The Rayleigh hypothesis and a related least‐squares solution to scattering problems for periodic surfaces and other scatterers
10.1029/rs008i008p00785
Crossref journal-article
American Geophysical Union (AGU)
Radio Science (13)
Abstract

The Rayleigh hypothesis is reviewed in relation to scattering by periodic surfaces, aperiodic surfaces, and bounded, two‐dimensional bodies. Conditions for its validity are described, and explicit results are quoted for a sinusoidal grating. Some methods to solve scattering problems for periodic surfaces are outlined. One particular procedure for periodic surfaces and bounded scatterers is examined in detail. This involves an expansion for the scattered field in terms of the same sets of elementary wavefunctions that occur in connection with the Rayleigh hypothesis. The coefficients are determined by satisfying the boundary condition in the least‐squares sense. It is shown that this solution converges uniformly to the scattered field at all points exterior to the boundary of the scatterer. Necessary completeness properties of the sets of wavefunctions are established in the appendices.

Bibliography

Millar, R. F. (1973). The Rayleigh hypothesis and a related least‐squares solution to scattering problems for periodic surfaces and other scatterers. Radio Science, 8(8–9), 785–796. Portico.

Authors 1
  1. R. F. Millar (first)
References 60 Referenced 248
  1. {'issue': '7', 'key': 'e_1_2_1_2_1', 'first-page': '71', 'article-title': 'Investigation of the scattering matrix of bodies of complex shape', 'author': 'Barantsev R. G.', 'year': '1968', 'journal-title': 'Vestn. Leningrad Univ.'} / Vestn. Leningrad Univ. / Investigation of the scattering matrix of bodies of complex shape by Barantsev R. G. (1968)
  2. {'issue': '7', 'key': 'e_1_2_1_3_1', 'first-page': '56', 'article-title': 'Concerning the Rayleigh hypothesis in the problem of scattering from finite bodies of arbitrary shapes', 'author': 'Barantsev R. G.', 'year': '1971', 'journal-title': 'Vestn. Leningrad Univ.'} / Vestn. Leningrad Univ. / Concerning the Rayleigh hypothesis in the problem of scattering from finite bodies of arbitrary shapes by Barantsev R. G. (1971)
  3. Barantsev R. G. G. V.Dubrovsky V. V.Kozachjek Separation of variables method development for noncanonical scattering problems URSI Symposium on Electromagnetic Wave Theory Tbilisi USSR 20–24Izdatel'stv. Nauka Sibirskoe Otdelenie Novosibirsk SSR 1971b.
  4. Beckmann P. A.Spizzichino The Scattering of Electromagnetic Waves from Rough Surfaces Int. Ser. Monogr. Electromagn. Waves 4 34–69 Pergamon New York 1963.
  5. 10.1070/PU1968v011n02ABEH003801
  6. 10.1063/1.1656800
  7. {'issue': '4', 'key': 'e_1_2_1_8_1', 'first-page': '500', 'article-title': 'On scattering of waves by many bodies', 'volume': '68', 'author': 'Burke J. E.', 'year': '1964', 'journal-title': 'J. Res. Nat. Bur. Stand., Sect. D'} / J. Res. Nat. Bur. Stand., Sect. D / On scattering of waves by many bodies by Burke J. E. (1964)
  8. {'key': 'e_1_2_1_9_1', 'first-page': '523', 'article-title': 'The multipole expansion of radiation fields', 'volume': '3', 'author': 'Calderon A. P.', 'year': '1954', 'journal-title': 'J. Ration. Mech. Anal.'} / J. Ration. Mech. Anal. / The multipole expansion of radiation fields by Calderon A. P. (1954)
  9. {'issue': '16', 'key': 'e_1_2_1_10_1', 'first-page': '1060', 'article-title': 'Etude numérique du champ diffracté par un réseau', 'volume': '268', 'author': 'Cerutti‐Maori G.', 'year': '1969', 'journal-title': 'C. R. Acad. Sci., Ser. B'} / C. R. Acad. Sci., Ser. B / Etude numérique du champ diffracté par un réseau by Cerutti‐Maori G. (1969)
  10. 10.1109/TMTT.1973.1127931
  11. {'issue': '3', 'key': 'e_1_2_1_12_1', 'first-page': '123', 'article-title': 'The sound absorption of an absorbing periodically uneven surface of rectangular profile', 'volume': '18', 'author': 'Bruijn A.', 'year': '1967', 'journal-title': 'Acustica'} / Acustica / The sound absorption of an absorbing periodically uneven surface of rectangular profile by Bruijn A. (1967)
