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Asymptotic Methods in Electromagnetics pp 233–373Cite as

Uniform Solutions

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Abstract

In this chapter we will present the uniform solutions in the context of GTD, and we begin by defining the properties of uniform asymptotic expansions in this section. Let f(X,ε) be a scalar or vectorial function of the variable X ∈ D, where D is a given domain, depending upon the small parameter ε∈R 0.

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References

Sects. 5.1 and 5.2

  1. D. S. Ahluwalia, “Uniform Asymptotic Theory Of Diffraction By The Edge Of A Three-Dimensional Body,” Siam J. Appi. Math., 18(2), 287–301, 1970.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Cole Perturbation methods in applied mathematics Blaisdell, 1968.

    MATH  Google Scholar 

  3. Y. A. Kravtsov, “Asymptotic Solutions Of Maxwell’s equations near a caustic,” Radiofizika, 7, 1049–1056 (IN RUSSIAN), 1964.

    Google Scholar 

  4. D. Ludwig, “Uniform Asymptotic Expansions At A Caustic,” Comm. Pure Appi. Math., 19, 215–250, 1966.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. H. Lewis and J. Boersma, “Uniform Asymptotic Theory Of Edge Diffraction,” J. Math. Phys., 10(12), 2291–2306, 1969.

    Article  ADS  Google Scholar 

  6. A. H. Nayfeh, Perturbation Methods, Chap. 4, Wiley-Interscience Publication, 1973.

    Google Scholar 

Sect. 5.3

  1. N. Bleistein, “Uniform asymptotic Expansions of Integrals with many nearby stationary points and algebraic singularities,” J. Math. Mech, 17, 533–559, 1967.

    MathSciNet  MATH  Google Scholar 

  2. V. A. Borovikov, “Diffraction By A Wedge With Curved Faces,” Sov. Phys. Acoust. 25(6), 465–471, 1979.

    MathSciNet  Google Scholar 

  3. J. Boersma and Y. Rahmat-Samii, “Comparison of two leading uniform theories of edge diffraction with the exact uniform asymptotic solution,” Radio Sci., 15(6)1179–1194, 1980.

    Article  ADS  Google Scholar 

  4. P. C. Clemmow, “Some extensions of the method of integration by steepest descents,” Quart. J. Mech. Appi. Math. 3, 241–256, 1950.

    Article  MathSciNet  MATH  Google Scholar 

  5. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, Englewood Cliffs, NJ, Prentice Hall, 1973.

    Google Scholar 

  6. C. Gennarelli and L. Palumbo, “A uniform asymptotic expression of a typical diffraction integral with many coalescing simple pole singularities and a first-order saddle point,” IEEE Trans. Ant. Prop., AP-32, 1122‱1124, 1984.

    Article  MathSciNet  ADS  Google Scholar 

  7. D. L. Hutchins and R. G. Kouyoumjian, “Asymptotic series describing the diffraction of a plane wave by a wedge,” Report 2183–3, ElectroScience Laboratory, Department of Electrical Engineering, The Ohio State University; prepared under Contract AF 19 (638)–5929 for Air Force Cambridge Research Laboratories.

    Google Scholar 

  8. Y. M. Hwang and R. G. Kouyoumjian, “A dyadic diffraction coefficient for an electromagnetic wave which is rapidly varying at an edge,” paper presented at USNC/URSI Annual Meeting, Boulder, Colorado, 1974.

    Google Scholar 

  9. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. AM., 52, 116–130, 1962.

    Article  ADS  Google Scholar 

  10. R. G. Kouyoumjian and P. H. Pathak, “A unifrom geometrical theory diffraction for an edge in a perfectly conducting surface,” Proc. IEEE, 62, 1448–1461, 1974.

    Article  ADS  Google Scholar 

  11. R. G. Kouyoumjian and P. H. Pathak, A uniform GTD approach to EM scattering and radiation in Acoustic Electromagnetic and Elastic Wave Scattering — High and Low Frequency Asymptotics, vol. II, edited by Varadan and Varadan, North Holland Publishers, 1986.

