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Research Papers

Number of Wavevectors for Each Frequency in a Periodic Structure

[+] Author and Article Information
Farhad Farzbod

Mem. ASME
Department of Mechanical Engineering,
University of Mississippi,
201A Carrier Hall,
University, MS 38677
e-mail: farzbod@olemiss.edu

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received September 6, 2016; final manuscript received March 16, 2017; published online June 22, 2017. Assoc. Editor: Mahmoud Hussein.

J. Vib. Acoust 139(5), 051006 (Jun 22, 2017) (8 pages) Paper No: VIB-16-1450; doi: 10.1115/1.4036466 History: Received September 06, 2016; Revised March 16, 2017

Periodic structures have interesting acoustic and vibration properties making them suitable for a wide variety of applications. In a periodic structure, the number of frequencies for each wavevector depends on the degrees-of-freedom of the unit cell. In this paper, we study the number of wavevectors available at each frequency in a band diagram. This analysis defines the upper bound for the maximum number of wavevectors for each frequency in a general periodic structure which might include damping. Investigation presented in this paper can also provide an insight for designing materials in which the interaction between unit cells is not limited to the closest neighbor. As an example application of this work, we investigate phonon dispersion curves in hexagonal form of boron nitride to show that first neighbor interaction is not sufficient to model dispersion curves with force-constant model.

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Figures

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Fig. 1

Dispersion curve for a general diatomic chain

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Fig. 2

A sample two-dimensional structure with out of plane motion and some internal masses

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Fig. 3

Two-dimensional structure with out of plane motion

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Fig. 4

Boron nitride in its hexagonal crystalline form (h-BN) and the unit cell displacements

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Fig. 5

Phonon dispersion curves of h-BN calculated by Wang et al. [45] (solid lines) and X-ray scattering (circles). The close-up of the box in the bottom left part of the graph is depicted on the right. The straight line at 0.32 THz shows an example value of ω for which we have more than six wavevectors.

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