Abstract
We propose a new approach to construct heuristic relations describing solutions of diffraction problems. Those relations are based on physical principles and allow one to interpret mathematically strict solutions. Since the heuristic relations possess high performance and accuracy, they can also be used along with any strict approach or experimental results for a significant improvement of efficiency of solutions of practical problems related to applications of the diffraction theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V.M. Babich and V. S. Buldyrev, Asymptotics Methods in Problems of Diffraction of Short Waves [in Russian], Nauka, Moscow (1972).
S. E. Bankov and I.V. Levchenko, “Equivalent boundary-value conditions for a band high-frequency array on the boundary separating two media,” Radiotech. Electron., 33, No. 10, 2045 (1988).
V. A. Biryukov, M.V. Muratov, I.B. Petrov, A. V. Sannikov, and A.V. Favorskaya, “Application of the net-characteristic method to unstructured tetrahedral nets for solution of direct problems in seismography of fissured strata,” Comput. Math. Math. Phys., 55, No. 10, 1762–1772 (2015).
V. A. Borovikov, Diffraction on Polygons and Polyhedrons [in Russian], Nauka, Moscow (1966).
V. A. Fok, Problems of Diffraction and Propagation of Magnetic Waves [in Russian], Sov. Radio, Moscow (1970).
Ph. Frank and R. Mises, Differential and Integral Equations of Mathematical Physics [Russian translation], ONTI, Moscow–Leningrad (1937).
W. B. Gordon, “Far field approximations to the Kirchhoff–Helmholtz representations of scattered fields,” IEEE Trans. Antennas Propag., AP-23, 590–592 (1975).
G. A. Grinberg, Selected Topics of Mathematical Theory of Electric and Magnetic Phenomena [in Russian], AN SSSR, Moscow (1948).
G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, Peter Peregrinus Ltd, London (1976).
J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am., 52, No. 2, 116–130 (1962).
Kh. Khenl, A. Maue, and K. Vestpfal’, Diffraction Theory [Russian translation], Mir, Moscow (1964).
Yu. A. Kravtsov and N. Ya. Zhu, Theory of Diffraction: Heuristic Approaches, Alpha Science Int. Ltd, Oxford (2010).
V. B. Levyant, I. B. Petrov, and M. V. Muratov, “Numeric simulation of wave responses from a system (cluster) of subvertical macrofissures,” Technol. Seismogr., 1, 5–21 (2012).
M. A. Lyalinov and N.Ya. Zhu, Scattering of Waves by Wedges and Cones with Impedance Boundary Conditions, SciTech Publishing Inc, Raileigh (2012).
G.T. Markov and A. F. Chaplin, Launching of Electromagnetic Waves [in Russian], Energiya, Moscow–Leningrad (1967).
E. I. Nefedov, Diffraction of Electromagnetic Waves on Dielectric Structures [in Russian], Nauka, Moscow (1979).
B. Nobl, The Wiener–Hopf Method for Solution of Partial Differential Equations [Russian translation], Mir, Moscow (1982).
G. Pelosi, Ya. Rahmat-Samii, and J. L. Volakis, “High-frequency techniques in diffraction theory: 50 years of achievements in GTD, PTD, and related approaches,” IEEE Antennas Propag. Mag., 55, No. 3, 16 (2013).
A. Sommerfeld, “Zur analytischen Theorie der W¨armeleitung,” Math. Ann., 45, 263–277 (1894).
A. Sommerfeld, “Mathematische Theorie der Diffraction,” Math. Ann., 47, 317–374 (1896).
M. Taylor, Pseudodifferential Operators [Russian translation], Mir, Moscow (1985).
P.Ya. Ufimtsev, Method of Edge Waves in Physical Theory of Diffraction [in Russian], Sov. Radio, Moscow (1962).
P.Ya. Ufimtsev, Essentials of Physical Theory of Diffraction [in Russian], BINOM. Laboratoriya znaniy, Moscow (2009).
P.Ya. Ufimtsev, Theory of diffractional edge waves in electrodynamics. Introduction to physical theory of diffraction [in Russian], BINOM. Laboratoriya znaniy, Moscow (2012).
L. A. Vaynshteyn, Electromagnetic Waves [in Russian], Radio i svyaz’, Moscow (1988).
M. V. Vesnik, “Using two-dimensional solutions in three-dimensional problems,” Radiotech. Electron., 38, No. 8, 1416–1423 (1993).
M.V. Vesnik, “Elimination of infinities in diffraction coefficients of physical optics current’s components for a shadow contour of a scatterer,” Proc. of the International Symposium on Electromagnetic Theory, St. Petersburg, 407–409 (1995).
