I'm studying Debye model for the specific heat capacity of solids. In class we found first of all that the number of modes per frequency is proportional to the frequency squared. We found that there is a maximum frequency, which is the Debye frequency. And then we derived the formula for the total internal energy, and studying its limit for low temperatures we found that $$c_{V}=\frac{12\pi^4R}{5}\left(\frac{T}{T_{D}}\right)^{3},$$ where $T_{D}$ is the Debye temperature. Of course we underlined that one of the hypothesis for this model is that the solid is considered as a continuum, so its vibrations propagate making the atoms vibrate together, therefore the wavelength has to be much longer than the distance between atoms.
After that, we analysed a one-dimensional lattice of atoms and derived the dispersion relation $$\omega^2=\frac{4b}{M}\sin^2\left(\frac{ka}{2}\right),$$ where $b$ is the elastic constant, $M$ is the mass of the atom, $k$ is the wave vector and a is the distance between two atoms. Of course if $k$ is small (meaning $\lambda$ is big, so we are in the hypothesis of the Debye model) this relation becomes $$\omega=\sqrt\frac{b}{M}ka.$$ We then concluded that $\omega$ is linear in $k$, as predicted by Debye model.
What I don't understand is why we can say this, where Debye model predicts that there is a linear relationship between $\omega$ and $k$.
