phase velocity question.

[gokul.er137]

i was intrigued by how to solve this.

 

the phase velocity of ocean waves is sqrt(g*lambda/2*pi) where g - acceleration due to gravity, lambda is the wavelength of the wave. The question is find the group velocity.

 

I tried many things but was not able to get to a proper answer. Any help?

[Klaynos]

Can you write down the equation for the phase velocity and the equation for the group velocity?

[gokul.er137]

vp -> phase velocity

vg -> group velocity

 

generally vp ->c^2(speed of light squared)/vg

 

Doing that and substituting does not give me a satisfactory answer.

 

On a more conceptual note -> group velocity is the rate of change of angular velocity('delta'omega) with the change in wave number('delta'k). Phase velocity is omega/k , that is at any particular instant. So to get the group velocity I could also differentiate omega with respect to k, but I do not know anything about them either.

[Klaynos]

Your second part is correct. The first is a special case.

 

[math] v_g = \frac {d \omega} {dk}[/math]

 

 

[math] v_p = \frac { \omega} {k} = \frac {\lambda} {T} [/math]

 

[math] v_p = \sqrt{g*\lambda/2*pi} [/math]

 

From that you can make some substitutions and find v_g