Dispersive Wave

Since the phase velocity in one dimension is given by $c_{p}={\frac {\omega }{k}}$ and the group velocity is given by $c_{g}={\frac {\partial \omega }{\partial k}}$, if $c_{g}=c_{p}$, then ${\frac {\partial \omega }{\partial k}}={\frac {\omega }{k}}$, so that $\omega =ck$ for some scalar $c$. Note that if $c_{p}=c_{g}=c$, then 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 ${\vec {c_{g}}}$ = $\nabla _{\vec {k}}\omega$. In this case the phase and group velocities may point many more directions than along the positive or negative $k$ axis.