[UPSC CSE 2002] Waves (Q61, 20M)

The phase velocity of the surface wave in a liquid of surface tension T and density \rho is given by u_p = \sqrt{\frac{g\lambda}{2\pi} + \frac{2\pi T}{\lambda\rho}} Show that the group velocity u_g of the surface wave is given by u_g = \frac{g + (12\pi^2 T) / \rho\lambda^2}{2\sqrt{\left(\frac{2\pi g}{\lambda} + \frac{8\pi^3 T}{\rho\lambda^3}\right)}}