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.

All of this was in the case of one dimension. In multiple dimensions we must turn to the definitions ${\vec {c_{p}}}$ = ${\frac {\omega }{|{\vec {k}}|}}{\hat {k}}$ and ${\vec {c_{g}}}$ = $\nabla _{\vec {k}}\omega$. We then define a dispersive wave as a wave for which ${\vec {c_{g}}}$ = ${\vec {c_{p}}}$.