For waves on the surface of water, the behavior of long wavelengths is dominated by gravit...

For waves on the surface of water, the behavior of long wavelengths is dominated by gravitational effects—a liquid “seeking its own level.” Short wavelengths are dominated by surface tension effects. Taking both into account, the dispersion relation is , where γ is the surface tension, p is the density of water, and g is, of course, the gravitational acceleration.

(a) Make a qualitative sketch of group velocity versus wave number. How does it behave for very large k? For veiy small k?

(b) Find the minimum possible group velocity and the Wavelength at which it occurs. (Use γ— 0.072 N/m, p — 103 kg/m3, and g = 9.8 m/s2.)