Exact solution and high frequency asymptotic methods in the wedge diffraction problem
DOI:
https://doi.org/10.18046/syt.v14i38.2285Keywords:
Geometrical Theory of Diffraction, Asymptotic Methods, Computational Electromagnetics.Abstract
Abstract
The Sommerfeld exact solution for canonical 2D wedge diffraction problem with perfectly conducting surfaces is presented. From the integral formulation of the problem, the Malyuzhinets solution is obtained and this result is extended to obtain the general impedance solution of canonical 2D wedge problem. Keller’s asymptotic solution is developed and the general formulation of exact solution it’s used to obtain general asymptotic methods for approximate solutions useful from the computational point of view. A simulation tool is used to compare numerical calculations of exact and asymptotic solutions. The numerical simulation of exact solution is compared to numerical simulation of an asymptoticmethod, and a satisfactory agreement found. Accuracy dependence with frequency is verified.
References
Aboserwal, N. & Balanis, C. (2014). Closed-form expression of the Maliuzhinets function using tanh-sinh quadrature rule. In: Antennas and Propagation Society International Symposium (APSURSI). IEEE.
Ahluwalia, D., Lewis, R., & Bodersma, J. (1968). Uniform asymptotic theory of diffraction by a plane screen. SIAM Journal on applied mathematics, 16(4), 783-807.
Ambaud, P. & Bergassoli, A. (1972). The problem of the wedge in acoustics. Acta Acustica united with Acustica, 27(5), 291-298.
Baibich, V., Lyalinov, M. & Grikurov, V. (2008). Diffraction theory, the Sommerfeld-Malyuzhinets technique [Alpha Science series on wave phenomena]. Oxford, UK: Alpha Science.
Balanis, C. (1989). Advanced engineering electromagnetic. New York, NY: John Wiley & Sons.
Bowman, J. & Senior, T. (1969). The wedge. In: Electromagnetic and acoustic scattering by simple shapes (pp. 252-283). Amsterdam, The Netherlands: North Holland.
Copson, E. (1965). Asymptotic expansions. Cambridge, UK: Cambridge University Press.
DeWitt-Morette, C., Low, S., Schulman, L., & Shiekh, A. (1986). Wedges I. Foundations of Physics 16(4), 311-349.
Erdglyi, A. (1956). Asymptotic expansions. New York, NY: Courier.
Felsen, L. & Marcuvitz, N. (1978). Radiation and scattering of waves. New York, NY: Prentice Hall.
Filippov, A. (1967). Investigation of solution of a nonstationary problem of diffraction of a plane wave by an impedance wedge, Zh. Vych. Matem. i Mat. Fiziki 7, 825-835.
Guevara, D., & Navarro, A. (2011). Estimación de parámetros de canal en entornos 3D [tesis]. Universidad Pontificia Bolivariana: Medellín, Colombia.
Hacivelioglu, F., Sevgi, L., & Ufimtsev, P. (2011). Electromagnetic wave scattering from a wedge with perfectly reflecting boundaries: Analysis of asymptotic techniques. IEEE Antennas and Propagation Magazine, 53(3), 232-253.
Hacivelioglu, F., Sevgi, L., & Ufimtsev, P. (2013). Numerical evaluations of diffraction formulas for the canonical wedge-scattering problem. IEEE Antennas and Propagation Magazine, 55(5), 257-272.
Hacivelioglu, F., Uslu, M., & Sevgi, L. (2011). A MATLAB-based virtual tool for the electromagnetic wave scattering from a perfectly reflecting wedge. IEEE Antennas and Propagation Magazine, 53(6), 234-243.
Hadden, W. & Pierce, A. (1981). Sound diffraction around screens and wedges for arbitrary point source locations. Journal of the Acoustical Society of America, 69(5), 1266-1276.
Herman, M., Volakis, J., & Senior, T. (1987). Analytic expressions for a function occurring in diffraction theory. IEEE Transactions on Antennas and Propagation, 35(9), 1083-1086.
