Difference between Material dispersion and waveguide dispersion?

[Cerkit]

Can someone please explain the difference between Material dispersion and waveguide dispersion?

 

[Meir Achuz]

"Dispersion" is determined by the relation between wave number
$$k(=2\pi/\lambda)$$ and frequency $$\omega(=2\pi f)$$.
In a medium, the relation is $$k=\sqrt{\epsilon\mu}\omega/c$$.
In a wave guide, it is $$k=[\sqrt{\epsilon\mu}/c]\sqrt{\omega^2-\omega^2_c}$$,
where $$\omega_c$$ depends on the dimensions of the wave guide.