Abstract
Propagation of modal solutions in a smoothly inhomogeneous Timoshenko beam is considered. An asymptotic description is given to the interaction of high-frequency bending and shear modes that occurs if their phase velocities intersect at some point at a nonzero angle. A similar problem on the propagation of discontinuities is also considered. The results are presented in the generalized form, so that they have applications to problems on the interaction of other types of waves, e.g., water waves, electromagnetic waves, etc. Bibliography: 19 titles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
K. F. Graff, Wave Motion in Elastic Solids, Clarendon Press, Oxford (1975).
A. Robinson, “Transient stresses in beams of variable characteristics,” Quart. J. Mech. Appl. Math., 10, No. 2, 148–159 (1957).
H. Buttler, “Theoretische Untersuchungen erzwungener elastischer Schuffsk orperschwingungen mit dem Ziel der Vorausbestimmung auftrender Amplituden in Rezonanzfall,” Schiffbauforschung, 5, No. 3–4, 209–214 (1966).
C. G. Yang, “Propagation of discontinuous waves in nonuniform Timoshenko beams,” Trans. ASME, E 33, 706–708 (1966).
L. D. Landau and E. M. Lifshitz, Quantum Mechanics. Non-relativistic Theory, Pergamon Press, London-Paris (1959).
V. S. Buslaev and L. A. Dmitrieva, "Adiabatic perturbations of a periodic potential. II,” Theor. Math. Phys., 73, No. 3, 1320–1329 (1987).
K. G. Budden, Radio Waves in the Ionosphere, Cambridge University Press, Cambridge (1961).
N. S. Bellyustin, “On a linear interaction of electromagnetic waves in an inhomogeneous magnetoactive plasma,” Radiofizika, 21, No. 11, 1563–1571 (1978).
T. I. Bichutskaya and V. V. Novikov, “Interaction of normal waves in a plane irregular waveguide in the presence of a degenerate point,” Radiofizika, 22, No. 7, 860–869 (1979).
M. V. Perel, “Overexcitation of modes in an anisotropic Earth-ionosphere waveguide on transequatorial paths in the presence of two degeneracy points,” Radiofizika, 33, No. 11, 882–889 (1990).
I. A. Molotkov and A. S. Starkov, “Local resonance interaction of waves in coupled waveguides,” Probl. Mat. Fiz., 10, 164–176 (1982).
M. M. Popov and I. N. Schitov, “What may happen if the velocities of qS waves in an anisotropic inhomogeneous medium coincide at a point?" in: Wave Inversion Technology, Report WITREPORT 98 (1999), pp. 287–291.
A. P. Kiselev and M. V. Perel, “The asymptotic theory of a local degeneration of Love and Rayleigh modes,” in: Proceedings of the Day on Diffraction 99 (1998), pp. 86–90.
V. V. Kucherenko, “Asymptotics of the solution of the system A(x,-ih ∂/∂x)u = 0 as h → 0 in the case of characteristics of variable multiplicity,” Izv. AN SSSR, Ser. Mat., 8, 631–666 (1974).
V. V. Kucherenko and Yu. V. Osipov, "Asymptotic solutions of ordinary differential equations with degenerating symbols,” Mat. Sb., 118, No. 1, 74–103 (1982).
V. M. Babič and N. Ya. Kirpičnikova, Boundary Layer Method in Diffraction Problems, Springer-Verlag, Berlin-New York (1979).
J. Hadamard, Le Probléme de Cauchy et les Equations en Derivées Partielles Linéares Hyperboliques, Hermann, Paris (1930).
G. D. Birkhoff, “On the asymptotic character of the solutions of certain linear differential equations containing a parameter,” Trans. Amer. Math. Soc., 9, No. 2, 219–231 (1908).
V. M. Babich and O. V. Izotova, On the Solution of Problems of Mathematical Physics in Generalized Functions, St. Petersburg University Press (1998).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Perel, M.V., Fialkovsky, I.V. & Kiselev, A.P. Resonance Interaction of Bending and Shear Modes in a Nonuniform Timoshenko Beam. Journal of Mathematical Sciences 111, 3775–3790 (2002). https://doi.org/10.1023/A:1016354430209
Issue Date:
DOI: https://doi.org/10.1023/A:1016354430209