendry, J. B., Schurig, D. & Smith, D. R. 2006 Controlling electromagnetic fields. Science 312, 17801782.
Pinkster, J. A.2011 A multi-domain approach in 3-D diffraction calculations 30th International Conference on Ocean, Offshore and Arctic Engineering, Rotterdam, Netherlands. OMAE2011-49414, pp. 355–364.
Porter, R.2011 Cloaking a cylinder in waves. In Proceedings of 26th International Workshop on Water Waves and Floating Bodies, Athens, Greece.
Porter, R. & Porter, D. 2001 Interaction of water waves with three-dimensional periodic topography. J. Fluid Mech. 434, 301335.
Ward, A. J. & Pendry, J. B. 1996 Refraction and geometry in Maxwell’s equations. J. Mod. Opt. 43 (4), 773793.
Wehausen, J. V. & Laitone, E. V. 1960 Surface waves. In Handbuch der Physik (ed. Flugge, S.), vol. 9, pp. 446778. Springer.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Cloaking of a vertical cylinder in waves using variable bathymetry

  • R. Porter (a1) and J. N. Newman (a2)

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