  12. {'issue': '6', 'key': 'e_1_2_1_13_1', 'first-page': '913', 'article-title': 'Equations for coefficients of wave reflections from a periodically uneven surface', 'volume': '87', 'author': 'Deriugin L. N.', 'year': '1952', 'journal-title': 'Dokl. Akad. Nauk. SSSR'} / Dokl. Akad. Nauk. SSSR / Equations for coefficients of wave reflections from a periodically uneven surface by Deriugin L. N. (1952)
  13. 10.1063/1.1665822
  14. 10.1109/TAP.1972.1140211
  15. 10.1121/1.1912022
  16. Holford R. L. (1971) Scattering of sound waves at a periodic pressure‐release surface: An exact solution Tech. Rep. 38 pp. contract N00014‐69‐C‐0074 Office of Naval Research Arlington Va..
  17. {'issue': '9', 'key': 'e_1_2_1_18_1', 'first-page': '1', 'article-title': 'Resonance in a slightly tilted waveguide', 'volume': '48', 'author': 'Itakura T.', 'year': '1965', 'journal-title': 'J. Inst. Elec. Commun. Eng. Jap.'} / J. Inst. Elec. Commun. Eng. Jap. / Resonance in a slightly tilted waveguide by Itakura T. (1965)
  18. 10.1109/TAP.1972.1140186
  19. {'key': 'e_1_2_1_20_1', 'volume-title': 'Approximate Methods of Higher Analysis', 'author': 'Kantorovich L. V.', 'year': '1964'} / Approximate Methods of Higher Analysis by Kantorovich L. V. (1964)
  20. 10.1364/JOSA.43.000408
  21. {'issue': '1', 'key': 'e_1_2_1_22_1', 'first-page': '1', 'article-title': 'Theory of the scattering of waves at periodically uneven surfaces', 'volume': '4', 'author': 'Lysanov Iu. P.', 'year': '1958', 'journal-title': 'Sov. Phys. Acoust.'} / Sov. Phys. Acoust. / Theory of the scattering of waves at periodically uneven surfaces by Lysanov Iu. P. (1958)
  22. 10.1121/1.1908654
  23. {'issue': '1', 'key': 'e_1_2_1_24_1', 'first-page': '79', 'article-title': 'Wave diffraction at a periodically uneven interface', 'volume': '15', 'author': 'Mashashvili E. S.', 'year': '1969', 'journal-title': 'Sov. Phys. Acoust.'} / Sov. Phys. Acoust. / Wave diffraction at a periodically uneven interface by Mashashvili E. S. (1969)
  24. {'issue': '2', 'key': 'e_1_2_1_25_1', 'first-page': '217', 'article-title': 'Diffraction of a plane wave by a periodic uneven interface between media', 'volume': '15', 'author': 'Mashashvili E. S.', 'year': '1969', 'journal-title': 'Sov. Phys. Acoust.'} / Sov. Phys. Acoust. / Diffraction of a plane wave by a periodic uneven interface between media by Mashashvili E. S. (1969)
  25. {'issue': '2', 'key': 'e_1_2_1_26_1', 'first-page': '323', 'article-title': 'On the use of the Kirchhoff approximation for the solution of reflection problems', 'volume': '5', 'author': 'Meecham W. C.', 'year': '1956', 'journal-title': 'J. Ration. Mech. Anal.'} / J. Ration. Mech. Anal. / On the use of the Kirchhoff approximation for the solution of reflection problems by Meecham W. C. (1956)
  26. 10.1063/1.1722378
  27. 10.1017/S0305004100003613
  28. 10.1049/el:19690314
  29. 10.1017/S0305004100046570
  30. 10.1017/S0305004100046545
  31. {'key': 'e_1_2_1_32_1', 'volume-title': 'Analytical Techniques in the Theory of Guided Waves', 'author': 'Mittra R.', 'year': '1971'} / Analytical Techniques in the Theory of Guided Waves by Mittra R. (1971)
  32. 10.1109/PROC.1969.7463
  33. 10.1016/0030-4018(70)90146-X
  34. 10.1016/0030-4018(71)90260-4
  35. 10.1016/0030-4018(71)90212-4
  36. 10.1080/713818324
  37. {'issue': '7', 'key': 'e_1_2_1_38_1', 'first-page': '468', 'article-title': "Sur la diffraction d'une onde plane par un reseau infiniment conducteur", 'volume': '262', 'author': 'Petit R.', 'year': '1966', 'journal-title': 'C. R. Acad. Sci., Ser. B'} / C. R. Acad. Sci., Ser. B / Sur la diffraction d'une onde plane par un reseau infiniment conducteur by Petit R. (1966)
  38. 10.1098/rspa.1907.0051
  39. Rayleigh Lord J. W.Strutt The Theory of Sound 2 89–96 Dover New York 1945.