Hongo, K. & Nakajima, E. (1986). Polynomial approximation of Malyuzhinets function. IEEE Transactions on Antennas and Propagation, 34(7), 942-947.
James, G. (1986). Geometrical theory of diffraction for electromagnetic waves [3rd ed.]. London, UK: Peter Peregrinus.
Keller, J. (1957). Diffraction by an aperture. J. Appl. Phys, 28(4), 426-444.
Keller, J. (1962). Geometrical theory of diffraction. J. Opt. Soc. Amer, 52(2), 116-130.
Keller, J. (1985). One hundred years of diffraction theory. IEEE Antennas and Propagation Magazine, 33(2), 123-126.
Kline, M. (1951). An asymptotic solution of Maxwell´s equations. Communications on Pure and Applied Mathematics, 4(2-3), 225-262.
Kouyoumjian, R. (1965). Asymptotic high frequency methods. Proceedings of the IEEE, 53(8), 864-876.
Kouyoumjian, R. & Pathal, P. (1974). A uniform geometrical theory of diffraction for an edge in a perfectly conducting Surface. Proceedings of the IEEE, 62(11), 1448-1461.
MacDonald, H., (1902). Electric waves. Cambridge, UK: Cambridge University Press.
MacNamara, D. (1990). Introduction to the uniform geometrical theory of diffraction. Norwood, MA: Artech House.
Malyuzhinets, D. (1955). Radiation of sound from the vibrating faces of an arbitrary wedge [part II]. Sov. Phys. Acoust, 1, 240-248.
Malyuzhinets, D. (1955). The radiation of sound by the vibrating boundaries of an arbitrary wedge [part I]. Sov. Phys. Acoust. 1, 152-174.
Malyuzhinets, D. (1958). Excitation, reflection and emission of surface waves from a wedge with given face impedances. Sov. Phys. Dokl, 3, 752-755.
Malyuzhinets, D. (1958). Inversion formula for the Sommerfeld integral. Sov. Phys. Dokl, 3, 52-56.
Malyuzhinets, D. (1958). Relation between the inversion formulas for the Sommerfeld integral and the formulas of Kontorovich–Lebedev. Sov. Phys. Dokl, 3, 266-268.
Malyuzhinets, G. (1950). Generalization of the reflection method in the theory of diffraction of sinusoidal waves [thesis]. Lebedev Physical Institute of the Russian Academy of Sciences: Moscow.
Morse, P. & Feshbach, H. (1953). Methods of theoretical physics. New York, NY: McGraw-Hill.
Murray, J. (1984). Asymptotic analysis. New York, NY: Springer.
Navarro A., Guevara, D., & Gómez, J. (2014). Modelado de canal inalámbrico utilizando técnicas de trazado de rayos: una revisión sistemática. Sistemas & Telemática, 12(30), 87-101.
Navarro, A., Guevara, D., & Africano, M. (2012). Calibración basada en medidas para modelos de trazado de rayos en 3D para ambientes exteriores urbanos andinos. Sistemas & Telemática, 10(21), 43-63.
Navarro, A., Guevara, D., & Londoño, S. (2014). Using 3D video game technology in channel modeling. IEEE Access, 2, 1652-1659.
Navarro, A., Guevara, D., Tami, D., Rego, C., & Moreira, F. (2015). Heuristic UTD coefficients applied for the channel characterization in an Andean scenario. In: Antennas and Propagation & USNC/URSI National Radio Science Meeting, 2015 IEEE International Symposium on. IEEE.
Navarro, A., Guevara, D., Escalante, D., Cruz, W., Gómez, J., Cardona, N., & Jimenez, J. (2016). Delay spread in mm wave bands for indoor using game engines 3D ray based tools. In: Antennas and Propagation (EuCAP), 2016 10th European Conference on. IEEE.
Oberhettinger, F. (1958). On the diffraction and reflection of waves and pulses by wedges and corners. Journal of Research of the National Bureau of Standards, 61(5), 343-365.
