# Harmonic1DMD:Background:Harmonic 1DMD Analytical Solution

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An analytical solution has been developed that describes Lattice Vibrations of 1D atom chains. The solution comes in the form:

${\displaystyle u(na,t)\propto e^{i(kna-\omega t)}}$

where ${\displaystyle u(na,t)}$ is displacement from position na at time t.

This equation can then be rewritten as:

${\displaystyle u\left(na,t\right)=\cos {(kna-\omega t)}+i\sin {(kna-\omega t)}}$

From this you can find the displacement of any atom, at position na and at time t, given k and ${\displaystyle \omega }$. By setting the proper conditions (periodic boundary), and assuming harmonic interactions (like springs), k and ${\displaystyle \omega }$ become the following:

${\displaystyle k={\frac {2\pi s}{aN}}}$ (s integer)

${\displaystyle \omega ={\sqrt {\frac {2K[1-\cos {(ka)}]}{M}}}={\sqrt {\frac {4K}{M}}}\mid \sin {(ka/2)\mid }}$