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I'm trying to understand dispersion relations in general.

I know that for a simple wave like a light wave there is a 'constant phase' so the dx/dt is equal to the ratio of the angular frequency (omega) by the wave vector (k).

However

**what does a 'constant phase' mean? How can I visualize the propagation of a wave with this characteristic, compared to other waves?**

On the other hand, then considering a crystal, the dispersion relation is defined as the 'delta_omega/delta_k'

**So I can't really visualize what this relation means.**

I'd appreciate any help in helping me intuitively understand these concepts.