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Fresnel-Type Transition Zones

  • ON THE 85TH ANNIVERSARY OF DMITRII SERGEEVICH LUKIN
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Abstract

A family of exact solutions of the two-dimensional Helmholtz equation is constructed, which are suitable for describing wavefields in transition zones arising in Fresnel-type diffraction. As examples, in addition to wedge diffraction, high-frequency asymptotics of the field in problems of diffraction on obstacles with non-smooth curvature are considered.

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Fig. 1.
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Notes

  1. Interpretation of mathematical expressions for wavefields implies the harmonic time-dependence \({\text{exp}}\left( { - i\omega t} \right)\), where \(t\) is time and \(\omega \) is angular frequency.

  2. Parabolic coordinates are useful in such problems, since in the region (1), (6) the phase difference \(kr - kx\) can be considered constant on parabolas \(x = C{{y}^{2}}\).

  3. The difference of phases of the waves is constant along these lines, \(r{\kern 1pt}^{'} - r = {\text{const}}\).

REFERENCES

  1. G. D. Malyuzhinets, Sov. Phys. Uspekhi 69, 749 (1960).

    MathSciNet  Google Scholar 

  2. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1969; Nauka, Moscow, 1973).

  3. V. A. Borovikov and B. E. Kinber, Geometric Theory of Diffraction (Svyaz’, Moscow, 1978; Institute of Electrical Engineers, London, 1994).

    MATH  Google Scholar 

  4. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953; Inostrannaya Literatura, Moscow, 1960), Vol. 2.

  5. N. V. Tsepelev, J. Sov. Math. 11 (3), 497 (1979).

    Article  Google Scholar 

  6. H. Bateman and A. Erdélyi, Higher Transcendental Functions (R. E. Krieger Pub. Co., Malabar, Fla., 1981; Nauka, Moscow, 1973), Vol. 2.

  7. A. Popov, A. Ladyzhensky (Brodskaya), and S. Khozioski, Russ. J. Math. Phys. 16 (2), 296 (2009).

    Article  MathSciNet  Google Scholar 

  8. P. Ya. Ufimtsev, Theory of Edge Diffraction in Electromagnetics (BINOM. Laboratoriya Znanii, Moscow, 2012; Tech Science Press, Encino, 2003).

  9. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Nauka, Moscow, 1967; Pergamon Press, Oxford, 1975).

  10. G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (Peter Peregrinus Ltd, 1986).

  11. L. Kaminetzky and J. B. Keller, SIAM J. Appl. Mat. 22 (1), 109 (1972).

    Article  Google Scholar 

  12. Z. M. Rogoff and A. P. Kiselev, Wave Motion 33 (2), 183 (2001).

    Article  Google Scholar 

  13. E. A. Zlobina and A. P. Kiselev, Wave Motion 96, 102571 (2020).

    Article  MathSciNet  Google Scholar 

  14. E. A. Zlobina and A. P. Kiselev, St. Petersburg Math. J. 33 (2), 207 (2022).

    Article  MathSciNet  Google Scholar 

  15. E. A. Zlobina, Zap. Nauchn. Sem. POMI 493, 169 (2020).

    MathSciNet  Google Scholar 

  16. E. A. Zlobina and A. P. Kiselev, J. Commun. Technol. Electron. 67 (2), 130 (2022).

    Article  Google Scholar 

  17. E. A. Zlobina, Zap. Nauchn. Sem. POMI 506, 43 (2021).

    MathSciNet  Google Scholar 

  18. A. V. Popov, Sov. Phys. Acoust. 19 (4), 375 (1974).

    Google Scholar 

  19. V. A. Fock, Electromagnetic diffraction and propagation problems (Sovetskoe Radio, Moscow, 1970; Pergamon Press, London, 1965).

  20. A. S. Kryukovskii, D. S. Lukin, E. A. Palkin, and D. S. Rastyagaev, J. Commun. Technol. Electron. 51, 1087 (2006).

    Article  Google Scholar 

  21. A. S. Kryukovskii, Uniform Asymptotic Theory of Edge and Corner Wave Catastrophes (Ros. Novyi Univ., Moscow, 2013) [in Russian].

  22. E. A. Zlobina, Mat. Zametki 114 (4), 347 (2023).

  23. V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Short-Wavelength Diffraction Theory (Nauka, Moscow, 1972; Alpha Science, Oxford, 2009) .

    Google Scholar 

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ACKNOWLEDGMENTS

The authors are indebted to the participants of the All-Russian Seminar “Mathematical Modeling of Wave Processes” (Russian New University, Moscow) and personally to D.S. Lukin, A.S. Kryukovsky, E.A. Palkin, V.T. Polyakov, and A.V. Popov for a helpful discussion of the results.

Funding

The work was supported by the Russian Science Foundation, grant no.: 22-21-00557.

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Authors and Affiliations

  1. St. Petersburg State University, 199034, St. Petersburg, Russia

    E. A. Zlobina & A. P. Kiselev

  2. St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences, 191023, St. Petersburg, Russia

    A. P. Kiselev

  3. Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, 199178, St. Petersburg, Russia

    A. P. Kiselev

Authors
  1. E. A. Zlobina
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  2. A. P. Kiselev
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Correspondence to E. A. Zlobina.

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Zlobina, E.A., Kiselev, A.P. Fresnel-Type Transition Zones. J. Commun. Technol. Electron. 68, 639–648 (2023). https://doi.org/10.1134/S1064226923060190

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