Osipov, A. & Norris, A. (1999). The Malyuzhinets theory for scattering from wedge boundaries: A review. Wave Motion, 29(4), 313-340.
Pathak, P. & Kouyoumjian, R. (1970). The dyadic diffraction coefficient for a perfectly conducting wedge [technical report 2183-4]. Columbus, OH: Ohio State University:
Pathak, P., & Kouyoumjian, R. (1974). An analysis for the radiation from apertures on curves surfaces by the geometrical theory of diffraction. Proceedings of the IEEE, 62(11), 1438-1447.
Petrashen, G., Nikolaev, B., & Kouzov, D. (1958). On the method of series in the theory of diffraction of waves from flat angular regions, Leningrad State University scientific reports: Math. Sci. 246, 5-165.
Pierce, A. (1989). Acoustics: An introduction to its physical principles and applications. New York, NY: Acoustical Society of America.
Sakharova, M. (1970). On asymptotic expansion of certain functions occurring in the wedge diffraction theory. Izv. Vuzov. Fizika, 11, 141-144.
Sevgi, L. (2014). Electromagnetic modelig and simulation. Hoboken, NJ: Wiley-IEEE.
Smith, L. (1953). Mathematical methods for scientists and engineers. New York, NY: Dover.
Sommerfeld, A. (1896). Mathematical theory of diffraction. Math. Ann. 47, 317-374.
Sommerfeld, A. (1901). Teoretisches über die beugung der röntgenstrahlen. Z. Mathem Phys. 46, 11-97.
Sommerfeld, A. (1935). Theorie der beugung. In: F. Frank & R.V. Mizes (Eds.), Die Dijfermtial- und lntegrolgleichungen der Mechanik und Physik [Vol. 2, Physical Part], (Cap. 20). Braunschweig, Germany: Friedr. Vieweg & Sobo.
Sommerfeld, A. (2004). Mathematical theory of diffraction. Boston, MA: Birkhäuser.
Thide, B. (2003). Electromagnetic field theory. Uppsala, Sweden: Upsilon.
Tuzhilin, A. (1963). New representations of diffraction fields in wedge-shaped regions with ideal boundaries. Sov. Phys. Acoust. 9(2), 168-172.
Tuzhilin, A. (1970). Theory of Malyuzhinets’ functional equations I: Homogeneous functional equations, general properties of their solutions, particular cases. Differencialnye Uravnenija, 6, 692-704.
Tuzhilin, A. (1970). Theory of Malyuzhinets’ functional equations II: The theory of Barnes’ infinite products. General solutions of homogeneous Malyuzhinets’ functional equations. Various representations of the basis solutions. Differencialnye Uravnenija, 6, 1048-1063.
Tuzhilin, A. (1971). Theory of Malyuzhinets’ functional equations III: Non-homogeneous functional equations, the general theory. Differencialnye Uravnenija, 7, 1077-1088.
Tuzhilin, A. (1973). Diffraction of a plane sound wave in an angular region with absolutely hard and slippery faces covered by thin elastic plates. Differencialnye Uravnenija, 9, 1875-1888.
Ufimtsev, P. (1957). Approximate computation of the diffraction of plane electromagnetic waves a certain metal bodies. Soviet Physics-Technical Physics, 2(8), 1708-1718.
Ufimtsev, P. (1958). Secondary diffraction o electromagnetic waves by a disk. Soviet Physics-Technical Physics, 3(3), 549-556.
Ufimtsev, P. (1971). Method of edge waves in the physical theory of diffraction [FTD-HC-23-259-71]. Dayton, OH: U.S. Air Force Systems Command - Foreign Technology Office. Available at: http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=AD0733203
Vico, F., Ferrando, M., Valero, A., Herranz, J., & Antonino, E. (2009). Computational electromagnetics and fast physical optics. Waves, 1, 155-161.
