# Dispersive Wave

Since the phase velocity in one dimension is given by and the group velocity is given by , if , then , so that for some scalar . Note that if , then all waves, no matter their wavelength, travel at the same speed. If the dispersion relation is some other function of , 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 = and = . In this case the phase and group velocities may point many more directions than along the positive or negative axis.