Sharma, Basant Lal On linear waveguides of square and triangular lattice strips: an application of Chebyshev polynomials. (English) ✄ ✂  ✁ Zbl 1390.78026 Sādhanā 42, No. 6, 901-927 (2017). Summary: An analysis of the linear waves in infinitely-long square and triangular lattice strips of identical particles with nearest neighbour interactions for all combinations of fixed and free boundary conditions, as well as the periodic boundary, is presented. Expressions for the dispersion relations and the associated normal modes in these waveguides are provided in the paper; some of which are expressed implicitly in terms of certain linear combinations of the Chebyshev polynomials. The effect of next-nearest-neighbour interaction is also included for the square lattice waveguides. It is found that localized propagating waves, so called surface wave modes, occur in the triangular lattice strips, as well as square lattice strips with next-nearest-neighbour interactions, when either or both boundaries are free. In this paper, the even and odd modes are also discussed separately, wherever applicable. Graphical illustrations of the dispersion curves are included for all waveguides. The discrete waveguides analysed in the paper have broad ap- plications in physics and engineering, including their merit in classical problems in elasticity, acoustics and electromagnetism, as well as recent technological issues involving various transport phenomena in quasi-one-dimensional nano-structures. MSC: 78A50 Antennas, waveguides in optics and electromagnetic theory 33C80 Connections of hypergeometric functions with groups and algebras, and related topics 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 37N35 Dynamical systems in control 37K60 Lattice dynamics; integrable lattice equations Cited in 8 Documents Keywords: dispersion; normal modes; square; triangular; Chebyshev; tridiagonal Full Text: DOI Link References: [1] Kolsky, H, Stress waves in solids, J. Sound Vibr., 1, 88-110, (1964) · Zbl 0203.27502 · doi:10.1016/0022-460X(64)90008-2 [2] Miklowitz J 1978 The theory of elastic waves and waveguides. \textit{Applied mathematics and mechanics.} New York: North Holland [3] Giebe, E; Blechschmidt, E, Experimentelle und theoretische untersuchungen uber dehnungseigenschwingungen von stben und rohren. II, Ann. Phys., 410, 457-485, (1933) · Zbl 59.1433.01 · doi:10.1002/andp.19334100502 [4] Rohrich, K, Ausbreitungsgeschwindigkeit ultraakustischer schwingungen in zylindrischen stben, Z. Phys., 73, 813-832, (1932)· doi:10.1007/BF01344227 [5] Schoeneck, H, Experimentelle untersuchungen der schwingungen zylindrischer einzelkristalle bei hohen elastischen frequenzen, Z. Phys., 92, 390-406, (1934)· doi:10.1007/BF01340823 [6] Chree, C, Longitudinal vibrations of a circular bar, Q. J. Pure Appl. Math., 21, 287-298, (1886) · Zbl 18.0968.01 [7] Pochhammer, L, Ueber die fortp anzungsgeschwindigkeiten kleiner schwingungen in einem unbegrenzten isotropen kreiscylin- der, J. di Reine Angew. Math., 81, 324-336, (1876) · Zbl 08.0641.02 [8] Rayleigh, L, On waves propagated along the plane surface of an elastic solid, Proc. London Math. Soc., s117, 4-11, (1885) · Zbl 17.0962.01 · doi:10.1112/plms/s1-17.1.4 [9] Rayleigh, L, On the free vibrations of an infinite plate of homogeneous isotropic elastic matter, Proc. London Math. Soc. S, 120, 225-237, (1888) · Zbl 21.1040.01 · doi:10.1112/plms/s1-20.1.225 [10] Lamb, H, On the propagation of tremors over the surface of an elastic solid. [abstract], R. Soc. London Pro. Ser. I, 72, 128-130, (1903) · Zbl 34.0859.01 [11] Lamb, H, On waves in an elastic plate, R. Soc. London Proc. Ser. A, 93, 114-128, (1917) · Zbl 46.1232.01 · doi:10.1098/rspa.1917.0008 [12] Brillouin L 1953 \textit{Wave propagation in periodic structures: electric filters and crystal lattices}. New York: Dover · Zbl Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2021 FIZ Karlsruhe GmbH Page 1