Electromagnetic Waves in an Inhomogeneous Medium

  • David Colton
  • Rainer Kress
Part of the Applied Mathematical Sciences book series (AMS, volume 93)


In the previous chapter, we considered the direct scattering problem for acoustic waves in an inhomogeneous medium. We now consider the case of electromagnetic waves. However, our aim is not to simply prove the electromagnetic analogue of each theorem in Chap.  8 but rather to select the basic ideas of the previous chapter, extend them when possible to the electromagnetic case, and then consider some themes that were not considered in Chap.  8, but ones that are particularly relevant to the case of electromagnetic waves. In particular, we shall consider two simple problems, one in which the electromagnetic field has no discontinuities across the boundary of the medium and the second where the medium is an imperfect conductor such that the electromagnetic field does not penetrate deeply into the body. This last problem is an approximation to the more complicated transmission problem for a piecewise constant medium and leads to what is called the exterior impedance problem for electromagnetic waves.


  1. 12.
    Angell, T.S., Colton, D., and Kress, R.: Far field patterns and inverse scattering problems for imperfectly conducting obstacles. Math. Proc. Camb. Phil. Soc. 106, 553–569 (1989).MathSciNetCrossRefGoogle Scholar
  2. 59.
    Cakoni, F., Colton, D., and Monk, P.: The Linear Sampling Method in Inverse Electromagnetic Scattering. SIAM Publications, Philadelphia, 2011.CrossRefGoogle Scholar
  3. 86.
    Colton, D.: Partial Differential Equations. Dover Publications, New York 2004.zbMATHGoogle Scholar
  4. 96.
    Colton, D., and Kress, R.: The impedance boundary value problem for the time harmonic Maxwell equations. Math. Meth. in the Appl. Sci. 3, 475–487 (1981).MathSciNetCrossRefGoogle Scholar
  5. 99.
    Colton, D., and Kress, R.: Time harmonic electromagnetic waves in an inhomogeneous medium. Proc. Royal Soc. Edinburgh 116 A, 279–293 (1990).Google Scholar
  6. 104.
    Colton, D., and Kress, R.: Integral Equation Methods in Scattering Theory. SIAM Publications, Philadelphia 2013.CrossRefGoogle Scholar
  7. 122.
    Colton, D., and L. Päivärinta, L.: Far-field patterns for electromagnetic waves in an inhomogeneous medium. SIAM J. Math. Anal. 21, 1537–1549 (1990).Google Scholar
  8. 170.
    Hähner, P.: Abbildungseigenschaften der Randwertoperatoren bei Randwertaufgaben für die Maxwellschen Gleichungen und die vektorielle Helmholtzgleichung in Hölder- und L 2-Räumen mit einer Anwendung auf vollständige Flächenfeldsysteme. Diplomarbeit, Göttingen 1987.Google Scholar
  9. 223.
    Jones, D.S.: Methods in Electromagnetic Wave Propagation. Clarendon Press, Oxford 1979.Google Scholar
  10. 241.
    Kirsch, A.: On the existence of transmission eigenvalues. Inverse Problems and Imaging 3, 155–172 (2009).MathSciNetCrossRefGoogle Scholar
  11. 293.
    Lebedev, N.N.: Special Functions and Their Applications. Prentice-Hall, Englewood Cliffs 1965.CrossRefGoogle Scholar
  12. 332.
    Müller, C.: Foundations of the Mathematical Theory of Electromagnetic Waves. Springer, Berlin 1969.CrossRefGoogle Scholar
  13. 343.
    Olver, F.W.J: Asymptotics and Special Functions. Academic Press, New York 1974.zbMATHGoogle Scholar
  14. 366.
    Protter, M.H., and Weinberger, H.F.: Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs 1967.zbMATHGoogle Scholar
  15. 375.
    Ringrose, J.R.: Compact Non–Self Adjoint Operators. Van Nostrand Reinhold, London 1971.zbMATHGoogle Scholar
  16. 411.
    van Bladel, J.: Electromagnetic Fields. Hemisphere Publishing Company, Washington 1985.Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • David Colton
    • 1
  • Rainer Kress
    • 2
  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  2. 2.Institut für Numerische und Angewandte MathematikGeorg-August-Universität GöttingenGöttingenGermany

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