    Google Scholar 

  12. S. W. Lee and G. A. Dexchamps, “A uniform asymptotic theory of electromagnetic diffraction by a curved wege,” IEEE Trans. Ant. Prop., AP-24, 25–34, 1976.

    ADS  Google Scholar 

  13. F. Oberhettinger, “An asymptotic series for functions occuring in the theory of diffraction of waves by wedges,” J. Math. Phys., 34, 245–255, 1955.

    MathSciNet  Google Scholar 

  14. H. Ott, “Die Sattelpunktsmethode in der Umgebung eines Poles mit Anwendung an die Wellenoptik und Akustik,” Annalen der Physik, 43, 393–403, 1943.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. W. Pauli, “On the asymptotic series for functions in the theory of the diffraction of light,” Phys. Rev., 54, 924–931, 1938.

    Article  ADS  MATH  Google Scholar 

  16. Y. Rahmat-Samii and R. Mittra, “A spectral domain interpretation of high frequencey phenomena,” IEEE Trans. Ant. Prop., AP-24, 676–687, 1977.

    Google Scholar 

  17. Y. Rahmat-Samii and R. Mittra, “Spectral analysis of high frequency diffraction of an arbitrary incident field by a half plane-comparison with four asymptotic techniques,” Radio Science, 13(1), 31–48, 1978.

    Google Scholar 

  18. R. G. Rojas, “Comparison between two asymptotic methods,” IEEE Trans. Ant. Prop., AP-35(12), 1489–1492, 1987.

    Google Scholar 

  19. Van der Waerden, “On the method of saddle points,” Appi. Sci. Research, B2, 33–45, 1951.

    Google Scholar 

Sects. 5.4

  1. J. George and H. Uberall, “Approximate methods to describe the reflections from cylinders and spheres with complex impedance,” J. Acoust., Soc. Am. 65(1), 15–24, 1979.

    Google Scholar 

  2. L. Kaminetzky and J. B. Keller, “Diffraction Coefficients For Higher Order Edges And Vertices,” SIAM J. Appi. Math., 22(1), 109–134, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. B. Keller, R. M. Lewis and B. D. Seckler, “Asymptotic solution of some diffraction problems,” Comm. Pure Appi. Math., 9, 207–265, 1956.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. A. Leontovitch, “Investigations of propagation of radio waves,” Soviet Radio, Moscow, 1948.

    Google Scholar 

  5. F. Molinet, Geometrical theory of diffraction, IEE Aps Newsletter, part II, pp. 5–16, 1987.

    Google Scholar 

  6. F. Molinet, “Etude de la diffraction par une ligne de discontinuité de la courbure,” Rapport MOTHESIM M 51, 1982.

    Google Scholar 

  7. F. Molinet, “Uniform asymtotic solution for the diffraction by a discontinuity in curvature,”Annales des TéLécom., 50 (5, 6), 523–535, 1995.

    Google Scholar 

  8. T. B. A. Senior, “The diffraction matrix for a discontinuity in curvature,” IEEE Trans. Ant. Prop., AP-20, 326–333, 1972.

    Google Scholar 

  9. A. Sommerfeld, Partial Differential Equation in Physics, Academic Press, New York, 1964.

    Google Scholar 

  10. V. H. Weston, “The effect of a discontinuity in curvature in high frequency scattering,” IRE Trans. Ant. Prop., AP-10, 775–780, 1962.

    Google Scholar 

Sects. 5.5

  1. V. M. Babich and N. Y. Kirpicnikova, The boundary-layer method in diffraction problems, Springer-Verlag, Berlin, Heidelberg, New York, 1979.

