Dispersive Wave

Since the [1] is ${\frac {\omega }{k}}$, if $\omega (k)=ck$ for some scalar $c$, then the phase velocity is $c$, independent of $k$, so that all waves, no matter their wavelength, travel at the same speed. But if the dispersion relation is some other function of $k$, waves of different wavelengths travel at different speeds, leading to dispersion.