Skip to main content
SpringerLink
Log in

Some Problems in Electromagnetics

  • Chapter
A Celebration of Mathematical Modeling

Abstract

This contribution surveys some of Joe Keller’s work in wave propagation and inverse problems, gives a snapshot of the current state of the field, and discusses some open problems in the area.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. K. Aygun, S. E. Fisher, A. A. Ergin, B. Shanker, and E. Michielssen,”Transient analysis of multielement wire antennas mounted on arbitrarily shaped perfectly conducting bodies, ” Radio Science, vol. 34, no. 4, pp. 781–796, April 1999.

    Article  Google Scholar 

  2. E. Altshuler and D.S. Linden, “Wire antenna design using genetic algorithms”, IEEE Antennas and Propagation Magazine 39 (1997) 33–43.

    Article  Google Scholar 

  3. T.S. Angell and A. Kirsch, “Multicriteria optimization in antenna design”, Math. Methods Appl. Sci. 15 (1992) 647–660.

    Article  MathSciNet  MATH  Google Scholar 

  4. T.S. Angell, R.E. Kleinman, and B. Vainberg, “Asymptotic methods for an optimal antenna problem”, SIAM J. Appl. Math. 59 (1999) 242–260.

    MathSciNet  MATH  Google Scholar 

  5. G.R. Benitz, “High-Defintion Vector Imaging,” Lincoln Laboratory Journal, vol. 10, pp. 147–169, 1997.

    Google Scholar 

  6. M. Cheney and B. Borden, “Microlocal structure of Inverse Synthetic Aperture Radar Data”, Inverse Problems 19, pp. 173–194, 2002.

    Article  MathSciNet  Google Scholar 

  7. M. Cheney and D. Isaacson, “Inverse problems for a perturbed dissipative half-space”, Inverse Problems, 11, 865–888 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Cooper and W. Strauss, “Scattering of waves by periodically moving bodies”, J. Functional Analysis 47 (1982) 180–229.

    Article  MathSciNet  MATH  Google Scholar 

  9. Y. Dermenjian and J.C. Guillot, “Scattering of elastic waves in a perturbed half-space with a free boundary. The limiting absorption principle”, Math. Meth. in the Appl. Sci. 10, 87–124 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  10. D. C. Dobson and F. Santosa, “Nondestructive evaluation of plates using eddy current methods”, International Journal of Engineering Science, 36 (1998), 395–409.

    Article  MathSciNet  MATH  Google Scholar 

  11. S. E. Fisher, D. S. Weile, and E. Michielssen,”Pareto genetic algorithm design of log-periodic monopole arrays mounted on realistic platforms, ” Journal of Electromagnetic Waves and Applications, vol. 13, no. 5, pp. 571–598, 1999.

    Article  MATH  Google Scholar 

  12. A. Karlsson and G. Kristensson, “Constitutive relations, dissipation and reciprocity for the Maxwell equations in the time domain”, J. Electr. Waves Appl. 6 (1992) 437–551.

    Google Scholar 

  13. J.B. Keller, H. Primakoff, M.J. Klein and E. Carstensen, Diffraction of Sound Around a Circular Disk, J.A.S.A., 19, 132–142, Jan. 1947; Math. Revs. 8, 545, 1947

    Google Scholar 

  14. J.B. Keller and A. Blank, Diffraction and Reflection of Pulses by Wedges and Corners, Comm. Pure Appl. Math., 4, 75–94, June 1951; Math. Revs., 12,564, 1951; 13, 304, 1952.

    Google Scholar 

  15. J.B. Keller, Diffraction of a shock or an electromagnetic pulse by a right-angled wedge, J. Appl. Phys., 23, 1267–1268, Nov. 1952.

    Article  Google Scholar 

  16. J.B. Keller, The Geometrical Theory of Diffraction,Proc. of the Symp. on Microwave Optics, Eaton Electronics Laboratory, McGill University, Montreal, Canada, June 1953, Vol. 1 (4 pages); Reprinted as, The Geometric Optics Theory of Diffraction, The McGill Symp. on Microwave Optics, B.S. Karasik and F.J. Zucker, eds., AFCRC, Bedford, MA, 1959, Vol. 2, 207–210.

    Google Scholar 

  17. J.B. Keller, R.M. Lewis, and B.D. Seckler, Asymptotic Solution of Some Diffraction Problems, 1956 Comm. Pure Appl. Math., 9, 207–265, June 1956; Math. Revs., 17, 41, 1956; 18, 43, 1957.

