Answer:
The group speed is [tex]\dfrac{v_{p}}{2}[/tex].
Explanation:
Given that,
Phase speed [tex]v_{p}=\dfrac{g}{\omega}[/tex]
[tex]\omega=\dfrac{2\pi}{T}[/tex]
Where, T = period of the wave
g =acceleration due to gravity
We need to calculate the group speed
Using formula of phase speed
[tex]v_{p}=\dfrac{\omega}{k}[/tex]
Put the value of [tex]v_{p}[/tex]
[tex]\dfrac{g}{\omega}=\dfrac{\omega}{k}[/tex]
[tex]\omega=\sqrt{gk}[/tex]....(I)
We need to calculate the group speed
Using formula of group speed
[tex]v_{g}=\dfrac{d\omega}{dk}[/tex]
[tex]v_{g}=\dfrac{d}{dk}(\sqrt{gk})[/tex]
On differentiating
[tex]v_{g}=\dfrac{1}{2}\sqrt{\dfrac{g}{k}}[/tex]
Put the value of k from equation (I)
[tex]v_{g}=\dfrac{1}{2}\sqrt{\dfrac{g}{\dfrac{\omega^2}{g}}}[/tex]
[tex]v_{g}=\dfrac{1}{2}\times\dfrac{g}{\omega}[/tex]
[tex]v_{g}=\dfrac{v_{p}}{2}[/tex]
Hence, The group speed is [tex]\dfrac{v_{p}}{2}[/tex].