    Google Scholar 

  2. D. Bouche, La méthode des courants asymptotiques, Thése de Docteur, Université de Bordeaux I, 1992.

    Google Scholar 

  3. J. A. Cullen, “Surface currents induced by short-wave length radiation,” Phys. Rev., 109, 1863–1867, 1958.

    Google Scholar 

  4. W. Franz and K. Klante, “Diffraction By Surfaces Of Variables Curvature,” IRE Trans. Ant. Prop., AP-7, 568–570, 1959.

    Google Scholar 

  5. V. A. Fock, “The field of a plane wave near the surface of a conducting body,” J. Phys. USSR, 10, 399–409, 1946.

    MathSciNet  Google Scholar 

  6. V. A. Fock, Electromagnetic Diffraction and Propagation, Pergamon, New York, 1965.

    Google Scholar 

  7. S. Hong, “Asymptotic theory of electromagnetic and a caustic diffraction by smooth convex surfaces of variable curvature,” J. Math. Phys., 8, 1223–1232, 1967.

    Article  ADS  Google Scholar 

  8. V. I. Ivanov, “Diffraction d’ondes électromagnétiques planes courtes en incidence oblique sur un cylindre convexe lisse,” Radiotecknica i ElectrÓnica, 5, 524–528, 1960.

    Google Scholar 

  9. O. Lafitte, Thése de Docteur, Université de Paris Sud, 1993.

    Google Scholar 

  10. B. R. Levy and J. B. Keller, “Diffraction by a smooth object,” Comm. Pure Appl. Math., 12, 159–209, 1959.

    Article  MathSciNet  Google Scholar 

  11. N. A. Logan and K. S. Yee, A mathematical model for diffraction by convex surfaces, in Electromagnetic Waves, edited by R. E. Langer, pp. 139–180, The University of Wisconsin Press, Madison, Wisconsin, 1962.

    Google Scholar 

  12. A. Michaeli, “High-frequency electromagnetic fields near a smooth convex surface in the shadow region,” Private Communication, 1991.

    Google Scholar 

  13. R. Mittra and Safavi-Naini, “Source radiation in the presence of smooth convex bodies,” Radio Science, 14, 217–237, 1979.

    Google Scholar 

  14. P. H. Pathak, “An asymptotic analysis of the scattering of plane waves by a smooth convex cylinder,” Radio Science, 14, 419–435, 1979.

    Google Scholar 

  15. L. W. Pearson, “A schema for automatic computation of Fock-type integrals,” IEEE Trans. Ant. Prop., AP-35(10), 1111–1118, 1987.

    Article  ADS  Google Scholar 

  16. P. H. Pathak, N. Wang, W. Burnside and R. Kouyoumjian, “A uniform GTD solution for the radiation from sources on a convex surface,” IEEE Trans Ant. Prop., AP-29, 609–622, 1981.

    Article  ADS  Google Scholar 

  17. J. R. Wait and A. M. Conda, “Pattern of an antenna on a curved loosy surface,” IRE Trans. Ant. Prop., AP-6, 348–359, 1958.

    Google Scholar 

Sect. 5.6

  1. N. C. Albertsen, “Diffraction of creeping waves,” Report LD 24, Electromagnetic Institute, Technical University Denmark, 1974.

    Google Scholar 

  2. N. C. Albertsen and P. L. Christiansen, “Hybrid diffraction coefficients for first and second order discontinuities of two-dimensional scatterers,” SIAM J. Appl. Math., 34 398–414, 1978.

    Article  MathSciNet  ADS  Google Scholar 

  3. J. M. L. Bernard, “Diffraction par un diedre ä faces courbes non parfaitement conducteur,” Rev. Tech. THOMSON-CSF, 23(2), 321–330, 1991.

    Google Scholar 

  4. V. A. Borovikov, “Diffraction by a wedge with curved faces,” AKUST. ZH. 25(6), 825–835.

    Google Scholar 

  5. C. W. Chuang and M. C. Liang, “A uniform asymptotic analysis of the diffraction by an edge in a curved screen,” RADIO SCIENCE23(5), 781–790, 1988.

    Article  ADS  Google Scholar 

  6. V. B. Filippov, “Diffraction by a curved half-plane,” Zap. Nauchn. Sem. LOMI (Leningrad), 42, 244, 1974.

    MATH  Google Scholar 

  7. K. C. Hill and P. H. Pathak, “A UTD analysis of the excitation of surface rays by an edge in an otherwise smooth perfectly-conducting convex surface,” URSI Radio Science meeting, Blacksburg, Virginia, 1987.