    Google Scholar 

  18. J.B. Keller, Diffraction by a Convex Cylinder,IRE Trans. on Antennas and Prop., Symp. on Electromagnetic Wave Theory, AP-4, 312–321, July 1956; Math. Revs., 19, 103, 1959.

    Google Scholar 

  19. J.B. Keller, Diffraction by an Aperture, J. Appl. Phys., 28,426–444, April 1957; Math. Revs., 20, 835, 1959; 21, 105, 1960.

    Google Scholar 

  20. J.B. Keller, R.M. Lewis, and B.D. Seckler, Diffraction by an Aperture II, J. Appl. Phys., 28, 570–579, May 1957; Math. Revs. 20, 835, 1959; 21, 105, 1960.

    Google Scholar 

  21. J.B. Keller, A Geometrical Theory of Diffraction, Calculus of Variations and Its Applications, Proceedings of Symposia in Applied Mathematics, 8, 27–52, McGraw-Hill, New York, 1958; Math. Revs. 20, 103, 1959.

    Google Scholar 

  22. J.B. Keller and B.D. Seckler, The Geometrical Theory of Diffraction in Inhomogeneous Media, J.A.S.A., 31, 192–205, Feb. 1959; Math. Revs. 20, 836 and 1132, 1959.

    Google Scholar 

  23. J.B. Keller, and B.D. Seckler, Asymptotic Theory of Diffraction in Inhomogeneous Media, J.A.S.A., 31, 206–216, Feb. 1959; Math. Revs. 20, 836 and 1132, 1959.

    Google Scholar 

  24. J.B. Keller and B.R. Levy, Diffraction by a Smooth Object, Comm. Pure Appl. Math., 12, 159–209, Feb. 1959; Math. Revs., 21, 212, 1960.

    Google Scholar 

  25. J.B. Keller, How Dark is the Shadow of a Round-Ended Screen ?, J. Appl. Phys., 30, 1452–1454, Sept. 1959.

    Article  Google Scholar 

  26. J.B. Keller and B.R. Levy, Diffraction by a Spheroid, Canadian J. of Phys., 38, 128–144, J.n. 1960; Math. Revs. 22 (2B), 228, 1961.

    Google Scholar 

  27. J.B. Keller and R.N. Buchal, Boundary Layer Problems in Diffraction Theory, Comm. Pure Appl. Math., 13, 85–114, Feb. 1960; Math. Revs. 22 (10B), 1797, 1961.

    Google Scholar 

  28. J.B. Keller, Backscattering from a Finite Cone, IRE Trans. on Antennas and Prop., AP-8, 175–182, Mar. 1960.

    Google Scholar 

  29. J.B. Keller and S.I. Rubinow, Shift of the Shadow Boundary and Scattering Cross Section of an Opaque Object, J. Appl. Phys., 32, 814–820, May 1961; Math. Revs. 22 (10B), 1797, 1961.

    Google Scholar 

  30. J.B. Keller, Backscattering from a Finite Cone-Comparison of Theory and Experiment, IRE Trans. Antennas and Prop., AP-9, 411–412, July 1961.

    Google Scholar 

  31. J.B. Keller and D.G. Magiros, Diffraction by a Semi-Infinite Screen with a Rounded End, Comm. Pure Appl. Math., 14, 475–471, 1961.

    MathSciNet  Google Scholar 

  32. J.B. Keller, Geometrical Theory of Diffraction, J. Opt. Soc. Am., 52,116–130, Feb. 1962; Math. Revs. 24, B 1115, 1962.

    Google Scholar 

  33. J.B. Keller A Survey of Short Wavelength Diffraction Theory, 1963 Geometrical Methods and Asymptotic Expansions in Wave Propagation, Symposium on Electromagnetic Theory and Antennas, Copenhagen, June 1962, Pergamon Press, New York, 1963, 3–9.

    Google Scholar 

  34. J.B. Keller and I. Kay, Asymptotic evaluation of the field at a caustic, J. Appl. Phys., 25, 876–883, 1954.

    Article  MathSciNet  MATH  Google Scholar 

  35. J.B. Keller and B.R. Levy, Scattering of Short Waves, Interdisciplinary Conference on Electromagnetic Scattering, August 1962, Milton Kerker, ed. Clarkson College of Technology, MacMillan, New York, 1963, 3–24.

    Google Scholar 

  36. J.B. Keller, Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems, Annals of Phys., 4, 180–188, June 1958; Math. Revs. 20, 934, 1959.

