Why Do We Need Wave Packets in Water Waves?

In summary: The KdV equation is a soliton equation which can be used to approximate water waves. The theory of solitons is a beautiful, intricate, and highly developed subject, so anyone who wants to know more should consult a good book since there is a lot to learn if you want to understand the basics. However, IMO it is not nearly as confusing as hanson makes out!
  • #1
hanson
319
0
Hi all.
Can someone explain me physically why we need to deal with wave packets in water waves?
I know the the nonlinear schrodinger equations deals with wave packets in water wave.
But why bother dealing with wave packets?
For the KdV equation, the concept of wave packets is not needed, why?
What so special about wave packets?
Please help.
 
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  • #2
Schroedinger equation deals with water? That's new to me.
 
  • #3
genneth said:
Schroedinger equation deals with water? That's new to me.

yes, nonlinear schrodinger equation can be used in water waves context.
Please help.
 
  • #4
They're special in that they represent particles?
 
  • #5
hanson said:
What so special about wave packets?
They are one of the very few analytical solutions to the NLSE.
 
  • #6
Recommend a good book

genneth said:
Schroedinger equation deals with water? That's new to me.

The theory of solitons is a beautiful, intricate, and highly developed subject, so anyone who wants to know more should consult a good book since there is a lot to learn if you want to understand the basics. However, IMO it is not nearly as confusing as hanson makes out!

I like the undergraduate level introduction by P. G. Drazin and R. S. Johnson, Solitons: an Introduction, Cambridge University Press, 1989.

If you follow this up, you will see how the usual Schroedinger equation plays a role in the famous inverse-scattering transform method of solving the KdV, a famous soliton equation, which arises as an approximation of water waves under certain circumstances. This spawned a great deal of work, including analysis of related PDEs, such as the mKdV, the BBM equation, the Camassa-Holm equation, etc. (the latter also arises as approximations of water waves and includes idealized "breaking of waves"; see math.AP/0709.0905).

The nonlinear Schroedinger equation is a nonlinear generalization of the Schroedinger equation which itself has some aspects of a soliton equation. The sine-Gordon equation is another well known nonlinear PDE which has some soliton-like solutions.

You can search the arXiv to find many recent papers discussing current research in this area. Needless to say, you will need a strong background in differential equations to follow this research.
 
  • #7
How does KdV equation deal with wave packets?
 

1. Why are wave packets important in water waves?

Wave packets are important in water waves because they allow for the transfer of energy and information through the water. Without wave packets, water waves would simply be a continuous oscillation with no distinct pattern or direction.

2. How do wave packets form in water waves?

Wave packets form in water waves through the process of superposition, where individual waves combine to create a larger wave with a distinct shape and direction. This is similar to how different musical notes can combine to form a chord.

3. What determines the size and shape of a wave packet in water waves?

The size and shape of a wave packet in water waves are determined by the characteristics of the individual waves that make up the packet, such as their amplitude, frequency, and wavelength. These factors can vary depending on the wind speed and depth of the water.

4. How do wave packets affect the movement of water?

Wave packets play a crucial role in the movement of water. As they travel through the water, they transfer energy and momentum, causing the water particles to move in a circular motion. This circular motion is what creates the characteristic rolling motion of water waves.

5. Can wave packets travel long distances in water?

Yes, wave packets can travel long distances in water. However, their shape and size may change as they encounter different conditions such as varying depths or wind speeds. Eventually, the wave packets will dissipate due to factors such as friction and energy loss.

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