DOI: 10.1017/s0305004100003613. On the Rayleigh assumption in scattering by a periodic surface

On the Rayleigh assumption in scattering by a periodic surface

10.1017/s0305004100003613

Crossref journal-article

Cambridge University Press (CUP)

Mathematical Proceedings of the Cambridge Philosophical Society (56)

Abstract

AbstractIn treating plane wave scattering by a periodic surface, Lord Rayleigh (10) assumed that the discrete, outgoing and evanescent plane wave representation for the scattered field was valid on the surface itself. Recently, this Rayleigh assumption has been questioned and criticized. For the surface y = b cos kx on which the total field vanishes, Petit and Cadilhac(8) have demonstrated its invalidity when Kb > 0·448. The present paper discusses scattering of a wave, incident from y > 0, by an analytic periodic surface y = f(x) ( – ∞ < x < ∞), and shows that the Rayleigh assumption is valid if and only if the solution can be continued analytically across the boundary at least to the line y = minf(x). Conformal mapping and results relating to the analytic continuation of solutions to elliptic partial differential equations reduce the problem to one involving the location of singularities and critical points of a potential Green's function. Provided that the perturbation of the surface from a plane is sufficiently gentle, the validity of the Rayleigh assumption is established. For the surface y = b cos kx it is shown that the assumption is valid if Kb < γ, where γ is a positive number no greater than 0·448, the precise value of which is unknown. Possible extensions of the analysis to different or more general situations are suggested.

Bibliography

Millar, R. F. (1969). On the Rayleigh assumption in scattering by a periodic surface. Mathematical Proceedings of the Cambridge Philosophical Society, 65(3), 773â791.

Authors 1

R. F. Millar (first)

References 14 Referenced 125

{'key': 'S0305004100003613_ref014', 'volume-title': 'The location of critical points of analytic and harmonic functions', 'volume': '34', 'author': 'Walsh', 'year': '1950'} / The location of critical points of analytic and harmonic functions by Walsh (1950)
10.1016/0003-4916(65)90269-1
10.1121/1.1918690
{'key': 'S0305004100003613_ref010', 'volume-title': 'The theory of sound', 'volume': 'II', 'author': 'Rayleigh', 'year': '1945'} / The theory of sound by Rayleigh (1945)
10.1121/1.1919253
{'key': 'S0305004100003613_ref006', 'first-page': '2135', 'article-title': 'Plane wave spectra in grating theory II', 'volume': '41', 'author': 'Millar', 'year': '1963', 'journal-title': 'Ganad. J. Phys.'} / Ganad. J. Phys. / Plane wave spectra in grating theory II by Millar (1963)
10.1364/JOSA.43.000408 / J. Opt. Soc. Amer. / Note on the theory of gratings by Lippmamst (1953)
{'key': 'S0305004100003613_ref003', 'volume-title': 'Foundations of potential theory', 'author': 'Kellogg', 'year': '1953'} / Foundations of potential theory by Kellogg (1953)
10.1098/rspa.1907.0051
{'key': 'S0305004100003613_ref008', 'first-page': '468', 'article-title': "Sur la diffraction d'une onde plane par un réseau infiniment conducteur", 'volume': '262', 'author': 'Petit', 'year': '1966', 'journal-title': 'C.R. Acad. Sci. Paris'} / C.R. Acad. Sci. Paris / Sur la diffraction d'une onde plane par un réseau infiniment conducteur by Petit (1966)
{'key': 'S0305004100003613_ref002', 'volume-title': 'Partial differential equations', 'author': 'Garabedian', 'year': '1964'} / Partial differential equations by Garabedian (1964)
10.1121/1.1918842
{'key': 'S0305004100003613_ref009', 'volume-title': 'Partial differential equations', 'author': 'Petrovsky', 'year': '1954'} / Partial differential equations by Petrovsky (1954)
{'key': 'S0305004100003613_ref001', 'volume-title': 'The scattering of electromagnetic waves from rough surfaces', 'volume': '4', 'author': 'Beckmann', 'year': '1963'} / The scattering of electromagnetic waves from rough surfaces by Beckmann (1963)

Dates

Type	When
Created	16 years, 9 months ago (Nov. 7, 2008, 10:32 a.m.)
Deposited	6 years, 2 months ago (June 1, 2019, 4:16 p.m.)
Indexed	3 months ago (May 20, 2025, 9:51 a.m.)
Issued	56 years, 3 months ago (May 1, 1969)
Published	56 years, 3 months ago (May 1, 1969)
Published Online	16 years, 9 months ago (Oct. 24, 2008)
Published Print	56 years, 3 months ago (May 1, 1969)

Funders 0

None

BibTeX

@article{Millar_1969, title={On the Rayleigh assumption in scattering by a periodic surface}, volume={65}, ISSN={1469-8064}, url={http://dx.doi.org/10.1017/s0305004100003613}, DOI={10.1017/s0305004100003613}, number={3}, journal={Mathematical Proceedings of the Cambridge Philosophical Society}, publisher={Cambridge University Press (CUP)}, author={Millar, R. F.}, year={1969}, month=may, pages={773–791} }

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