    Google Scholar 

  8. M. Idemen and L. B. Felsen, “Diffraction of a whispering gallery mode by the edge of a thin concave cylindrically curved surface,” IEEE Trans. Ant. Prop., AP- 29 571–579,1981.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. J.B. Keller “Diffraction by an aperture,” J. Appl. Phys., 28,426–444, 1957.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. R. G. Kouyoumjian and P. H. Pathak, “A geometrical theory of diffraction for an edge in a perfectly conducting surface,” PROC. IEEE, 621448–1474, 1974.

    Google Scholar 

  11. M. C. Liang, A Generalized Uniform Gtd Ray Solution For The Diffraction By A Perfectly Conducting Wedge With Convex FacesPh. D Thesis, Ohio State University, 1988.

    Google Scholar 

  12. S. N. Lee and G. A. Deschamps, “A uniform asymptotic theory of electromagnetic diffraction by a curved wedge,” IEEE Trans. Ant. Prop., AP- 24, 25-34, 1976.

    Article  MathSciNet  ADS  Google Scholar 

  13. M. C. Liang, P. H. Pathak and C. W. Chuang, “A generalized uniform GTD ray solution for the diffraction by a wedge with convex faces,” Congre,′s URSI, Prague, Aoüt 1990.

    Google Scholar 

  14. A. Michaeli, “Transition functions for high-frequency diffraction by a curved perfectly conducting wedge; Part I: Canonical solution for a curved sheet; Part II: A partially uniform solution for a general wedge angle; Part III: Extension to overlapping transition regions,” IEEE Trans. Ant. Prop., 27, 1073–1092, 1989.

    Article  MathSciNet  ADS  Google Scholar 

  15. F. Molinet, “Diffraction d’une onde rampante par une ligne de discontinuité du plan tangent,” ANNALES DES TÉLÉCOM32(5–6), 197, 1977.

    Google Scholar 

  16. P. H. Pathak, “An asymptotic analysis of the scattering of plane waves by a smooth convex cylinder, ” RADIO SCIENCE14(3), 419 – 435, 1979.

    Article  ADS  Google Scholar 

  17. P. H. Pathak and R. G. Kouyoumjian, “On the diffraction of edge excited surface rays,” Paper presented at the 1977 USNC/URSI Meeting, Stanford University, Stanford, CA, 22 – 24, June 1977.

    Google Scholar 

  18. A. Sommerfeld, “Mathematische Theorie der Diffraktion,” MATH. ANN. 47, 317–374, 1986. Sect. 5.7

    Article  MathSciNet  Google Scholar 

Sect. 5.7

  1. Y. A. Kravtsov and Y. I. Orlov, “Caustics, catastrophes and wavefields,” SOV. PHYS. USP.26, 1983.

    Google Scholar 

  2. Y. A. Kravtsov, Radiofizika7, 664, 1964.

    Google Scholar 

  3. Y. A. KravtsovRadiofizika7, 1049, 1964.

    Google Scholar 

  4. D. Ludwig, Comm. Pure Appl. Math,19, 215, 1966.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. CEA, CELV, F-94195, Villeneuve St. Georges, France

    Dr. Daniel Bouche

  2. Mothesim, La Boursidière, F-92357, Le Plessis, Robinson, France

    Dr. Frédéric Molinet

  3. Electrical and Computer Engineering Department, University of Illinois, 1406 W. Green St., 61801-2991, Urbana, IL, USA

    Professor Raj Mittra

Authors
  1. Dr. Daniel Bouche
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  2. Dr. Frédéric Molinet
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  3. Professor Raj Mittra
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© 1997 Springer-Verlag Berlin Heidelberg

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Bouche, D., Molinet, F., Mittra, R. (1997). Uniform Solutions. In: Asymptotic Methods in Electromagnetics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60517-8_5

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  • DOI: https://doi.org/10.1007/978-3-642-60517-8_5

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  • Print ISBN: 978-3-642-64440-5

  • Online ISBN: 978-3-642-60517-8

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