Significance of Group velocity Equation

$$v_g\equiv\frac{\partial\omega}{\partial k}.$$ The above equation is for Group velocity of waves, so what is the Physical Interpretation of this Equation? As we know $$k$$ is wave number which shows how many times the wave repeats itself and $$\omega$$ is the angular frequency which means the number of rotations per second. So how can my group velocity changes when I increase the wave-number?

• Your question is not clear. Have you tried applying it to, say, waves on deep water where $\omega=\sqrt{gk}$? Here the phase velocity is $\omega/k= \sqrt{g/k}$ while the group velicity $(1/2) \sqrt{g/k}$ is one half of this. If you throw a stone in pond you can easily see the difference as you watch the waves spread. – mike stone Oct 11 '20 at 12:27
• More on group velocity. – Qmechanic Oct 11 '20 at 13:08

The phase velocity, $$v_{ph} = \omega/k$$, relates the period and the wave length, whereas the group velocity, $$v_{g} = \partial\omega/\partial k$$, can be shown to describe (approximately) the speed of propagation of a wave packet, and therefore the speed with which information propagates (since communicating information can be understood as modulation of the wave.)
• Should the the second $v_{ph}$ be $v_{g}$? – user45664 Oct 11 '20 at 17:19