# 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.