Water-waves: Group vs phase velocity

In summary, when calculating the time it takes for a wave to reach shore, it is important to use the group velocity rather than the phase velocity. This is because the group velocity represents the speed at which the wave and/or packet of information travels. While phase velocity is useful for considering interference between waves, it does not accurately reflect the movement of the wave itself. The group velocity should be calculated using the wavelength of the predominant waves in the wave-group.
  • #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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  • #2
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.
 
  • #3
ah, I didn't know that. Thanks! :)
 

FAQ: Water-waves: Group vs phase velocity

1. What is the difference between group velocity and phase velocity?

Group velocity refers to the speed at which the energy of a wave travels, while phase velocity refers to the speed at which the actual wave crests move. In other words, group velocity is the speed at which the wave packet moves, while phase velocity is the speed at which individual wave crests move.

2. How are group and phase velocities related?

Group velocity is always less than or equal to the phase velocity. This means that the wave packet, which is made up of individual wave crests, will travel at a slower speed than the individual wave crests themselves.

3. What factors affect the group and phase velocities of water waves?

The group and phase velocities of water waves are affected by several factors, including the depth of the water, the wavelength of the waves, and the shape and steepness of the wave. These factors can cause changes in the wave's speed and direction.

4. Can group and phase velocities be different for different types of water waves?

Yes, the group and phase velocities can vary for different types of water waves. For example, in deep water, the group and phase velocities of long, low-energy waves (such as ocean swells) are nearly equal, while in shallow water, the group velocity is significantly less than the phase velocity.

5. Why is it important to understand the difference between group and phase velocities of water waves?

Understanding the difference between group and phase velocities is crucial for predicting the behavior of water waves and their potential impact on coastlines and structures. It is also important for various applications such as ocean engineering and coastal protection.

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