Skip to main content
SpringerLink
Log in

Asymptotic Methods for Partial Differential Equations: The Reduced Wave Equation and Maxwell’s Equations

  • Chapter
Surveys in Applied Mathematics

Part of the book series: Surveys in Applied Mathematics ((SUAM))

Abstract

The short wavelength or high frequency asymptotic theory of the reduced wave equation and of Maxwell’s equations is presented. The theory is applied to representative problems involving reflection, transmission, and diffraction in both homogeneous and inhomogeneous media. It is a slightly revised version of a report1 written in 1964 by the authors. Despite its age, it still provides a convenient introduction to the formal asymptotic theory and to the geometrical theory of diffraction.

Deceased.

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

Access this chapter

Log in via an institution
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.

References

  1. J. Bazer and O. Fleischman, “Propagation of weak hydromagnetic discontinuities,” Physics of Fluids, 2: 366 (1956).

    Article  MathSciNet  Google Scholar 

  2. J. Bazer and J. Hurley, Geometrical Hydromagnetics, Research Report EM-176, N.Y.U. (1962).

    Google Scholar 

  3. R. N. Buchal and J. B. Keller, “Boundary layer problems in diffraction theory,” Comm. Pure Appl. Math., 13: 85 (1960).

    Article  MATH  MathSciNet  Google Scholar 

  4. C. R. Chester and J. B. Keller, “Asymptotic solution of systems of linear ordinary differential equations with discontinuous coefficients,” J. Math. and Mech., 10: 557, (1961).

    MATH  MathSciNet  Google Scholar 

  5. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II, Interscience Publishers (1962).

    Google Scholar 

  6. A. Erdélyi, Asymptotic Expansions, Dover Publications (1956).

    Google Scholar 

  7. F. G. Friedlander and J. B. Keller, “Asymptotic expansions of solutions of (∇2 + k 2)u = 0,” Comm. Pure Appl. Math., 8: 387 (1955).

    Article  MATH  MathSciNet  Google Scholar 

  8. F. G. Friedlander, Sound Pulses, Cambridge University Press (1958).

    Google Scholar 

  9. K. O. Friedrichs and J. B. Keller, “Geometrical acoustics II. Diffraction, reflection and refraction of a weak spherical or cylindrical shock at a plane interface,” J. Appl. Phys., 26: 961 (1955).

    Article  MATH  MathSciNet  Google Scholar 

  10. D. S. Jones and M. Kline, “Asymptotic expansion of multiple integrals and the method of stationary phase,” J. Math. and Phys., 37: 1 (1958).

    MATH  MathSciNet  Google Scholar 

  11. F. C. Karal and J. B. Keller, “Elastic wave propagation in homogeneous and inhomogeneous media,” J. Acoust. Soc. Amer., 31: 694 (1959).

    Article  MathSciNet  Google Scholar 

  12. I. Kay and J. B. Keller, “Asymptotic evaluation of the field at a caustic,” J. Appl. Phys., 25: 876 (1954).

    Article  MATH  MathSciNet  Google Scholar 

  13. I. Kay, “New equation for the asymptotic field amplitude in a two-dimensional inhomogeneous medium,” J. Acoust. Soc. Amer., 33: 1085 (1961).

    Article  MathSciNet  Google Scholar 

  14. J. B. Keller and H. B. Keller, “Determination of reflected and transmitted fields by geometrical optics,” J. Opt. Soc. Amer., 40: 48 (1950).

    Article  MathSciNet  Google Scholar 

  15. J. B. Keller and S. Prieser, Determination of Reflected and Transmitted Fields by Geometrical Optics II, Research Report, E.M. 20, N.Y.U. (1950).

    Google Scholar 

  16. J. B. Keller, “The geometrical theory of diffraction,” Proceedings of the Symposium on Microwave Optics, McGill University (1953).

    Google Scholar 

  17. J. B. Keller J. Opt. Soc. Amer., 52: 116, (1962).

    Article  MathSciNet  Google Scholar 

  18. J. B. Keller, “Geometrical acoustics I. The theory of weak shocks,” J. Appl. Phys., 25: 993 (1954).

    Google Scholar 

  19. J. B. Keller, R. M. Lewis, and B. D. Seckler, “Asymptotic solution of some diffraction problems,” Comm. Pure Appl. Math., 9: 2 (1956).

    MathSciNet  Google Scholar 

  20. J. B. Keller, Diffraction by a Convex Cylinder, I.R.E. Transactions, AP-4: 312 (1956).

    Google Scholar 

  21. J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys., 28: 426 (1957).

    Article  MATH  MathSciNet  Google Scholar 

  22. J. B. Keller, R. M. Lewis, and B. D. Seckler, “Diffraction by an aperture II,” J. Appl. Phys., 28: 570 (1957).

    Article  MATH  MathSciNet  Google Scholar 

  23. J. B. Keller, “Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems,” Annals of Physics, 4: 180(1958).

    Article  MATH  MathSciNet  Google Scholar 

  24. J. B. Keller and B. R. Levy, Decay Exponents and Diffraction Coefficients for Surface Waves on Surfaces of Nonconstant Curvature, I.R.E. Transactions, AP-7: Special Supplement (1959).

