In this derivation of phonon solutions, everywhere, we are forcefully assuming the wavelike characteristics along the length of the chain. While all we can deduce for finding out the fundamental frequencies is that the solution will be periodic in time, and solution should be of the form $\exp(i\omega t)$, I am not getting how the derivers arrive directly at $\exp(ikx - i\omega t)$. That one is in Kittel.
In the figure below, Here also, somehow, $q$ has been deliberately has been involved linking it to distance. PS: $n$ is a measure of distance along the chain.