Dispersive Wave

Since the phase velocity is given by $c_{p}={\frac {\omega }{k}}$ and the group velocity is given by $c_{g}=\partial _{k}\omega$, if $\omega (k)=ck$ for some scalar $c$, then $c_{p}=c_{g}=c$ independent of $k$, so that all waves, no matter their wavelength, travel at the same speed. If the dispersion relation is some other function of $k$, waves of different wavelengths travel at different speeds, leading to dispersion.