    Google Scholar 

  25. J. B. Keller, Backscattering from a Finite Cone, I.R.E. Transactions, AP-8: 175 (1960).

    Google Scholar 

  26. J. B. Keller and F. C. Karal, “Surface wave excitation and propagation,” J. Appl. Phys., 31: 1039, (1960).

    Article  MathSciNet  Google Scholar 

  27. J. B. Keller and S. I. Rubinow, “Asymptotic solutions of eigenvalue problems,” Annals of Physics, 9: 24 (1960).

    Article  MATH  Google Scholar 

  28. M. Kline, “An asymptotic solution of Maxwell’s equations,” Comm. Pure Appl. Math., 4: 225 (1951).

    Article  MATH  MathSciNet  Google Scholar 

  29. M. Kline, “Asymptotic solutions of linear hyperbolic partial differential equations,” J. Rat. Mech. Anal, 3: 315 (1954).

    MATH  MathSciNet  Google Scholar 

  30. B. R. Levy and J. B. Keller, “Diffraction by a smooth object,” Comm. Pure Appl Math., 12: 159 (1959).

    Article  MathSciNet  Google Scholar 

  31. B. R. Levy and J. B. Keller, “Diffraction by a spheroid,” Can. J. Phys., 38: 128 (1960).

    Article  MATH  MathSciNet  Google Scholar 

  32. B. R. Levy, “Diffraction by an elliptic cylinder,” J. Math. and Meck., 9: 147 (1960).

    MATH  Google Scholar 

  33. R. M. Lewis, “Asymptotic expansion of steady-state solutions of symmetric hyperbolic linear differential equations,” J. Math, and Mech., 7: 593 (1958).

    MATH  MathSciNet  Google Scholar 

  34. D. Ludwig, “Exact and asymptotic solutions of the Cauchy problem,” Comm. Pure Appl. Math., 13: 473 (1960).

    Article  MATH  MathSciNet  Google Scholar 

  35. R. K. Luneburg, The Mathematical Theory of Optics, Lecture Notes, Brown University, 1944; re-published by Univ. Calif. Press, Los Angeles 1964. Propagation of Electromagnetic Waves, Lecture Notes, New York University (1948).

    Google Scholar 

  36. C. Schensted, “Electromagnetic and acoustic scattering by a semi-infinite body of revolution” J. Appl. Phys., 26: 306 (1955).

    Article  MATH  MathSciNet  Google Scholar 

  37. B. D. Seckler and J. B. Keller, “Geometrical theory of diffraction in inhomogeneous media,” J. Acoust. Soc. Amer., 31: 192 (1959).

    Article  MathSciNet  Google Scholar 

  38. B. D. Seckler and J. B. Keller, “Asymptotic theory of diffraction in inhomogeneous media,” J. Acoust. Soc. Amer., 31: 206(1959).

    Article  MathSciNet  Google Scholar 

  39. J. A. Stratton, Electromagnetic Theory, McGraw-Hill, New York (1941).

    MATH  Google Scholar 

  40. G. B. Whitham, “Group velocity and energy propagation for three dimensional waves,” Comm. Pure Appl. Math., 14: 675 (1961).

    Article  MATH  MathSciNet  Google Scholar 

  41. M. Kline, Electromagnetic Theory and Geometrical Optics, N.Y.U., EM Div., Research Report EM-171 (1962).

    Google Scholar 

  42. E. Zauderer, Wave Propagation around a Smooth Object, N.Y.U., EM Div., Research Report (EL-6).

    Google Scholar 

  43. A. J. W. Sommerfeld, Optics: Academic Press, Inc., New York (1954).

    MATH  Google Scholar 

  44. P. D. Lax, “Asymptotic solutions of oscillatory initial value problems,” Duke Math. J., 24: 627, (1957).

    Article  MATH  MathSciNet  Google Scholar 

  45. J. J. Stoker, Differential Geometry, Wiley-Interscience, New York, (1969).

    MATH  Google Scholar 

  46. R. C. Hansen, ed., Geometric Theory of Diffraction, IEEE Press, New York (1981).

    Google Scholar 

  47. S. Cornbleet, “Geometrical optics reviewed: a new light on an old subject,” Proc. IEEE, 71: 471 (1986).

    Article  Google Scholar 

  48. J. M. Arnold, “Geometrical theories of wave propagation: a contemporary review,” Proc. IEEE (J), 133: 165 (1986).

    Google Scholar 

  49. Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media, Springer, New York, (1990).

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departments of Mathematics and Mechanical Engineering, Stanford University, Stanford, California, 94305, USA

    Joseph B. Keller

  2. Courant Institute of Mathematical Sciences, New York University, New York, New York, 10012, USA

    Robert M. Lewis

Authors
  1. Joseph B. Keller
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert M. Lewis
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Departments of Mathematics and Mechanical Engineering, Stanford University, Stanford, California, 94305, USA

    Joseph B. Keller

  2. Courant Institute of Mathematical Sciences, New York University, New York, New York, 10012, USA

    David W. McLaughlin

  3. Department of Mathematics, Stanford University, Stanford, California, 94305, USA

    George C. Papanicolaou

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer Science+Business Media New York

About this chapter

Cite this chapter

Keller, J.B., Lewis, R.M. (1995). Asymptotic Methods for Partial Differential Equations: The Reduced Wave Equation and Maxwell’s Equations. In: Keller, J.B., McLaughlin, D.W., Papanicolaou, G.C. (eds) Surveys in Applied Mathematics. Surveys in Applied Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-0436-2_1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4899-0436-2_1

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4899-0438-6

  • Online ISBN: 978-1-4899-0436-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation