This work reports some recent advances in diffraction theory by canonical shapes like wedges or cones with impedancetype boundary conditions. Our basic aim in the present paper is to demonstrate that functional difference equations of the second order deliver a very natural and efficient tool to study such a kind of problems (For a thorough and up-to-date overview of the scattering and diffraction in general the readers are referred to a special section of the journal ``Radio Science'' edited by Uslenghi .). To this end we consider two problems: diffraction of a normally incident plane electromagnetic wave by an impedance wedge whose exterior is divided into two parts by a semi-infinite impedance sheet and diffraction of a plane acoustic wave by a right-circular impedance cone. In both cases the problems can be formulated in a traditional fashion as boundary-value problems of the scattering theory. For the first problem the Sommerfeld-Malyuzhinets technique enables one to reduce it to a problem for a vectorial system of functional Malyuzhinets equations. Then the system is transformed to uncoupled second-order functional difference-equations (SOFDE) for each of the unknown spectra. In the second problem the incomplete separation of variables leads directly to a functional difference-equation of the second order. Hence, it is remarkable that in both cases the key mathematical tool is an SOFDE which is an analog of a second-order differential equation with variable coefficients. The latter is reducible to an integral equation which is known to be the most traditional tool for its solution. It has recently been recognised that reducing SOFDEs to integral equations is also one of the most efficient approaches for their study. The integral equations which are developed for the problems at hand are both of the second kind and obey Fredholm property. In the problem of diffraction by a wedge the generalised Malyuzhinets function is exploited on the preliminary step then ``inversion'' of a simple difference operator with constant coefficients leads to an integral equation of the second kind. The corresponding integral operator is represented as a sum of the identical operator and a compact one . However, in the second problem the situation is slightly different: the integral operator can be represented by a sum of the boundedlyinvertible (Dixon's operator) and compact operators. This situation was earlier considered by Bernard in his study of diffraction by an impedance cone, and important advances have been made (see [3-6]). The Fredholm property is crucial for the elaboration of different numerical schemes. In our cases we exploited direct numerical approaches based on the quadrature formulae and computed the farfield asymptotics for the problems at hand. Various numerical results are demonstrated and discussed.
2. Lyalinov, M. A. and N. Y. Zhu, "A solution procedure for second-order difference equations and its application to electromagneticwave diffraction in a wedge-shaped region," Proc. R. Soc. Lond. A, Vol. 459, No. 2040, 3159-3180, 2003.
3. Bernard, J. M. L., Methode analytique et transformees fonctionnelles pour la diffraction d’ondes par une singularite conique: equation integrale de noyau non oscillant pour le cas d’impedance constante, rapport CEA-R-5764, Editions Dist-Saclay, 1997.
4. Bernard, J. M. L. and M. A. Lyalinov, "Diffraction of acoustic waves by an impedance cone of an arbitrary cross-section," Wave Motion, Vol. 33, 155-181, 2001.
5. Antipov, Y. A., "Diffraction of a plane wave by a circular cone with an impedance boundary condition," SIAM J. Appl. Math., Vol. 62, No. 4, 1122-1152, 2002.
6. Lyalinov, M. A., "Diffraction of a plane wave by an impedance cone," J. Math. Sci., Vol. 127, No. 6, 2446-2460, 2005.
7. Zhu, N. Y. and M. A. Lyalinov, "Diffraction of a normally incident plane wave by an impedance wedge with its exterior bisected by a semi-infinite impedance sheet," IEEE Trans. Antennas Propagat., Vol. 52, No. 10, 2753-2758, 2004.
8. Buldyrev, V. S. and M. A. Lyalinov, Mathematical Methods in Modern Electromagnetic Diffraction, Science House, Tokyo, 2001.
9. Maliuzhinets, G. D., "Excitation, reflection and emission of surface waves from a wedge with given face impedances," Sov. Phys. Dokl., Vol. 3, No. 10, 752-755, 1958.
10. Babich, V. M., M. A. Lyalinov, and V. E. Grikurov, Diffraction Theory:The Sommerfeld-Malyuzhinets Technique, Alpha Science, Oxford, U.K., 2007.
11. Bobrovnikov, M. S. and V. V. Fisanov, Diffraction of Waves in Angular Regions, Tomsk University Press, Tomsk, Russia, 1988.
12. Avdeev, A. D., "On a special function in the problem of diffraction by a wedge in an anisotropic plasma," J. Commun. Tech. Electric., Vol. 39, No. 1, 70-78, 1994.
13. Lyalinov, M. A. and N. Y. Zhu, "Diffraction of a skew incident plane wave by an anisotropic impedance wedge --- A class of exactly solvable cases," Wave Motion, Vol. 30, No. 3, 275-288, 1999.
14. Tuzhilin, A. A., "The theory of Malyuzhinets’ inhomogeneous functional equations," Diff. Uravn., Vol. 9, 1875-1888, 1973.
15. Zhu, N. Y. and M. A. Lyalinov, "Numerical study of diffraction of a normally incident plane wave at a hollow wedge with different impedance sheet faces by the method of parabolic equation," Radio Science, Vol. 36, No. 4, 507-515, 2001.
16. Lyalinov, M. A. and N. Y. Zhu, "Acoustic scattering by a circular semi-transparent conical surface," J. Eng. Math., Vol. 59, No. 4, 385-398, 2007.
17. Lyalinov, M. A. and N. Y. Zhu, "Diffraction of a skew incident plane electromagnetic wave by an impedance wedge," Wave Motion, Vol. 44, No. 1, 21-43, 2006.
18. Lyalinov, M. A. and N. Y. Zhu, "Diffraction of a skew incident plane electromagnetic wave by a wedge with anisotropic impedance faces," Radio Science, Vol. 42, 2007.
19. Bernard, J. M. L., M. A. Lyalinov, and N. Y. Zhu, "Analytical-numerical calculation of diffraction coefficients for a circular impedance cone," IEEE Trans. Antennas Propagat., submitted, 2007.
20. Lyalinov, M. A., V. P. Smyshlyaev, and N. Y. Zhu, "Scattering of a plane electromagnetic wave by a circular hollow cone with thin semi-transparent walls,", under preparation, 2007.