Skip to main content
SpringerLink
Log in

Mathematics of Wavefields

  • Chapter
  • First Online:
Applied Mathematical Analysis: Theory, Methods, and Applications

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 177))

  • 728 Accesses

Abstract

Wave propagation and scattering occupy a large part of physical, mathematical and engineering sciences. The purpose of this chapter is to present the basic mathematical theory of certain aspects of wavefields, that is, waves and fields, as they occur under various physical situations. These are considered in both scalar or acoustical and vector or electromagnetic media, that is, in the context of Helmholtz’s and Maxwell’s equations. The major emphasis is on the mathematical aspects of Green’s functions, tensors and operators. In particular, the singularities involved are discussed at length. The basic mathematical concepts, tools and techniques, necessary for the presentation, are summarized in the beginning. It is shown that mathematical analyses reveal many subtleties hidden in the wavefields that would otherwise have gone unnoticed. Detailed derivations of the equations are provided whenever possible and necessary. Also, if there are alternative ways of solving a problem, these have been presented. Finally, copious remarks and notes are included for better explaining certain points.

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 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 249.99
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

References

  1. Chadan, K., Sabatier, P.C.: Inverse Problems in Quantum Scattering, Springer, New York (1989). See also Ghosh Roy, D.N.: Method of Inverse Problems in Physics and Imaging Sciences. CRC Press, Boca Raton, Fl (1990)

    Google Scholar 

  2. Roman, P.: An Advanced Quantum Theory, Pergamon Press, New York

    Google Scholar 

  3. Buchanan, J.L., Gilbert, R.P., Wirgin, A., Xu, Y.S.: Marine Acoustics: Direct and Inverse Problems. SIAM, Philadelphia (2004)

    Book  Google Scholar 

  4. Hansen, T.B., Yaghjian, A.D.: Plane-Wave Theory of Time-Domain Fields. IEEE Press (1999)

    Google Scholar 

  5. Chew, W.C.: Waves and Fields in Inhomogeneous Media. IEEE Press, New York (1995)

    Book  Google Scholar 

  6. Jones, D.S.: Acoustic and Electromagnetic Waves. Oxford University Press, New York (1986)

    Google Scholar 

  7. Colton, D., Kress, R.: Inverse Acousti and Electromagnetic Scattering Theory. Springer, Berlin (1992)

    Book  Google Scholar 

  8. Stakgold, I.: Green’s Functions and Boundary Value Problems, 2nd edn. Wiley, New York (1999)

    MATH  Google Scholar 

  9. Taylor, A.E.: General Theory of Functions and Integration. Dover, New York (1985)

    Google Scholar 

  10. Zorich, V.A.: Mathematical Analysis I and II. Springer, Berlin (2004)

    MATH  Google Scholar 

  11. Zuily, C.: Problems in Distributions and Partial Differential Equations. North-Holland, Amsterdam (1988)

    MATH  Google Scholar 

  12. van Kranendonk, J., Sipe, J.E.: Progress in Optics. In: Wolf, E. (ed.) vol. XV, 245, North-Holland, Amsterdam

    Google Scholar 

  13. Hanson, G.W., Yakovlev, A.B.: Operator Theory for Electromagnetics. Springer, New York (2002)

    Book  Google Scholar 

  14. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover (1973)

    Google Scholar 

  15. Fleming, W.H.: Functions of Several Variables. Addison-Wesley, Reading, MA (1965). Also Kaplan, W.: Advanced Calculus. Addison-Wesley, Cambridge, MA (1952)

    Google Scholar 

  16. Friedman, B.: Principles and Techniques of Applied Mathematics. John Wiley, New York (1956)

    MATH  Google Scholar 

  17. Flanders, H.: Differentiation under the integral sign, AMS 80, 617. See also Silberstien, M.: Applications of a generalized Leibnitz rule for calculating electromagnetic fields within continuous source regions. Radio Sci. 26, 183 (1991)

    Article  Google Scholar 

  18. Eringen, A.C.: Mechanics of Continua. Wiley, New York (1967)

    MATH  Google Scholar 

  19. Bonnet, M.: Boundary Integral Equation Methods for Solids and Fluids. John Wiley, Chichester (1995)

    Google Scholar 

  20. Haug, E.J., Choi, K.K., Komkov, V.: Design Sensitivity Analysis of Structured Systems. Academic Press, Orlando (1986)

