Waves on the ocean have different velocities, depending on their depth. Long-wavelength wa...

Waves on the ocean have different velocities, depending on their depth. Long-wavelength waves, traveling deep in the ocean, have a speed given approximately by

where g is the acceleration of gravity. Short-wavelength waves, corresponding to surface ripples, have a velocity given approximately by

where ρ is the density and T is the surface tension. Show that the group velocity for long-wavelength waves is 1/2 their phase velocity and the group velocity for short-wavelength waves is 3/2 their phase velocity.