    Google Scholar 

  37. J.B. Keller and F.C. Karal, Jr., Effective Dielectric Constant, Permeability, and Conductivity of a Random Medium and the Velocity and Attenuation Coefficient of Coherent Waves, J. Math. Phys., 7, 661–670, 1966.

    Article  MathSciNet  Google Scholar 

  38. J.B. Keller, Effective Conductivity, Dielectric Constant and Permeability of a Dilute Suspension, Philips Res. Repts., 30, 83–90, 1975.

    Google Scholar 

  39. J.B. Keller, Effective Behavior of Heterogeneous Media, Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, ed., Plenum, New York, 1977, 631–644.

    Book  Google Scholar 

  40. J.B. Keller, Effective conductivities of reciprocal media, Random Media, ed. G. Papanicolaou, Springer-Verlag, New York, 1987, pp. 183–188, 1987.

    Google Scholar 

  41. J.B. Keller and J. Nevard, Homogenization of rough boundaries and interfaces, SIAM J. Appl. Math., 57, 1660–1686, 1997.

    MathSciNet  MATH  Google Scholar 

  42. J.B. Keller, Wave Propagation in Random Media, Proc. Symp. in Appl. Math., 13, Hydrodynamic Instability, Am. Math. Soc., 227–246, 1962.

    Google Scholar 

  43. J.B. Keller and F.C. Karal, Jr., Elastic, Electromagnetic, and Other Waves in a Random Medium, J. Math. Phys., 5, 537–547, 1964.

    Article  MathSciNet  MATH  Google Scholar 

  44. J.B. Keller, Stochastic Equations and Wave Propagation in Random Media, Proc. Symp. Appl. Math., 16,145–170, Am. Math. Soc., McGraw-Hill, New York, 1964.

    Google Scholar 

  45. J.B. Keller, The Velocity and Attenuation of Waves in a Random Medium, Electromagnetic Scattering, R.L. Rowell and R.S. Stein, eds., Proceedings of ICES II, Gordon and Breach Science Publishers, New York, 1967, 823–834.

    Google Scholar 

  46. J.B. Keller, A Survey of the Theory of Wave Propagation in Continuous Random Media,Symposium on Turbulence of Fluids and Plasmas, Polytechnic Institute of Brooklyn, N.Y., 1968, 131–142.

    Google Scholar 

  47. J.B. Keller, P. Chose, I. Kupiec, L.B. Felsen, and S. Rosenbaum, Reflection and Transmission by a Random Medium, Radio Science, 4, 1067–1077, 1969.

    Article  Google Scholar 

  48. J.B. Keller, J.A. Morrison, and G.C. Papanicolaou, Mean Power Transmission Through a Slab of Random Medium, Comm. Pure Appl. Math., 24, 473–489, 1971.

    MathSciNet  MATH  Google Scholar 

  49. J.B. Keller and G.C. Papanicolaou, Stochastic Differential Equations with Applications to Random Harmonic Oscillators and Wave Propagation in Random Media, SIAM J. Appl. Math., 21, 287–305, 1971.

    MathSciNet  Google Scholar 

  50. J.B. Keller and P.-L. Chow, Wave Propagation in a Random Lattice I, J. Math. Phys., 13, 1404–1411, 1972.

    Article  MATH  Google Scholar 

  51. J.B. Keller and J.G. Watson, Reflection, Scattering, and Absorption of Acoustic Waves by Rough Surfaces,. Acoust. Soc. Amer., 74, 1887–1894, 1983.

    Article  MATH  Google Scholar 

  52. J.B. Keller and J.G. Watson, Rough Surface Scattering via the Smoothing Method, J. Acoust. Soc. Am., 75, 1705–1708, 1984.

    Article  MATH  Google Scholar 

  53. J.B. Keller and W. Boyse, Short acoustic, electromagnetic and elastic waves in random media, J. Opt. Soc. Amer. A, 12, 380–389, 1995.

    Article  MathSciNet  Google Scholar 

  54. J.B. Keller, G. Papanicolaou, and L. Ryzhik, Transport equations for elastic and other waves in random media, Wave Motion, 24, 327–370, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  55. J.B. Keller, Inverse Problems, American Math. Monthly, 83, 107–118, 1976.

    Article  Google Scholar 

  56. J.B. Keller, The Inverse Scattering Problem in G.O. and the Design of Reflectors, IRE Trans. on Antennas and Prop., AP-7, 146–149, April 1959.

    Google Scholar 

  57. J.B. Keller, The Inverse Problem of Electromagnetic Scattering by a Metallic Object, Proc.First GISAT Symp., Mitre Corp., Bedford, MA., 1965, 13–21.

