The multidimensional stationary-phase method for an asymptotic estimate of edge effects in the multiple Kirchhoff diffraction by curved surfaces
Authors:
Edoardo Scarpetta and Mezhlum A. Sumbatyan
Journal:
Quart. Appl. Math. 72 (2014), 731-745
MSC (2010):
Primary 78A45
DOI:
https://doi.org/10.1090/S0033-569X-2014-01365-3
Published electronically:
September 25, 2014
MathSciNet review:
3291825
Full-text PDF Free Access
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Abstract: An analytical approach is proposed to study the contribution of edge effects in the multiple high-frequency diffraction, according to guidelines of classical Kirchhoff theory in (scalar) wave propagation. We start from a suitable asymptotic analysis of the Kirchhoff diffraction integral, here set up in a generalized (iterated) form to describe the multiple reflections from an arbitrary sequence of curved reflecting (smooth) surfaces. The explicit formula obtained for a concrete example of double reflection is compared with the results from direct numerical simulation.
References
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- Gary M. Jebsen and Herman Medwin, On the failure of the Kirchhoff assumption in backscatter, J. Acoust. Soc. Amer. 72 (1982), no. 5, 1607–1611. MR 677408, DOI https://doi.org/10.1121/1.388496
- Joachim Focke, Asymptotische Entwicklungen mittels der Methode der stationären Phase, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Nat. Kl. 101 (1954), no. 3, 48 (German). MR 68650
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- M. V. Fedoryuk, The stationary phase method and pseudodifferential operators, Russian Math. Surveys 26 (1971), 65-115.
- J. C. Cooke, Stationary phase in two dimensions, IMA J. Appl. Math. 29 (1982), no. 1, 25–37. MR 667780, DOI https://doi.org/10.1093/imamat/29.1.25
- E. Scarpetta and M. A. Sumbatyan, Explicit analytical representations in the multiple high-frequency reflection of acoustic waves from curved surfaces: the leading asymptotic term, Acta Acustica united with Acustica 97 (2011), 115-127.
- M. A. Sumbatyan and N. V. Boyev, High-frequency diffraction by non-convex obstacles, J. Acoust. Soc. Amer. 95 (1994), 2346-2353.
- Edoardo Scarpetta and Mezhlum A. Sumbatyan, An asymptotic estimate of the edge effects in the high frequency Kirchhoff diffraction theory for 3-d problems, Wave Motion 48 (2011), no. 5, 408–422. MR 2793773, DOI https://doi.org/10.1016/j.wavemoti.2011.02.003
- E. Scarpetta and M. A. Sumbatyan, High-frequency multiple Kirchhoff diffraction by flat reflectors with a change of ray-path planes, Acta Acustica united with Acustica 98 (2012), 700-712.
- E. Scarpetta and M. A. Sumbatyan, The contribution of the edge effects in the multiple high-frequency Kirchhoff diffraction by plane surfaces, Wave Motion 50 (2013), 210–225.
- M. V. Fedorjuk, The method of stationary phase in the multidimensional case. Contribution from the boundary of the domain, Ž. Vyčisl. Mat i Mat. Fiz. 10 (1970), 286–299 (Russian). MR 273263
References
- H. Hönl, A. W. Maue, and R. Westpfahl, Theory of Diffraction, Naval Intelligence Support Center, New York, 1978.
- L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, John Wiley, Hoboken, New Jersey, 2003.
- V. A. Borovikov and B. Ye. Kinber, Geometrical theory of diffraction, IEE Electromagnetic Waves Series, vol. 37, Institution of Electrical Engineers (IEE), London, 1994. Translated and revised from the Russian original. MR 1292857 (95j:78003)
- M. Born and E. Wolf, Principles of Optics (7th ed.), Cambridge University Press, 1999.
- R. Guenther, Diffraction and Gaussian Beams. In: Modern Optics, John Wiley, New York, 1990, 323-360.
- V. M. Babich, V. S. Buldyrev, Asymptotic Methods in Short-Wavelength Diffraction Theory, Springer-Verlag, Berlin / Heidelberg, 1989.
- H. Kuttruff, Room Acoustics, Applied Science, London, 1973.
