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

Acoustically invisible gateways

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.
Citation: Brian Sleeman, Evgeny Lakshtanov. Acoustically invisible gateways. Inverse Problems and 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.

[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.

[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.

[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.

[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).

[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.

[7]

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

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[14]

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

[15]

I. Newton, "Philosophiae Naturalis Principia Mathematica," 1687.

[16]

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

[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.

[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).

[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.

[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.

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.

[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.

[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.

[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.

[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).

[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.

[7]

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

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[14]

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

[15]

I. Newton, "Philosophiae Naturalis Principia Mathematica," 1687.

[16]

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

[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.

[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).

[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.

[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.

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