Wave Propagation in Linear and Nonlinear Periodic Media Analysis and
The contributions in this volume present both the theoretical background and an overview of the state-of-the art in wave propagation in linear and nonlinear periodic media in a consistent format. They combine the material issued from a variety of engineer
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Series Editors: The Rectors Friedrich Pfeiffer - Munich Franz G. Rammerstorfer - Wien Jean Salençon - Palaiseau
The Secretary General
Executive Editor
The series presents lecture notes, monographs, edited works and and Applied Mathematics. ! and technical community results obtained in some of the activities " #$ % % &
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2%( 2 9 0)*' + )+ + '%7 ) %(9 ;/* *% ' E/ρ. This special case corresponds to a defect resonance mode, for which the upper and lower bars within the interface do not move, whereas the middle bar vibrates while being connected to the upper and lower bars via elastic links of stiffness γ. The corresponding equation of motion for such a defect mode leads to (Ek 2 + 2γ − ρω 2 )U2 = 0,
(80)
which is fully consistent with the observed peak in transmission. The computations in Fig. 12 are based on the algorithm of Section 0.6, and, in particular, equations (65), (64), (79) and (60). In these numerical computations, we have taken the linear mass density ρ = 1000 kg/m and the longitudinal stiffness E = 5 × 106 kN for the bars, the shear stiffness γ = 0.05 GPa for the vertical elastic links and the mass density 3 ρamb = 1000 kg/m , Poisson ratio νamb = 0.3 and Young’s modulus Eamb = 1 GPa for the ambient elastic medium Ω+ ∪ Ω− ; the pressure
Waves and Defect Modes in Structured Media
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Figure 12. The reflected and transmitted energies ER and ET for a threebar interface as functions of the wave number k and the apparent velocity c. Diagrams (a) and (c) show the surface plots of ER and ET . Diagrams (b) and (d) present the corresponding contour plots. The material parameters for the interface are ρ = 1000 kg/m, E = 5 × 106 kN. The ambient medium is of the mass density ρamb = 1000 kg/m3 , Poisson ratio νamb = 0.3 and Young’s modulus Eamb = 1 GPa; the horizontal stiffness of the vertical links is γ = 0.05 GPa/m2.
and shear wave speeds in the ambient medium are respectively α = 1160.24 m/s and β = 620.174 m/s. In conclusion we emphasise on the connection between physical problems of different origins. Dispersion of waves in media with boundaries or built-in micro-structure is a fundamental phenomenon, which occurs in problems of acoustics, models of water waves, simple systems involving onedimensional harmonic oscillators, as well as complex elastic systems leading
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A.B. Movchan, M. Brun, N.V. Movchan
to analysis of full vector problems of elasticity both for discrete and continuous systems. Indeed, if the system is infinite and periodic, the analysis is reduced to an elementary cell and dispersion properties of Bloch-Floquet waves can be represented via dispersion diagrams, where stop bands may exist and hence one c
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