Understanding the dispersion of waves

In summary: If you take a (temporal) trace of that wave at some point, it will have a certain shape. Move along the path a bit and the (temporal) shape will not change if the two waves have the same speed (c). When c varies, the shape of the wave will change and that's dispersion.In summary, dispersion is a change in the shape of a wave, caused by the different speeds of different waves travelling through a medium.
  • #1
stephen8686
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I am trying to learn about the dispersion of waves and used one of Walter Lewin's lectures (see below) as a source. I understand phase and group velocity and dispersion relations, but I don't understand when/what kinds of waves are prone to dispersion.
For example, a simple wave in the form $$y(x,t)=A_{0}sin(kx-wt)$$ will never disperse no matter what medium it's in because there are no "groups" to have a group velocity, right?

As I understand it, to have any dispersion you need a wave in the form $$y(x,t)=A_{0}sin(k_{1}x-w_{1}t)+A_{0}sin(k_{2}x-w_{2}t)=2A_{0}sin(k_{3}x-w_{3}t)cos(k_{4}x-w_{4}t)$$
But this is just the interference of two waves, so can you only have dispersion when you have more than one wave (of different frequency) interfering? So do pulses disperse because, looking at it from a Fourier analysis perspective, they are built from a bunch of waves of different frequencies?

thanks
 
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  • #2
Only a pure frequency harmonic wave of infinite extent is a "simple" wave. For finite extent, there are more wavelengths involved (indeed they conspire to suppress the wave envelope at the ends) We always deal with finite extent, particularly in communications where it is vitally important.
The dispersion is as you describe it and Prof. Lewin can tell you the rest far better than I.
 
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  • #3
My understanding is that waves disperse because of dissipative processes like viscosity that are present in all real fluids.
 
  • #4
Chestermiller said:
My understanding is that waves disperse because of dissipative processes like viscosity that are present in all real fluids.
I don't think that is strictly true.
The proximate theoretical cause is that ω=ω(k) or equivalently that different wavelenths move at different speeds. The most common example is deep water waves where v=√(λg /2π). Also the dispersion in optical glass (which causes chromatic aberration) is present without concomitant dissipation.
Dissipative process can cause dispersion, but are not the most typical cause.
 
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  • #5
hutchphd said:
concomitant
I had to look that word up.
As an adjective " natural, or associated".
I will have to use it 5 times, as they say, to burn it into my memory.
 
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stephen8686 said:
As I understand it, to have any dispersion you need a wave in the form
y(x,t)=A0sin(k1x−w1t)+A0sin(k2x−w2t)=2A0sin(k3x−w3t)cos(k4x−w4t)​
y(x,t)=A0sin(k1x−w1t)+A0sin(k2x−w2t)=2A0sin(k3x−w3t)cos(k4x−w4t)y(x,t)=A_{0}sin(k_{1}x-w_{1}t)+A_{0}sin(k_{2}x-w_{2}t)=2A_{0}sin(k_{3}x-w_{3}t)cos(k_{4}x-w_{4}t)
But this is just the interference of two waves, so can you only have dispersion when you have more than one wave (of different frequency) interfering?
If you take a (temporal) trace of that wave at some point, it will have a certain shape. Move along the path a bit and the (temporal) shape will not change if the two waves have the same speed (c). When c varies, the shape of the wave will change and that's dispersion. I think the devil is in the detail of what must be happening to the relative phases of the two waves as they progress in a non-dispersive medium ( the k's and the ω's). That's a sort of reality check with the evidence that pulse shapes don't change with distance without a dispersive medium.
 
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1. What is the dispersion of waves?

The dispersion of waves refers to the phenomenon where the speed of a wave changes depending on its frequency. This means that different frequencies of a wave will travel at different speeds, causing the wave to spread out or disperse.

2. How does dispersion affect wave behavior?

Dispersion can affect the behavior of waves in several ways. In some cases, it can cause a wave to break up into smaller waves, or it can cause waves to travel in different directions. It can also affect the shape and size of a wave, as well as its energy and intensity.

3. What factors affect the dispersion of waves?

The dispersion of waves can be affected by several factors, including the properties of the medium through which the wave is traveling, such as its density and elasticity. The shape and size of the wave can also play a role, as well as the frequency and amplitude of the wave.

4. How is the dispersion of waves measured?

The dispersion of waves can be measured using a variety of methods, depending on the type of wave and the specific properties being studied. For example, in oceanography, dispersion can be measured using buoys and other instruments that track the movement of waves over time. In optics, dispersion can be measured using spectrometers to analyze the wavelengths of light.

5. Why is understanding the dispersion of waves important?

Understanding the dispersion of waves is important for a variety of reasons. It can help us predict and understand the behavior of waves in different environments, such as in the ocean or in different materials. It also has practical applications, such as in the development of new technologies that rely on wave behavior, such as fiber optics and telecommunications. Additionally, understanding dispersion can also lead to advancements in fields such as medicine and environmental science.

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