  40. 10.1007/978-3-642-46077-7_8
  41. {'key': 'e_1_2_1_42_1', 'volume-title': 'The Theory of Functions', 'author': 'Titchmarsh E. C.', 'year': '1939'} / The Theory of Functions by Titchmarsh E. C. (1939)
  42. {'key': 'e_1_2_1_43_1', 'first-page': '90', 'volume-title': 'Integral Equations', 'author': 'Tricomi F. G.', 'year': '1957'} / Integral Equations by Tricomi F. G. (1957)
  43. {'key': 'e_1_2_1_44_1', 'first-page': '715', 'article-title': 'On multiple scattering of waves', 'volume': '64', 'author': 'Twersky V.', 'year': '1960', 'journal-title': 'J. Res. Nat. Bur. Stand., Sect. D'} / J. Res. Nat. Bur. Stand., Sect. D / On multiple scattering of waves by Twersky V. (1960)
  44. 10.1016/0003-4916(65)90269-1
  45. {'issue': '3', 'key': 'e_1_2_1_46_1', 'first-page': '287', 'article-title': 'Diffraction by a periodic surface', 'volume': '10', 'author': 'Urusovskii I. A.', 'year': '1965', 'journal-title': 'Sov. Phys. Acoust.'} / Sov. Phys. Acoust. / Diffraction by a periodic surface by Urusovskii I. A. (1965)
  46. {'issue': '1', 'key': 'e_1_2_1_47_1', 'first-page': '75', 'article-title': 'Wave diffraction by a sinusoidal surface', 'volume': '11', 'author': 'Urusovskii I. A.', 'year': '1965', 'journal-title': 'Sov. Phys. Acoust.'} / Sov. Phys. Acoust. / Wave diffraction by a sinusoidal surface by Urusovskii I. A. (1965)
  47. 10.1007/BF00411719
  48. {'issue': '5', 'key': 'e_1_2_1_49_1', 'first-page': '715', 'article-title': 'On completeness of a system of metaharmonic functions', 'volume': '90', 'author': 'Vekua I. N.', 'year': '1953', 'journal-title': 'Dokl. Akad. Nauk. SSSR'} / Dokl. Akad. Nauk. SSSR / On completeness of a system of metaharmonic functions by Vekua I. N. (1953)
  49. {'key': 'e_1_2_1_50_1', 'first-page': '84', 'volume-title': 'New Methods for Solving Elliptic Equations', 'author': 'Vekua I. N.', 'year': '1967'} / New Methods for Solving Elliptic Equations by Vekua I. N. (1967)
  50. Walsh J. L. Interpolation and Approximation by Rational Functions in the Complex Domain Amer. Math. Soc. Colloq. Publ. 20 American Mathematical Society Providence R. I. 1956.
  51. 10.1121/1.1911619
  52. 10.1103/PhysRevD.3.825
  53. 10.1090/S0002-9939-1956-0078912-4
  54. 10.1121/1.1919357
  55. 10.1109/TAP.1972.1140187
  56. {'issue': '6', 'key': 'e_1_2_1_57_1', 'first-page': '901', 'article-title': 'New approximate method of eigenvalues for waveguide', 'volume': '44', 'author': 'Yasuura K.', 'year': '1961', 'journal-title': 'J. Inst. Elec. Commun. Eng. Jap.'} / J. Inst. Elec. Commun. Eng. Jap. / New approximate method of eigenvalues for waveguide by Yasuura K. (1961)
  57. {'key': 'e_1_2_1_58_1', 'first-page': '257', 'volume-title': 'Radio Waves and Circuits; Radio Electronics', 'author': 'Yasuura K.', 'year': '1971'} / Radio Waves and Circuits; Radio Electronics by Yasuura K. (1971)
  58. Yasuura K. H.Ikuno On the modified Rayleigh hypothesis and the mode‐matching method International Symposium on Antennas and Propagation Sendai Japan 173–174Institute of Electronics and Communication Engineers of Japan Tokyo Japan 1971.
  59. 10.1109/TAP.1971.1139908
  60. 10.1109/TAP.1971.1140031
Dates
Type When
Created 17 years, 6 months ago (Feb. 6, 2008, 10:06 a.m.)
Deposited 1 year, 10 months ago (Sept. 22, 2023, 1:47 p.m.)
Indexed 11 months, 1 week ago (Sept. 13, 2024, 12:30 a.m.)
Issued 52 years ago (Aug. 1, 1973)
Published 52 years ago (Aug. 1, 1973)
Published Online 12 years, 8 months ago (Dec. 7, 2012)
Published Print 52 years ago (Aug. 1, 1973)
Funders 0

None

@article{Millar_1973, title={The Rayleigh hypothesis and a related least‐squares solution to scattering problems for periodic surfaces and other scatterers}, volume={8}, ISSN={1944-799X}, url={http://dx.doi.org/10.1029/rs008i008p00785}, DOI={10.1029/rs008i008p00785}, number={8–9}, journal={Radio Science}, publisher={American Geophysical Union (AGU)}, author={Millar, R. F.}, year={1973}, month=aug, pages={785–796} }