American Institute of Mathematical Sciences

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February  2011, 5(1): 203-217. doi: 10.3934/ipi.2011.5.203

Acoustically invisible gateways

Brian Sleeman 1, and  Evgeny Lakshtanov 2,

1. 

School of Mathematics, University of Leeds, Leeds, LS2 9JT
2. 

Department of Mathematics, Aveiro University, Aveiro 3810, Portugal

Received  May 2010 Revised  September 2010 Published  February 2011

In recent years considerable interest has been directed at devising obstacles which appear "invisible" to various types of wave propagation. That is; suppose for example we can construct an obstacle coated with an appropriate material such that when illuminated by an incident field (e.g plane wave) the wave scattered by the obstacle has zero cross section (equivalently radiation pattern)for all incident directions and frequencies then the obstacle appears to have no effect on the illuminating wave and the obstacle can be considered invisible. Such an electromagnetic cloaking device has been constructed by Schurig et. al [18]. Motivated by recent work [1] concerning the problem of a parallel flow of particles falling on a body with piecewise smooth boundary and leaving no trace the analogous problem of acoustic scattering of a plane wave illuminating the same body, a gateway, is considered. It is shown that at high frequencies, with use of the Kirchoff approximation and the geometrical theory of diffraction, the scattered far field in a range of observed directions, e. g the back scattered direction, is zero for a discrete set of wave numbers. That is the gateway is acoustically "invisible" in that direction.
Keywords: Geometrical Theory of Diffraction., Acoustic Scattering, Kirchoff Approximation.
Mathematics Subject Classification: Primary: 74J20, 78A45; Secondary: 34L2.
Citation: Brian Sleeman, Evgeny Lakshtanov. Acoustically invisible gateways. Inverse Problems & Imaging, 2011, 5 (1) : 203-217. doi: 10.3934/ipi.2011.5.203
References:
[1]

A. Aleksenko and A. Plakhov, Bodies of zero resistance and bodies invisible in one direction, Nonlinearity, 22 (2009), 1247-1258. doi: 10.1088/0951-7715/22/6/001.  Google Scholar

[2]

H. Chen, C. T. Chan, S. Liu and Z. Lin, A simple route to a tunable electromagnetic gateway, New J. Physics, 11 (2009), 083012. doi: 10.1088/1367-2630/11/8/083012.  Google Scholar

[3]

H. Chen, B-L. Wu, B. Zhang and J. A. Kong, Electromagnetic wave interactions with a metamaterial cloak, Phys. Rev. Letters, 99 (2007), 063903-1,4. Google Scholar

[4]

S. A. Cummer, B. Popa, D. Schurig, D. R. Smith and J. Pendry, Full-wave simulations of electromagnetic cloaking structures, Phys. Rev. E, 74 (2006), 036621. doi: 10.1103/PhysRevE.74.036621.  Google Scholar

[5]

S. A. Cummer, B. Popa, D. Schurig, D. R. Smith, J. Pendry, M. Rahm and A. Starr, Scattering theory derivation of a 3D acoustic cloaking shell, Phys. Rev. Letters, 100 (2008). Google Scholar

[6]

B. Dietz and U. Smilansky, A scattering approach to the quantization of billiards-The inside-outside duality, Chaos, 3 (1993), 581-589. doi: 10.1063/1.165962.  Google Scholar

[7]

A. Greenleaf, M. Lassas and G. Uhlmann, On nonuniqueness for Calderon's inverse problem, Math. Res. Letters, 10 (2003), 685-693.  Google Scholar

[8]

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Full-wave invisibility of active devices at all frequencies, Comm. Math. Phys., 275 (2007), 749-789. doi: 10.1007/s00220-007-0311-6.  Google Scholar

[9]

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Comment on "Scattering Theory Derivation of a 3D Acoustic Cloaking Shell," preprint, 2008, available from http://arXiv.org/abs/0801.3279.  Google Scholar

[10]

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Invisibility and inverse problems, Bull. Amer. Math. Soc, 46 (2009), 55-97. doi: 10.1090/S0273-0979-08-01232-9.  Google Scholar

[11]

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Cloaking devices, electromagnetic wormholes and transformation optics, SIAM Rev., 51 (2009), 3-33. doi: 10.1137/080716827.  Google Scholar

[12]

J. B. Keller, Geometrical theory of diffraction, J. Opt. Soc. Amer., 52 (1962), 116-130. doi: 10.1364/JOSA.52.000116.  Google Scholar

[13]

U. Leonhardt, Optical conformal mapping, Science, 312 (2006), 1777-1780. doi: 10.1126/science.1126493.  Google Scholar

[14]

U. Leonhardt and T. Tyc, Broadband invisibility by non-euclidean cloaking, Science, 323 (2009), 110-112. doi: 10.1126/science.1166332.  Google Scholar

[15]

I. Newton, "Philosophiae Naturalis Principia Mathematica,", 1687., ().   Google Scholar

[16]

J. B. Pendry, D. Schurig and D. R. Smith, Controlling electromagnetic fields, Science, 312 (2006), 1780. doi: 10.1126/science.1125907.  Google Scholar

