The Green's function for the two-dimensional Helmholtz equation in periodic domains

Journal of Engineering Mathematics

Volumn 33, Issue 4, 1998, Pages 377-401

  Linton, C M ^a  

^a LOUGHBOROUGH UNIVERSITY   (United Kingdom)

Author keywords

Green's functions; Helmholtz equation; Periodic domains; Waveguides

Indexed keywords

ACOUSTIC DEVICES; INTEGRAL EQUATIONS; MATHEMATICAL TRANSFORMATIONS; NUMERICAL ANALYSIS; WAVEGUIDES;

HELMHOLTZ EQUATION; PARALLEL PLATE ACOUSTIC WAVEGUIDES;

GREEN'S FUNCTION;

EID: 0032066150     PISSN: 00220833     EISSN: None     Source Type: Journal    
DOI: 10.1023/a:1004377501747     Document Type: Article

References (42)

1. R. Petit (ed), Electromagnetic Theory of Gratings, volume 22 of Topics in Current Physics. Berlin: Springer-Verlag (1980) 284 pp.
2. C. H. Wilcox, Scattering Theory for Diffraction Gratings. New York: Springer-Verlag (1984) 163 pp.
3. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover (1965) 1046 pp.
4. D. Shanks, Non-linear transformations of divergent and slowly-convergent sequences. J. Math. Phys. 34 (1955) 1-42.
5. P. Wynn, On the convergence and stability of the epsilon algorithm. SIAM J. Num. Anal. 3 (1966) 91-122.
6. Noyan Kinayman and M. I. Aksun, Comparative study of acceleration techniques for integrals and series in electromagnetic problems. Radio Science 30 (1995) 1713-1722.
7. G. Barton, Elements of Green's Functions and Propagation. Potentials, Diffusion, and Waves. Oxford: Clarendon Press (1989) 465 pp.
8. D. V. Evans and C. M. Linton, Edge waves along periodic coastlines. Q. Jl. Mech. Appl. Math. 46 (1993) 642-656.
9. V. Twersky, On the scattering of waves by an infinite grating. IRE Trans. on Antennas and Propagation 4 (1956) 330-345.
10. S. Singh and R. Singh, On the use of ρ-algorithm in series acceleration. IEEE Trans. Antennas Propagat. 39 (1991) 1514-1517.
11. C. M. Linton and D. V Evans, The radiation and scattering of surface waves by a vertical circular cylinder in a channel. Phil. Trans. R. Soc. Lond. A 338 (1992) 325-357.
12. James R. Wait, Reflection from a wire grid parallel to a conducting plane. Can J. Phys. 32 (1954) 571-579.
13. R. Lampe, P. Klock and P. Mayes, Integral transforms useful for the accelerated summation of periodic, free-space Green's functions. IEEE Trans. Microwave Theory Tech. 33 (1985) 734-736.
14. J. D. Achenbach and Z. L. Li, Reflection and transmission of scalar waves by a periodic array of screens. Wave Motion 8 (1986) 225-234.
15. A. E. Heins, The Green's function for periodic structures in diffraction theory with an application to parallel plate media, I. J. Math. Mech. 6 (1957) 401-426.
16. P. M. Van den Berg, Diffraction theory of a reflection grating. Appl. Sci. Res. 24 (1971) 261-293.
17. Roy E. Jorgenson and Raj Mittra, Efficient calculation of the free-space periodic Green's function. IEEE Trans. Antennas Propagat. 38 (1990) 633-642.
18. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products. New York: Academic Press (1965) 1160pp.
19. N. A. Nicorovici, R. C. McPhedran and R. Petit, Efficient calculation of the Green's function for electromagnetic scattering by gratings. Phys. Rev. E 49 (1994) 4563-4577.
20. L. Lewin, Polylogarithms and Associated Functions. New York: North Holland (1981) 359 pp.
21. F. Zolla and R. Petit, Sur le calcul de la fonction de Green en théorie des résaux. In: OHD'95. University of Zaragosa (1995).
22. S. Singh, W. F. Richards, J. R. Zinecker, and D. R. Wilton, Accelerating the convergence of series representing the free space periodic Green's function. IEEE Trans. Antennas Propagat. 38 (1990) 1958-1962.
23. R. M. Shubair and Y. L. Chow, Efficient computation of the periodic Green's function in layered dielectric media. IEEE Trans. Microwave Theory Tech. 41 (1993) 498-502.
24. R. W. Yeung and S. H. Sphaier, Wave-interference effects on a truncated cylinder in a channel. J. Engng. Math. 23 (1989) 95-117.
25. A. W. Mathis and A.F. Peterson, A comparison of acceleration procedures for the two-dimensional periodic Green's function. IEEE Trans. Antennas Propagat. 44 (1996) 567-571.
26. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms, volume 1. New York: McGraw-Hill (1954) 391 pp.
27. C. M. Linton and D. V. Evans, The interaction of waves with a row of circular cylinders. J. Fluid Mech. 251 (1993) 687-708.
28. P. McIver and G. S. Bennett, Scattering of water waves by axisymmetric bodies in a channel. J. Engng. Math. 27 (1993) 1-29.
29. R. E. Crandall and J. F. Delford, The potential within a crystal-lattice. J. Phys. A 20 (1987) 2279-2292.
30. R. Petit, From functional analysis to the method of ficticious sources in E.M. diffraction. In: MMET'94. Kharkov (1994).
31. N. A. Nicorovici and R. C. McPhedran, Lattice sums for off-axis electromagnetic scattering by gratings. Phys. Rev. E 50 (1994) 3143-3160.
32. V. Twersky, Elementary function representation of Schlömilch series. Arch. Rational Mech. Anal. 8 (1961) 323-332.
33. P. P. Ewald, Die Berechnung optischer und elektrostatischen Gitterpotentiale. Ann. Phys. 64 (1921) 253-268.
34. M. Born and Kun Huang, Dynamical Theory of Crystal Lattices. The International Series of Monographs on Physics. Oxford: Clarendon Press (1954).
35. K. E. Jordan, G. R. Richter and Ping Sheng, An efficient numerical evaluation of the Green's function for the Helmholtz operator on periodic structures. J. Comput. Phys. 63 (1986) 222-235.
36. A. Sugiyama, Method for rapid convergence of lattice sums by the use of Fourier transforms. J. Phys. Soc. Jpn 53 (1984) 1624-1633.
37. F. S. Han and B. Segall, Energy bands in periodic lattices-Green's function method. Phys. Rev. 124 (1961) 1786-1796.
38. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. Cambridge: Cambridge University Press, 4th edition (1927) 608 pp.
39. W. Gautschi, Efficient computation of the complex error function. SIAM J. Num. Anal. 7 (1970) 187-198.
40. D. V. Evans and M. Fernyhough, Edge waves along periodic coastlines. Part 2. J. Fluid Mech. 297 (1995) 301-325.
41. C. M. Linton and D. V. Evans, Integral equations for a class of problems concerning obstacles in waveguides. J. Fluid Mech. 245 (1992) 349-365.
42. J. N. Newman, Approximation of free-surface Green functions. In: P. A. Martin and G. R. Wickham (eds.), Wave Asymptotics. Cambridge University Press (1992) pp. 107-135.