- #1
Nikitin
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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?
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