Volakis, V. & Senior, T. (1985). Simple expressions for a function occurring in diffraction theory. IEEE Transactions on Antennas and Propagation, 33(6), 678-680
Weinberger, H. (1965). A first course in partial differentials equations with complex variable and transform methods. New York, NY: Dover.
Zavadskii, V. & Sakharova, M. (1961). Tables of the special function 8.z/. Moscow, Russia: Akust. Inst. Rep.
Zavadskii, V., & Sakharova, M. (1967). Application of the special function 8 in problems of wave diffraction in wedge-shaped regions, Sov. Phys. Acoust, 13, 48-54.
Ahluwalia, D., Lewis, R., & Bodersma, J. (1968). Uniform asymptotic theory of diffraction by a plane screen. SIAM Journal on applied mathematics, 16(4), 783-807.
Ambaud, P. & Bergassoli, A. (1972). The problem of the wedge in acoustics. Acta Acustica united with Acustica, 27(5), 291-298.
Baibich, V., Lyalinov, M. & Grikurov, V. (2008). Diffraction theory, the Sommerfeld-Malyuzhinets technique [Alpha Science series on wave phenomena]. Oxford, UK: Alpha Science.
Balanis, C. (1989). Advanced engineering electromagnetic. New York, NY: John Wiley & Sons.
Bowman, J. & Senior, T. (1969). The wedge. In: Electromagnetic and acoustic scattering by simple shapes (pp. 252-283). Amsterdam, The Netherlands: North Holland.
Copson, E. (1965). Asymptotic expansions. Cambridge, UK: Cambridge University Press.
DeWitt-Morette, C., Low, S., Schulman, L., & Shiekh, A. (1986). Wedges I. Foundations of Physics 16(4), 311-349.
Erdglyi, A. (1956). Asymptotic expansions. New York, NY: Courier.
Felsen, L. & Marcuvitz, N. (1978). Radiation and scattering of waves. New York, NY: Prentice Hall.
Filippov, A. (1967). Investigation of solution of a nonstationary problem of diffraction of a plane wave by an impedance wedge, Zh. Vych. Matem. i Mat. Fiziki 7, 825-835.
Guevara, D., & Navarro, A. (2011). Estimación de parámetros de canal en entornos 3D [tesis]. Universidad Pontificia Bolivariana: Medellín, Colombia.
Hacivelioglu, F., Sevgi, L., & Ufimtsev, P. (2011). Electromagnetic wave scattering from a wedge with perfectly reflecting boundaries: Analysis of asymptotic techniques. IEEE Antennas and Propagation Magazine, 53(3), 232-253.
Hacivelioglu, F., Sevgi, L., & Ufimtsev, P. (2013). Numerical evaluations of diffraction formulas for the canonical wedge-scattering problem. IEEE Antennas and Propagation Magazine, 55(5), 257-272.
Hacivelioglu, F., Uslu, M., & Sevgi, L. (2011). A MATLAB-based virtual tool for the electromagnetic wave scattering from a perfectly reflecting wedge. IEEE Antennas and Propagation Magazine, 53(6), 234-243.
Hadden, W. & Pierce, A. (1981). Sound diffraction around screens and wedges for arbitrary point source locations. Journal of the Acoustical Society of America, 69(5), 1266-1276.
Herman, M., Volakis, J., & Senior, T. (1987). Analytic expressions for a function occurring in diffraction theory. IEEE Transactions on Antennas and Propagation, 35(9), 1083-1086.
Hongo, K. & Nakajima, E. (1986). Polynomial approximation of Malyuzhinets function. IEEE Transactions on Antennas and Propagation, 34(7), 942-947.
James, G. (1986). Geometrical theory of diffraction for electromagnetic waves [3rd ed.]. London, UK: Peter Peregrinus.
Keller, J. (1957). Diffraction by an aperture. J. Appl. Phys, 28(4), 426-444.
Keller, J. (1962). Geometrical theory of diffraction. J. Opt. Soc. Amer, 52(2), 116-130.
