Question

The phase speed vp for deep ocean waves is given by vp = g/ω where ω = 2π/T and T is the period of the wave; g is the acceleration due to gravity. What is the group speed vg?

Answer 1

Answer:

The group speed is $\dfrac{v_{p}}{2}$.

Explanation:

Given that,

Phase speed $v_{p}=\dfrac{g}{\omega}$

$\omega=\dfrac{2\pi}{T}$

Where, T = period of the wave

g =acceleration due to gravity

We need to calculate the group speed

Using formula of phase speed

$v_{p}=\dfrac{\omega}{k}$

Put the value of $v_{p}$

$\dfrac{g}{\omega}=\dfrac{\omega}{k}$

$\omega=\sqrt{gk}$....(I)

We need to calculate the group speed

Using formula of group speed

$v_{g}=\dfrac{d\omega}{dk}$

$v_{g}=\dfrac{d}{dk}(\sqrt{gk})$

On differentiating

$v_{g}=\dfrac{1}{2}\sqrt{\dfrac{g}{k}}$

Put the value of k from equation (I)

$v_{g}=\dfrac{1}{2}\sqrt{\dfrac{g}{\dfrac{\omega^2}{g}}}$

$v_{g}=\dfrac{1}{2}\times\dfrac{g}{\omega}$

$v_{g}=\dfrac{v_{p}}{2}$

Hence, The group speed is $\dfrac{v_{p}}{2}$.