[17]

Z. Ruan, M. Yan, C. W. Neff and M. Qiu, Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations, Phys. Rev. Letters, (2007), 113903. doi: 10.1103/PhysRevLett.99.113903.  Google Scholar

[18]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F Starr and D. R. Smith, Science Express, 19 (2006). Google Scholar

[19]

B. D. Sleeman, The inverse acoustic obstacle scattering problem and its interior dual, Inverse Problems and Imaging, 3 (2009), 211-229. doi: 10.3934/ipi.2009.3.211.  Google Scholar

[20]

B. D. Sleeman, Acoustic Scattering by Obstacles Invisible in the Backscattered Direction, in "Advanced Topics in Scattering Theory and Biomedical Engineering" (eds A.Charalambopoulos, D. I. Fotiadis and D. Polyzos), World Scientific, (2010), 187-201. Google Scholar

show all references

References:
[1]

A. Aleksenko and A. Plakhov, Bodies of zero resistance and bodies invisible in one direction, Nonlinearity, 22 (2009), 1247-1258. doi: 10.1088/0951-7715/22/6/001.  Google Scholar

[2]

H. Chen, C. T. Chan, S. Liu and Z. Lin, A simple route to a tunable electromagnetic gateway, New J. Physics, 11 (2009), 083012. doi: 10.1088/1367-2630/11/8/083012.  Google Scholar

[3]

H. Chen, B-L. Wu, B. Zhang and J. A. Kong, Electromagnetic wave interactions with a metamaterial cloak, Phys. Rev. Letters, 99 (2007), 063903-1,4. Google Scholar

[4]

S. A. Cummer, B. Popa, D. Schurig, D. R. Smith and J. Pendry, Full-wave simulations of electromagnetic cloaking structures, Phys. Rev. E, 74 (2006), 036621. doi: 10.1103/PhysRevE.74.036621.  Google Scholar

[5]

S. A. Cummer, B. Popa, D. Schurig, D. R. Smith, J. Pendry, M. Rahm and A. Starr, Scattering theory derivation of a 3D acoustic cloaking shell, Phys. Rev. Letters, 100 (2008). Google Scholar

[6]

B. Dietz and U. Smilansky, A scattering approach to the quantization of billiards-The inside-outside duality, Chaos, 3 (1993), 581-589. doi: 10.1063/1.165962.  Google Scholar

[7]

A. Greenleaf, M. Lassas and G. Uhlmann, On nonuniqueness for Calderon's inverse problem, Math. Res. Letters, 10 (2003), 685-693.  Google Scholar

[8]

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Full-wave invisibility of active devices at all frequencies, Comm. Math. Phys., 275 (2007), 749-789. doi: 10.1007/s00220-007-0311-6.  Google Scholar

[9]

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Comment on "Scattering Theory Derivation of a 3D Acoustic Cloaking Shell," preprint, 2008, available from http://arXiv.org/abs/0801.3279.  Google Scholar

[10]

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Invisibility and inverse problems, Bull. Amer. Math. Soc, 46 (2009), 55-97. doi: 10.1090/S0273-0979-08-01232-9.  Google Scholar

[11]

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Cloaking devices, electromagnetic wormholes and transformation optics, SIAM Rev., 51 (2009), 3-33. doi: 10.1137/080716827.  Google Scholar

[12]

J. B. Keller, Geometrical theory of diffraction, J. Opt. Soc. Amer., 52 (1962), 116-130. doi: 10.1364/JOSA.52.000116.  Google Scholar

[13]

U. Leonhardt, Optical conformal mapping, Science, 312 (2006), 1777-1780. doi: 10.1126/science.1126493.  Google Scholar

[14]

U. Leonhardt and T. Tyc, Broadband invisibility by non-euclidean cloaking, Science, 323 (2009), 110-112. doi: 10.1126/science.1166332.  Google Scholar

[15]

I. Newton, "Philosophiae Naturalis Principia Mathematica,", 1687., ().   Google Scholar

[16]

J. B. Pendry, D. Schurig and D. R. Smith, Controlling electromagnetic fields, Science, 312 (2006), 1780. doi: 10.1126/science.1125907.  Google Scholar

[17]

Z. Ruan, M. Yan, C. W. Neff and M. Qiu, Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations, Phys. Rev. Letters, (2007), 113903. doi: 10.1103/PhysRevLett.99.113903.  Google Scholar

[18]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F Starr and D. R. Smith, Science Express, 19 (2006). Google Scholar

[19]

B. D. Sleeman, The inverse acoustic obstacle scattering problem and its interior dual, Inverse Problems and Imaging, 3 (2009), 211-229. doi: 10.3934/ipi.2009.3.211.  Google Scholar

[20]

B. D. Sleeman, Acoustic Scattering by Obstacles Invisible in the Backscattered Direction, in "Advanced Topics in Scattering Theory and Biomedical Engineering" (eds A.Charalambopoulos, D. I. Fotiadis and D. Polyzos), World Scientific, (2010), 187-201. Google Scholar

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