with m being the mass of each atom, κ κ the spring constant for the harmonic oscillator, and a still being the spacing between atoms in the ground state. After plotting this relation, Debye's estimation of the cut-off wavelength based on the linear assumption remains accurate, because for every wavenumber bigger than π π / a (that is, for λ λ is smaller than 2 a ), a wavenumber that is smaller than π π / a could be found with the same angular frequency. This means the resulting physical manifestation for the mode with the larger wavenumber is indistinguishable from the one with the smaller wavenumber. Therefore, the study of the dispersion relation can be limited to the first Brillouin zone k ∈ ∈ [ − − π π a , π π a ] without any loss of accuracy or information. This is possible because the system consists of discretized points, as is demonstrated in the animated picture. Dividing the dispersion relation by k and inserting π π / a for k , we find the speed of a wave with k = π π / a to be v s ( k = π π / a ) = 2 a π π κ κ m .