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...
Saved in:
| Main Authors: | , |
|---|---|
| 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 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832560403361562624 |
|---|---|
| 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. |
| format | Article |
| id | doaj-art-952ab32b8a7f4d75b54de9d177182170 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1986-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| 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 |
| work_keys_str_mv | AT jnboyd agrouptheoreticapproachtogeneralizedharmonicvibrationsinaonedimensionallattice AT pnraychowdhury agrouptheoreticapproachtogeneralizedharmonicvibrationsinaonedimensionallattice AT jnboyd grouptheoreticapproachtogeneralizedharmonicvibrationsinaonedimensionallattice AT pnraychowdhury grouptheoreticapproachtogeneralizedharmonicvibrationsinaonedimensionallattice |