    Google Scholar 

  58. R.M. Lewis, “Asymptotic theory of wave-propagation”, Arch. Rational Mech. Anal. 20 (1965) 191–250.

    Article  MathSciNet  MATH  Google Scholar 

  59. P. Lax and R. Phillips, Scattering Theory, Academic Press, New York, 1967.

    MATH  Google Scholar 

  60. R.K. Luneberg, Mathematical Theory of Optics. Brown University, Advanced Instruction and Research in Mechanics, Providence, R. I., 1944.

    Google Scholar 

  61. B. Luong and F. Santosa, “Quantitative imaging of corrosion in plates by eddy current methods”, SIAM Journal on Applied Mathematics, 58 (1998), pages 1509–1531.

    Article  MathSciNet  MATH  Google Scholar 

  62. A. Melin and J. Sjöstrand, “Fourier integral operators with complex-valued phase functions”, in Fourier integral operators and partial differential equations, pp. 120–223, Lecture Notes in Math. vol. 459 ( Springer, Berlin, 1975 )

    Google Scholar 

  63. A. Melin and J. Sj¨ostrand. Sj¨ostrand, “Fourier integral operators with complex phase functions and parametrix for an interior boundary value problem”, Comm. Partial Differential Equations 1 (1976) 313–400.

    Article  MathSciNet  MATH  Google Scholar 

  64. C.J. Nolan and M. Cheney, Synthetic aperture inversion Inverse Problems 18 (2002) 221–36.

    Article  MathSciNet  MATH  Google Scholar 

  65. S.M. Nair and J.H. Rose, “Reconstruction of three-dimensional conductivity variations from eddy current (electromagnetic induction) data”, Inverse Problems 6 (1990) 1007–1030.

    Article  MATH  Google Scholar 

  66. K.E. Oughstun and G.C. Sherman, Electromagnetic Pulse Propagation in Causal Deielectrics, Springer, Berlin, 1994.

    Book  Google Scholar 

  67. R. Pike and P. Sabbatier, eds. Scattering and inverse scattering in pure and applied science. Academic Press, Inc., San Diego, CA, 2002.

    Google Scholar 

  68. M. Reed and B. Simon, Functional Analysis, Methods of Modern Mathematical Physics, volume 1, Academic Press.

    Google Scholar 

  69. M. Reed and B. Simon, Scattering Theory, Methods of Modern Mathematical Physics, volume 3, Academic Press.

    Google Scholar 

  70. E. Somersalo and M. Cheney, “Estimates for wave propagation in inhomogenous acoustic media”, J. Math. Anal. and Appls. 162 (1991) 410– 429.

    Google Scholar 

  71. K. Sobczyk, Stochastic Wave Propagation, Polish Scientific Publishers, Warsaw, 1985 (distributed by Elsevier).

    Google Scholar 

  72. M. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, NJ (1981).

    MATH  Google Scholar 

  73. L. Tsang, J.A. Kong, and R.T. Shin, Theory of Microwave Remote Sensing, Wiley, New York, 1985.

    Google Scholar 

  74. D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Schultz, “Composite media with simultaneously negative permeability and permittivity”, Phys. Rev. Lett. 84 (2000) 4184–4187.

    Article  Google Scholar 

  75. P. Stefanov and G. Uhlmann, “Inverse backscattering for the acoustic equation”, SIAM J. Math. Anal. 28 (1997) 1191–1204.

    MathSciNet  MATH  Google Scholar 

  76. V.G. Veselago, Sov. Phys. USPEKHI 10 (1968) 509.

    Article  Google Scholar 

  77. J. Van Bladel, Electromagnetic Fields, Hemisphere Publishing Corporation (Taylor & Francis group), 1985.

    Google Scholar 

  78. R. Weder, Spectral and Scattering Theory for Wave Propagation in Perturbed Stratified Media, Springer, New York, 1991.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA

    Margaret Cheney

Authors
  1. Margaret Cheney
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Aerospace Engineering, Technion — Israel Institute of Technology, Haifa, Israel

    Dan Givoli

  2. Department of Mathematics, University of Basel, Basel, Switzerland

    Marcus J. Grote

  3. Department of Mathematics, Stanford University, Stanford, California, USA

    George C. Papanicolaou

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Cheney, M. (2004). Some Problems in Electromagnetics. In: Givoli, D., Grote, M.J., Papanicolaou, G.C. (eds) A Celebration of Mathematical Modeling. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0427-4_2

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-0427-4_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-6526-1

  • Online ISBN: 978-94-017-0427-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation