Vol. 82
Latest Volume
All Volumes
PIERB 106 [2024] PIERB 105 [2024] PIERB 104 [2024] PIERB 103 [2023] PIERB 102 [2023] PIERB 101 [2023] PIERB 100 [2023] PIERB 99 [2023] PIERB 98 [2023] PIERB 97 [2022] PIERB 96 [2022] PIERB 95 [2022] PIERB 94 [2021] PIERB 93 [2021] PIERB 92 [2021] PIERB 91 [2021] PIERB 90 [2021] PIERB 89 [2020] PIERB 88 [2020] PIERB 87 [2020] PIERB 86 [2020] PIERB 85 [2019] PIERB 84 [2019] PIERB 83 [2019] PIERB 82 [2018] PIERB 81 [2018] PIERB 80 [2018] PIERB 79 [2017] PIERB 78 [2017] PIERB 77 [2017] PIERB 76 [2017] PIERB 75 [2017] PIERB 74 [2017] PIERB 73 [2017] PIERB 72 [2017] PIERB 71 [2016] PIERB 70 [2016] PIERB 69 [2016] PIERB 68 [2016] PIERB 67 [2016] PIERB 66 [2016] PIERB 65 [2016] PIERB 64 [2015] PIERB 63 [2015] PIERB 62 [2015] PIERB 61 [2014] PIERB 60 [2014] PIERB 59 [2014] PIERB 58 [2014] PIERB 57 [2014] PIERB 56 [2013] PIERB 55 [2013] PIERB 54 [2013] PIERB 53 [2013] PIERB 52 [2013] PIERB 51 [2013] PIERB 50 [2013] PIERB 49 [2013] PIERB 48 [2013] PIERB 47 [2013] PIERB 46 [2013] PIERB 45 [2012] PIERB 44 [2012] PIERB 43 [2012] PIERB 42 [2012] PIERB 41 [2012] PIERB 40 [2012] PIERB 39 [2012] PIERB 38 [2012] PIERB 37 [2012] PIERB 36 [2012] PIERB 35 [2011] PIERB 34 [2011] PIERB 33 [2011] PIERB 32 [2011] PIERB 31 [2011] PIERB 30 [2011] PIERB 29 [2011] PIERB 28 [2011] PIERB 27 [2011] PIERB 26 [2010] PIERB 25 [2010] PIERB 24 [2010] PIERB 23 [2010] PIERB 22 [2010] PIERB 21 [2010] PIERB 20 [2010] PIERB 19 [2010] PIERB 18 [2009] PIERB 17 [2009] PIERB 16 [2009] PIERB 15 [2009] PIERB 14 [2009] PIERB 13 [2009] PIERB 12 [2009] PIERB 11 [2009] PIERB 10 [2008] PIERB 9 [2008] PIERB 8 [2008] PIERB 7 [2008] PIERB 6 [2008] PIERB 5 [2008] PIERB 4 [2008] PIERB 3 [2008] PIERB 2 [2008] PIERB 1 [2008]
2018-09-09
Wedge Diffraction as an Instance of Radiative Shielding
By
Progress In Electromagnetics Research B, Vol. 82, 1-16, 2018
Abstract
The celebrated Sommerfeld wedge diffraction solution is reexamined from a null interior field perspective. Exact surface currents provided by that solution, when considered as disembodied half-plane laminae radiating into an ambient, uniform space both inside and outside the wedge proper, do succeed in reconstituting both a specular, mirror field above the exposed face, and a shielding plane-wave field of a sign opposite to that of the incoming excitation which, under superposition, creates both the classical, geometric-optics shadow, and a strictly null interior field at the dominant, plane-wave level. Both mirror and shadow radiated fields are controlled by the residue at just one simple pole encountered during a spectral radiative field assembly, fixed in place by incidence direction φ0 as measured from the exposed face. The radiated fields further provide diffractive contributions drawn from two saddle points that track observation angle φ: Even these, more or less asymptotic contributions, are found to cancel exactly within the wedge interior, while, on the outside, they recover in its every detail the canonical structure lying at the base of GTD (geometric theory of diffraction). It is earnestly hoped that this revised scattering viewpoint, while leaving intact all details of the existing solution, will impart to it a fresh, physically robust meaning. Moreover, inasmuch as this viewpoint confirms, admittedly in an extreme limit, the concept of field self-consistency (known in rather more picturesque language as Ewald-Oseen extinction), perhaps such explicit vindication may yet encourage efforts to seek exact solutions to scattering/diffraction by electromagnetically permeable (i.e., dielectric) wedges, efforts that harness integral equations with polarization/ohmic currents distributed throughout wedge volumes as sources radiating into an ambient, uniform reference medium.
Citation
Jan Alexander Grzesik, "Wedge Diffraction as an Instance of Radiative Shielding," Progress In Electromagnetics Research B, Vol. 82, 1-16, 2018.
doi:10.2528/PIERB18011202
References

1. Grzesik, J., "Field matching through volume suppression," IEE Proceedings, Part H (Antennas and Optics), Vol. 127, No. 1, 20-26, Feb. 1980.

2. Sommerfeld, A., "Mathematische theorie der diffraction," Mathematische Annalen, Vol. 16, 317-374, 1896.
doi:10.1007/BF01447273

3. MacDonald, H. M., "Appendix D," Electric Waves, 186-198, Cambridge University Press, Cambridge, UK, 1902.

4. Sommerfeld, A., P. Frank, and R. von Mises, eds., "Theorie der Beugung," Die Differential- und Integralgleichungen der Mechanik und Physik, Zweiter physikalische teil, Chap. 20, 808{830, Friedrich Vieweg & Sohn, Braunchshweig, Deutschland, 1935 (Mary S. Rosenberg, WWII publisher, New York, NY, USA, 1943).

5. Sommerfeld, A., Optics, Lectures on Theoretical Physics, Vol. IV, 249-272, Academic Press, New York, NY, USA, Translated from the German by O. Laporte and P. A. Moldauer, 1954.

6. Pauli, W., "On asymptotic series for functions in the theory of diffraction of light," Phys. Rev., Vol. 54, 924-931, Dec. 1, 1938.

7. Landau, L. D. and E. M. Lifshitz, "Electrodynamics of continuous media," Course of Theoretical Physics, Vol. 8, 304-312, Addison-Wesley Publishing Company, Reading, Mass., USA, Translated from the Russian by J. B. Sykes and J. S. Bell, 1960.

8. Baker, B. B. and E. T. Copson, "`Sommerfeld's theory of diffraction,' and `Diffraction by a plane screen'," The Mathematical Theory of Huygens Principle, Chap. 4, 124-152, and Chap. 5, 153-189, Clarendon Press, Oxford, UK, 1949.

9. Stratton, J. A., Electromagnetic Theory, 364-369, McGraw-Hill Book Company, New York, NY, USA, 1941.

10. Mittra, R. and S. W. Lee, Analytical Techniques in the Theory of Guided Waves, (preferred Riemann sheet) 20{23, (Wiener-Hopf) 82ff, (Jones method) 97ff, The MacMillan Company, New York, NY, USA, 1971.

11. Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd Ed., (formulae (2) and (3)) 405, Cambridge University Press, Cambridge, UK, 1966.

12. Clemmow, P. C., "Some extensions of the method of steepest descents," QJMAM, Vol. 3, No. 2, 924-931, 1950.

13. Clemmow, P. C., The Plane Wave Spectrum Representation of Electromagnetic Fields, 43-58, (with particular attention to 56{58), IEEE, Inc., New York, NY, USA, 1996.
doi:10.1109/9780470546598

14. Ciarkowski, A., J. Boersma, and R. Mittra, "Plane-wave diffraction by a wedge --- A spectral domain approach," IEEE Trans. Antennas Propag., Vol. 32, No. 1, 20-29, Jan. 1, 1984.
doi:10.1109/TAP.1984.1143190

15. Haciveliogliu, F., L. Sevgi, and P. Y. U mtsev, "Electromagnetic wave scattering from a wedge with perfectly re ecting boundaries: Analysis of asymptotic techniques," IEEE Antennas Propag. Mag., Vol. 53, No. 3, 232-253, Jun. 2011.
doi:10.1109/MAP.2011.6028472

16. U mtsev, P. Y., Fundamentals of the Physical Theory of Diffraction, John Wiley and Sons, Inc., Hoboken, NJ, USA, 2007.
doi:10.1002/0470109017

17. Bateman, H., "Diffraction problems," Partial Differential Equations of Mathematical Physics, Chap. 11, 476{490, Dover Publications, New York, NY, USA, 1959.

18. Rubinowicz, A., "Beugung an einer Halbebene," Die Beugungswelle in der Kirchhoffschen Theorie der Beugung, Chap. 4, 125-149, Panstwowe Wydawnictwo Naukowe, Warsaw, Poland, 1957.

19. Lamb, H., "On Sommerfeld's diffraction problem, and on re ection by a parabolic mirror," Proc. London Math. Soc., Vol. 4, 190ff, 1906.

20. Lamb, S. H., Hydrodynamics, 6th Ed., Sec. 308, 538-541, Dover Publications, New York, NY, USA, 1945.

21. MacDonald, K. T., "Sommerfeld's diffraction problem," Physics Examples, 1-18, Dept. of Physics, Princeton University, Princeton, NJ, USA, Jun. 25, 2014.

22. Noble, B., Methods Based on the Wiener-Hopf Technique, Pergamon Press, New York, NY, USA, 1958.

23. Schwinger, J., L. L. De Raad, Jr., K. A. Milton, and W.-Y. Tsai, "Exact solution for current," Classical Electrodynamics, Sec. 48, 512{516, Perseus Books, Reading, Mass., USA, 1998.

24. Born, M. and E. Wolf, "Rigorous diffraction theory," Principles of Optics, 7th (expanded) Edition, Chap. 11 (contributed by P. C. Clemmow), 633{672, Cambridge University Press, Cambridge, UK, 2003.

25. Carrier, G. F., M. Krook, and C. E. Pearson, "Dual integral equations," Functions of a Complex Variable, Sec. 8-5, 399-404, McGraw-Hill Book Company, New York, NY, USA, 1966.

26. Friedlander, F. G., "The diffraction of a pulse by a wedge," Sound Pulses, Chap. 5, 108-146, Cambridge University Press, Cambridge, UK, 1958.

27. Grinberg, G. A., Selected Problems in the Mathematical Theory of Electric and Magnetic Phenomena, Chap. 22, USSR Academy of Sciences, Moscow, 1948.

28. Sneddon, I. N., "The Kontorovich-Lebedev transform," The Use of Integral Transforms, Chap. 6, 353-368, McGraw-Hill Book Company, New York, NY, USA, 1972.

29. Nikoskinen, K. I. and I. V. Lindell, "Image solution for poisson's equation in wedge geometry," IEEE Trans. Antennas Propag., Vol. 43, No. 2, 179-187, Feb. 1995.
doi:10.1109/8.366380

30. Scharstein, R. W., "Green's function for the harmonic potential of the three-dimensional wedge transmission problem," IEEE Trans. Antenn Propag., Vol. 52, No. 2, 452-460, Feb. 2004.
doi:10.1109/TAP.2004.823949

31. "Session 6, scattering by wedges I," Symposium Digest, Volume Two, IEEE AP-S International Symposium and USNC/URSI National Radio Science Meeting, 1067ff., Newport Beach, CA, USA, Jun. 18-Jun. 23, 1995.

32. "Session 9, scattering by wedges II," Symposium Digest, Volume Two, IEEE AP-S International Symposium and USNC/URSI National Radio Science Meeting, 1347ff., Newport Beach, CA, USA, Jun. 18-Jun. 23, 1995.

33. Lyalinov, M. A. and N. Y. Zhu, "Scattering of wedges and cones by impedance boundary conditions," The Mario Boella Series on Electromagnetism in Information and Communication, SciTech Publishing, The Institution of Engineering and Technology, Edison, NJ, 2013.

34. Daniele, V. G. and G. Lombardi, "Wiener-Hopf solution for impenetrable wedges at skew incidence," IEEE Trans. Antennas Propag., Vol. 54, No. 9, 2472-2485, Sep. 2006.
doi:10.1109/TAP.2006.880723

35. Daniele, V. G. and R. S. Zich, "The Wiener-Hopf method in electromagnetics," The Mario Boella Series on Electromagnetism in Information and Communication,The Mario Boella Series on Electromagnetism in Information and Communication, SciTech Publishing, The Institution of Engineering and Technology, Edison, NJ, 2014.