A group theoretic approach to generalized harmonic vibrations in a one dimensional lattice - J. N. Boyd; P. N. Raychowdhury (International Journal of Mathematics and Mathematical Sciences)
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.