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Cloaking in shallow-water waves via nonlinear medium transformation

Published online by Cambridge University Press:  30 July 2015

Ahmad Zareei
Affiliation:
Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA
Mohammad-Reza Alam*
Affiliation:
Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA
*
Email address for correspondence: reza.alam@berkeley.edu

Abstract

A major obstacle in designing a perfect cloak for objects in shallow-water waves is that the linear transformation media scheme (also known as transformation optics) requires spatial variations of two independent medium properties. In the Maxwell’s equation and for the well-studied problem of electromagnetic cloaking, these two properties are permittivity and permeability. Designing an anisotropic material with both variable permittivity and variable permeability, while challenging, is achievable. On the other hand, for long gravity waves, whose governing equation maps one-to-one to the single polarization Maxwell’s equations, the two required spatially variable properties are the water depth and the gravitational acceleration; in this case changing the gravitational acceleration is simply impossible. Here we present a nonlinear transformation that only requires the change in one of the medium properties, which, in the case of shallow-water waves, is the water depth, while keeping the gravitational acceleration constant. This transformation keeps the governing equation perfectly intact and, if the cloak is large enough, asymptotically satisfies the necessary boundary conditions. We show that with this nonlinear transformation an object can be cloaked from any wave that merely satisfies the long-wave assumption. The presented transformation can be applied as well for the design of non-magnetic optical cloaks for electromagnetic waves.

Type
Papers
Copyright
© 2015 Cambridge University Press 

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References

Alam, M.-R. 2012 Broadband cloaking in stratified seas. Phys. Rev. Lett. 108, 084502.CrossRefGoogle ScholarPubMed
Alam, M.-R., Liu, Y. & Yue, D. K. P. 2009a Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part I. Perturbation analysis. J. Fluid Mech. 624, 191224.CrossRefGoogle Scholar
Alam, M.-R., Liu, Y. & Yue, D. K. P. 2009b Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part II. Numerical simulation. J. Fluid Mech. 624, 225253.CrossRefGoogle Scholar
Bangerth, W., Hartmann, R. & Kanschat, G. 2007 Deal.II—a general-purpose object-oriented finite element library. ACM Trans. Math. Softw. 33 (4), 24.CrossRefGoogle Scholar
Bangerth, W. & Kanschat, G.1999 Concepts for Object-oriented Finite Element Software: The Deal. II Library. Preprint 43, SFB 359; http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.136.7882.Google Scholar
Berraquero, C. P., Maurel, A., Petitjeans, P. & Pagneux, V. 2013 Experimental realization of a water-wave metamaterial shifter. Phys. Rev. E 88, 051002.CrossRefGoogle ScholarPubMed
Brûlé, S., Javelaud, E. H., Enoch, S. & Guenneau, S. 2014 Experiments on seismic metamaterials: molding surface waves. Phys. Rev. Lett. 112, 133901.CrossRefGoogle ScholarPubMed
Cai, W., Chettiar, U. K., Kildishev, A. V. & Shalaev, V. M. 2007 Optical cloaking with metamaterials. Nat. Photonics 1 (4), 224227.CrossRefGoogle Scholar
Carraro, T., Goll, C., Marciniak-Czochra, A. & Mikelić, A. 2013 Pressure jump interface law for the Stokes–Darcy coupling: confirmation by direct numerical simulations. J. Fluid Mech. 732, 510536.Google Scholar
Chen, H. & Chan, C. T. 2007 Acoustic cloaking in three dimensions using acoustic metamaterials. Appl. Phys. Lett. 91 (18); doi:10.1063/1.2803315.Google Scholar
Chen, H., Yang, J., Zi, J. & Chan, C. T. 2009 Transformation media for linear liquid surface waves. Europhys. Lett. 85 (2), 24004.CrossRefGoogle Scholar
Cummer, S. A., Popa, B.-I., Schurig, D., Smith, D. R. & Pendry, J. 2006 Full-wave simulations of electromagnetic cloaking structures. Phys. Rev. E 74, 036621.CrossRefGoogle ScholarPubMed
Cummer, S. A. & Schurig, D. 2007 One path to acoustic cloaking. New J. Phys. 9 (3), 45.CrossRefGoogle 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, 134501.CrossRefGoogle Scholar
Farhat, M., Guenneau, S., Enoch, S. & Movchan, A. B. 2009 Cloaking bending waves propagating in thin elastic plates. Phys. Rev. B 79, 033102.CrossRefGoogle Scholar
Huang, X., Zhong, S. & Liu, X. 2014 Acoustic invisibility in turbulent fluids by optimised cloaking. J. Fluid Mech. 749, 460477.CrossRefGoogle Scholar
Jikov, V. V., Oleinik, O. A. & Kozlov, S. M. 1994 Homogenization of Differential Operators and Integral Functionals. Springer.CrossRefGoogle Scholar
Leonhardt, U. 2006 Optical conformal mapping. Science 312 (5781), 17771780.Google Scholar
Mei, C. C., Stiassnie, M. & Yue, D. K.-P. 2005 Theory and Applications of Ocean Surface Waves: Linear Aspects. vol. 23. World Scientific.Google Scholar
Newman, J. N. 2014 Cloaking a circular cylinder in water waves. Eur. J. Mech. (B/Fluids) 47 (0), 145150; enok Palm Memorial Volume.CrossRefGoogle Scholar
Pendry, J. B., Schurig, D. & Smith, D. R. 2006 Controlling electromagnetic fields. Science 312 (5781), 17801782.CrossRefGoogle ScholarPubMed
Porter, R. & Newman, J. N. 2014 Cloaking of a vertical cylinder in waves using variable bathymetry. J. Fluid Mech. 750, 124143.CrossRefGoogle Scholar
Riedlbauer, D., Drexler, M., Drummer, D., Steinmann, P. & Mergheim, J. 2014 Modelling, simulation and experimental validation of heat transfer in selective laser melting of the polymeric material {PA12}. Comput. Mater. Sci. 93 (0), 239248.CrossRefGoogle Scholar
Schurig, D., Mock, J. J., Justice, B. J., Cummer, S. A., Pendry, J. B., Starr, A. F. & Smith, D. R. 2006 Metamaterial electromagnetic cloak at microwave frequencies. Science 314 (5801), 977980.CrossRefGoogle ScholarPubMed
Yan, M., Ruan, Z. & Qiu, M. 2007 Cylindrical invisibility cloak with simplified material parameters is inherently visible. Phys. Rev. Lett. 99, 233901.CrossRefGoogle ScholarPubMed
Young, E. C. 1992 Vector and Tensor Analysis. CRC Press.Google Scholar
Zhang, S., Genov, D. A., Sun, C. & Zhang, X. 2008 Cloaking of matter waves. Phys. Rev. Lett. 100, 123002.Google Scholar
Zhang, S., Xia, C. & Fang, N. 2011 Broadband acoustic cloak for ultrasound waves. Phys. Rev. Lett. 106, 024301.Google Scholar