Keller, J. (1985). One hundred years of diffraction theory. IEEE Antennas and Propagation Magazine, 33(2), 123-126.
Kline, M. (1951). An asymptotic solution of Maxwell´s equations. Communications on Pure and Applied Mathematics, 4(2-3), 225-262.
Kouyoumjian, R. (1965). Asymptotic high frequency methods. Proceedings of the IEEE, 53(8), 864-876.
Kouyoumjian, R. & Pathal, P. (1974). A uniform geometrical theory of diffraction for an edge in a perfectly conducting Surface. Proceedings of the IEEE, 62(11), 1448-1461.
MacDonald, H., (1902). Electric waves. Cambridge, UK: Cambridge University Press.
MacNamara, D. (1990). Introduction to the uniform geometrical theory of diffraction. Norwood, MA: Artech House.
Malyuzhinets, D. (1955). Radiation of sound from the vibrating faces of an arbitrary wedge [part II]. Sov. Phys. Acoust, 1, 240-248.
Malyuzhinets, D. (1955). The radiation of sound by the vibrating boundaries of an arbitrary wedge [part I]. Sov. Phys. Acoust. 1, 152-174.
Malyuzhinets, D. (1958). Excitation, reflection and emission of surface waves from a wedge with given face impedances. Sov. Phys. Dokl, 3, 752-755.
Malyuzhinets, D. (1958). Inversion formula for the Sommerfeld integral. Sov. Phys. Dokl, 3, 52-56.
Malyuzhinets, D. (1958). Relation between the inversion formulas for the Sommerfeld integral and the formulas of Kontorovich–Lebedev. Sov. Phys. Dokl, 3, 266-268.
Malyuzhinets, G. (1950). Generalization of the reflection method in the theory of diffraction of sinusoidal waves [thesis]. Lebedev Physical Institute of the Russian Academy of Sciences: Moscow.
Morse, P. & Feshbach, H. (1953). Methods of theoretical physics. New York, NY: McGraw-Hill.
Murray, J. (1984). Asymptotic analysis. New York, NY: Springer.
Navarro A., Guevara, D., & Gómez, J. (2014). Modelado de canal inalámbrico utilizando técnicas de trazado de rayos: una revisión sistemática. Sistemas & Telemática, 12(30), 87-101.
Navarro, A., Guevara, D., & Africano, M. (2012). Calibración basada en medidas para modelos de trazado de rayos en 3D para ambientes exteriores urbanos andinos. Sistemas & Telemática, 10(21), 43-63.
Navarro, A., Guevara, D., & Londoño, S. (2014). Using 3D video game technology in channel modeling. IEEE Access, 2, 1652-1659.
Navarro, A., Guevara, D., Tami, D., Rego, C., & Moreira, F. (2015). Heuristic UTD coefficients applied for the channel characterization in an Andean scenario. In: Antennas and Propagation & USNC/URSI National Radio Science Meeting, 2015 IEEE International Symposium on. IEEE.
Navarro, A., Guevara, D., Escalante, D., Cruz, W., Gómez, J., Cardona, N., & Jimenez, J. (2016). Delay spread in mm wave bands for indoor using game engines 3D ray based tools. In: Antennas and Propagation (EuCAP), 2016 10th European Conference on. IEEE.
Oberhettinger, F. (1958). On the diffraction and reflection of waves and pulses by wedges and corners. Journal of Research of the National Bureau of Standards, 61(5), 343-365.
Osipov, A. & Norris, A. (1999). The Malyuzhinets theory for scattering from wedge boundaries: A review. Wave Motion, 29(4), 313-340.
Pathak, P. & Kouyoumjian, R. (1970). The dyadic diffraction coefficient for a perfectly conducting wedge [technical report 2183-4]. Columbus, OH: Ohio State University:
Pathak, P., & Kouyoumjian, R. (1974). An analysis for the radiation from apertures on curves surfaces by the geometrical theory of diffraction. Proceedings of the IEEE, 62(11), 1438-1447.
