Question

wave traveling on the surface of water is composed of elementary harmonic waves of the type: A ex)where the phase velocity is proportional to the square root of its wavelength (provided that λ is less than the depth of the water) In terms of the wave-number k, find the phase velocity and the group velocity of this wave.

Answer 1

$$v_p=\frac{\omega}{k}=A\sqrt{\lambda}=\frac{A}{\sqrt{k}}$$

$$v_g=v_p+k\frac{dv_p}{dk}=\frac{A}{\sqrt{k}}+k\frac{d}{dk}\frac{A}{\sqrt{k}}$$

$$v_g=\frac{A}{\sqrt{k}}+k\frac{-A}{2k\sqrt{k}}=\frac{A}{2\sqrt{k}}=\frac{v_p}{2}$$