While studying the brillouin zone I came across the dispersion relation and the group velocity. The group velocity is given by v=dω/dκ, I understand this in the sense of beats where it is Δω/Δκ and I understand that the group velocity is the propagation speed of the envelope function. However I don't understand why it can be said to be the rate of change of frequency as a function of a change in the wave vector (using the dispersion relation). The units work out but I just can't seem to picture it. However the phase velocity makes perfect sense to me. Any way to intuitively understand this would be appreciated. Maybe some kind of mathematical proof without referring to beats would help. Thanks a lot!