M.V. Vesnik, “Analytic solution of the boundary-value problem for the Helmholtz equation,” Radiotech. Electron., 45, No. 1, 66–76 (2000).
M. V. Vesnik, “The analytical solution for the electromagnetic diffraction on 2-D perfectly conducting scatterers of arbitrary shape,” IEEE Trans. Antennas and Propagation, AP-49, No. 12, 1638–1644 (2001).
M.V. Vesnik, “Analytic solution of boundary-value problems in the diffraction theory by the method of generalized eikonal,” Radiotech. Electron., 48, No. 9, 1078–1084 (2003).
M.V. Vesnik, “Analytic solution of boundary-value problems for the wave equation with variable wave number by the method of generalized eikonal,” Nonlinear World, 1, No. 1-2, 59–63 (2003).
M.V. Vesnik, “Computing diffraction coefficients for two-dimensional semi-infinite perfectly conductive scatterer by means of the generalized eikonal method,” Electromag. Waves Electron. Syst., 9, No. 11, 23–29 (2004).
M. V. Vesnik, “Method of generalized eikonal as a new approach to diffraction process description,” Abstr. of the International Seminar “Days on Diffraction,” St. Petersburg, 79-80 (2006).
M. V. Vesnik, “Analytic solution of the diffraction problem for an electromagnetic wave on twodimensional perfectly conductive half-plate by the method of generalized eikonal,” Radiotech. Electron., 53, No. 2, 144–156 (2008).
M. V. Vesnik, “On construction of the refined heuristic solution in the diffraction problem for a flat angular sector,” Radiotech. Electron., 56, No. 5, 573–586 (2011).
M. V. Vesnik, “Analytic solution of two-dimensional diffraction problem for an electromagnetic wave on a truncated wedge,” Radiotech. Electron., 57, No. 10, 1053–1065 (2012).
M. V. Vesnik, “Analytical heuristic solution for the problem of elastic wave diffraction by a polygonal flat 3D scatterer,” Abstr. of the International Conference “Days on Diffraction,” St. Petersburg, 89 (2013).
M. V. Vesnik, “Efficiency of different heuristic approaches to calculation of electromagnetic diffraction by polyhedrons and other scatterers,” Radio Science, 49, No. 10, 945–953 (2014).
M. V. Vesnik, “Construction of heuristic diffraction coefficients in analytic solutions of scattering problems for wave fields of various physical nature on flat polygonal plates with compound boundary-value conditions,” Radiotech. Electron., 59, No. 6, 543–551 (2014).
M. V. Vesnik, “Refinement of physical optics approximations in the diffraction problems on threedimensional objects,” Proc. of the Second All-Russian Microwave Conference, Moscow, 443–448 (2014).
M. V. Vesnik, The method of the Generalized Eikonal. New Approaches in the Diffraction Theory, Walter de Gruyter GmbH, Berlin–Boston (2015).
M. V. Vesnik, “Deterministic theory of radio-waves propagation in urban environment,” Proc. of the International Scientific Conference “Radiation and Scattering of Electromagnetic Waves,” Rostov-on-Don, 378–382 (2015).
M. V. Vesnik, “Heuristic expression for the diffraction coefficient of a semitransparent half-plane,” Proc. of the Third All-Russian Microwave Conference, Moscow, 281–285 (2015).
M. Vesnik and Yu. A. Kravtsov, “Diffraction by bodies with wedges: method of generalized eikonal (MGE),” Theory of Diffraction: Heuristic Approaches, Alpha Science Int. Ltd., Oxford, 243–246 (2010).
M. V. Vesnik and P.Y. Ufimtsev, “An asymptotic feature of corner waves scattered by polygonal plates,” Electromagnetics, 12, No. 3-4, 265–272 (1992).
N. N. Voytovich, B. Z. Katsenelenbaum, E.N. Korshunova, L. I. Pangonis, M. L. Pereyaslavets, A.N. Sivov, and A.D. Shatrov, Electrodynamics of Antennas with Semitransparent Surfaces: Methods of Constructive Synthesis [in Russian], Nauka, Moscow (1989).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 62, Differential and Functional Differential Equations, 2016.
Rights and permissions
About this article
Cite this article
Vesnik, M.V. Physical Interpretation of Strict Solutions of Diffraction Problems by Heuristic Relations. J Math Sci 239, 751–770 (2019). https://doi.org/10.1007/s10958-019-04324-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-019-04324-8