A group theoretic approach to generalized harmonic vibrations in a one dimensional lattice

Beginning with a group theoretical simplification of the equations of motion for harmonically coupled point masses moving on a fixed circle, we obtain the natural frequencies of motion for the array. By taking the number of vibrating point masses to be very large, we obtain the natural frequencies o...

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Main Authors: J. N. Boyd, P. N. Raychowdhury
Format: Article
Language:English
Published: Wiley 1986-01-01
Series:International Journal of Mathematics and Mathematical Sciences
Subjects:
Online Access:http://dx.doi.org/10.1155/S0161171286000169
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author J. N. Boyd
P. N. Raychowdhury
author_facet J. N. Boyd
P. N. Raychowdhury
author_sort J. N. Boyd
collection DOAJ
description Beginning with a group theoretical simplification of the equations of motion for harmonically coupled point masses moving on a fixed circle, we obtain the natural frequencies of motion for the array. By taking the number of vibrating point masses to be very large, we obtain the natural frequencies of vibration for any arbitrary, but symmetric, harmonic coupling of the masses in a one dimensional lattice. The result is a cosine series for the square of the frequency, fj2=1π2∑ℓ=0sa(ℓ)cosℓβ where 0<β=2πjN≤2π, j∈{1,2,3,…,N} and a(ℓ) depends upon the attractive force constant between the j-th and (j+ℓ)-th masses. Lastly, we show that these frequencies will be propagated by wave forms in the lattice.
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spelling doaj-art-952ab32b8a7f4d75b54de9d1771821702025-02-03T01:27:47ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251986-01-019113113610.1155/S0161171286000169A group theoretic approach to generalized harmonic vibrations in a one dimensional latticeJ. N. Boyd0P. N. Raychowdhury1Department of Mathematical Sciences, Virginia Commonwealth University, Richmond 23284, Virginia, USADepartment of Mathematical Sciences, Virginia Commonwealth University, Richmond 23284, Virginia, USABeginning with a group theoretical simplification of the equations of motion for harmonically coupled point masses moving on a fixed circle, we obtain the natural frequencies of motion for the array. By taking the number of vibrating point masses to be very large, we obtain the natural frequencies of vibration for any arbitrary, but symmetric, harmonic coupling of the masses in a one dimensional lattice. The result is a cosine series for the square of the frequency, fj2=1π2∑ℓ=0sa(ℓ)cosℓβ where 0<β=2πjN≤2π, j∈{1,2,3,…,N} and a(ℓ) depends upon the attractive force constant between the j-th and (j+ℓ)-th masses. Lastly, we show that these frequencies will be propagated by wave forms in the lattice.http://dx.doi.org/10.1155/S0161171286000169harmonic couplingfrequencies of motionwave formslattice.
spellingShingle J. N. Boyd
P. N. Raychowdhury
A group theoretic approach to generalized harmonic vibrations in a one dimensional lattice
International Journal of Mathematics and Mathematical Sciences
harmonic coupling
frequencies of motion
wave forms
lattice.
title A group theoretic approach to generalized harmonic vibrations in a one dimensional lattice
title_full A group theoretic approach to generalized harmonic vibrations in a one dimensional lattice
title_fullStr A group theoretic approach to generalized harmonic vibrations in a one dimensional lattice
title_full_unstemmed A group theoretic approach to generalized harmonic vibrations in a one dimensional lattice
title_short A group theoretic approach to generalized harmonic vibrations in a one dimensional lattice
title_sort group theoretic approach to generalized harmonic vibrations in a one dimensional lattice
topic harmonic coupling
frequencies of motion
wave forms
lattice.
url http://dx.doi.org/10.1155/S0161171286000169
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