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Two Families of Modal Waves for Periodic Structures with Two Field Functions: A Cayleigh-Hamilton Approach

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This paper aims at providing an analytical study of periodic structures depending on two field functions, using the Cayleigh-Hamilton theorem. This approach allows to emphasize the role played in the constitution of modal waves (i.e. guided and surface modal waves), on one hand by the period cell, and on the other hand by the number of periods of the structure. Introducing the formalism of transfer matrix and the theory of Floquet waves leads, for guided modal waves, to two families of modal waves: the structure modal waves, which take place in the passing bands and which are strongly dependent on the number of periods in the structure, and the period modal waves, which take place in the stopping bands and which depend only on the period cell and on the nature of the external condition. In the case of semi-infinite periodic structures, surface modal waves are found, despite of the fact that all the layers are fluid media. Some results are also given when the two external boundary conditions are different one from the other, in particular, in the case of structures for which the period cell of the last period is modified (one layer added or suppressed). At low frequencies, the "effective" continuous medium (homogenization technique) appears to be anisotropic. In order to make it easier to understand, the presentation is focused on the case of fluid-type periodically layered structures, but the same conclusions can be drawn for any periodic structure described with two field functions (2nd-order transfer matrix): one-dimensional periodic lattice such as diatomic chain of coupled spheres for example, SH waves in isotropic layers, fluid layers, etc. In the case of one-dimensional structures, these results concern eigen vibrations.
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Document Type: Research Article

Publication date: November 1, 2007

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  • Acta Acustica united with Acustica, published together with the European Acoustics Association (EAA), is an international, peer-reviewed journal on acoustics. It publishes original articles on all subjects in the field of acoustics, such as general linear acoustics, nonlinear acoustics, macrosonics, flow acoustics, atmospheric sound, underwater sound, ultrasonics, physical acoustics, structural acoustics, noise control, active control, environmental noise, building acoustics, room acoustics, acoustic materials, acoustic signal processing, computational and numerical acoustics, hearing, audiology and psychoacoustics, speech, musical acoustics, electroacoustics, auditory quality of systems. It reports on original scientific research in acoustics and on engineering applications. The journal considers scientific papers, technical and applied papers, book reviews, short communications, doctoral thesis abstracts, etc. In irregular intervals also special issues and review articles are published.
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