Petrashen, G., Nikolaev, B., & Kouzov, D. (1958). On the method of series in the theory of diffraction of waves from flat angular regions, Leningrad State University scientific reports: Math. Sci. 246, 5-165.
Pierce, A. (1989). Acoustics: An introduction to its physical principles and applications. New York, NY: Acoustical Society of America.
Sakharova, M. (1970). On asymptotic expansion of certain functions occurring in the wedge diffraction theory. Izv. Vuzov. Fizika, 11, 141-144.
Sevgi, L. (2014). Electromagnetic modelig and simulation. Hoboken, NJ: Wiley-IEEE.
Smith, L. (1953). Mathematical methods for scientists and engineers. New York, NY: Dover.
Sommerfeld, A. (1896). Mathematical theory of diffraction. Math. Ann. 47, 317-374.
Sommerfeld, A. (1901). Teoretisches über die beugung der röntgenstrahlen. Z. Mathem Phys. 46, 11-97.
Sommerfeld, A. (1935). Theorie der beugung. In: F. Frank & R.V. Mizes (Eds.), Die Dijfermtial- und lntegrolgleichungen der Mechanik und Physik [Vol. 2, Physical Part], (Cap. 20). Braunschweig, Germany: Friedr. Vieweg & Sobo.
Sommerfeld, A. (2004). Mathematical theory of diffraction. Boston, MA: Birkhäuser.
Thide, B. (2003). Electromagnetic field theory. Uppsala, Sweden: Upsilon.
Tuzhilin, A. (1963). New representations of diffraction fields in wedge-shaped regions with ideal boundaries. Sov. Phys. Acoust. 9(2), 168-172.
Tuzhilin, A. (1970). Theory of Malyuzhinets’ functional equations I: Homogeneous functional equations, general properties of their solutions, particular cases. Differencialnye Uravnenija, 6, 692-704.
Tuzhilin, A. (1970). Theory of Malyuzhinets’ functional equations II: The theory of Barnes’ infinite products. General solutions of homogeneous Malyuzhinets’ functional equations. Various representations of the basis solutions. Differencialnye Uravnenija, 6, 1048-1063.
Tuzhilin, A. (1971). Theory of Malyuzhinets’ functional equations III: Non-homogeneous functional equations, the general theory. Differencialnye Uravnenija, 7, 1077-1088.
Tuzhilin, A. (1973). Diffraction of a plane sound wave in an angular region with absolutely hard and slippery faces covered by thin elastic plates. Differencialnye Uravnenija, 9, 1875-1888.
Ufimtsev, P. (1957). Approximate computation of the diffraction of plane electromagnetic waves a certain metal bodies. Soviet Physics-Technical Physics, 2(8), 1708-1718.
Ufimtsev, P. (1958). Secondary diffraction o electromagnetic waves by a disk. Soviet Physics-Technical Physics, 3(3), 549-556.
Ufimtsev, P. (1971). Method of edge waves in the physical theory of diffraction [FTD-HC-23-259-71]. Dayton, OH: U.S. Air Force Systems Command - Foreign Technology Office. Available at: http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=AD0733203
Vico, F., Ferrando, M., Valero, A., Herranz, J., & Antonino, E. (2009). Computational electromagnetics and fast physical optics. Waves, 1, 155-161.
Volakis, V. & Senior, T. (1985). Simple expressions for a function occurring in diffraction theory. IEEE Transactions on Antennas and Propagation, 33(6), 678-680
Weinberger, H. (1965). A first course in partial differentials equations with complex variable and transform methods. New York, NY: Dover.
Zavadskii, V. & Sakharova, M. (1961). Tables of the special function 8.z/. Moscow, Russia: Akust. Inst. Rep.
Zavadskii, V., & Sakharova, M. (1967). Application of the special function 8 in problems of wave diffraction in wedge-shaped regions, Sov. Phys. Acoust, 13, 48-54.
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