Water-waves: Group vs phase velocity

[Nikitin]

Homework Statement

Say you have a small boat moving through water, and creating a short wave-group which is a superposition of waves in the range of 0.2m-2m. If the shore 50meters away, how long will it take the fastest of the wave-components to reach shore? {assume the depth is constantly very deep, and the wave-group is traveling directly in the direction of the shore}

Homework Equations

$v_g = \frac{d \omega}{dk}$, $v_p=\frac{\omega}{k}$

The Attempt at a Solution

I assumed that the velocity of the fastest wave-components (the ones with wavelength of 2 meters) would be their phase velocity, but I am wrong according to the solutions manual. The actual velocity is their group velocity,,, for some reason.

I am confused. Isn't the group velocity the velocity of the entire wave-group? Or do all the wavelengths make their own "groups", which is then added together into a swiftly dispersing "mega-group"? Why is it wrong to simply use the phase velocity to calculate the time it takes for the wave to reach shore?

Heck, this brings up an interesting question: How can I calculate the group-velocity of a wave-group? Ie, what value for wavelength should I insert into it? The wavelength of the predominant waves?

 

[greenlight88]

The reason you need to use the group velocity is that information (the wave and/or the packet) travels at this speed. The phase velocity just tells you what happens to a particular phase (say the crest) through space - it dies away at the end of your wave, it has no meaning outside of the wave packet. Phase velocity is useful if you have to think of interference between waves, but it is the group velocity that tells you how the wave moves. So since it is the wave that reaches the shore, we need the group velocity here.

 

[Nikitin]

ah, I didn't know that. Thanks! :)