- Joseph B. Keller, Geometrical theory of diffraction, J. Opt. Soc. Amer. 52 (1962), 116–130. MR 0135064 (24 \#B1115)
- D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to the uniform geometrical theory of diffraction, The Artech House Antennas and Propagation Library, Artech House Inc., Boston, MA, 1990. MR 1118379 (92m:78017)
- Eugen Skudrzyk, The foundations of acoustics, Springer-Verlag, New York, 1971. Basic mathematics and basic acoustics. MR 0502736 (58 \#19668)
- I. Tolstoy, Wave Propagation, McGraw-Hill, New York, 1973.
- Michael E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, vol. 34, Princeton University Press, Princeton, N.J., 1981. MR 618463 (82i:35172)
- Mezhlum A. Sumbatyan and Antonio Scalia, Equations of mathematical diffraction theory, Differential and Integral Equations and Their Applications, vol. 5, Chapman & Hall/CRC, Boca Raton, FL, 2005. MR 2099746 (2005i:74044)
- Gary M. Jebsen and Herman Medwin, On the failure of the Kirchhoff assumption in backscatter, J. Acoust. Soc. Amer. 72 (1982), no. 5, 1607–1611. MR 677408 (83m:73030), DOI https://doi.org/10.1121/1.388496
- Joachim Focke, Asymptotische Entwicklungen mittels der Methode der stationären Phase, Ber. Verh. Sächs. Akad. Wiss. Leipzig. Math.-Nat. Kl. 101 (1954), no. 3, 48 (German). MR 0068650 (16,919a)
- Nicholas Chako, Asymptotic expansions of double and multiple integrals occurring in diffraction theory, J. Inst. Math. Appl. 1 (1965), 372–422. MR 0204944 (34 \#4779)
- M. V. Fedoryuk, The stationary phase method and pseudodifferential operators, Russian Math. Surveys 26 (1971), 65-115.
- J. C. Cooke, Stationary phase in two dimensions, IMA J. Appl. Math. 29 (1982), no. 1, 25–37. MR 667780 (84d:41056), DOI https://doi.org/10.1093/imamat/29.1.25
- E. Scarpetta and M. A. Sumbatyan, Explicit analytical representations in the multiple high-frequency reflection of acoustic waves from curved surfaces: the leading asymptotic term, Acta Acustica united with Acustica 97 (2011), 115-127.
- M. A. Sumbatyan and N. V. Boyev, High-frequency diffraction by non-convex obstacles, J. Acoust. Soc. Amer. 95 (1994), 2346-2353.
- Edoardo Scarpetta and Mezhlum A. Sumbatyan, An asymptotic estimate of the edge effects in the high frequency Kirchhoff diffraction theory for 3-d problems, Wave Motion 48 (2011), no. 5, 408–422. MR 2793773 (2012j:35298), DOI https://doi.org/10.1016/j.wavemoti.2011.02.003
- E. Scarpetta and M. A. Sumbatyan, High-frequency multiple Kirchhoff diffraction by flat reflectors with a change of ray-path planes, Acta Acustica united with Acustica 98 (2012), 700-712.
- E. Scarpetta and M. A. Sumbatyan, The contribution of the edge effects in the multiple high-frequency Kirchhoff diffraction by plane surfaces, Wave Motion 50 (2013), 210–225.
- M. V. Fedorjuk, The method of stationary phase in the multidimensional case. Contribution from the boundary of the domain, Ž. Vyčisl. Mat. i Mat. Fiz. 10 (1970), 286–299 (Russian). MR 0273263 (42 \#8144)
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Additional Information
Edoardo Scarpetta
Affiliation:
Department of Industrial Engineering, University of Salerno, 84084 Fisciano (SA), Italy
Email:
escarpetta@unisa.it
Mezhlum A. Sumbatyan
Affiliation:
Faculty of Mathematics, Mechanics and Computer Science, Southern Federal University, Milchakova Street 8a, 344090 Rostov-on-Don, Russia
Email:
sumbat@math.rsu.ru
Received by editor(s):
November 27, 2012
Published electronically:
September 25, 2014
Additional Notes:
The authors thank the reviewer for his useful suggestions
Article copyright:
© Copyright 2014
Brown University
