Dispersive Wave

From Fluids Wiki
Jump to: navigation, search

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.