Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-20T07:01:44.309Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Rayleigh assumption in scattering by a periodic surface. II

Published online by Cambridge University Press:  24 October 2008

R. F. Millar
R. F. Millar
Affiliation:
Radio and Electrical Engineering Division, National Research Council, Ottawa, Canada
Get access Rights & Permissions [Opens in a new window]

Abstract

The Rayleigh assumption, which concerns the domain of validity of a representation for the field scattered by a periodic surface under time-harmonic plane-wave excitation, is re-examined. The analysis employs a technique developed to locate singularities of solutions to the Helmholtz equation. When applied to the surface profile v = b cos κu (-∞ < u < ∞) considered originally by Lord Rayleigh, it is found that the Rayleigh assumption is valid if 0 ≤ κb < 0·448 and is not valid if κb > 0·448; this is more precise than an earlier result. It is shown how these findings may be reconciled with the work of others who have suggested, or concluded, that the Rayleigh assumption is never valid.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 69 , Issue 1 , January 1971 , pp. 217 - 225
Copyright
Copyright © Cambridge Philosophical Society 1971

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Lippmann, B. A.Note on the theory of gratings. J. Opt. Soc. Amer. 43 (1953), 408.CrossRefGoogle Scholar
(2)Millar, R. F.Plane wave spectra in grating theory. II. Scattering by an infinite grating of identical cylinders. Canad. J. Phys. 41 (1963), 2135–2154.CrossRefGoogle Scholar
(3)Millar, R. F.On the Rayleigh assumption in scattering by a periodic surface. Proc. Cambridge Philos. Soc. 65 (1969), 773–791.CrossRefGoogle Scholar
(4)Millar, R. F.The location of singularities of two-dimensional harmonic functions. I. Theory. SIAM J. Math. Anal. 1 (1970), 333–344.CrossRefGoogle Scholar
(5)Millar, R. F.The location of singularities of two-dimensional harmonic functions. II. Applications. SIAM J. Math. Anal. 1 (1970), 345–353.CrossRefGoogle Scholar
(6)Millar, R. F.Singularities of two-dimensional exterior solutions of the Helmholtz equation. Proc. Cambridge Philos. Soc. 69 (1970), 167–188.Google Scholar
(7)Mullar, C.Zur Methode der Strahlungskapazität von H. Weyl. Math. Z. 56 (1952), 80–83.CrossRefGoogle Scholar
(8)Murphy, S. R. and Lord, G. E.Scattering from a sinusoidal surface—a direct comparison of the results of Marsh and Uretsky. J. Acoust. Soc. Amer. 36 (1964), 1598–1599.CrossRefGoogle Scholar
(9)Pavageau, J.Sur la méthode des spectres d'ondes planes dans les problèmes de diffraction. C.R. Acad. Sci. Paris. Sér. B 266 (1968), 1352–1355.Google Scholar
(10)Petit, R. and Cadilhac, M.Étude théorique de la diffraction par un réseau. C.R. Acad. Sci. Paris 259 (1964), 2077–2080.Google Scholar
(11)Petit, R. and Cadilhac, M.Sur la diffraction d'une onde plane par un réseau infiniment conducteur. C.R. Acad. Sci. Paris Sér. B 262 (1966), 468–471.Google Scholar
(12)Petit, R. and Cadilhac, M.Sur la diffraction d'une onde plane par un réseau infiniment conducteur. C.R. Acad. Sci. Paris Sér. B 264 (1967), 1441–1444.Google Scholar
(13)Rayleigh, Lord.The theory of sound, vol. II (Dover Publications; New York, 1945).Google Scholar
(14)Rayleigh, Lord.On the dynamical theory of gratings. Proc. Roy. Soc. Ser. A 79 (1907), 399–416.Google Scholar
(15)Uretsky, J. L.The scattering of plane waves from periodic surfaces. Ann. Physics 33 (1965), 400–427.CrossRefGoogle Scholar
(16)Wilcox, C. H.A generalization of theorems of Rellich and Atkinson. Proc. Amer. Math. Soc. 7 (1956), 271–276.CrossRefGoogle Scholar