    MATH  Google Scholar 

  21. Tai, C.T.: Generalized vector and dyadic analysis. In: Ghosh Roy, D.N., Couchman, L., Shirron, J. (eds.) Inverse Obstacle Transmission Problem in Acoustics, Inverse Problems, 1998, vol. 14, pp. 903. IEEE Press, New York (1997)

    Google Scholar 

  22. Dorn, O., Miller, E.L., Rappaport, C.M.: A shape reconstruction method for electromagnetic tomography using adjoint fields and level set. Inverse Prob. 16, 1119–1156 (2000)

    Article  MathSciNet  Google Scholar 

  23. Norton, S.J.: Iterative inverse scattering algorithms: methods for computing frechet derivative. JASA 106, 2653 (1999)

    Article  Google Scholar 

  24. Ghosh Roy, D.N., Mudalier, S.: Domain derivatives in dielectric rough surface scattering. IEEE Trans. AP. Also Ghosh Roy, D.N., Couchman, L., Warner, J.: Scattering and inverse scattering via shape deformation. Inverse Probl. 13, 585 (1997)

    Google Scholar 

  25. Gel’fund, I.M., Shilov, G.E.: Generalized Functions, vol. 1. Academic Press, New York (1964)

    Google Scholar 

  26. Kanwal, R.P.: Generalized Functions. 3rd ed., Birkh\(\ddot{a}\)ser, Boston (2004)

    Book  Google Scholar 

  27. Estrada, R., Kanwal, R.P.: A Distributional Approach to Asymptotics, 2nd edn., Birkh\(\ddot{a}\)ser, Boston (2002)

    Book  Google Scholar 

  28. Sch\(\ddot{u}\)ker, T.: Distributions: fourier transforms and some of their applications. World Scientif. Singapore (1991)

    Google Scholar 

  29. Renardy, M., Rogers, R.C.: An Introduction to Partial Differential Equations. Springer, New York (1993)

    MATH  Google Scholar 

  30. Idziaszek, D., Calero, T.: Pseudopotential method for higher partial wave scattering. Phys. Rev. Lett. 96, 013201 (2006)

    Article  Google Scholar 

  31. Dacol, D.K., Ghosh Roy, D.N.: Wave scattering in waveguides. J. Math. Phys. 44, 2133 (2003)

    Article  MathSciNet  Google Scholar 

  32. Stampfer, F., Wagner, P.: J. Math. Anal. Appl. 342, 202 (2008)

    Article  MathSciNet  Google Scholar 

  33. Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. Springer, Berlin (1996)

    Book  Google Scholar 

  34. Morse, P.M., Ingaard, K.U.: Theoretical Acoustics. Princeton University Press, Princeton, NJ (1968)

    Google Scholar 

  35. Johnson, S.A., Stenger, F., Wilcox, C., Ball, J., Berggren, M.J.: Wave equations and inverse solutions for soft tissue. Acoustic. Imag. 11, 409 (1981)

    Article  Google Scholar 

  36. Reid, W.T.: Ordinary Differential Equations. Wiley, New York (1971)

    MATH  Google Scholar 

  37. Stratton, J.: Electromagnetic Theory. McGraw-Hill, New York (1941)

    MATH  Google Scholar 

  38. Jackson, J.D.: Classical Electrodynamics. John Wiley, New York (1998)

    MATH  Google Scholar 

  39. Keller, O.: Attached and radiated electromagnetic fields of an electric point dipole. JOSA B 16, 835. See also Keller, O., Wolf, E. (eds.) Progress in Optics XXXVII. North-Holland, Amsterdam

    Google Scholar 

  40. Nieto-Vesperinas, M.: Scattering and Diffraction in Physical Optics, Wiley, New York (1997). See also Set\(\ddot{a}\)l\(\ddot{a}\), T., Kaivola, M., Friberg, A.T.: Decomposition of the point-dipole field into homogeneous and evanescent parts. Phys. Rev. E 59(1), 1200 (1990). See also Mandel, L., Wolf, E.: Optical Coherence and Quantum Optics, Cambridge University Press, New York (1995). Roseau, M.: Asymptotic Wave Theory. North-Holland, Amsterdam (1976)

