Editing Debye model (section)

===Alternative derivation===
[[File:CPT-sound-nyquist-thereom-1.5percycle.svg|thumb|250px|The physical result of two waves can be identical when at least one of them has a wavelength that is bigger than twice the initial distance between the masses.]]

For a one-dimensional chain, the formula for the Debye frequency can also be reproduced using a theorem for describing [[aliasing]]. The [[Nyquist–Shannon sampling theorem]] is used for this derivation, the main difference being that in the case of a one-dimensional chain, the discretization is not in time, but in space.

The cut-off frequency can be determined from the cut-off wavelength. From the sampling theorem, we know that for wavelengths smaller than <math> 2a </math>, or twice the sampling distance, every mode is a repeat of a mode with wavelength larger than <math> 2a </math>, so the cut-off wavelength should be at <math> \lambda_{\rm D} = 2 a </math>. This results again in <math> k_{\rm D} = \frac{2 \pi}{\lambda_D} = \pi / a </math>, rendering
<math display="block"> \omega_{\rm D} = \frac {\pi v_{\rm s}}{a} .</math>

It does not matter which dispersion relation is used, as the same cut-off frequency would be calculated.