    Google Scholar 

  41. Mikki, S., Antar, Y.: New Foundations For Applied Electromagnetics. Artech House, Boston (2016)

    Google Scholar 

  42. Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, D., Photons and Atoms, Introduction to Quantum Electrodynamics, Wiley, New York. See also Brill O.L., Goodman, B.: Causality in the Coulomb gauge. Am. J. Phys. 35, 832 (1967)

    Google Scholar 

  43. Pierce, A.D.: Acoustics. McGraw-Hill, New York (1981)

    Google Scholar 

  44. Williams, E.: Fourier Acoustics. Academic Press, San Diego (1999)

    Google Scholar 

  45. Hecht, E., Sejac, A.: Optics, 2nd edn. Addison-Wesley, Reading, MA (1987)

    Google Scholar 

  46. Bose, J.C.: On the influence of the thickness of the air-space on total reflection of electric radiation. Proc. Roy. Soc. London 62, 300 (1894)

    Google Scholar 

  47. de Fornel, F.: Evanescent Waves. Springer, New York (2001)

    Book  Google Scholar 

  48. Wolf, E., Foley, J.T.: Opt. Lett. 23, 16 (1998)

    Article  Google Scholar 

  49. Yaghjian, A.D.: Electric dyadic Green’s functions in the source region. IEEE Proc. 68, 248 (1980). See also Yaghjian, A.D.: Maxwellian and cavity electromagnetic fields within sources. Am J. Phys. 53, 859 (1985)

    Article  Google Scholar 

  50. Frahm, C.P.: Some novel delta-function identities. Am. J. Phys. 51, 826 (1983)

    Article  Google Scholar 

  51. Farassat, F.: Introduction to Generalized Functions With Applications in Aerodynamics and Aeroacoustics, p. 3428. NASA Tech, Paper (1994)

    Google Scholar 

  52. Bohren, C.F., Huffmann, D.R.: Absorption and Scattering of Light by Small Particles. John Wiley, New York (1983)

    Google Scholar 

  53. Hnizdo, V.: Generalized second-order derivatives of 1/r. Eur. J. Phys. 32, 287 (2011)

    Article  Google Scholar 

  54. Weigelhofer, W.: Delta-function identities and electromagnetic field singularities. Am. J. Phys. 57, 455 (1989)

    Article  Google Scholar 

  55. Lee, S.W.: Singularity in Green’s function and its numerical evaluation. IEEE Trans. Micro. Theor. Tech. 36,1289 (1980). See also Van Bladel, J.: Singular Electromagnetic Fields and Sources, Clarendon Press, Oxford (1991). Also Azvestas, J. S. et al.: Comments on Singularity in Green’s function and its numerical evaluation. IEEE Trans. Ant. Prop. AP 31, 174 (1983)

    Google Scholar 

  56. Moroz, A.: Depolarization field of spheroidal particles. Opt. Soc. Am. B 26, 517 (2009)

    Article  MathSciNet  Google Scholar 

  57. Silberstien, M.: Applications of a generalized Leibnits rule for calculating electromagnetic elds within continuous source regions. Radio Sci. 26, 183 (1991)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Utah Center for Advanced Imaging Research, 741 Arapeen Drive, Salt Lake City, UT, 84111, USA

    D. N. Ghosh Roy

  2. Riverside Research Institute, Dayton Research Center, 2640 Hibiscus way, Dayton, OH, 45431, USA

    D. N. Ghosh Roy

Authors
  1. D. N. Ghosh Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. N. Ghosh Roy .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Gauhati University, Guwahati, India

    Hemen Dutta

  2. Computational Intelligence Laboratory, Department of Electrical & Computer Engineering, University of Manitoba, Winnipeg, MB, Canada

    James F. Peters

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Ghosh Roy, D.N. (2020). Mathematics of Wavefields. In: Dutta, H., Peters, J. (eds) Applied Mathematical Analysis: Theory, Methods, and Applications. Studies in Systems, Decision and Control, vol 177. Springer, Cham. https://doi.org/10.1007/978-3-319-99918-0_7

Download citation

Publish with us

Policies and ethics

Search

Navigation